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EE 6201
CIRCUIT THEARY
A Course Material on
CIRCUIT THEORY
By
Mr.SHAJAHAN M.E.,
ASSISTANT PROFESSOR
DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING
M.E.T ENGINEERING COLLEGE
1
METU
ELECTRICAL AND ELECTRONICS ENGINEERING
Chapter 1 Basic Electrical Terms
Any basic electrical or electronic circuit consists of three separate but very much related quantities, Voltage, (V),
Current, ( I ) and Resistance, ( Ω ).
1.1 Voltage.
Voltage is the potential energy of an electrical supply stored in the form of an electrical charge, and the greater the
voltage the greater is its ability to produce an electrical current flowing through a given circuit. As energy has the abilit y to
do work this potential energy can be described as the work required in joules to move the electrical current around a
circuit from one point or node to another. The difference in voltage between any two nodes in a circuit is known as the
Potential Difference, p.d. or sometimes called Electromotive Force, (EMF) and is measured in Volts with the circuit
symbol V, or lowercase "v", although Energy, E lowercase "e" is sometimes used.
A constant voltage source is called a DC Voltage with a voltage that varies periodically with time is called an AC voltage.
Voltage is measured in volts, and one volt can be defined as the electrical pressure required to force an electrical current
of one ampere through a resistance of one Ohm. Voltages are generally expressed in Volts with prefixes used to denote
microvolts (μV = 10-6V), millivolts (mV = 10-3V) or kilovolts (kV = 103V). Voltage can be either positive or negative.
Batteries or power supplies are mostly used to produce a steady D.C. (direct current) voltage source such as 5v, 12v, 24v
etc in electronic circuits and systems. While A.C. (alternating current) voltage sources are available for domestic house
and industrial power and lighting as well as power transmission. The mains voltage supply in the United Kingdom is
currently 230 volts a.c. and 110 volts a.c. in the USA with general electronic circuits operating on a voltage supply of
between 1.5V and 24V d.c. The circuit symbol for a constant voltage source usually given as a battery symbol with a
positive, + and negative, - sign indicating the direction of the polarity. The circuit symbol for an alternating voltage source
is a circle with a sine wave inside.
A simple relationship can be made between a tank of water and a voltage supply. The higher the water tank above the
outlet the greater the pressure of the water as more energy is released, the higher the voltage the greater the potential
energy as more electrons are released. Voltage is always measured as the difference between any two points in a circuit
and the voltage between these two points is generally referred to as the "Voltage drop". Any voltage source whether DC
or AC likes an open or semi-open circuit condition but hates any short circuit condition as this can destroy it.
1.2 Electric Current.
Electrical Current is the movement of electrical charge and is measured in Amperes, symbol I, for Intensity). It is the
continuous and uniform flow of electrons (negative particles of an atom) around a circuit that are being "pushed" by the
voltage source. In reality, electrons flow from the negative (-ve) terminal to the positive (+ve) terminal of the supply and for
ease of circuit understanding conventional current flow assumes that the current flows from the positive to the negative
terminal. Generally in circuit diagrams the flow of current through the circuit usually has an arrow associated with the
symbol, I, or lowercase I to indicate the actual direction of the current flow. However, this arrow usually indicates the
direction of conventional current flow and not necessarily the direction of the actual flow.
In electronic circuits, a current source is a circuit element that provides a specified amount of current for example, 1A, 5A
10Amps etc, with the circuit symbol for a constant current source given as a circle with an arrow inside indicating its
direction. Current is measured in Amps and an amp or ampere is defined as the number of electrons or charge ( Q in
Coulombs) passing a certain point in the circuit in one second, ( t in Seconds). Current is generally expressed in Amps
with prefixes used to denote micro amps (μA = 10-6A) or milli amps (mA = 10-3A). Electrical current can be either
positive or negative.
Current that flows in a single direction is called Direct Current, or D.C. and current that alternates back and forth through
the circuit is known as Alternating Current, or A.C.. Whether AC or DC current only flows through a circuit when a
voltage source is connected to it with its "flow" being limited to both the resistance of the circuit and the voltage source
pushing it. Also, as AC currents (and voltages) are periodic and vary with time the "effective" or "RMS", (Root Mean
Squared) value given as Irms produces the same average power loss equivalent to a DC current Iaverage . Current sources
are the opposite to voltage sources in that they like short or closed circuit conditions but hate open circuit conditions as no
current will flow.
Using the tank of water relationship, current is the equivalent of the flow of water through the pipe with the flow being the
same throughout the pipe. The faster the flow of water the greater the current. Any current source whether DC or AC likes a
short or semi-short circuit condition but hates any open circuit condition as this prevents it from flowing.
1.3 Resistance.
The Resistance of a circuit is its ability to resist or prevent the flow of current (electron flow) through it making it
necessary to apply a bigger voltage to the circuit to cause the current to flow again. Resistance is measured in Ohms,
Greek symbol (Ω, Omega) with prefixes used to denote Kilo-ohms (kΩ = 103Ω) and Mega-ohms (MΩ = 106Ω).
Resistance cannot be negative only positive.
The amount of resistance determines whether the circuit is a "good conductor" - low resistance, or a "bad conductor" high resistance. Low resistance, for example 1Ω or less implies that the circuit is a good conductor made from materials
such as copper, aluminium or carbon while a high resistance, 1MΩ or more implies the circuit is a bad conductor made
from insulating materials such as glass, porcelain or plastic. A "semiconductor" on the other hand is a material whose
resistance is half way between that of a good conductor and a good insulator such as silicon and germanium and is used
to make Diodes and Transistors etc.
Resistance in a circuit prevents short circuits (unless its very low) by limiting and controlling the amount of current flowing
in a circuit by the voltage supply connected to it and therefore the transfer of power from source to load. Resistance is not
affected by frequency and the AC impedance of a pure resistance is equal to its DC resistance. Resistance also has the
ability to change the characteristics of a circuit by the effect of load resistance or by temperature which changes its
resistivity.
For very low values of resistance, for example milli-ohms, (mΩ´s) it is sometimes more easier to use the reciprocal of
resistance (1/R) rather than resistance (R) itself. The reciprocal of resistance is called Conductance, symbol (G) and it is
the ability of a conductor or device to conduct electricity with high values of conductance implying a good conductor and
low values of conductance implying a bad conductor. The unit of conductance is the Siemen, symbol (S).
Again, using the water relationship, resistance is the diameter or the length of the pipe the water fl ows through. The
smaller the diameter of the pipe the larger the resistance to the flow of water, and therefore the larger the resistance.
Relationship between Voltage and Current in a circuit of constant resistance.
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Summary
This relationship between Voltage, Current and Resistance forms the basis of Ohms Law which is discussed in the
next tutorial and a basic summary of the three units is given below.
•
Voltage or potential difference is the measure of potential energy between two points in a circuit and is
commonly referred to as its "volt drop".
•
When a voltage source is connected to a closed loop circuit the voltage will produce a current flowing around
the circuit.
•
In D.C. voltage sources the symbols +ve (positive) and -ve (negative) are used to denote the polarity of the
voltage supply.
•
Voltage is measured in "Volts" and has the symbol "V" for voltage or "E" for energy.
•
Current flow is a combination of electron flow and hole flow through a circuit.
•
Current is the continuous and uniform flow of charge around the circuit and is measured in " Amperes" or
"Amps" and has the symbol "I".
•
The effective (rms) value of an AC current has the same average power loss equivalent to a DC current
flowing through a resistive element.
•
Resistance is the opposition to current flowing around a circuit.
•
Low values of resistance implies a conductor and high values of resistance implies an insulator.
•
Resistance is measured in "Ohms" and has the Greek symbol "Ω" or the letter "R".
Quantity
Symbol
Unit
of
Measure
Abbreviation
Voltage
V or E
Volt
V
Current
I
Amp
A
Resistance
R
Ohms
Ω
Chapter 2 Basic Electrical Laws And Theorems
2.1 Ohms Law
The relationship between Voltage, Current and Resistance in any DC electrical circuit was firstly discovered by the
German physicist Georg Ohm, (1787 - 1854). Georg Ohm found that, at a constant temperature, the electrical current
flowing through a fixed linear resistance is directly proportional to the voltage applied across it, and also inversely
proportional to the resistance. This relationship between the Voltage, Current and Resistance forms the bases of
Ohms Law and is shown below.
Ohms Law Relationship
By knowing any two values of the Voltage, Current or Resistance quantities we can use Ohms Law to find the third
missing value. Ohms Law is used extensively in electronics formulas and calculations so it is "very important to
understand and accurately remember these formulas".
3
To find Voltage (V)
[V = I x R]
V (volts) = I (amps) x R (Ω)
To find Current (I)
[I = V ÷ R]
I (amps) = V (volts) ÷ R (Ω)
To find Resistance (R)
[R = V ÷ I]
R (Ω) = V (volts) ÷ I (amps)
It is sometimes easier to remember Ohms law relationship by using pictures. Here the three quantities have been
superimposed into a triangle giving voltage at the top with current and resistance at the bottom. This arrangement
represents the position of each quantity in the ohm's law formulas.
2.2 Ohms Law Triangle
Then by using Ohms Law we can see that a voltage of 1V applied to a resistor of 1Ω will cause a current of 1A to flow and
the greater the resistance, the less current will flow for any applied voltage. Any Electrical device or component that obeys
"Ohms Law" that is, the current flowing through it is proportional to the voltage across it ( I α V), such as resistors or
cables, are said to be "Ohmic" in nature, and devices that do not, such as transistors or diodes, are said to be "Nonohmic" devices.
2.2.1 Power in Electrical Circuits
Electrical Power, (P) in a circuit is the amount of energy that is absorbed or produced within the circuit. A source of
energy such as a voltage will produce or deliver power while the connected load absorbs it. The quantity symbol for power
4
is P and is the product of voltage multiplied by the current with the unit of measurement being the Watt (W) with prefixes
used to denote milliwatts (mW = 10-3W ) or kilowatts (kW = 103W ). By using Ohm's law and substituting for V, I and R
the formula for electrical power can be found as:
To find Power (P)
[P = V x I]
P (watts) = V (volts) x I (amps)
Also,
[P = V2 ÷ R]
P (watts) = V2 (volts) ÷ R (Ω)
Also,
[P = I2 x R]
P (watts) = I2 (amps) x R (Ω)
2.3 The Power Triangle
One other point about Power, if the calculated power is positive in value for any formula the component absorbs the
power, but if the calculated power is negative in value the component produces power, in other words it is a source of
electrical energy. Also, we now know that the unit of power is the WATT but some electrical devices such as electric
motors have a power rating in Horsepower or hp. The relationship between horsepower and watts is given as: 1hp =
746W .
2.4 Ohms Law Equations
We can now take all the equations from above for finding Voltage, Current, Resistance and Power and condense them
into a simple Ohms Law pie chart for use in DC circuits.
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Ohms Law Pie Chart
Example No1
For the circuit shown below find the Voltage V, the Current I, the Resistance R and the Power P.
Voltage [ V = I x R ] = 2 x 12Ω = 24V
Current [ I = V ÷ R ] = 24 ÷ 12Ω = 2A
Resistance [ R = V ÷ I ] = 24 ÷ 2 = 12 Ω
Power [ P = V x I ] = 24 x 2 = 48W
Power within an electrical circuit is only present when BOTH voltage and current are present for example, In an
Opencircuit condition, Voltage is present but there is no current flow I = 0 (zero), therefore V x 0 is 0 so the power
dissipated within the circuit must also be 0. Likewise, if we have a Short-circuit condition, current flow is present but
there is no voltage V = 0, therefore 0 x I = 0 so again the power dissipated within the circuit is 0.
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As electrical power is the product of V x I, the power dissipated in a circuit is the same whether the circuit contains high
voltage and low current or low voltage and high current flow. Generally, power is dissipated in the form of Heat (heaters),
Mechanical Work such as motors, etc or Energy in the form of radiated (Lamps) or stored energy (Batteries).
2.4.1 Energy in Electrical Circuits
Electrical Energy that is either absorbed or produced is the product of the electrical power measu red in Watts and the
time in Seconds with the unit of energy given as Watt-seconds or Joules.
Although electrical energy is measured in Joules it can become a very large value when used to calculate the energy
consumed by a component. For example, a single 100 W light bulb connected for one hour will consume a total of 100
watts x 3600 sec = 360,000 Joules. So prefixes such as kilojoules (kJ = 103J) or megajoules (MJ = 106J) are used
instead. If the electrical power is measured in "kilowatts" and the time is given in hours then the unit of energy is in
kilowatt-hours or kWh which is commonly called a "Unit of Electricity" and is what consumers purchase from their
electricity suppliers
2.4.2 Electrical Units of Measure
The standard SI units used for the measurement of voltage, current and resistance are the Volt [V], Ampere [A] and Ohm
[Ω] respectively. Sometimes in electrical or electronic circuits and systems it is necessary to use multiples or sub-multiples
(fractions) of these standard units when the quantities being measured are very large or very small. The following table
gives a list of some of the prefixes used in electrical formulas and component values.
Prefix
Symbol
Multiplier
Power of Ten
Voltage
V
1
100
Current
I
1
100
Resistance
Ω
1
100
Capacitance
F
1
100
Inductance
H
1
100
Frequency
Hz
1
100
Power
W
1
100
impedance
Z
1
100
Giga
G
1,000,000,000
109
Mega
M
1,000,000
106
kilo
k
1,000
103
milli
m
1/1,000
10-3
micro
µ
1/1,000,000
10-6
nano
n
1/1,000,000,000
10-9
pico
p
1/1,000,000,000,000
10-12
7
As well as the "Standard" units of measurements shown above, other units are also used in electrical engineering to
denote other values and quantities such as:
• Wh − The Watt-Hour, The amount of electrical energy consumed in the circuit by a load of one watt
drawing power for one hour, eg a Light Bulb. It is commonly used in the form of kWh
(Kilowatt-hour) which is 1,000 watt-hours or MWh (Megawatt-hour) which is 1,000,000 watthours.
• dB − The Decibel, The decibel is a one tenth unit of the Bel (symbol B) and is used to represent
gain either in voltage, current or power. It is a logarithmic unit expressed in dB and is
commonly used to represent the ratio of input to output in amplifier, audio circuits or
loudspeaker
systems.
For example, the dB ratio of an input voltage (Vin) to an output voltage (Vout) is expressed as
20log10 (Vout/Vin). The value in dB can be either positive (20dB) representing gain or negative
(-20dB) representing loss with unity, ie input
= output expressed as
0dB.
• θ−
Phase Angle, The Phase Angle is the difference in degrees between the voltage waveform
and the current waveform having the same periodic time. It is a time difference or time shift
and depending upon the circuit element can have a "leading" or "lagging" value. The phase
angle
of
a
waveform
is
measured
in
degrees
or
radians.
• ω − Angular Frequency, Another unit which is mainly used in a.c. circuits to represent the Phasor
Relationship between two or more waveforms is called Angular Frequency, symbol ω. This is
a rotational unit of angular frequency 2πƒ with units in radians per second, rads/s. The
complete revolution of one cycle is 360 degrees or 2π, therefore, half a revolution is given as
180
degrees
or
π
rad.
• τ−
Time Constant, The Time Constant of an impedance circuit or linear first-order system is the
time it takes for the output to reach 63.7% of its maximum or minimum output value when
subjected to a Step Response input. It is a measure of reaction time.
2.5 Kirchoffs Circuit Laws
We saw in the Resistors tutorial that a single equivalent resistance, ( RT ) can be found when two or more resistors are
connected together in either series, parallel or combinations of both, and that these circuits obey Ohm's Law. However,
sometimes in complex circuits such as bridge or T networks, we can not simply use Ohm's Law alone to find the voltages
or currents circulating within a circuit. For these types of calculations we need certain rules which allow us to obtain the
circuit equations and for this we can use Kirchoffs Circuit Laws.
In 1845, a German physicist, Gustav Kirchoff developed a pair of rules or laws which deal with the conservation of
current and energy in electrical circuits, one which deals with current flow, Kirchoffs Current Law, (KCL) and one which
deals with voltage, Kirchoffs Voltage Law, (KVL).
2.6 Kirchoff's First Law - The Current Law, (KCL)
Kirchoff´s current law states that the "total current or charge entering a junction or node is exactly equal to the charge
leaving the node as it has no other place to go except to leave, as no charge is lost within the node ". In other words the
algebraic sum of ALL the currents entering and leaving a node must be equal to zero, I(exiting) + I(entering) = 0. This idea is
known as the "Conservation of Charge".
Kirchoff's Current Law
8
Here, the 3 currents entering the node, I1, I2, I3 are all positive in value and the 2 currents leaving the node, I4 and I5 are
negative in value. Then this means we can also rewrite the equation as;
I1 + I2 + I3 - I4 - I5 = 0
The term Node in an electrical circuit generally refers to a connection or junction of two or more current carrying paths or
conductors such as cables and components. Also for current to flow either in or out of a node a closed circuit path must
exist.
2.7 Kirchoff's Second Law - The Voltage Law, (KVL)
Kirchoff´s voltage or loop law states that "in any closed loop network, the total voltage around the loop is equal to the sum of
all the voltage drops within the same loop" which is also equal to zero. In other words the algebraic sum of all voltages
within the loop must be equal to zero. This idea is known as the "Conservation of Energy".
Kirchoff's Voltage Law
Starting at any point in the loop continue in the same direction noting the direction of all the voltage drops, either positive
or negative, and returning back to the same starting point. It is important to maintain the same direction either clockwise or
anti-clockwise or the final voltage sum will not be equal to zero.
Example No1
Find the current flowing in the 40Ω Resistor, R3
9
The circuit has 3 branches, 2 nodes (A and B) and 2 independent loops.
Using Kirchoffs Current Law, KCL the equations are given as;
At node A :
I1 + I2 = I3
At node B :
I3 = I1 + I2
Using Kirchoffs Voltage Law, KVL the equations are given as;
Loop 1 is given as :
10 = R1 x I1 + R3 x I3 = 10I1 + 40I3
Loop 2 is given as :
20 = R2 x I2 + R3 x I3 = 20I2 + 40I3
Loop 3 is given as :
10 - 20 = 10I1 - 20I2
As I3 is the sum of I1 + I2 we can rewrite the equations as;
Eq. No 1 :
10 = 10I1 + 40(I1 + I2) = 50I1 + 40I2
Eq. No 2 :
20 = 20I1 + 40(I1 + I2) = 40I1 + 60I2
We now have two "Simultaneous Equations" that can be reduced to give us the value of both
Substitution of I1 in terms of I2 gives us the value of I1 as -0.143
Substitution of
As :
Amps
I2 in terms of I1 gives us the value of I2 as +0.429 Amps
I3 = I1 + I2
The current flowing in resistor R3 is given as :
-0.143 + 0.429 = 0.286 Amps
10
I1 and I2
and the voltage across the resistor R3 is given as :
0.286 x 40 = 11.44 volts
The negative sign for I1 means that the direction of current flow initially chosen was wrong, but never the less still valid. In
fact, the 20v battery is charging the 10v battery.
2.8 Application of Kirchoffs Circuit Laws
These two laws enable the Currents and Voltages in a circuit to be found, ie, the circuit is said to be "Analysed", and the
basic procedure for using Kirchoffs Circuit Laws is as follows:
•
•
•
•
•
1. Assume all voltage sources and resistances are given. (If not label them V1, V2 ..., R1, R2 etc)
2. Label each branch with a branch current. (I1, I2, I3 etc)
3. Find Kirchoffs first law equations for each node.
4. Find Kirchoffs second law equations for each of the independent loops of the circuit.
5. Use Linear simultaneous equations as required to find the unknown currents.
Chapter 3 Basic Electrical Circuit Analysis
3.1 Circuit Analysis
In the previous tutorial we saw that complex circuits such as bridge or T-networks can be solved using "Kirchoff's Circuit
Laws". While Kirchoff´s Laws give us the basic method for analysing any complex electrical circuit, there are ways of
improving upon this method by using Mesh Current Analysis or Nodal Voltage Analysis that results in a lessening of
the math's involved and when large networks are involved this reduction in maths can be a big advantage.
For example, consider the circuit from the previous section.
3.1.1 Mesh Analysis Circuit
One simple method of reducing the amount of math's involved is to do Kirchoff´s First Current Law equations to determine
the currents, I1 and I2 flowing in the two resistors and there is no need to calculate the current I3 as its just the sum of I1
and I2. Then the Second Voltage Law simply becomes:
•
•
Equation No 1 :
Equation No 2 :
10 = 50I1 + 40I2
20 = 40I1 + 60I2
therefore, one line of math's calculation have been saved.
3.1.2 Mesh Current Analysis
11
A more easier method of solving the above circuit is by using Mesh Current Analysis or Loop Analysis which is also
sometimes called "Maxwell´s Circulating Currents" method. Instead of labelling the branch currents we need to label
each "closed loop" with a circulating current. As a general rule of thumb, only label inside loops in a clockwise direction
with circulating currents as the aim is to cover all the elements of the circuit at least once. Any required branch current
may be found from the appropriate loop or mesh currents as before using Kirchoff´s method.
For example: :
i1 = I1 , i2 = -I2 and I3 = I1 - I2
We now write Kirchoff´s second voltage law equations in the same way as before to solve them but the advantage of this
method is that it ensures that the information obtained from the circuit equations is the minimum required to solve the
circuit as the information is more general and can easily be put into a matrix form.
For example, consider the circuit from the previous section.
These equations can be solved quite quickly by using a single mesh impedance matrix Z. Each element ON the principal
diagonal will be "positive" and is the total impedance of each mesh. Where as, each element OFF the principal diagonal
will either be "zero" or "negative" and represents the circuit element connecting all the appropriate meshes. This then
gives us a matrix of:
Where:
•
•
•
[ V ] gives the total battery voltage for loop 1 and then loop 2.
[ I ] states the names of the loop currents which we are trying to find.
[ R ] is called the resistance matrix.
and this gives I1 as -0.143 Amps and I2 as -0.429 Amps
As :
I3 = I1 - I2
12
The current I3 is therefore given as :
-0.143 - (-0.429) = 0.286 Amps
which is the same value of 0.286 amps, we found using Kirchoff´s circuit law in the previous tutorial.
Page Summary.
This "look-see" method of circuit analysis is probably the best of all the circuit analysis methods with the basic procedure
for solving Mesh Current Anaysis equations is as follows:
•
•
•
•
•
1. Label all the internal loops with circulating currents. ( I1, I2, ...IL etc)
2. Write the [ L x 1 ] column matrix [ V ] giving the sum of all voltage sources in each loop.
o
o
o
•
3. Write the [ L x L ] matrix, [ R ] for all the resistances in the circuit as follows;
R11 = the total resistance in the first loop.
Rnn = the total resistance in the Nth loop.
RJK = the resistance which directly joins loop J to Loop K.
4. Write the matrix or vector equation [V] = [R] x [I] where [I] is the list of currents to be found.
3.2 Nodal Voltage Analysis
As well as using Mesh Analysis to solve the currents flowing around complex circuits it is also possible to use Nodal
Analysis methods too. Nodal Voltage Analysis complements Mesh Analysis in that it is equally powerful and based on
the same concepts of matrix analysis. As its name implies, Nodal Voltage Analysis uses the "Nodal" equations of
Kirchoff´s First Law to find the voltage potentials around the circuit. By adding together all these nodal voltage the net
result will be equal to zero. Then, if there are "N" nodes in the circuit there will be "N-1" independent nodal equations and
these alone are sufficient to describe and hence solve the circuit.
At each node point write down Kirchoff´s first law equation, that is: "the currents Entering a node are exactly equal to the
currents Leaving the node" then express each current in terms of the voltage across the branch. For "N" nodes, one node
will be used as the reference node and all the other voltages will be referenced or measured with respect to this common
node.
For example, consider the circuit from the previous section.
Nodal Analysis Circuit
In the above circuit, node D is chosen as the reference node and the other three nodes are assumed to have voltages,
Va, Vb and Vc with respect to node D. For example;
13
As Va = 10v and Vc = 20v , Vb can be easily found by:
again is the same value of 0.286 amps, we found using Kirchoff´s circuit law in the previous tutorial.
From both Mesh and Nodal Analysis methods we have looked at so far, this is the simplest method of solving this
particular circuit. Generally, Nodal voltage analysis is more appropriate when there are a larger number of current sources
around. The network is then defined as: [ I ] = [ Y ] [ V ] where [ I ] are the driving current sources, [ V ] are the nodal
voltages to be found and [ Y ] is the admittance matrix of the network which operates on [ V ] to give [ I ].
Nodal Analysis Summary.
The basic procedure for solving Nodal Analysis equations is as follows:
•
1. Write down the current vectors, assuming currents into a node are positive. ie, a (N x 1) matrices
for "N" independent nodes.
•
2. Write the admittance matrix [Y] of the network where:
o
Y11 = the total admittance of the first node.
o
Y22 = the total admittance of the second node.
o
RJK = the total admittance joining node J to node K.
•
3. For a network with "N" independent nodes, [ Y] will be an ( N x N) matrix and that Ynn will be
positive and Yjk will be negative or zero value.
•
•
4. The voltage vector will be (N x L) and will list the "N" voltages to be found.
3.3 Thevenins Theorem
In the previous 3 tutorials we have looked at solving complex electrical circuits using Kirchoff´s Circuit Laws, Mesh
Analysis and finally Nodal Analysis but there are many more "Circuit Analysis Theorems" available to calculate the
14
currents and voltages at any point in a circuit. In this tutorial we will look at one of the more common circuit analysis
theorems (next to Kirchoff´s) that has been developed, Thevenins Theorem.
Thevenins Theorem states that "Any linear circuit containing several voltages and resistances can be replaced by just a
Single Voltage in series with a Single Resistor". In other words, it is possible to simplify any "Linear" circuit, no matter how
complex, to an equivalent circuit with just a single voltage source in series with a resistance connected to a load as shown
below. Thevenins Theorem is especially useful in analyzing power or battery systems and other interconnected circuits
where it will have an effect on the adjoining part of the circuit.
Thevenins equivalent circuit.
As far as the load resistor RL is concerned, any "One-port" network consisting of resistive circuit elements and energy
sources can be replaced by one single equivalent resistance Rs and voltage Vs, where Rs is the source resistance value
looking back into the circuit and Vs the open circuit voltage at the terminals.
For example, consider the circuit from the previous section.
Firstly, we have to remove the centre 40Ω resistor and short out (not physically as this would be dangerous) all the emf´s
connected to the circuit, or open circuit any current sources. The value of resistor Rs is found by calculating the total
resistance at the terminals A and B with all the emf´s removed, and the value of the voltage required Vs is the total
voltage across terminals A and B with an open circuit and no load resistor Rs connected. Then, we get the following
circuit.
15
Find the Equivalent Resistance (Rs)
Find the Equivalent Voltage (Vs)
We now need to reconnect the two voltages back into the circuit, and as
is calculated as:
so the voltage drop across the 20Ω resistor can be calculated as:
16
VS = VAB the current flowing around the loop
VAB = 20 - (20Ω x 0.33amps) =
13.33 volts.
Then the Thevenins Equivalent circuit is shown below with the 40Ω resistor connected.
and from this the current flowing in the circuit is given as:
which again, is the same value of 0.286
amps, we found using Kirchoff´s circuit law in the previous tutorial.
Thevenins theorem can be used as a circuit analysis method and is particularly useful if the load is to take a series of
different values. It is not as powerful as Mesh or Nodal analysis in larger networks because the use of Mesh or Nodal
analysis is usually necessary in any Thevenin exercise, so it might as well be used from the start. However, Thevenins
equivalent circuits of Transistors, Voltage Sources such as batteries etc, are very useful in circuit design.
Page Summary.
The basic procedure for solving Thevenins Analysis equations is as follows:
•
•
•
•
1. Remove the load resistor RL or component concerned.
2. Find RS by shorting all voltage sources or by open circuiting all the current sources.
3. Find VS by the usual circuit analysis methods.
4. Find the current flowing through the load resistor RL.
3.4 Nortons Theorem
In some ways Norton's Theorem can be thought of as the opposite to "Thevenins Theorem", in that Thevenin reduces
his circuit down to a single resistance in series with a single voltage. Norton on the other hand reduces his circuit down to
a single resistance in parallel with a constant current source. Nortons Theorem states that "Any linear circuit containing
several energy sources and resistances can be replaced by a single Constant Current generator in parallel with a Single
Resistor". As far as the load resistance, RL is concerned this single resistance, RS is the value of the resistance looking
back into the network with all the current sources open circuited and IS is the short circuit current at the output terminals
as shown below.
17
Nortons equivalent circuit.
The value of this "Constant Current" is one which would flow if the two output terminals where shorted together while the
source resistance would be measured looking back into the terminals, (the same as Thevenin).
For example, consider our now familiar circuit from the previous section.
To find the Nortons equivalent of the above circuit we firstly have to remove the centre 40Ω load resistor and short out the
terminals A and B to give us the following circuit.
When the terminals A and B are shorted together the two resistors are connected in parallel across their two respective
voltage sources and the currents flowing through each resistor as well as the total short circuit current can n ow be
calculated as:
18
with A-B Shorted Out
If we short-out the two voltage sources and open circuit terminals A and B, the two resistors are now effectively
connected together in parallel. The value of the internal resistor Rs is found by calculating the total resistance at the
terminals A and B giving us the following circuit.
Find the Equivalent Resistance (Rs)
Having found both the short circuit current, Is and equivalent internal resistance, Rs this then gives us the following
Nortons equivalent circuit.
Nortons equivalent circuit.
19
Ok, so far so good, but we now have to solve with the original 40Ω load resistor connected across terminals A and B as
shown below.
Again, the two resistors are connected in parallel across the terminals A and B which gives us a total resistance of:
The voltage across the terminals A and B with the load resistor connected is given as:
Then the current flowing in the 40Ω load resistor can be found as:
which again, is the same value of 0.286
amps, we found using Kirchoff´s circuit law in the previous tutorials.
Nortons Analysis Summary.
20
The basic procedure for solving Nortons Analysis equations is as follows:
•
•
•
•
1. Remove the load resistor RL or component concerned.
2. Find RS by shorting all voltage sources or by open circuiting all the current sources.
3. Find IS by placing a shorting link on the output terminals A and B.
4. Find the current flowing through the load resistor RL.
3.5 The Maximum Power Transfer Theorem
•
•
•
•
•
We have seen in the previous tutorials that any complex circuit or network can be replaced by a single energy
source in series with a single internal source resistance, RS. Generally, this source resistance or even
impedance if inductors or capacitors are involved is of a fixed value in Ohm´s. However, when we connect a load
resistance, RL across the output terminals of the power source, the impedance of the load will vary from an
open-circuit state to a short-circuit state resulting in the power being absorbed by the load becoming dependent
on the impedance of the actual power source. Then for the load resistance to absorb the maximum power
possible it has to be "Matched" to the impedance of the power source and this forms the basis of the Maximum
Power Transfer Theorem.
The Maximum Power Transfer Theorem is another useful analysis method to ensure that the maximum
amount of power will be dissipated in the load resistance when the value of the load resistance is exactly equal
to the resistance of the power source. The relationship between the load impedance and the internal impedance
of the energy source will give the power in the load. Consider the circuit below.
Thevenin's Equivalent Circuit.
In our Thevenin equivalent circuit above, the Maximum Power Transfer Theorem states that "the maximum
amount of power will be dissipated in the load resistance if it is equal in value to the Thevenin or Norton source
resistance of the network supplying the power" in other words, the load resistance resulting in greatest power
dissipation must be equal in value to the equivalent Thevenin source resistance, then RL = RS but if the load
resistance is lower or higher in value than the Thevenin source resistance of the network, its dissipated power
will be less than maximum. For example, find the value of the load resistance, RL that will give the maximum
power transfer in the following circuit.
Example No1.
21
Where:
RS = 25Ω
RL is variable between
VS = 100v
•
•
•
•
0
-
100Ω
Then by using the following Ohm's Law equations:
We can now complete the following table to determine the current and power in the circuit for different values of
load resistance.
Table of Current against Power
RL
0
5
10
15
20
I
0
3.3
2.8
2.5
2.2
P
0
55
78
93
97
RL
25
30
40
60
100
I
2.0
1.8
1.5
1.2
0.8
P
100
97
94
83
64
•
Using the data from the table above, we can plot a graph of load resistance, RL against power, P for different
values of load resistance. Also notice that power is zero for an open-circuit (zero current condition) and also for a
short-circuit (zero voltage condition).
•
Graph of Power against Load Resistance
22
•
From the above table and graph we can see that the Maximum Power Transfer occurs in the load when the
load resistance, RL is equal to the source resistance, RS so then: RS = RL = 25Ω. This is called a "Matched
condition" and as a general rule, maximum power is transferred from an active device such as a power supply or
battery to an external device occurs when the impedance of the external device matches that of the source.
Improper impedance matching can lead to excessive power use and dissipation.
3.6 Transformer Impedance Matching
•
•
•
One very useful application of impedance matching to provide maximum power transfer is in the output stages of
amplifier circuits, where the speakers impedance is matched to the amplifier output impedance to obtain
maximum sound power output. This is achieved by using a Matching Transformer to couple the load to the
amplifiers output as shown below.
Transformer Coupling
Maximum power transfer can be obtained even if the output impedance is not the same as the load impedance.
This can be done using a suitable "turns ratio" on the transformer with the corresponding ratio of load
impedance, ZLOAD to output impedance, ZOUT matches that of the ratio of the transformers primary turns to
secondary turns as a resistance on one side of the transformer becomes a different value on the other. If the
load impedance, ZLOAD is purely resistive and the source impedance is purely resistive, ZOUT then the equation
for finding the maximum power transfer is given as:
23
•
•
Where: NP is the number of primary turns and NS the number of secondary turns on the transformer. Then by
varying the value of the transformers turns ratio the output impedance can be "matched" to the source
impedance to achieve maximum power transfer. For example,
•
Example No2.
•
If an 8Ω loudspeaker is to be connected to an amplifier with an output impedance of 1000Ω, calculate the turns
ratio of the matching transformer required to provide maximum power transfer of the audio signal. Assume the
amplifier source impedance is Z1, the load impedance is Z2 and the turns ratio is given as N.
•
Generally, small transformers used in low power audio amplifiers are usually regarded as ideal so any losses can
be ignored.
24
3.7 Star and Delta Transforms
We can now solve simple series, parallel or bridge type resistive networks using Kirchoff´s Circuit Laws , Meshcurrent Analysis or Nodal-voltage Analysis techniques but in a balanced 3-phase circuit we can use different mathematical
techniques to simplify the analysis of the circuit and thereby reduce the amount of math's involved which in itself is a good
thing. Standard 3-phase circuits or networks take on two major forms with names that represent the way in which the
resistances are connected, a Star connected Network which has the symbol of the letter, Υ (wye) and a Delta connected
Network which has the symbol of a triangle, Δ (delta). If a 3-phase, 3-wire supply or even a 3-phase load is connected in
one type of configuration, it can be easily transformed or changed it into an equivalent configuration of the other type by
using either the Star to Delta Transformation or Delta to Star Transformation process.
A resistive network consisting of three impedances can be connected together to form a T or "Tee" configuration but the
network can also be redrawn to form a Star or Υ type network as shown below.
T-connected and Star-connected Resistor Network.
As we have already seen, we can redraw the T resistor network to produce an equivalent Star or Υ type network. But we
can also convert a Pi or π type resistor network into an equivalent Delta or Δ type network as shown below.
Pi-connected and Delta-connected Resistor Network.
25
Having now defined exactly what is a Star and Delta connected network it is possible to transform the Υ into an
equivalent Δ network and also to convert a Δ into an equivalent Υ network using a Transformation process. This
process allows us to produce a mathematical relationship between the various resistors and their equivalents measured
between the terminals 1-2, 1-3 or 2-3 for either a Star or Delta connected circuit. However, the resulting networks are
only equivalent for voltages and currents external to the Star or Delta networks, as internally the voltages and currents are
different but each network will consume the same amount of power and have the same power factor to each other.
Delta-Star Transformation
To convert a Delta network to an equivalent Star network we need to derive a transformation formula for equating the
various resistors to each other between the various terminals. Consider the circuit below.
Delta to Star Network.
Compare the resistances between terminals 1 and 2.
26
Resistance between the terminals 2 and 3.
Resistance between the terminals 1 and 3.
This now gives us three equations and taking equation 3 from equation 2 gives:
Then, re-writing Equation 1 will give us:
Adding together equation 1 and the result above of equation 3 minus equation 2 gives:
27
From which gives us the final equation for resistor P as:
Then to summarize a little the above maths, we can now say that resistor P in a Star network can be found as Equation 1
plus (Equation 3 minus Equation 2) or Eq1 + (Eq3 - Eq2).
Similarly, to find resistor Q in a Star network, is equation 2 plus the result of equation 1 minus equation 3 or Eq2 + (Eq1
- Eq3) and this gives us the transformation of Q as:
And again, to find resistor R in a Star network, is equation 3 plus the result of equation 2 minus equation 1 or Eq3 +
(Eq2 - Eq1) and this gives us the transformation of R as:
When converting a Delta network into a Star network the denominators of all of the transformation formulas are the same: A
+ B + C, and which is the sum of ALL the Delta resistances. Then to convert any Delta connected network to an
equivalent Star network we can summarized the above transformation equations as:
Delta to Star Transformations Equations
28
Example No1
Convert the following Delta Resistive Network into an equivalent Star Network.
3.7.1 Star-Delta Transformation
We have seen above that when converting from a Delta network to an equivalent Star network that the resistor connected
to one terminal is the product of the two Delta resistances connected to the same terminal, for example resistor P is the
product of resistors A and B connected to terminal 1. By re-writing the previous formulas a little we can also find the
transformation formulas for converting a resistive Star network to an equivalent Delta network as shown below.
Star to Delta Network.
29
The value of the resistor on any one side of the Delta, Δ network is the sum of all the two-product combinations of
resistors in the Star network divide by the Star resistor located "directly opposite" the Delta resistor being found. For
example, resistor A is given as:
with respect to terminal 3 and resistor B is given as:
with respect to terminal 2 with resistor C given as:
with respect to terminal 1.
By dividing out each equation by the value of the denominator we end up with three separate transformation formulas that
can be used to convert any Delta resistive network into an equivalent Star network as given below.
Star to Delta Transformations Equations
One final point about converting a Star resistive network to an equivalent Delta network. If all the resistors in the Star
network are equal in value then the resultant resistors in the equivalent Delta network will be three times the value of the
Star resistors and equal, giving: RDELTA = 3RSTAR
3.8 Capacitive Reactance
In the RC Network tutorial we saw that when a DC voltage is applied to a capacitor the capacitor draws a charging
current from the supply and charges up. Likewise, when the supply voltage is reduced the charge stored in the capacitor
also reduces, the capacitor discharges. In a circuit were the applied voltage signal is continually changing from a positive
cycle to a negative cycle at a rate that is determined by the frequency (a sine wave voltage), so the capacitor is either
being charged or discharged on a continuous basis. As the capacitor charges or discharges a current flows through it
30
which itself is restricted by the internal resistance of the capacitor. This internal resistance is known as its Capacitive
Reactance and is given the symbol XC in ohms.
Unlike resistance which has a fixed value, ie 100Ωs, 1kΩ, 10kΩ etc (this is because resistance obeys Ohms Law),
Capacitive Reactance varies with frequency. As the frequency applied to the capacitor increases its reactance
(measured in ohms) decreases and as the frequency decreases its reactance increases. This is because the electrons in
the form of an electrical charge on the capacitor plates, pass from one plate to the other more rapidly with respect to the
varying frequency. As the frequency increases, the capacitor passes more charge across the plates in a given time
resulting in a greater current flow through the capacitor appearing as if the internal resistance of the capacitor has
decreased. Therefore, a capacitor connected in an AC circuit can be said to be "Frequency Dependant".
Capacitive Reactance has the electrical symbol "Xc" and has units measured in Ohms the same as resistance, ( R). It is
calculated using the following formula:
Capacitive Reactance
Capacitive
Formula
•
•
•
•
•
Reactance
Where:
Xc = Capacitive Reactance in Ohms, (Ω)
π (pi) = 3.142
ƒ = Frequency in Hertz, (Hz)
C = Capacitance in Farads, (F)
Example No1
Calculate the capacitive reactance of a 220nF capacitor at a frequency of 1kHz and again at 20kHz.
At a frequency of 1kHz,
Again at a frequency of 20kHz,
where: ƒ = frequency in Hertz and C = capacitance in Farads
31
It can be seen that as the frequency applied to our 220nF capacitor increases from 1kHz to 20kHz, its reactance
decreases from approx 723Ωs to just 36Ωs. For any given value of capacitance the reactance of a capacitor can be
plotted against the frequency as shown below.
Capacitive Reactance against Frequency
By re-arranging the reactance formula above, we can also find at what frequency a capacitor gives a particular reactance
value.
For example. - At which frequency would a 2.2uF Capacitor have a reactance value of 200Ωs?
Or we can find the value of the capacitor in Farads by knowing the applied frequency and its reactance value at that
frequency.
For example. - What will be the value of a Capacitor when it has a reactance of 200Ω when connected to a Frequency of
50Hz.
We can see from the above examples that a capacitor when connected to an AC circuit acts a bit like a frequency
controlled variable resistor and at very low frequencies, such as 1Hz our 220nF capacitor has a reactance of approx
723KΩs (giving the effect of an open circuit), and at very high frequencies such as 1Mhz the capacitor will have a
reactance of just 0.7 ohms (giving the effect of a short circuit). At zero frequency or DC the capacitor has infinite
reactance looking like an "open-circuit" and blocking any flow of current. This property makes capacitors ideal for use in
power supply smoothing circuits to reduce the effects of any unwanted Ripple Voltage as the capacitor applies an AC
signal short circuit path to any unwanted frequency signals on the output terminals.
Summary
32
So, we can summarize the behaviour of a capacitor in a circuit with a varying voltage signal as being a sort of frequency
controlled resistor that has a high Reactance value (open circuit condition) at very low frequencies and low Reactance
value (short circuit condition) at very high frequencies as shown in the graph above. It is important to remember these two
conditions when using capacitors in RC filter circuits.
Chapter 4 Solution Of Simple Electrical Circuits
4.1 Connecting Resistors in Series
Resistors can be connected together in either a series connection, a parallel connection or combinations of both series
and parallel together, to produce more complex networks whose overall resistance is a combination of the individual
resistors. Whatever the combination, all resistors obey Ohm's Law and Kirchoff's Circuit Laws.
Resistors in Series.
Resistors are said to be connected in "Series", when they are daisy chained together in a single line. Since all the current
flowing through the first resistor has no other way to go it must also pass through the second resistor and the third and so
on. Then, resistors in series have a Common Current flowing through them, for example:
IR1 = IR2 = IR3 = IAB = 1mA
In the following example the resistors R1, R2 and R3 are all connected together in series between points A and B.
Series Resistor Circuit
As the resistors are connected together in series the same current passes through each resistor in the chain and the total
resistance, RT of the circuit must be equal to the sum of all the individual resistors added together. That is
RT = R1 + R2 + R3
and by taking the individual values of the resistors in our simple example above, the total resistance is given as:
RT = R1 + R2 + R3 = 1kΩ + 2kΩ + 6kΩ = 9kΩ
Therefore, we can replace all 3 resistors above with just one single resistor with a value of 9kΩ.
Where 4, 5 or even more resistors are all connected together in series, the total resistance of the series circuit RT would
still be the sum of all the individual resistors connected together. This total resistance is generally known as the
Equivalent Resistance and can be defined as; " a single value of resistance that can replace any number of resistors
without altering the values of the current or the voltage in the circuit". Then the equation given for calculating total
resistance of the circuit when resistors are connected together in series is given as:
Series Resistor Equation
33
Rtotal = R1 + R2 + R3 + ..... Rn etc.
One important point to remember about resistors in series circuits, the total resistance ( RT) of any two or more resistors
connected together in series will always be GREATER than the value of the largest resistor in the chain and in our
example above RT = 9kΩ were as the largest value resistor is only 6kΩ.
The voltage across each resistor connected in series follows different rules to that of the current. We know from the above
circuit that the total supply voltage across the resistors is equal to the sum of the potential differences across
R1 , R2 and R3 , VAB = VR1 + VR2 + VR3 = 9V.
Using Ohm's Law, the voltage across the individual resistors can be calculated as:
Voltage across R1 = IR1 = 1mA x 1kΩ = 1V
Voltage across R2 = IR2 = 1mA x 2kΩ = 2V
Voltage across R3 = IR3 = 1mA x 6kΩ = 6V
giving a total voltage VAB of ( 1V + 2V + 6V ) = 9V which is equal to the value of the supply voltage.
The equation given for calculating the total voltage in a series circuit which is the sum of all the individual voltages adde d
together is given as:
Vtotal = V1 + V2 + V3 + ..... Vn
Then series resistor networks can also be thought of as "voltage dividers" and a series resistor circuit having N resistive
components will have N-different voltages across it while maintaining a common current.
By using Ohm's Law, either the voltage, current or resistance of any series connected circuit can easily be found.
4.2 The Potential Divider.
Connecting resistors in series like this across a single DC supply voltage has one major advantage, different voltages
appear across each resistor producing a circuit called a Potential or Voltage Divider Network. The circuit shown above
is a simple potential divider where three voltages 1V, 2V and 6V are produced from a single 9V supply. Kirchoff's
voltage laws states that "the supply voltage in a closed circuit is equal to the sum of all the voltage drops (IR) around
the circuit" and this can be used to good effect.
The basic circuit for a potential divider network (also known as a voltage divider) for resistors in series is shown below.
Potential Divider Network
34
In this circuit the two resistors are connected in series across Vin, which is the power supply voltage connected to the
resistor, Rtop, where the output voltage Vout is the voltage across the resistor Rbottom which is given by the formula. If more
resistors are connected in series to the circuit then different voltages will appear across each resistor with regards to their
individual resistance R (Ohms law IxR) providing different voltage points from a single supply. However, care must be
taken when using this type of network as the impedance of any load connected to it can affect the output voltage. For
example,
Suppose you only have a 12V DC supply and your circuit which has an impedance of 50Ω requires a 6V supply.
Connecting two equal value resistors, of say 50Ω each, together as a potential divider network across the 12V will do this
very nicely until you connect your load circuit to the network. This is demonstrated below.
Example No1.
Calculate the voltage across X and Y.
a) Without RL connected
b) With RL connected
35
As you can see from above, the output voltage Vout without the load resistor connected gives us the required output
voltage of 6V but the same output voltage at Vout when the load is connected drops to only 4V, ( Resistors in Parallel).
Then the output voltage Vout is determined by the ratio of Vtop to Vbottom with the effect of reducing signal or voltage levels
being known as Attenuation so care must be taken when using a potential divider networks. The higher the load
impedance the less is the loading effect on the output.
A variable resistor, potentiometer or Pot as it is more commonly called, is a good example of a multi-resistor potential
divider within a single package. Here a fixed voltage is applied across the two outer fixed connections and the variable
output voltage is taken from the wiper terminal. Multi-turn pots allow for a more accurate output voltage control.
Uses for a Potential Divider Circuit
If we replace one of the resistors in the potential divider circuit above with a Sensor such as a thermistor, Light
Dependant Resistor (LDR) or even a switch, we can convert the analogue quantity being sensed into a suitable electrical
signal which is capable of being measured.
For example, the following thermistor circuit has a resistance of 10KΩ at 25°C and a resistance of 100Ω at 100°C.
Calculate the output voltage (Vout) for both temperatures.
36
At 25°C
At 100°C
By changing the fixed 1KΩ resistor, R2 in our simple circuit above to a variable resistor or potentiometer, a particular
output voltage set point can be obtained over a wider temperature range.
4.3 Resistors in Parallel
Resistors are said to be connected together in "Parallel" when both of their terminals are respectively connected to each
terminal of the other resistor or resistors. The voltage drop across all of the resistors in parallel is the same. Then,
Resistors in Parallel have a Common Voltage across them and in our example below the voltage across the resistors is
given as:
VR1 = VR2 = VR3 = VAB = 12V
In the following circuit the resistors R1, R2 and R3 are all connected together in parallel between the two points A and B.
Parallel Resistor Circuit
37
In the previous series resistor circuit we saw that the total resistance, RT of the circuit was equal to the sum of all the
individual resistors added together. For resistors in parallel the equivalent circuit resistance RT is calculated differently.
Parallel Resistor Equation
Here, the reciprocal ( 1/Rn ) value of the individual resistances are all added together instead of the resistances
themselves. This gives us a value known as Conductance, symbol G with the units of conductance being the Siemens,
symbol S. Conductance is therefore the reciprocal or the inverse of resistance, ( G = 1/R ). To convert this conductance
sum back into a resistance value we need to take the reciprocal of the conductance giving us then the total resistance, RT
of the resistors in parallel.
Example No1
For example, find the total resistance of the following parallel network
Then the total resistance RT across the two terminals A and B is calculated as:
38
This method of calculation can be used for calculating any number of individual resistances connected together within a
single parallel network. If however, there are only two individual resistors in parallel then a much simpler and quicker
formula can be used to find the total resistance value, and this is given as:
Example No2
Consider the following circuit with the two resistors in parallel combination.
Using our two resistor formula above we can calculate the total circuit resistance, RT as:
One important point to remember about resistors in parallel, is that the total circuit resistance ( RT) of any two resistors
connected together in parallel will always be LESS than the value of the smallest resistor and in our example above
39
RT = 14.9kΩ were as the value of the smallest resistor is only 22kΩ. Also, in the case of R1 being equal to the value of
R2, ( R1 = R2 ) the total resistance of the network will be exactly half the value of one of the resistors, R/2.
Consider the two resistors in parallel above. The current that flows through each of the resistors ( IR1 and IR2 ) connected
together in parallel is not necessarily the same value as it depends upon the resistive value of the resistor. However, we
do know that the current that enters the circuit at point A must also exit the circuit at point B. Kirchoff's Current Laws.
states that "the total current leaving a circuit is equal to that entering the circuit - no current is lost". Thus, the total current
flowing in the circuit is given as:
IT = IR1 + IR2
Then by using Ohm's Law, the current flowing through each resistor can be calculated as:
Current flowing in R1 = V/R1 = 12V ÷ 22kΩ = 0.545mA
Current flowing in R2 = V/R2 = 12V ÷ 47kΩ = 0.255mA
giving us a total current IT flowing around the circuit as: IT =
0.545mA + 0.255mA = 0.8mA or 800uA.
The equation given for calculating the total current flowing in a parallel resistor circuit which is the sum of all the individual
currents added together is given as:
Itotal = I1 + I2+ I3 + ..... In
Then parallel resistor networks can also be thought of as "current dividers" because the current splits or divides between
the various branches and a parallel resistor circuit having N resistive networks will have N-different current paths while
maintaining a common voltage.
4.4 Resistor Combinations
In the previous two tutorials we have learnt how to connect resistors together to form either a Series circuit or a Parallel
circuit, but what if we want to connect them together in "BOTH" parallel and series within the same circuit to produce a
resistor combination, how do we calculate the combined circuit resistance for this.
Resistor circuits that combine series and parallel resistors circuits together are generally known as Resistor
Combination circuits and the method of calculating their combined resistance is the same as that for any individual series
or parallel circuit. For example:
For example, Calculate the total current (I) taken from the 12v supply.
40
At first glance this may seem a difficult task, but if we look a little closer we can see that the two resistors, R2 and R3 are
both connected together in a "SERIES" combination so we can add them together and the resultant resistance for this
would be,
R2 + R3 = 8 Ω + 4 Ω = 12 Ω
So now we can replace both the resistors R2 and R3 with a single resistor of resistance value 12 Ω
Now we have single resistor RT in "PARALLEL" with the resistor R4, (resistors in parallel) and again we can reduce this
combination to a single resistor value of R(combination) using the formula for two parallel connected resistors as follows.
41
The resultant circuit now looks something like this:
The two remaining resistances, R1 and R(comb) are connected together in a "SERIES" combination and again they can be
added together so the total circuit resistance between points A and B is therefore given as:
R( A - B ) = Rcomb + R1 = 6 Ω + 6 Ω = 12 Ω.
and a single resistance of just 12 Ω can be used to replace the original 4 resistor combinations circuit above.
Now by using Ohm´s Law, the value of the circuit current (I) is simply calculated as:
In any Series-Parallel Resistor Combination circuit by firstly identifying simple series or parallel connected resistors and
replacing them with individual resistors of the equivalent value, it is easily possible to reduce any complex circuit to that of a
single equivalent resistance. However, calculations of complex Bridge and T networks which cannot be reduced to a
simple parallel or series circuit using equivalent resistances need to be solved using Kirchoff's Current Law , and
Kirchoff's Voltage Law which will be dealt with in another tutorial.
4.4.1 Potential Difference
The voltage difference between any two points in a circuit is known as the Potential Difference or pd between these two
points and is what makes the current in the circuit flow. The greater the potential difference across a component the
bigger will be the current flowing through it. For example, if the voltage at one side of a resistor measures 8V and the
other side of the resistor it measures 5V, then the potential difference across the resistor would be 3V (8 - 5). The voltage
at any point in a circuit is measured with respect to a common point. For electrical circuits, the earth or ground potential is
usually taken to be at zero volts (0v) and everything is referenced to that point. This is similar to measuring height. We
measure the height of hills in a similar way by saying that the sea level is at zero feet and then compare other points to
that level. In the same way we call the lowest voltage in a circuit zero volts and give it the name of ground, zero volts or
earth, then all other voltage points in the circuit are compared or referenced to that ground point. For example
Potential Difference
42
As the units of measure for Potential Difference are volts, potential difference is mainly called Voltage. Individual
voltages connected in series can be added together to give us a "Total Voltage" sum of the circuit, but voltages across
components that are connected in parallel will always be of the same value, for example.
for series connected voltages,
for parallel connected voltages,
Example No1
By using Ohm's Law, the current flowing through a resistor can be calculated. For example, Calculate the current
flowing through a 100Ω resistor that has one of its terminals connected to 50 volts and the other terminal connected to 30
volts.
Voltage at terminal A is equal to 50v and the voltage at terminal B is equal to 30v. Therefore, the voltage across the
resistor is given as:
VA = 50v, VB = 30v, therefore, VA - VB = 50 - 30 = 20v
The voltage across the resistor is 20v therefore, the current flowing in the resistor is given as: I
= VAB ÷ R = 20V ÷ 100Ω = 200mA
4.4.2 Voltage Divider
43
We know from the previous tutorials that by connecting together resistors in series across a potential difference we can
produce a voltage divider circuit giving ratios of voltages with respect to the supply voltage across the series combination.
This then produces a Voltage Divider network that only applies to resistors in series as parallel resistors produce a
current divider network. Consider the circuit below.
Voltage Division
The circuit shows the principal of a voltage divider circuit where the output voltage drops across each resistor, R1, R2,
R3 and R4 are referenced to a common point. For any number of resistors connected together in series the total
resistance, RT of the circuit divided by the supply voltage Vs will give the circuit current as I = Vs/RT, Ohm's Law. Then
the individual voltage drops across each resistor can be simply calculated as: V = IxR.
The voltage at each point, P1, P2, P3 etc increases according to the sum of the voltages at each point up to the supply
voltage, Vs and we can also calculate the individual voltage drops at any point without firstly calculating the circuit current
by using the following formula.
Voltage Divider Equation
Where, V(x) is the voltage to be found, R(x) is the resistance producing the voltage, RT is the total series resistance and VS is
the supply voltage.
Then by using this equation we can say that the voltage dropped across any resistor in a series circuit is proportional to
the magnitude of the resistor and the total voltage dropped across all the resistors must equal the voltage source as
defined by Kirchoff's Voltage Law . So by using the Voltage Divider Equation, for any number of series resistors the
voltage drop across any individual resistor can be found.
4.5 Resistors in AC Circuits
44
In the previous tutorials we have looked at Resistors, their connections and used Ohm's Law to calculate the voltage,
current and power associated with them. In all cases both the voltage and current has been assumed to be of a constant
polarity, flow and direction, in other words Direct Current or DC. But there is another type of supply known as
Alternating Current or AC whose voltage switches polarity from positive to negative and back again over time and also
whose current with respect to the voltage oscillates back and forth. The shape of an AC supply follows the mathematical
form of a "Sine wave" and is commonly called a Sinusoidal Wave Form and whose voltage is given as V(t) =
Vmax sin ωt.
When using pure resistors in AC circuits that have negligible values of inductance or capacitance, the same principals of
Ohm's Law , circuit rules for voltage, current and power (and even Kirchoff's Laws) apply as they do for DC resistive
circuits the only difference is in the use of a "peak-to-peak" or "rms" quantities. When working with such rules it is usual to
use only "rms" values. Also the symbol used for defining an AC voltage source is that of a "wavy" line as opposed to a
battery symbol for DC and this is shown below.
Symbol Representation of DC and AC Supplies
Resistors are "passive" devices, that is they do not produce or consume any electrical energy, but convert electrical
energy into heat. In DC circuits the ratio of Voltage to Current in a resistor is called its resistance. However, in AC circuits
this ratio of voltage to current depends upon the frequency and phase difference or phase angle ( φ ) of the supply. So
when using resistors in AC circuits the term Impedance, symbol Z is the generally used and we can say that DC
resistance = AC impedance, R = Z.
For resistors in AC circuits the current flowing through them will rise and fall as the voltage rises and falls. The current and
voltage reach maximum, fall through zero and reach minimum at exactly the same time. i.e, they rise and fall
simultaneously and are said to be "in-phase" as shown below.
V-I Phase Relationship and Vector Diagram
We can see that at any point along the horizontal axis that the instantaneous voltage and current are in-phase, that is their
phase angle φ is 0o. Then these instantaneous values of voltage and current can be compared to give the ohmic value of
the resistance simply by using ohms law. In purely resistive series AC circuits, all the voltage drops across the resistors
can be added together to find the total circuit voltage as all the voltages are in-phase with each other. Likewise, in a purely
resistive parallel AC circuit, all theindividual branch currents can be added together to find the total circuit current because
all the branch currents are in-phase with each other.
45
Since for resistors in AC circuits the phase angle φ between the voltage and the current is zero, then the power factor of
the circuit is given as cos 0o = 1.0. The power in the circuit at any instant in time can be found by multiplying the voltage
and current at that instant. Then the power (P), consumed by the circuit is given as P = Vrms Ι cosΦ in watt's. But since
cosΦ = 1 in a purely resistive circuit, the power consumed is simply given as, P = Vrms Ι the same as for Ohm's Law .
This then gives us the "Power" waveform and which is shown below as a series of positive pulses because when the
voltage and current are both in their positive half of the cycle the resultant power is positive. When the voltage and current
are both negative, the product of the two negative values gives a positive power pulse.
Power Waveform in a Pure Resistance
Then for a purely resistive load fed from an AC supply:
Where,
•
•
•
•
P
V
I
R
is the Average Power in Watts
is the RMS Supply Voltage in Volts
is the RMS Supply Current in Amps
is the resistance of the resistor in Ohm´s (Ω)
Any resistive heating element such as Electric Fires, Toasters, Kettles, Irons, Water Heaters etc can be classed as a
resistive AC circuit and we use resistors in AC circuits to heat our homes and water.
Example No1
A 1000W heating element is connected to a 250v AC supply voltage. Calculate the resistance of the element when it is
hot and the amount of current taken from the supply.
46
Example No2
Calculate the power being consumed by a 100Ω resistor connected to a 240v supply.
As there is only one component connected to the supply, the resistor, then
VR = VS
Resistors Summary
•
The job of Resistors are to limit current flow through an electrical circuit.
•
Resistance is measured in Ohm's and is given the symbol Ω
•
Carbon, Film and Wirewound are all types of resistors.
•
Resistor colour codes are used to identify the resistance and tolerance rating of small resistors.
•
The BS1852 Standard is used to identify large size resistors.
•
Tolerance is the percentage measure of the accuracy of a resistor and the E6 (20%), E12 (10%), E24 (5%)
and E96 (1%) Series of tolerance values are available.
Series Resistors
•
Resistors that are daisy chained together in a single line are said to be connected in SERIES.
•
Series connected resistors have a common Current flowing through them.
•
Itotal = I1 = I2 = I3 .... etc
•
•
The total circuit resistance of Series resistors is equal to.
Rtotal = R1 + R2 + R3 .... etc
•
•
Total circuit Voltage is equal to the sum of all the individual voltage drops.
Vtotal = V1 + V2 + V3.... etc
•
The total resistance of a series connected circuit will always be greater than the highest value resistor.
Parallel Resistors
47
•
•
•
Resistors that have both of their respective terminals connected to each terminal of another resistor or
resistors are said to be connected in PARALLEL.
Parallel resistors have a common Voltage across them.
VS = V1 = V2 = V3 .... etc
•
•
Total resistance of a parallel circuit is equal to.
Itotal = 1/( 1/R1 + 1/R2 + 1/R3 ... etc)
•
•
Total circuit current flow is equal to the sum of all the individual branch currents added together.
Itotal = I1 + I2 + I3 .... etc
•
The total resistance of a parallel circuit will always be less than the value of the smallest resistor.
Power Rating
•
•
The larger the power rating, the greater the physical size of the resistor.
All resistors have a maximum power rating and if exceeded will result in the resistor overheating and
becoming damaged.
•
Standard resistor power rating sizes are 1/8 W, 1/4 W, 1/2 W, 1 W, and 2 W .
•
Low ohmic value power resistors are generally used for current sensing or power supply applications.
•
The power rating of resistors can be calculated using the formula.
•
In AC Circuits the voltage and current flowing in a pure resistor are always "in-phase" producing 0 o phase
shift..
•
When used in AC Circuits the AC impedance of a resistor is equal to its DC Resistance.
•
The AC circuit impedance for resistors is given the symbol
Z.
4.6 Capacitors in Parallel
Capacitors are said to be connected "in parallel" when both of their terminals are respectively connected to each terminal
of the other capacitor or capacitors. The voltage ( Vc) across all the capacitors connected in parallel is THE SAME. Then,
parallel capacitors have a Common Voltage supply across them and:
VC1 = VC2 = VC3 = VAB = 12V
In the following circuit capacitor, C1 and capacitor, C2 are connected in parallel between points A and B.
48
When capacitors are connected in parallel the total capacitance in the circuit is equal to the sum of all the individual
capacitors added together. That is:
Parallel Capacitors Equation
Where all the capacitors are given in the same capacitance units, either uF, nF or pF.
Example No1
Taking the three capacitor values from the above, we can calculate the total circuit capacitance as:
CT = C1 + C2 + C3 = 0.1uF + 0.2uF + 0.3uF = 0.6uF
One important point to remember about parallel connected capacitor circuits, the total capacitance ( CT) of any two or
more capacitors connected together in parallel will always be GREATER than the value of the largest capacitor in the
chain and in our example above CT = 0.6uF were as the largest value capacitor is only 0.3uF.
When 4, 5, 6 or even more capacitors are connected together the total capacitance of the circuit CT would still be the sum of
all the individual capacitors added together and the total capacitance of a parallel circuit is always greater than the
highest value capacitor.
The total charge stored by the circuit is also the sum of the individual values and is given as:
QT = Q1 + Q2 + Q3 + etc ...
Where; Q
= CV , Capacitance x Voltage.
Also the total current flowing the circuit is also the sum of the individual currents and is given as:
IT = I1 + I2 + I3 + etc ...
49
Example No2.
Calculate the combined capacitance of the following capacitors when they are connected together in parallel
combination: a) 2 capacitors, each of 47nF and b) A capacitor of 470nF connected to a capacitor of 1uF.
a) Total Capacitance,
CT = C1 + C2 = 47nF + 47nF = 94nF or 0.094uF
b) Total Capacitance,
CT = C1 + C2 = 470nF + 1uF
therefore, CT = 470nF + 1000nF = 1470nF or 1.47uF
4.7 Capacitors in Series
Capacitors are said to be "connected in series" when they are effectively "daisy chained" together in a single line. The
charging current ( Ic) flowing through the capacitors is THE SAME for the same amount of time so each capacitor stores
the same amount of charge regardless of its capacitance and:
QT = Q1 = Q2 = Q3 etc ....
In the following circuit, capacitors, C1, C2 and C3 are connected together in series between points A and B.
In the parallel circuit we saw that the total capacitance, CT of the circuit was equal to the sum of all the individual
capacitors added together. In a series connected circuit the equivalent capacitance CT is calculated differently. Here, the
reciprocal (1/C) of the individual capacitors are added together (just like resistors in parallel) instead of the capacitors
themselves, for example.
Series Capacitors Equation
50
Example No1
Taking the three capacitor values from the above, we can calculate the total circuit capacitance as:
One important point to remember about series connected capacitor circuits, is that the total circuit capacitance (CT) of any
capacitors connected together in series will always be LESS than the value of the smallest capacitor and in our example
above CT = 0.055uF were as the value of the smallest capacitor is only 0.1uF.
This reciprocal method of calculation can be used for calculating any number of capacitors connected together in a single
series network. If however, there are only two capacitors connected together in series, then a much simpler and quicker
formula can be used and is given as:
In the above series circuit the DC supply voltage, VT between points A and B is divided between all three capacitors with
the smaller capacitor having the largest voltage drop across it and vice versa. The voltage across the capacitors, V1, V2
and V3 is inversely proportional to its capacitance and for the three capacitors in our example above is given as:
Where QT is the total charge across the circuit and is given as: CT x VT, (which for above is 0.66 micro-coulombs).
Example No2.
Find the overall capacitance and the individual voltages of two capacitors connected together in series when connected to a
12V d.c. supply.
a) 2 capacitors, each of 47nF.
51
b) A capacitor of 470nF and a capacitor of 1uF.
a) Total Capacitance,
Voltage across Capacitors,
b) Total Capacitance,
Voltage across Capacitors,
4.8 Capacitors in AC Circuits
When capacitors are connected across a direct current DC supply voltage they become charged to the value of the
applied voltage, acting like temporary storage devices and maintain or hold this charge indefinitely as long as the supply
voltage is present. During this charging process, a charging current, I will flow into the capacitor opposing any changes to
the voltage at a rate that is equal to the rate of change of the electrical charge on the plates. This charging current can be
defined as: I = CdV/dt. Once the capacitor is "fully-charged" the capacitor blocks the flow of any more electrons onto its
plates as they have become saturated. However, if we apply an alternating current or AC supply, the capacitor will
alternately charge and discharge at a rate determined by the frequency of the supply. Then capacitors in AC circuits are
constantly charging and discharging.
We know that the flow of electrons through the capacitor is directly proportional to the rate of change of the voltage across
the plates. Then, we can see that capacitors in AC circuits like to pass current when the voltage across itsplates is
constantly changing with respect to time such as in AC signals, but it does not like to pass current when the applied
voltage is of a constant value such as in DC signals. Consider the circuit below.
AC Capacitor Circuit
52
In the purely capacitive circuit above, the capacitor is connected directly across the AC supply voltage. As the supply
voltage increases and decreases, the capacitor charges and discharges with respect to this change. We know that the
charging current is directly proportional to the rate of change of the voltage across the plates with this rate of change at its
greatest as the supply voltage crosses over from its positive half cycle to its negative half cycle or vice versa at points, 0o
and 180o along the sine wave. Consequently, the least voltage change occurs when the AC sine wave crosses over at its
maximum or minimum peak voltage level, ( Vm ). At these positions in the cycle the maximum or minimum currents are
flowing through the capacitor circuit and this is shown below.
AC Capacitor Phasor Diagram
At 0o the rate of change of the supply voltage is increasing in a positive direction resulting in a maximum charging current
at that instant in time. As the applied voltage reaches its maximum peak value at 90 o for a very brief instant in time the
supply voltage is neither increasing or decreasing so there is no current flowing through the circuit. As the applied voltage
begins to decrease to zero at 180 o, the slope of the voltage is negative so the capacitor discharges in the negative
direction. At the 180 o point along the line the rate of change of the voltage is at its maximum again so maximum current
flows at that instant and so on. Then we can say that for capacitors in AC circuits the instantaneous current is at its
minimum or zero whenever the applied voltage is at its maximum and likewise the instantaneous value of the current is at
its maximum or peak value when the applied voltage is at its minimum or zero. From the waveform above, we can see
that the current is leading the voltage by 1/4 cycle or 90 o as shown by the vector diagram. Then we can say that in a
purely capacitive circuit the alternating voltage lags the current by 90o.
We know that the current flowing through the capacitor is in opposition to the rate of change of the applied voltage but just
like resistors, capacitors also offer some form of resistance against the flow of current through the circuit, but with
capacitors in AC circuits this AC resistance is known as Reactance or more commonly in capacitor circuits, Capacitive
Reactance, so capacitors in AC circuits suffer from Capacitive Reactance.
4.9 Capacitive Reactance
53
Capacitive Reactance in a purely capacitive circuit is the opposition to current flow in AC circuits only. Like resistance,
reactance is also measured in Ohm's but is given the symbol X to distinguish it from a purely resistive value. As
reactance can also be applied to Inductors as well as Capacitors it is more commonly known as Capacitive Reactance
for capacitors in AC circuits and is given the symbol Xc so we can actually say that Capacitive Reactance is Resistance
that varies with frequency. Also, capacitive reactance depends on the value of the capacitor in Farads as well as the
frequency of the AC waveform and the formula used to define capacitive reactance is given as:
Capacitive Reactance
Where:
F
is
in
Hertz
and
C
is
in
Farads.
2πF can also be expressed collectively as the Greek letter Omega, ω to denote an angular frequency.
From the capacitive reactance formula above, it can be seen that if either of the Frequency or Capacitance where to be
increased the overall capacitive reactance would decrease. As the frequency approaches infinity the capacitors reactance
would reduce to zero acting like a perfect conductor. However, as the frequency approaches zero or DC, the capacitors
reactance would increase up to infinity, acting like a very large resistance. This means then that capacitive reactance is
"Inversely proportional" to frequency for any given value of Capacitance and this shown below:
Capacitive Reactance against Frequency
The Reactance value of a Capacitor
decreases as the Frequency across
it increases.
Example No1.
Find the current flowing in a circuit when a 4uF capacitor is connected across a 880v, 60Hz supply.
54
Summary of Capacitors
•
A Capacitor consists of two metal plates separated by a dielectric.
•
The dielectric can be made of many insulating materials such as air, glass, paper, plastic etc.
•
A Capacitor is capable of storing electrical charge and energy.
•
The higher the value of capacitance, the more charge the Capacitor can store.
•
The larger the area of the plates or the smaller their separation the more charge the Capacitor can store.
•
A Capacitor is said to be Fully Charged when the voltage across its plates equals the supply voltage.
•
The symbol for electrical charge is Q and its unit is the Coulomb.
•
Electrolytic Capacitors are polarized. They have a +ve and a -ve terminal.
•
•
•
Capacitance is measured in Farads, which is a very large unit so Microfarad (uF), Nanofarad (nF) and
Picofarad (pF) are generally used.
Capacitors that are daisy chained together in a line are said to be connected in Series.
Capacitors that have both of their respective terminals connected to each terminal of another capacitor are
said to be connected in Parallel.
•
Parallel connected Capacitors have a common supply voltage across them.
•
Series connected Capacitors have a common current flowing through them.
•
Capacitive reactance is the opposition to current flow in AC circuits.
•
In AC capacitive circuits the voltage "lags" the current by 90 o.
Chapter 5 Semiconductor Physics
5.1 Semiconductor Basics
If Resistors are the most basic passive component in electrical or electronic circuits, then the Signal Diode can be
classed as the most basic "Active" component. It is a very simple non-linear electronic device that will only allow current to
flow through it in one direction only, acting more like a one way electrical valve. But, before we have a look at how signal
diodes work we first need to understand their basic construction and concept.
Diodes are made from a single piece of Semiconductor material which has a positive " P-region" at one end and a
negative "N-region" at the other, and has a resistivity somewhere between that of a conductor and an insulator. But what is
a "Semiconductor" material?, firstly let's look at what makes something either a Conductor or an Insulator.
5.2 Resistivity
55
The electrical Resistance of an electrical or electronic component or device is generally defined as being the ratio of the
Voltage difference across it to the Current flowing through it, basic Ohm´s Law principals. The problem with using
resistance as a measurement is that it depends very much on the physical size of the material being measured as well as
the material out of which it is made. For example, If we were to increase the length of the material (making it longer) its
resistance would also increase. Likewise, if we increased its diameter (making it fatter) its resistance would then
decrease. So we want to be able to define the material in such a way as to indicate its ability to either conduct or oppose
the flow of electrical current through it. The quantity that is used to indicate this is called Resistivity and is generally given
the Greek symbol ρ, (Rho) measured in Ohms per metre, (Ω's/m).
If the resistivity of various materials is compared, they can be classified into three main groups, Conductors, Insulators
and Semi-conductors as shown below.
Resistivity Chart
Notice also that there is a small margin between the
resistivity of conductors, eg. Silver, Gold etc,
compared to the large margin for the resistivity of
insulators, eg. Glass and Quartz. The resistivity of all
the materials also depends upon their temperature.
Conductors
From above we now know that Conductors are materials that have a low value of resistivity allowing them to easily pass
an electrical current due to there being plenty of free electrons floating about within their basic atom structure. When a
positive voltage potential is applied to the material these "free electrons" leave their parent atom and travel together
through the material forming an electron or current flow. Examples of good conductors are generally metals such as
Copper, Aluminium, Silver or non metals such as Carbon because these materials have very few electrons in their outer
"Valence Shell" or ring, resulting in them being easily knocked out of the atom's orbit. This allows them to flow freely
56
through the material until they join up with other atoms, producing a "Domino Effect" through the material thereby creating
an electrical current.
Generally speaking, most metals are good conductors of electricity, as they have very small resistance values, usually in
the region of micro-ohms per metre with the resistivity of conductors increasing with temperature because metals are also
generally good conductors of heat.
Insulators
Insulators on the other hand are the exact opposite of conductors. They are made of materials, generally non-metals,
that have very few or no "free electrons" floating about within their basic atom structure because the electrons in the outer
valence shell are strongly attracted by the positively charged inner nucleus. So if a potential voltage is applied to the
material no current will flow as there are no electrons to move and which gives these materials their insulati ng properties.
Insulators also have very high resistances, millions of ohms per metre, and are generally not affected by normal
temperature changes (although at very high temperatures wood becomes charcoal and changes from an insulator to a
conductor). Examples of good insulators are marble, fused quartz, p.v.c. plastics, rubber etc.
Insulators play a very important role within electrical and electronic circuits, because without them electrical circuits would
short together and not work. For example, insulators made of glass or porcelain are used for insulating and supporting
overhead transmission cables while epoxy-glass resin materials are used to make printed circuit boards, PCB's etc.
Semi-conductors
Semi-conductors materials such as Silicon and Germanium, have electrical properties somewhere in the middle,
between those of a "Conductor" and an "Insulator". They are not good conductors nor good insulators (hence their name
semi-conductors). They have very few "fee electrons" because their atoms are closely grouped together in a crystalline
pattern called a "Crystal Lattice". However, their ability to conduct electricity can be greatly improved by adding certain
"Impurities" to this crystalline structure thereby, producing more free electrons than holes or vice versa. By controlling th e
amount of impurities added to the semiconductor material it is possible to control its conductivity. This process of adding
impurity atoms to semiconductor atoms (the order of 1 impurity atom per 10 million (or more) atoms of the semiconductor)
is called Doping.
The most commonly used semiconductor material is Silicon. It has four valence electrons in its outer most shell which it
shares with its adjacent atoms in forming covalent bonds. The structure of the bond between two silicon atoms is such
that each atom shares one electron with its neighbour making the bond very stable. As there are very few free electrons
available to move from place to place producing an electrical current, crystals of pure silicon (or germanium) are therefore
good insulators, or at the very least very high value resistors. Silicon atoms are arranged in a definite symmetrical pattern
making them a crystalline solid structure. A crystal of pure silicon (silicon dioxide or glass) is generally said to be an
intrinsic crystal.
57
The diagram above shows the structure and lattice of a 'normal' pure crystal of Silicon.
5.3 N-type Semiconductor.
In order for our silicon crystal to conduct electricity, we need to introduce an impurity atom such as Arsenic, Antimony or
Phosphorus into the crystalline structure. These atoms have five outer electrons in their outermost co-valent bond to
share with other atoms and are commonly called "Pentavalent" impurities. This allows four of the five electrons to bond
with its neighbouring silicon atoms leaving one "free electron" to move about when an electrical voltage is applied
(electron flow). As each impurity atom "donates" one electron, pentavalent atoms are generally known as "Donors".
Antimony (symbol Sb) is frequently used as a pentavalent additive as it has 51 electrons arranged in 5 shells around the
nucleus. The resulting semiconductor material has an excess of current-carrying electrons, each with a negative charge,
and is therefore referred to as "N-type" material with the electrons called "Majority Carriers" and the resultant holes
"Minority Carriers".
The diagram above shows the structure and lattice of the donor impurity atom Antimony.
58
5.4 P-Type Semiconductor.
If we go the other way, and introduce a "Trivalent" (3-electron) impurity into the crystal structure, such as Aluminium,
Boron or Indium, only three valence electrons are available in the outermost covalent bond meaning that the fourth bond
cannot be formed. Therefore, a complete connection is not possible, giving the semiconductor material an abundance of
positively charged carriers known as "holes" in the structure of the crystal. As there is a hole an adjoining free electron is
attracted to it and will try to move into the hole to fill it. However, the electron filling the hole leaves another hole beh ind it
as it moves. This in turn attracts another electron which in turn creates another hole behind, and so forth giving the
appearance that the holes are moving as a positive charge through the crystal structure (conventional current flow). As
each impurity atom generates a hole, trivalent impurities are generally known as "Acceptors" as they are continually
"accepting" extra electrons. Boron (symbol B) is frequently used as a trivalent additive as it has only 5 electrons arranged
in 3 shells around the nucleus. Addition of Boron causes conduction to consist mainly of positive charge carriers results in
a "P-type" material and the positive holes are called "Majority Carriers" while the free electrons are called "Minority
Carriers".
The diagram above shows the structure and lattice of the acceptor impurity atom Boron.
Summary
N-type (e.g. add Antimony)
These are materials which have Pentavalent impurity atoms (Donors) added and conduct by "electron" movement and
are called, N-type Semiconductors.
In these types of materials are:
•
•
•
•
o
o
•
1. The Donors are positively charged.
2. There are a large number of free electrons.
3. A small number of holes in relation to the number of free electrons.
4. Doping gives:
positively charged donors.
negatively charged free electrons.
5. Supply of energy gives:
59
o
o
negatively charged free electrons.
positively charged holes.
P-type (e.g. add Boron)
These are materials which have Trivalent impurity atoms (Acceptors) added and conduct by "hole" movement and are
called, P-type Semiconductors.
In these types of materials are:
•
•
•
•
o
o
•
o
o
1. The Acceptors are negatively charged.
2. There are a large number of holes.
3. A small number of free electrons in relation to the number of holes.
4. Doping gives:
negatively charged acceptors.
positively charged holes.
5. Supply of energy gives:
positively charged holes.
negatively charged free electrons.
and both P and N-types as a whole, are electrically neutral.
5.5 The PN-junction
In the previous tutorial we saw how to make an N-type Semiconductor material by doping it with Antimony and also how to
make a P-type Semiconductor material by doping that with Boron. This is all well and good, but these semiconductor N and
P-type materials do very little on their own as they are electrically neutral, but when we join (or fuse) together these two
materials they behave in a very different way producing what is generally known as a P-N Junction.
When the N and P-type semiconductor materials are first brought together some of the free electrons move across the
junction to fill up the holes in the P-type material producing negative ions, but because the electrons have moved they
leave behind positive ions on the negative N-side and the holes move across the junction in the opposite direction into the
region where there are large numbers of free electrons. This movement of electrons and holes across the junction is
known as diffusion. This process continues until the number of electrons which have crossed the junction have a large
enough electrical charge to repel or prevent any more carriers from crossing the junction. Eventually a state of equilibrium
(electrically neutral situation) will occur producing a "Potential Barrier" zone around the area of the junction as the donor
atoms repel the holes and the acceptor atoms repel the electrons. Since no free charge carriers can rest in a position
where there is a potential barrier it is therefore "depleted" of any free mobile carriers, and this area around the junction is
now called the Depletion Layer.
The PN-junction
60
As the N-type material has lost electrons and the P-type has lost holes, the N-type material has become positive with
respect to the P-type. The external voltage required to overcome this barrier potential that now exists and allow electrons
to move freely across the junction is very much dependent upon the type of semiconductor material used and its actual
temperature, and for Silicon this is about 0.6 - 0.7 volts and for Germanium it is about 0.3 - 0.35 volts. This potential
barrier will always exist even if the device is not connected to any external power source.
The significance of this built-in potential is that it opposes both the flow of holes and electrons across the junction and is
why it is called the potential barrier. In practice, a PN-junction is formed within a single crystal of material rather than just
simply joining or fusing together two separate pieces. Electrical contacts are also fused onto either side of the crystal to
enable an electrical connection to be made to an external circuit.
Then the resulting device that has been made is called a PN-junction Diode or Rectifier Diode.
Chapter 6 Semiconductor Diodes and Types
6.1 The PN Junction Diode
The effect described in the previous tutorial is achieved without any external voltage potential being applied to the actual
PN-junction. However, if we were to make electrical connections at the ends of both the N-type and the P-type materials
and then connect them to a battery in the appropriate direction, the depletion layer around the junction can be increased
or decreased thereby increasing or decreasing the effective resistance of the junction itself. The behaviour of the PNjunction with regards to the potential barrier size produces an asymmetrical conducting device, better known as the
Junction Diode.
6.2 The Basic Diode Symbol and Static I-V Characteristics.
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But before we can use the PN-junction as a practical device or as a rectifying device we need to firstly "Bias" the junction,
ie connect a voltage potential across it. On the voltage axis above "Reverse Bias" refers to an external voltage potential
which increases the potential barrier. An external voltage which decreases the potential barrier is said to act in the
"Forward Bias" direction.
There are 3 possible "biasing" conditions for the standard Junction Diode and these are:
1.Zero Bias - No external voltage potential is applied to the PN-junction.
2. Reverse Bias - The voltage potential is connected negative,
(-ve) to the P-type material and
positive, (+ve) to the N-type material across the diode which has the effect of Increasing the
PN-junction width.
3. Forward Bias - The voltage potential is connected positive, (+ve) to the P-type material and
negative, (-ve) to the N-type material across the diode which has the effect of Decreasing the
PN-junction width.
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Zero Bias.
When a diode is connected in a Zero Bias condition, no external potential energy is applied to the PN-junction. However if
the diodes terminals are shorted together, a few holes (majority carriers) in the P-type material with enough energy to
overcome the potential barrier will move across the junction against this barrier potential. This is known as the "Forward
Current" and is referenced as IF
Likewise, holes generated in the N-type material (minority carriers), find this situation favourable and move across the
junction in the opposite direction. This is known as the "Reverse Current" and is referenced as IR, as shown below.
Zero Biased Diode.
An "Equilibrium" or balance will be established when the two currents are equal and both moving in opposite directions, so
that the net result is zero current flowing in the circuit. When this occurs the junction is said to be in a state of " Dynamic
Equilibrium".
This state of equilibrium can be broken by raising the temperature of the PN-junction causing an increase in the
generation of minority carriers, thereby resulting in an increase in leakage current.
Reverse Bias.
When a diode is connected in a Reverse Bias condition, a positive voltage is applied to the N-type material and a
negative voltage is applied to the P-type material. The positive voltage applied to the N-type material attracts electrons
towards the positive electrode and away from the junction, while the holes in the P-type end are also attracted away from
the junction towards the negative electrode. The net result is that the depletion layer grows wider due to a lack of
electrons and holes and presents a high impedance path, almost an insulator. The result is that a high potential barrier is
created thus preventing current from flowing through the semiconductor material.
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A Reverse Biased Junction showing the Increase in the Depletion Layer.
This condition represents the high resistance direction of a PN-junction and practically zero current flows through the
diode with an increase in bias voltage. However, a very small leakage current does flow through the junction which can
be measured in microamperes, (μA). One final point, if the reverse bias voltage Vr applied to the junction is increased to
a sufficiently high enough value, it will cause the PN-junction to overheat and fail due to the avalanche effect around the
junction. This may cause the diode to become shorted and will result in maximum circuit current to flow, Ohm's Law and
this shown in the reverse characteristics curve below.
Reverse Characteristics Curve for a Diode.
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Sometimes this avalanche effect has practical applications in voltage stabilising circuits where a series limiting resistor i s
used with the diode to limit this reverse breakdown current to a preset maximum value thereby producing a fixed voltage
output across the diode. These types of diodes are commonly known as Zener Diodes and are discussed in a later
tutorial.
Forward Bias.
When a diode is connected in a Forward Bias condition, a negative voltage is applied to the N-type material and a
positive voltage is applied to the P-type material. If this external voltage becomes greater than the value of the potential
barrier, 0.7 volts for Silicon and 0.3 volts for Germanium, the potential barriers opposition will be overcome and current will
start to flow as the negative voltage pushes or repels electrons towards the junction giving them the energy to cross over
and combine with the holes being pushed in the opposite direction towards the junction by the positive voltage. This
results in a characteristics curve of zero current flowing up to this "knee" voltage and high current flow through the diode
with little increase in the external voltage as shown below.
Forward Characteristics Curve for a Diode.
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This results in the depletion layer becoming very thin and narrow and which now represents a low impedance path
thereby producing a very small potential barrier and allowing high currents to flow. The point at which this takes pla ce is
represented on the static I-V characteristics curve above as the "knee" point.
Forward Biased Junction Diode showing a Reduction in the Depletion Layer.
This condition represents the low resistance direction in a PN-junction allowing very large currents to flow through the
diode with only a small increase in bias voltage. The actual potential difference across the junction or diode is kept
constant by the action of the depletion layer at about 0.3v for Germanium and about 0.7v for Silicon diodes. Since the
diode can conduct "infinite" current above this knee point as it effectively becomes a short circuit, resistors are used in
series with the device to limit its current flow. Exceeding its maximum forward current specification causes the device to
dissipate more power in the form of heat than it was designed for resulting in failure of the device.
Summary
66
1). Semiconductors contain two types of mobile charge carriers, Holes and Electrons.
2). The holes are positively charged while the electrons negatively charged.
3). A semiconductor may be doped with donor impurities such as Antimony (N-type doping), so that it contains mobile
charges which are primarily electrons.
4). A semiconductor may be doped with acceptor impurities such as Boron (P-type doping), so that it contains mobile
charges which are mainly holes.
5). When a diode is Zero Biased no external energy source is applied and a natural Potential Barrier is developed
across a PN-junction which is about 0.7v for Silicon diodes and about 0.3v for Germanium diodes.
6). When a diode is Forward Biased the PN-junction is "reduced" and current flows through the diode.
7). When a diode is Reverse Biased the PN-junction is "increased" and zero current flows, (only a very small leakage
current).
6.3 The Zener Diode
In the previous Signal Diode tutorial we saw that a "reverse biased" diode passes very little current but will suffer
breakdown or damage if the reverse voltage applied across it is made to high. However, Zener Diodes or "Breakdown
Diodes" as they are sometimes called, are basically the same as the standard junction diode but are specially made to
have a low pre-determined Reverse Breakdown Voltage, called the "Zener Voltage" ( Vz). In the forward direction it
behaves just like a normal signal diode passing current, but when the reverse voltage applied to it exceeds the selected
reverse breakdown voltage a process called Avalanche Breakdown occurs in the depletion layer and the current through
the diode increases to the maximum circuit value, which is usually limited by a series resistor. The point at which current
flows can be very accurately controlled (to less than 1% tolerance) in the doping stage of the diodes construction giving it
a specific Zener Breakdown voltage (Vz) ranging from a few volts up to a few hundred volts.
Zener Diode I-V Characteristics
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Zener Diodes are used in the "REVERSE" bias mode, i.e. the anode connects to the negative supply, and from its I -V
characteristics curve above, we can see that the Zener diode has a region in its reverse bias characteristics of almost a
constant voltage regardless of the current flowing through the diode. This voltage across the diode (it's Zener Voltage, Vz)
remains nearly constant even with large changes in current through the diode caused by variations in the supply voltage
or load. This ability to control itself can be used to great effect to regulate or stabilise a voltage source against supply or
load variations. The diode will continue to regulate until the diode current falls below the minimum Iz value in the reverse
breakdown region.
6.4 The Zener Regulator
Zener Diodes can be used to produce a stabilised voltage output by passing a small current through it from a voltage
source via a suitable current limiting resistor, ( RS). We remember from the previous tutorials that the DC output voltage
from the half or full-wave rectifiers contains ripple superimposed onto the DC voltage and that as the load value changes
so to does the average output voltage. By connecting a simple Zener stabiliser circuit as shown below across the output of
the rectifier a more stable reference voltage can be produced.
Zener Diode Stabiliser
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The resistor, RS is connected in series with the Zener diode to limit the current flow with the output from the voltage
source, VS being connected across the combination while the stabilised output voltage Voutis taken from across the Zener
diode. The Zener diode is connected with its Cathode terminal connected to the positive rail of the DC supply so it is
reverse biased and will be operating in its breakdown condition. Resistor RS is selected so to limit the maximum current
flowing in the circuit. When no load resistance, RL is connected to the circuit, no load current ( IL = 0), is drawn and all the
circuit current passes through the Zener diode which dissipates its maximum power. Care must be taken when selecting
the appropriate value of resistance that the Zener maximum power rating is not exceeded under this "no-load" condition.
There is a minimum Zener current for which the stabilization of the voltage is effective and the Zener current must stay
above this value operating within its breakdown region at all times. The upper limit of current is of course dependant upon
the power rating of the device.
Example No1.
A 5.0v stabilised power supply is required from a 12v d.c. input source. The maximum power rating of the
Zener diode is 2W. Using the circuit above calculate:
a) The maximum current flowing in the Zener Diode.
b) The value of the series resistor, RS
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c) The load current IL if a load resistor of 1kΩ is connected across the Zener diode.
d) The total supply current IS
Zener Values
As well as producing a single stabilised voltage output, Zener diodes can also be joined together in series along with
normal silicon signal diodes to produce a variety of different reference voltage values as shown below.
Zener Diodes Connected in Series
The values of the individual Zener diodes can be chosen to suit the application while the silicon diode drops about 0.7v in
the forward bias condition. The supply voltage, Vin must of course be higher than the largest output reference voltage and
in our example above this is 19v.
A typical Zener diode is the 500mW BZX55 series or the larger 1.3W BZX85 series were the Zener voltage is given as for
example, C7V5 for a 7.5V device giving a diode reference of BZX55C7V5. The individual voltage values for these small
but very useful diodes are given in the table below.
BZX55 Zener Diode Power Rating 500mW
2.4V
2.7V
3.0V
3.3V
3.6V
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3.9V
4.3V
4.7V
5.1V
5.6V
6.2V
6.8V
7.5V
8.2V
9.1V
10V
11V
12V
13V
15V
16V
18V
20V
22V
24V
27V
30V
33V
36V
39V
43V
47V
BZX85 Zener Diode Power Rating 1.3W
3.3V
3.6V
3.9V
4.3V
4.7V
5.1V
5.6
6.2V
6.8V
7.5V
8.2V
9.1V
10V
11V
12V
13V
15V
16V
18V
20V
22V
24V
27V
30V
33V
36V
39V
43V
47V
51V
56V
62V
6.5 Light Emitting Diodes
Light Emitting Diodes or LED´s, are among the most widely used of all the types of diodes available. They are the most
visible type of diode, that emits a fairly narrow bandwidth of either visible coloured light, invisible infra -red or laser type
light when a forward current is passed through them. A "Light Emitting Diode" or LED as it is more commonly called, is
basically just a specialised type of PN-junction diode, made from a very thin layer of fairly heavily doped semiconductor
material. When the diode is Forward Biased, electrons from the semiconductors conduction band combine with holes from
the valence band, releasing sufficient energy to produce photons of light. Because of this thin layer a reasonable number
of these photons can leave the junction and radiate away producing a coloured light output.
Typical LED Characteristics
Semiconductor
Material
Wavelength Colour
VF @ 20mA
Unlike normal diodes which are made for detection or
power rectification, and which are generally made from GaAs
850-940nm
Infra-Red 1.2v
either Germanium or Silicon semiconductor material,
Light Emitting Diodes are made from compound type GaAsP
630-660nm
Red
1.8v
semiconductor materials such as Gallium Arsenide
(GaAs), Gallium Phosphide (GaP), Gallium Arsenide GaAsP
605-620nm
Amber
2.0v
Phosphide (GaAsP), Silicon Carbide (SiC) or Gallium
Indium Nitride (GaInN). The exact choice of the GaAsP:N
585-595nm
Yellow
2.2v
semiconductor material used will determine the overall
550-570nm
Green
3.5v
430-505nm
Blue
3.6v
450nm
White
4.0v
wavelength of the photon light emissions and therefore GaP
the resulting colour of the light emitted, as in the case of
the visible light coloured LEDs, (RED, AMBER, GREEN SiC
etc)
GaInN
From the table above we can see that the main P-type dopant used in the manufacture of Light Emitting Diodes is
Gallium(Ga, atomic number 31) and the main N-type dopant used is Arsenic (As, atomic number 31) giving the resulting
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Gallium Arsenide (GaAs) crystal structure, which has the characteristics of radiating significant amounts of infrared
radiation from its junction when a forward current is flowing through it. By also adding Phosphorus (P, atomic number 15),
as a third dopant the overall wavelength of the emitted radiation is reduced to give visible red light to the human eye.
Further refinements in the doping process of the PN-junction have resulted in a range of colours available from red,
orange and amber through to yellow, and the recently developed blue LED which is achieved by injecting nitrogen atoms
into the crystal structure during the doping process.
6.6 Light Emitting Diodes I-V Characteristics.
Light Emitting Diode (LED) Schematic symbol and its I-V Characteristics Curves showing the different colours
available.
Before a light emitting diode can "emit" any form of light it needs a current to flow through it, as it is a current dependan t
device. As the LED is to be connected in a forward bias condition across a power supply it should be Current Limited
using a series resistor to protect it from excessive current flow. From the table above we can see that each LED has its
own forward voltage drop across the PN-junction and this parameter which is determined by the semiconductor material
used is the forward voltage drop for a given amount of forward conduction current, typically for a forward current of 20mA.
In most cases LEDs are operated from a low voltage DC supply, with a series resistor to limit the forward current to a
suitable value from say 5mA for a simple LED indicator to 30mA or more where a high brightness light output is needed.
6.6.1 LED Series Resistance.
The series resistor value RS is calculated by simply using Ohm´s Law, knowing the required forward current IF, the
supply voltage VS and the expected forward voltage drop of the LED, VF at this current level as shown below.
LED Series Resistor Circuit
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Example No1.
An amber coloured LED is to be connected to a 5.0v stabilised power supply. Using the circuit above calculate the series
resistor required to limit the forward current to less than 10mA.
We remember from the Resistors tutorials, that resistors come in standard preferred values. Our calculation shows to
limit the current flowing through the LED to 10mA exactly, we would require a 300Ω resistor. In the E12 series of
resistors there is no 300Ω resistor so we would need to choose the next highest value, which is 330Ω. A quick
recalculation shows the new forward current value is now 9.1mA, and this is ok.
The brightness of a LED cannot be controlled by simply varying the current flowing through it. Allowing more current to
flow through the LED will make it glow brighter but will also cause it to dissipate more heat. LEDs are designed to produce a
set amount of light operating with a forward current of about 10 to 20mA. In situations where power savings are
important, less current may be possible. However, reducing the current to below say 5mA may dim its light output to much
or even turn the LED "OFF" completely. A much better way to control the brightness of LEDs is to use a control process
known as "Pulse Width Modulation" or PWM, in which the LED is turned "ON" and "OFF" continuously at varying
frequencies depending upon the required light intensity.
When higher light output is required, a pulsed current with a fairly short duty cycle ("ON-OFF" Ratio) allows the current
and therefore the light output to be increased significantly during the actual pulses, while still keeping the LEDs av erage
current level and power dissipation within its limits. The human eyes fills in the gaps between the "ON" and "OFF" light
pulses, providing the pulse frequency is high enough, making it appear as a continuous light output. So pulses at a
frequency of 100Hz or more actually appear brighter to the eye than continuous light of the same average intensity.
6.6.2 Multi-LEDs
LEDs are available in a wide range of shapes, colours and various sizes with different light output intensities available,
with the most common (and cheapest to produce) being the standard 5mm Red LED. LED's are also available in various
"packages" arranged to produce both letters and numbers with the most common being that of a "Seven-Segment
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Display" arrangement. Nowadays, full colour flat screen LED displays are available with a large number of dedicated I.C.s
available for driving the displays directly.
Most LEDs produce a single output colour however, multi-colour LEDs are now available that can produce a variety of
colours within a single device. These are actually 2 LEDs fabricated within a single package as shown below.
A Bi-colour LED
Terminal A
LED
Selected
AC
+
-
LED 1
ON
OFF
ON
LED 2
OFF
ON
ON
Colour
Green
Red
Yellow
Here the LEDs are connected in "inverse parallel", so that the colour red is emitted when the device is connected in one
direction and the colour green is emitted when it is biased in the other direction. This type of arrangement is useful for
giving polarity indication, for example correct connection of batteries, power supplies etc. Also, both LEDs would take it in
turn to light if the device was connected (via suitable resistor) to a low voltage, low frequency AC supply.
A Multi or Tri-colour LED
Output
Colour
LED
1
Current
LED
Red
Orange
Yellow
Green
0
5mA
9.5mA
15mA
10mA
6.5mA
3.5mA
0
2
Current
This multicoloured LED comprises of a single Red and Green LED with the cathode terminals connected together. This
device gives out a Red or a Green colour by turning "ON" only one LED at a time. It can also generate additional shades
of colours such as Orange or Yellow by turning "ON" the two LEDs in different ratios of forward current as shown in the
table thereby generating 4 different colours from just two diode junctions.
6.6.3 LED Displays
As well as individual colour or multi-colour LEDs, light emitting diodes can be combined together in a single package to
produce displays such as bargraphs, strips, arrays and 7-segment displays. A 7-segment LED display provides a very
convenient way of displaying information or digital data in the form of Numbers, Letters or even Alpha-numerical
characters and as their name suggests, they consist of 7 individual LEDs (the segments), within one single display
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package. In order to produce the required numbers or characters from 0 to 9 and A to F respectively, on the display the
correct combination of LED segments need to be illuminated. A standard 7 -segment LED display generally has 8 input
connections, one for each LED segment and one that acts as a common terminal or connection for all the internal
segments.
There are two important types of 7-segment LED digital display.
The Common Cathode Display (CCD)
In the common cathode display, all the cathode connections of the LEDs are joined together
and the individual segments are illuminated by application of a HIGH, logic "1" signal.
The Common Anode Display (CAD)
In the common anode display, all the anode connections of the LEDs are joined together
and the individual segments are illuminated by connecting the terminals to a LOW, logic "0"
signal.
Typical 7-segment Display
Chapter 7 Current Controlled Transistors
7.1 Bipolar Transistor Basics
In the Diode tutorials we saw that simple diodes are made up from two pieces of semiconductor material, either Silicon or
Geranium to form a simple PN-junction and we also learnt about their properties and characteristics. If we now join
together two individual diodes end to end giving two PN-junctions connected together in series, we now have a three
layer, two junction, three terminal device forming the basis of a Bipolar Junction Transistor, or BJT for short. This type
of transistor is generally known as a Bipolar Transistor, because its basic construction consists of two PN-junctions with
each terminal or connection being given a name to identify it and these are known as the Emitter, Base and Collector
respectively.
The word Transistor is an acronym, and is a combination of the words Transfer Varistor used to describe their mode of
operation way back in their early days of development. There are two basic types of bipolar transistor construction, NPN
and PNP, which basically describes the physical arrangement of the P-type and N-type semiconductor materials from
which they are made. Bipolar Transistors are "CURRENT" Amplifying or current regulating devices that control the
amount of current flowing through them in proportion to the amount of biasing current applied to their base terminal. The
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principle of operation of the two transistor types NPN and PNP, is exactly the same the only difference being in the
biasing (base current) and the polarity of the power supply for each type.
7.2 Bipolar Transistor Construction
The construction and circuit symbols for both the NPN and PNP bipolar transistor are shown above with the arrow in the
circuit symbol always showing the direction of conventional current flow between the base terminal and its emitter
terminal, with the direction of the arrow pointing from the positive P-type region to the negative N-type region, exactly the
same as for the standard diode symbol.
There are basically three possible ways to connect a Bipolar Transistor within an electronic circuit with each method of
connection responding differently to its input signal as the static characteristics of the transistor vary with each circuit
arrangement.
•
1. Common Base Configuration - has Voltage Gain but no Current Gain.
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•
•
•
•
2. Common Emitter Configuration - has both Current and Voltage Gain.
3. Common Collector Configuration - has Current Gain but no Voltage Gain.
7.3 The Common Base Configuration.
As its name suggests, in the Common Base or Grounded Base configuration, the BASE connection is common to both
the input signal AND the output signal with the input signal being applied between the base and the emitter terminals. The
corresponding output signal is taken from between the base and the collector terminals as shown with the base terminal
grounded or connected to a fixed reference voltage point. The input current flowing into the emitter is quite large as its the
sum of both the base current and collector current respectively therefore, the collector current output is less than the
emitter current input resulting in a Current Gain for this type of circuit of less than "1", or in other words it "Attenuates" the
signal.
The Common Base Amplifier Circuit
This type of amplifier configuration is a non-inverting voltage amplifier circuit, in that the signal voltages Vin and Vout are
In-Phase. This type of arrangement is not very common due to its unusually high voltage gain characteristics. Its Output
characteristics represent that of a forward biased diode while the Input characteristics represent that of an illuminated
photo-diode. Also this type of configuration has a high ratio of Output to Input resistance or more importantly "Load"
resistance (RL) to "Input" resistance ( Rin) giving it a value of "Resistance Gain". Then the Voltage Gain for a common
base can therefore be given as:
Common Base Voltage Gain
The Common Base circuit is generally only used in single stage amplifier circuits such as microphone pre-amplifier or RF
radio amplifiers due to its very good high frequency response.
7.4 The Common Emitter Configuration.
In the Common Emitter or Grounded Emitter configuration, the input signal is applied between the base, while the output
is taken from between the collector and the emitter as shown. This type of configuration is the most commonly used circuit
for transistor based amplifiers and which represents the "normal" method of connection. The common emitter amplifier
configuration produces the highest voltage, current and power gain of all the three bipolar transistor configurations. This is
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mainly because the input impedance is LOW as its connected to a forward-biased junction, while the output impedance is
HIGH as it is taken from a reverse-biased junction.
The Common Emitter Amplifier Circuit
In this type of configuration, the current flowing out of the transistor must be equal to the currents flowing into the
transistor as the emitter current is given as Ie = Ic + Ib. Also, as the load resistance ( RL) is connected in series with the
collector, the Current gain of the Common Emitter Transistor Amplifier is quite large as it is the ratio of Ic/Ib and is given
the symbol of Beta, (β). Since the relationship between these three currents is determined by the transistor itself, any
small change in the base current will result in a large change in the collector current. Then, small changes in base current
will thus control the current in the Emitter/Collector circuit.
By combining the expressions for both Alpha, α and Beta, β the mathematical relationship between these parameters
and therefore the current gain of the amplifier can be given as:
Where: "Ic" is the current flowing into the collector terminal, "Ib" is the current flowing into the base terminal and "Ie" is the
current flowing out of the emitter terminal.
Then to summarise, this type of bipolar transistor configuration has a greater input impedance, Current gain and Power
gain than that of the common base configuration but its Voltage gain is much lower. The common emitter is an inverting
amplifier circuit resulting in the output signal being 180o out of phase with the input voltage signal.
7.5 The Common Collector Configuration.
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In the Common Collector or Grounded Collector configuration, the collector is now common and the input signal is
connected to the base, while the output is taken from the emitter load as shown. This type of configuration is commonly
known as a Voltage Follower or Emitter Follower circuit. The emitter follower configuration is very useful for impedance
matching applications because of the very high input impedance, in the region of hundreds of thousands of Ohms, and it
has relatively low output impedance.
The Common Collector Amplifier Circuit
The common emitter configuration has a current gain equal to the β value of the transistor itself. In the common collector
configuration the load resistance is situated in series with the emitter so its current is equal to that of the emitter curre nt.
As the emitter current is the combination of the collector AND base currents combined, the load resistance in this type of
amplifier configuration also has both the collector current and the input current of the base flowing through it. Then the
current gain of the circuit is given as:
This type of bipolar transistor configuration is a non-inverting amplifier circuit in that the signal voltages of Vin and Vout
are "In-Phase". It has a voltage gain that is always less than "1" (unity). The load resistance of the common collector
amplifier configuration receives both the base and collector currents giving a large current gain (as with the Common
Emitter configuration) therefore, providing good current amplification with very little voltage gain.
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Bipolar Transistor Summary.
The behaviour of the bipolar transistor in each one of the above circuit configurations is very different and produces
different circuit characteristics with regards to Input impedance, Output impedance and Gain and this is summarised in the
table below.
Transistor Characteristics
The static characteristics for Bipolar Transistor amplifiers can be divided into the following main groups.
Input Characteristics:-
Output Characteristics:-
Common Base -
IE ÷ VEB
Common Emitter -
IB ÷ VBE
Common Base -
IC ÷ VC
Common Emitter -
I C ÷ VC
Transfer Characteristics:- Common Base Common Emitter -
IE ÷ IC
IB ÷ IC
with the characteristics of the different transistor configurations given in the following table:
Characteristic
Common
Base
Common
Emitter
Common
Collector
Input impedance
Low
Medium
High
Output impedance
Very High
High
Low
Phase Angle
0o
180o
0o
Voltage Gain
High
Medium
Low
Current Gain
Low
Medium
High
Power Gain
Low
Very High
Medium
7.5.1 NPN Transistors
In the previous tutorial we saw that the standard Bipolar Transistor or BJT, comes in two basic forms. An NPN
(Negative-Positive-Negative) type and a PNP (Positive-Negative-Positive) type, with the most commonly used transistor
type being NPN Transistors. We also learnt that the transistor junctions can be biased in one of three different ways Common Base, Common Emitter and Common Collector. In this tutorial we will look more closely at the "Common
Emitter" configuration using NPN Transistors and an example of its current flow characteristics is given below.
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NPN Transistor Configuration
Note: Conventional current flow.
We know that the transistor is a "CURRENT" operated device and that a large current ( Ic) flows freely through the device
between the collector and the emitter terminals. However, this only happens when a small biasing current ( Ib) is flowing
into the base terminal of the transistor thus allowing the base to act as a sort of current control input. The ratio of these
two currents (Ic/Ib) is called the DC Current Gain of the device and is given the symbol of hfe or nowadays Beta, (β).
Beta has no units as it is a ratio. Also, the current gain from the emitter to the collector terminal, Ic/Ie, is called Alpha,
(α), and is a function of the transistor itself. As the emitter current Ie is the product of a very small base current to a very
large collector current the value of this parameter α is very close to unity, and for a typical low-power signal transistor this
value ranges from about 0.950 to 0.999.
α and β Relationships
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By combining the two parameters α and β we can produce two mathematical expressions that gives the relationship
between the different currents flowing in the transistor.
The values of Beta vary from about 20 for high current power transistors to well over 1000 for high frequency low power
type bipolar transistors. The equation for Beta can also be re-arranged to make Ic as the subject, and with zero base
current (Ib = 0) the resultant collector current Ic will also be zero, (β x 0). Also when the base current is high the
corresponding collector current will also be high resulting in the base current controlling the collector current. One of the
most important properties of the Bipolar Junction Transistor is that a small base current can control a much larger
collector current. Consider the following example.
Example No1.
An NPN Transistor has a DC current gain, (Beta) value of 200. Calculate the base current Ib required to switch a resistive
load of 4mA.
Therefore, β = 200, Ic = 4mA and Ib = 20µA.
One other point to remember about NPN Transistors. The collector voltage, (Vc) must be greater than the emitter
voltage, (Ve) to allow current to flow through the device between the collector-emitter junction. Also, there is a voltage
drop between the base and the emitter terminal of about 0.7v for silicon devices as the input characteristics of an NPN
Transistor are of a forward biased diode. Then the base voltage, ( Vbe) of an NPN Transistor must be greater than this 0.7
V otherwise the transistor will not conduct with the base current given as.
Where: Ib is the base current, Vb is the base bias voltage, Vce is the base-emitter volt drop (0.7v) and Rb is the base
input resistor.
Example No2.
An NPN Transistor has a DC base bias voltage, Vb of 10v and an input base resistor, Rb of 100kΩ. What will be the
value of the base current into the transistor.
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Therefore, Ib = 93µA.
7.5.2 The Common Emitter Configuration.
As well as being used as a switch to turn load currents "ON" or "OFF" by controlling the Base signal to the transistor, NPN
Transistors can also be used to produce a circuit which will also amplify any small AC signal applied to its Base terminal. If
a suitable DC "biasing" voltage is firstly applied to the transistors Base terminal thus allowing it to always operate within its
linear active region, an inverting amplifier circuit called a Common Emitter Amplifier is produced.
One such Common Emitter Amplifier configuration is called a Class A Amplifier. A Class A Amplifier operation is one
where the transistors Base terminal is biased in such a way that the transistor is always operating halfway between its cutoff
and saturation points, thereby allowing the transistor amplifier to accurately reproduce the positive and negative halves of
the AC input signal superimposed upon the DC Biasing voltage. Without this "Bias Voltage" only the positive half of the
input waveform would be amplified. This type of amplifier has many applications but is commonly used in audio circuits
such as pre-amplifier and power amplifier stages.
With reference to the common emitter configuration shown below, a family of curves known commonly as the Output
Characteristics Curves, relates the output collector current, ( Ic) to the collector voltage, (Vce) when different values of
base current, (Ib) are applied to the transistor for transistors with the same β value. A DC "Load Line" can also be drawn
onto the output characteristics curves to show all the possible operating points when different values of base current are
applied. It is necessary to set the initial value of Vce correctly to allow the output voltage to vary both up and down when
amplifying AC input signals and this is called setting the operating point or Quiescent Point, Q-point for short and this is
shown below.
The Common Emitter Amplifier Circuit
Output Characteristics Curves for a Typical Bipolar Transistor
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The most important factor to notice is the effect of Vce upon the collector current Ic when Vce is greater than about 1.0
volts. You can see that Ic is largely unaffected by changes in Vce above this value and instead it is almost entirely
controlled by the base current, Ib. When this happens we can say then that the output circuit represents that of a
"Constant Current Source". It can also be seen from the common emitter circuit above that the emitter current Ie is the
sum of the collector current, Ic and the base current, Ib, added together so we can also say that " Ie = Ic + Ib " for the
common emitter configuration.
By using the output characteristics curves in our example above and also Ohm´s Law, the current flowing through the load
resistor, (RL), is equal to the collector current, Ic entering the transistor which inturn corresponds to the supply voltage,
(Vcc) minus the voltage drop between the collector and the emitter terminals, ( Vce) and is given as:
Also, a Load Line can be drawn directly onto the graph of curves above from the point of "Saturation" when Vce = 0 to
the point of "Cut-off" when Ic = 0 giving us the "Operating" or Q-point of the transistor. These two points are calculated
as:
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Then, the collector or output characteristics curves for Common Emitter NPN Transistors can be used to predict the
Collector current, Ic, when given Vce and the Base current, Ib. A Load Line can also be constructed onto the curves to
determine a suitable Operating or Q-point which can be set by adjustment of the base current.
7.5.3 PNP Transistors
PNP Transistors are the exact opposite to the NPN Transistors device we looked at previously. Basically, in this type of
transistor construction the two diodes are reversed with respect to the NPN type, with the arrow, which also defines the
Emitter terminal this time pointing inwards in the transistor symbol. Also, all the polarities are reversed which means that
PNP Transistors "sink" current as opposed to the NPN transistor which "sources" current. Then, PNP Transistors use a
small output base current and a negative base voltage to control a much larger emitter-collector current. The construction
of a PNP transistor consists of two P-type semiconductor materials either side of the N-type material as shown below.
PNP Transistor Configuration
Note: Conventional current flow.
PNP Transistors have very similar characteristics to their NPN bipolar cousins, except that the polarities (or biasing) of
the current and voltage directions are reversed for any one of the possible three configurations looked at in the first
tutorial, Common Base, Common Emitter and Common Collector. Generally, PNP Transistors require a negative ( -ve)
voltage at their Collector terminal with the flow of current through the emitter-collector terminals being Holes as opposed
to Electrons for the NPN types. Because the movement of holes across the depletion layer tends to be slower than for
electrons, PNP transistors are generally more slower than their equivalent NPN counterparts when operating.
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To cause the Base current to flow in a PNP transistor the Base needs to be more negative than the Emitter (current must
leave the base) by approx 0.7 volts for a silicon device or 0.3 volts for a germanium device with the formulas used to
calculate the Base resistor, Base current or Collector current are the same as those used for an equivalent NPN transistor
and is given as.
Generally, PNP transistors can replace NPN transistors in electronic circuits, the only difference is the polarities of the
voltages, and the directions of the current flow. PNP Transistors can also be used as switching devices and an example of a
PNP transistor switch is shown below.
7.5.4 PNP Transistor Circuit
The Output Characteristics Curves for a PNP transistor look very similar to those for an equivalent NPN transistor
except that they are rotated by 180o to take account of the reverse polarity voltages and currents, (the currents flowing out
of the Base and Collector in a PNP transistor are negative).
7.5.5 Transistor Matching
You may think what is the point of having PNP Transistors, when there are plenty of NPN Transistors available?. Well,
having two different types of transistors PNP & NPN, can be an advantage when designing amplifier circuits such as
Class B Amplifiers that use "Complementary" or "Matched Pair" transistors or for reversible H-Bridge motor control
circuits. A pair of corresponding NPN and PNP transistors with near identical characteristics to each other are called
Complementary Transistors for example, a TIP3055 (NPN), TIP2955 (PNP) are good examples of complementary or
matched pair silicon power transistors. They have a DC current gain, Beta, (Ic / Ib) matched to within 10% and high
Collector current of about 15A making them suitable for general motor control or robotic applications.
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7.6 Identifying PNP Transistors
We saw in the first tutorial of this Transistors section, that transistors are basically made up of two Diodes connected
together back-to-back. We can use this analogy to determine whether a transistor is of the type PNP or NPN by testing its
Resistance between the three different leads, Emitter, Base and Collector. By testing each pair of transistor leads in
both directions will result in six tests in total with the expected resistance values in Ohm's given below.
•
•
•
•
•
1. Emitter-Base Terminals - The Emitter to Base should act like a normal diode and conduct one way
only.
2. Collector-Base Terminals - The Collector-Base junction should act like a normal diode and conduct
one way only.
3. Emitter-Collector Terminals - The Emitter-Collector should not conduct in either direction.
Transistor Resistance Values for PNP and NPN types
Between Transistor Terminals
PNP
NPN
Collector
Emitter
RHIGH
RHIGH
Collector
Base
RLOW
RHIGH
Emitter
Collector
RHIGH
RHIGH
Emitter
Base
RLOW
RHIGH
Base
Collector
RHIGH
RLOW
Base
Emitter
RHIGH
RLOW
7.7 The Transistor as a Switch
When used as an AC signal amplifier, the transistors Base biasing voltage is applied so that it operates within its "Active"
region and the linear part of the output characteristics curves are used. However, both the NPN & PNP type bipolar
transistors can be made to operate as an "ON/OFF" type solid state switch for controlling high power devices such as
motors, solenoids or lamps. If the circuit uses the Transistor as a Switch, then the biasing is arranged to operate in the
output characteristics curves seen previously in the areas known as the "Saturation" and "Cut-off" regions as shown
below.
7.7.1 Transistor Curves
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The pink shaded area at the bottom represents the "Cut-off" region. Here the operating conditions of the transistor are
zero input base current (Ib), zero output collector current (Ic) and maximum collector voltage (Vce) which results in a large
depletion layer and no current flows through the device. The transistor is switched "Fully-OFF". The lighter blue area to
the left represents the "Saturation" region. Here the transistor will be biased so that the maximum amount of base current
is applied, resulting in maximum collector current flow and minimum collector emitter voltage which results in the depletion
layer being as small as possible and maximum current flows through the device. The transistor is switched "Fully -ON".
Then we can summarize this as:
•
•
•
1. Cut-off Region - Both junctions are Reverse-biased, Base current is zero or very small resulting in zero
Collector current flowing, the device is switched fully "OFF".
2. Saturation Region - Both junctions are Forward-biased, Base current is high enough to give a CollectorEmitter voltage of 0v resulting in maximum Collector current flowing, the device is switched fully "ON".
An example of an NPN Transistor as a switch being used to operate a relay is given below. With inductive loads such as
relays or solenoids a flywheel diode is placed across the load to dissipate the back EMF generated by the inductive load
when the transistor switches "OFF" and so protect the transistor from damage. If the load is of a very high current or
voltage nature, such as motors, heaters etc, then the load current can be controlled via a suitable relay as shown.
7.7.2 Transistor Switching Circuit
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The circuit resembles that of the Common Emitter circuit we looked at in the previous tutorials. The difference this time is
that to operate the transistor as a switch the transistor needs to be turned either fully "OFF" (Cut-off) or fully "ON"
(Saturated). An ideal transistor switch would have an infinite resistance when turned "OFF" resulting in zero current flow
and zero resistance when turned "ON", resulting in maximum current flow. In practice when turned "OFF", small leakage
currents flow through the transistor and when fully "ON" the device has a low resistance value causing a small saturation
voltage (Vce) across it. In both the Cut-off and Saturation regions the power dissipated by the transistor is at its minimum.
To make the Base current flow, the Base input terminal must be made more positive than the Emitter by increasing it
above the 0.7 volts needed for a silicon device. By varying the Base-Emitter voltage Vbe, the Base current is altered and
which in turn controls the amount of Collector current flowing through the transistor as previously discussed. When
maximum Collector current flows the transistor is said to be Saturated. The value of the Base resistor determines how
much input voltage is required and corresponding Base current to switch the transistor fully "ON".
Example No1.
For example, using the transistor values from the previous tutorials of: β = 200, Ic = 4mA and Ib = 20uA, find the
value of the Base resistor (Rb) required to switch the load "ON" when the input terminal voltage exceeds 2.5v.
Example No2.
Again using the same values, find the minimum Base current required to turn the transistor fully "ON" (Saturated) for a
load that requires 200mA of current.
Transistor switches are used for a wide variety of applications such as interfacing large current or high voltage devices
like motors, relays or lamps to low voltage digital logic IC's or gates like AND Gates or OR Gates. Here, the output from a
digital logic gate is only +5v but the device to be controlled may require a 12 or even 24 volts supply. Or the load such as
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a DC Motor may need to have its speed controlled using a series of pulses (Pulse Width Modulation) and transistor
switches will allow us to do this faster and more easily than with conventional mechanical switches.
7.7.3 Digital Logic Transistor Switch
The base resistor, Rb is required to limit the output current of the logic gate.
7.8 Darlington Transistors
Sometimes the DC current gain of the bipolar transistor is too low to directly switch the load current or voltage, so multiple
switching transistors are used. Here, one small input transistor is used to switch "ON" or "OFF" a much larger current
handling output transistor. To maximise the signal gain the two transistors are connected in a "Complementary Gain
Compounding Configuration" or what is generally called a "Darlington Configuration" where the amplification factor is
the product of the two individual transistors.
Darlington Transistors simply contain two individual bipolar NPN or PNP type transistors connected together so that the
current gain of the first transistor is multiplied with that of the current gain of the second transistor to produce a device
which acts like a single transistor with a very high current gain. The overall current gain Beta (β) or Hfe value of a
Darlington device is the product of the two individual gains of the transistors and is given as:
So Darlington Transistors with very high β values and high Collector currents are possible compared to a single transistor.
An example of the two basic types of Darlington transistor are given below.
7.8.1 Darlington Transistor Configurations
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The above NPN Darlington transistor configuration shows the Collectors of the two transistors connected together with the
Emitter of the first transistor connected to the Base of the second transistor therefore, the Emitter current of the first
transistor becomes the Base current of the second transistor. The first or "input" transistor receives an input signal,
amplifies it and uses it to drive the second or "output" transistors which amplifies it again resulting in a very high curren t
gain. As well as its high increased current and voltage switching capabilities, another advantage of a Darlington transistor
is in its high switching speeds making them ideal for use in Inverter circuits and DC motor or stepper motor control
applications.
One difference to consider when using Darlington transistors over the conventional single bipolar transistor type is that the
Base-Emitter input voltage Vbe needs to be higher at approx 1.4v for silicon devices, due to the series connection of the
two PN junctions.
Then to summarise when using a Transistor as a Switch.
•
Transistor switches can be used to switch and control lamps, relays or even motors.
•
When using bipolar transistors as switches they must be fully "OFF" or fully "ON".
•
Transistors that are fully "ON" are said to be in their Saturation region.
•
Transistors that are fully "OFF" are said to be in their Cut-off region.
•
In a transistor switch a small Base current controls a much larger Collector current.
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•
When using transistors to switch inductive relay loads a "Flywheel Diode" is required.
•
When large currents or voltages need to be controlled, Darlington Transistors are used
Chapter 8 Voltage Controlled Transistors
8.1 The Field Effect Transistor
In the Bipolar Junction Transistor tutorials, we saw that the output Collector current is determined by the amount of
current flowing into the Base terminal of the device and thereby making the Bipolar Transistor a CURRENT operated
device. The Field Effect Transistor, or simply FET however, use the voltage that is applied to their input terminal to
control the output current, since their operation relies on the electric field (hence the name field effect) generated by the
input voltage. This then makes the Field Effect Transistor a VOLTAGE operated device.
The Field Effect Transistor is a unipolar device that has very similar properties to those of the Bipolar Transistor ie, high
efficiency, instant operation, robust and cheap, and they can be used in most circuit applications that use the equivalent
Bipolar Junction Transistors, (BJT). They can be made much smaller than an equivalent BJT transistor and along with
their low power consumption and dissipation make them ideal for use in integrated circuits such as the CMOS range of
chips.
We remember from the previous tutorials that there are two basic types of Bipolar Transistor construction, NPN and
PNP, which basically describes the physical arrangement of the P-type and N-type semiconductor materials from which
they are made. There are also two basic types of Field Effect Transistor, N-channel and P-channel. As their name
implies, Bipolar Transistors are "Bipolar" devices because they operate with both types of charge carriers, Holes and
Electrons. The Field Effect Transistor on the other hand is a "Unipolar" device that depends only on the conduction of
Electrons (N-channel) or Holes (P-channel).
The Field Effect Transistor has one major advantage over its standard bipolar transistor cousins, in that their input
impedance is very high, (Thousands of Ohms) making them very sensitive to input signals, but this high sensitivity also
means that they can be easily damaged by static electricity. There are two main types of field effect transistor, the
Junction Field Effect Transistor or JFET and the Insulated-gate Field Effect Transistor or IGFET), which is more
commonly known as the standard Metal Oxide Semiconductor Field Effect Transistor or MOSFET for short.
8.2 The Junction Field Effect Transistor
We saw previously that a bipolar junction transistor is constructed using two PN junctions in the main current path
between the Emitter and the Collector terminals. The Field Effect Transistor has no junctions but instead has a narrow
"Channel" of N-type or P-type silicon with electrical connections at either end commonly called the DRAIN and the
SOURCE respectively. Both P-channel and N-channel FET's are available. Within this channel there is a third connection
which is called the GATE and this can also be a P or N-type material forming a PN junction and these connections are
compared below.
Bipolar Transistor
Field Effect Transistor
Emitter - (E)
Source - (S)
Base - (B)
Gate - (G)
Collector - (C)
Drain - (D)
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The semiconductor "Channel" of the Junction Field Effect Transistor is a resistive path through which a voltage Vds
causes a current Id to flow. A voltage gradient is thus formed down the length of the channel with this voltage becoming
less positive as we go from the drain terminal to the source terminal. The PN junction therefore has a high reverse bias
at the drain terminal and a lower reverse bias at the source terminal. This bias causes a "depletion layer" to be formed
within the channel and whose width increases with the bias. FET's control the current flow through them between the
drain and source terminals by controlling the voltage applied to the gate terminal. In an N-channel JFET this gate
voltage is negative while for a P-channel JFET the gate voltage is positive.
8.3 Bias arrangement for an N-channel JFET and corresponding circuit symbols.
The cross sectional diagram above shows an N-type semiconductor channel with a P-type region called the gate diffused
into the N-type channel forming a reverse biased PN junction and its this junction which forms the depletion layer around
the gate area. This depletion layer restricts the current flow through the channel by reducing its effective width and thus
increasing the overall resistance of the channel.
When the gate voltage Vg is equal to 0V and a small external voltage ( Vds) is applied between the drain and the source
maximum current (Id) will flow through the channel slightly restricted by the small depletion layer. If a negative voltage
(Vgs) is now applied to the gate the size of the depletion layer begins to increase reducing the overall effective area of the
channel and thus reducing the current flowing through it, a sort of "squeezing" effect. As the gate voltage (Vgs) is made
more negative, the width of the channel decreases until no more current flows between the drain and the source and the
FET is said to be "pinched-off". In this pinch-off region the gate voltage, Vgs controls the channel current and Vds has
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little or no effect. The result is that the FET acts more like a voltage controlled resistor which has zero resistance when
Vgs = 0 and maximum "ON" resistance (Rds) when the gate voltage is very negative.
8.3.1 Output characteristic voltage-current curves of a typical junction FET.
The voltage Vgs applied to the gate controls the current flowing between the drain and the source terminals. Vgs refers to
the voltage applied between the gate and the source while Vds refers to the voltage applied between the drain and the
source. Because a Field Effect Transistor is a VOLTAGE controlled device, "NO current flows into the gate!" then the
source current (Is) flowing out of the device equals the drain current flowing into it and therefore (Id = Is).
The characteristics curves example shown above, shows the four different regions of operation for a JFET and these are
given as:
•
•
•
•
Ohmic Region - The depletion layer of the channel is very small and the JFET acts like a variable resistor.
Cut-off Region - The gate voltage is sufficient to cause the JFET to act as an open circuit as the channel
resistance is at maximum.
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•
•
•
Saturation or Active Region - The JFET becomes a good conductor and is controlled by the gate-source
voltage, (Vgs) while the drain-source voltage, (Vds) has little or no effect.
Breakdown Region - The voltage between the drain and source, (Vds) is high enough to causes the
JFET's resistive channel to break down and pass current.
The control of the drain current by a negative gate potential makes the Junction Field Effect Transistor useful as a
switch and it is essential that the gate voltage is never positive for an N-channel JFET as the channel current will flow to
the gate and not the drain resulting in damage to the JFET. The principals of operation for a P-channel JFET are the
same as for the N-channel JFET, except that the polarity of the voltages need to be reversed.
8.4 The MOSFET
As well as the Junction Field Effect Transistor, their is another type of Field Effect Transistor available whose Gate input
is insulated from the main current carrying channel. The most common type of insulated gate FET or IGFET is the Metal
Oxide Semiconductor Field Effect Transistor or MOSFET for short.
The MOSFET type of field effect transistor has a "Metal Oxide" gate (usually silicon dioxide commonly known as glass),
which is electrically insulated from the main semiconductor N-channel or P-channel. This isolation of the controlling gate
makes the input resistance of the MOSFET extremely high in the Mega-ohms region and almost infinite. As the gate
terminal is isolated from the main current carrying channel ""NO current flows into the gate"" and like the JFET, the
MOSFET also acts like a voltage controlled resistor. Also like the JFET, this very high input resistance can easily
accumulate large static charges resulting in the MOSFET becoming easily damaged unless carefully handled or
protected.
8.4.1 Basic MOSFET Structure and Symbol
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We also saw previously that the gate of a JFET must be biased in such a way as to forward-bias the PN junction but in a
MOSFET device no such limitations applies so it is possible to bias the gate in either polarity. This makes MOSFET's
specially valuable as electronic switches or to make logic gates because with no bias they are normally non -conducting
and the high gate resistance means that very little control current is needed. Both the P-channel and the N-channel
MOSFET is available in two basic forms, the Enhancement type and the Depletion type.
8.5 Depletion-mode MOSFET
The Depletion-mode MOSFET, which is less common than the enhancement types is normally switched "ON" without a
gate bias voltage but requires a gate to source voltage (Vgs) to switch the device "OFF". Similar to the JFET types. For
N-channel MOSFET's a "Positive" gate voltage widens the channel, increasing the flow of the drain current and
decreasing the drain current as the gate voltage goes more negative. The opposite is also true for the P-channel types.
The depletion mode MOSFET is equivalent to a "Normally Closed" switch.
8.5.1 Depletion-mode N-Channel MOSFET and circuit Symbols
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Depletion-mode MOSFET's are constructed similar to their JFET transistor counterparts where the drain-source channel is
inherently conductive with electrons and holes already present within the N-type or P-type channel. This doping of the
channel produces a conducting path of low resistance between the drain and source with zero gate bias.
8.6 Enhancement-mode MOSFET
The more common Enhancement-mode MOSFET is the reverse of the depletion-mode type. Here the conducting
channel is lightly doped or even undoped making it non-conductive. This results in the device being normally "OFF" when
the gate bias voltage is equal to zero.
A drain current will only flow when a gate voltage (Vgs) is applied to the gate terminal. This positive voltage creates an
electrical field within the channel attracting electrons towards the oxide layer and thereby reducing the overall resistance
of the channel allowing current to flow. Increasing this positive gate voltage will cause an increase in the drain current, Id
through the channel. Then, the Enhancement-mode device is equivalent to a "Normally Open" switch.
8.6.1 Enhancement-mode N-Channel MOSFET and circuit Symbols
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Enhancement-mode MOSFET's make excellent electronics switches due to their low "ON" resistance and extremely high
"OFF" resistance and extremely high gate resistance. Enhancement-mode MOSFET's are used in integrated circuits to
produce CMOS type Logic Gates and power switching circuits as they can be driven by digital logic levels.
MOSFET Summary
The MOSFET has an extremely high input gate resistance and as such a easily damaged by static electricity if not
carefully protected. MOSFET's are ideal for use as electronic switches or common-source amplifiers as their power
consumption is very small. Typical applications for MOSFET's are in Microprocessors, Memories, Calculators and Logic
Gates etc. Also, notice that the broken lines within the symbol indicates a normally "OFF" Enhancement type showing that
"NO" current can flow through the channel when zero gate voltage is applied and a continuous line within the symbol
indicates a normally "ON" Depletion type showing that current "CAN" flow through the channel with zero gate voltage. For
P-Channel types the symbols are exactly the same for both types except that the arrow points outwards.
This can be summarised in the following switching table.
MOSFET type
Vgs = +ve
Vgs = 0
Vgs = -ve
N-Channel Depletion
ON
ON
OFF
N-Channel Enhancement
ON
OFF
OFF
P-Channel Depletion
OFF
ON
ON
P-Channel Enhancement
OFF
OFF
ON
8.7 The MOSFET as a Switch
We saw previously, that the N-channel, Enhancement-mode MOSFET operates using a positive input voltage and has an
extremely high input resistance (almost infinite) making it possible to interface with nearly any logic gate or driver capable
of producing a positive output. Also, due to this very high input ( gate) resistance we can parallel together many different
MOSFET's until we achieve the current handling limit required. While connecting together various MOSFET's may enable
us to switch high current or high voltage loads, doing so becomes expensive and impractical in both components and
circuit board space. To overcome this problem Power Field Effect Transistors or Power FET's where developed.
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We now know that there are two main differences between FET's, Depletion-mode for JFET's and Enhancement-mode for
MOSFET's and on this page we will look at using the Enhancement-mode MOSFET as a Switch.
By applying a suitable drive voltage to the gate of an FET the resistance of the drain-source channel can be varied from
an "OFF-resistance" of many hundreds of kΩ's, effectively an open circuit, to an "ON-resistance" of less than 1Ω,
effectively a short circuit. We can also drive the MOSFET to turn "ON" fast or slow, or to pass high currents or low
currents. This ability to turn the power MOSFET "ON" and "OFF" allows the device to be used as a very efficient switch
with switching speeds much faster than standard bipolar junction transistors.
8.7.1 An example using the MOSFET as a switch
In this circuit arrangement an Enhancement-mode N-channel
MOSFET is being used to switch a simple lamp "ON" and
"OFF" (could also be an LED). The gate input voltage VGS is
taken to an appropriate positive voltage level to turn the device
and the lamp either fully "ON", (VGS = +ve) or a zero voltage
level to turn the device fully "OFF",
(VGS = 0).
If the resistive load of the lamp was to be replaced by an
inductive load such as a coil or solenoid, a "Flywheel" diode
would be required in parallel with the load to protect the
MOSFET from any back-emf.
Above shows a very simple circuit for switching a resistive load such as a lamp or LED. But when using power MOSFET's
to switch either inductive or capacitive loads some form of protection is required to prevent the MOSFET device from
becoming damaged. Driving an inductive load has the opposite effect from driving a capacitive load. For example, a
capacitor without an electrical charge is a short circuit, resulting in a high "inrush" of current and when we remove the
voltage from an inductive load we have a large reverse voltage build up as the magnetic field collapses, resulting in an
induced back-emf in the windings of the inductor.
For the power MOSFET to operate as an analogue switching device, it needs to be switched between its "Cut-off Region"
where VGS = 0 and its "Saturation Region" where VGS(on) = +ve. The power dissipated in the MOSFET ( PD depends upon
the current flowing through the channel ID at saturation and also the "ON-resistance" of the channel given as RDS(on). For
example.
Example No1
Lets assume that the lamp is rated at 6v, 24W and is fully "ON", the MOSFET has a channel "ON -resistance" ( RDS(on) )
value of 0.1ohms. Calculate the power dissipated in the MOSFET switch.
The current flowing through the lamp is calculated as:
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The power dissipated in the FET is given as:
You may think, well so what!, but when using the MOSFET as a switch to control DC motors or high inrush current
devices the "ON" channel resistance ( RDS(on) ) is very important. For example, motor control MOSFET's are subjected to
high in-rush currents as the motor first begins to rotate. A high RDS(on) channel resistance value would simply result in
power being dissipated within the MOSFET resulting in an excessive temperature rise which could result in the MOSFET
becoming too hot and damaged due to thermal overload. A low RDS(on) value on the other hand is also desirable to help
reduce the effective saturation voltage VDS(sat) = ID x RDS(on)) across the MOSFET. When using MOSFET´s or any type of
Field Effect Transistor as a switch, it is always advisable to use ones with very low RDS(on) values or to mount them onto
suitable heatsinks to help reduce any thermal runaway and damage.
8.8 Power MOSFET Motor Control
Because of the extremely high input gate resistance of MOSFET´s, its very fast switching speeds and the ease at which
they can be driven makes them ideal to interface with op-amps or standard logic gates. However, care must be taken to
ensure that the gate-source input voltage is correctly chosen because when using the MOSFET as a switch the device
must obtain a low RDS(on) channel resistance in proportion to this input gate voltage. For example, do not apply a 12v
signal if a 5v signal voltage is required. Power MOSFET´s can be used to control the movement of DC motors or
brushless stepper motors directly from computer logic or Pulse-width Modulation (PWM) type controllers. As a DC motor
offers high starting torque and which is also proportional to the armature current, MOSFET switches along with a PWM
can be used as a very good speed controller that would provide smooth and quiet motor operation.
8.8.1 Simple Power MOSFET Motor Controller
As the motor load is inductive, a simple "Free-wheeling" diode
is connected across the load to dissipate any back emf
generated by the motor when the MOSFET turns it "OFF".
The Zener diode is used to prevent excessive gate-source
input voltages.
100
Summary of Bipolar Junction Transistors
•
The Bipolar Junction Transistor (BJT) is a three layer device constructed form two semiconductor diode
junctions joined together, one forward biased and one reverse biased.
•
There are two main types of bipolar junction transistors, the NPN and the gate transistor.
•
Transistors are "Current Operated Devices" where a much smaller Base current causes a larger Emitter to
Collector current, which themselves are nearly equal, to flow.
•
The most common transistor connection is the Common-emitter configuration.
•
Requires a Biasing voltage for AC amplifier operation.
•
The Collector or output characteristics curves can be used to find either Ib, Ic or β to which a load line can
be constructed to determine a suitable operating point, Q with variations in base current determining the operating
range.
•
A transistor can also be used as an electronic switch to control devices such as lamps, motors and solenoids
etc.
•
Inductive loads such as DC motors, relays and solenoids require a reverse biased "Flywheel" diode placed
across the load. This helps prevent any induced back emf's generated when the load is switched "OFF" from
damaging the transistor.
•
The NPN transistor requires the Base to be more positive than the Emitter while the PNP type requires that
the Emitter is more positive than the Base.
Summary of Field Effect Transistors
•
Field Effect Transistors, or FET's are "Voltage Operated Devices" and can be divided into two main types:
Junction-gate devices called JFET's and Insulated-gate devices called IGFET´s or more commonly known as
MOSFET's.
•
Insulated-gate devices can also be sub-divided into Enhancement types and Depletion types. All forms
are available in both N-channel and P-channel versions.
•
FET's have very high input resistances so very little or no current (MOSFET types) flows into the input
terminal making them ideal for use as electronic switches.
•
The input impedance of the MOSFET is even higher than that of the JFET due to the insulating oxide layer
and therefore static electricity can easily damage MOSFET devices so care needs to be taken when handling
them.
•
FET's have very large current gain compared to junction transistors.
•
They can be used as ideal switches due to their very high channel "OFF" resistance, low "ON" resistance.
8.9 The Field Effect Transistor Family-tree
101
Field Effect Transistors can be used to replace normal Bipolar Junction Transistors in electronic circuits and a simple
comparison between FET's and transistors stating both their advantages and their disadvantages is given below.
Field Effect Transistor (FET)
Bipolar Junction Transistor (BJT)
1
Low voltage gain
High voltage gain
2
High current gain
Low current gain
3
Very input impedance
Low input impedance
4
High output impedance
Low output impedance
5
Low noise generation
Medium noise generation
6
Fast switching time
Medium switching time
7
Easily damaged by static
Robust
8
Some require an input to turn it "OFF"
Requires zero input to turn it "OFF"
9
Voltage controlled device
Current controlled device
10
Exhibits the properties of a Resistor
11
More expensive than bipolar
Cheap
12
Difficult to bias
Easy to bias
Chapter 9 MISCELLANEOUS And OPTO ELECTRONIC DEVICES
9.1 The Thermistor.
102
A Thermistor on the other hand is a THERM-ally sensitive res-ISTOR which changes its physical resistance with
temperature. They are generally made from ceramic type semiconductor materials such as oxides of nickel, manganese
or cobalt coated in glass which makes them easily damaged. Most types of thermistor's have a Negative Temperature
Coefficient of resistance or (NTC), that is their resistance value goes DOWN with an increase in the temperature but
some with a Positive Temperature Coefficient, (PTC), their resistance value goes UP with an increase in temperature
are also available. Their main advantage is their speed of response to any changes in temperature, accuracy and
repeatability.
Thermistors are ceramic type semiconductors made from Metal Oxide technology that are generally formed into small
glass beads or balls which gives a relatively fast response to any changes in temperature. They are rated by their resistive
value at room temperature (usually at 25 oC) and their power rating with respect to current flow. Thermistors are available
with resistances at room temperature from 10´s of Megaohms down to just a few Ohms, but for sensing purposes those
types with values in the kilo-ohms are generally used.
Thermistors are passive resistive devices which means we need to pass a current through it to produce a measurable
voltage output. Then thermistors are generally connected in series with a suitable biasing resistor to form a potential
divider network and the choice of resistor gives a voltage output at some pre-determined temperature point or value for
example:
Example No1
The following thermistor has a resistance value of 10KΩ at 25 oC and a resistance value of 100Ω at 100 oC. Calculate the
voltage drop across the thermistor and hence its output voltage (Vout) for both temperatures when connected in series
with a 1kΩ resistor across a 12v power supply.
At 25oC
At 100oC
103
by changing the fixed resistor value of R2 (in our example 1kΩ) to a potentiometer or preset, a voltage output can be
obtained at a predetermined temperature set point for example, 5v output at 60 oC and by varying the potentiometer a
particular output voltage level can be obtained over a wider temperature range.
It needs to be noted however, that thermistor's are non-linear devices and their standard resistance values at room
temperature is different between different thermistor's, which is due mainly to the materials they are made of. Thermistor's
are therefore given a Beta temperature constant (B) which can be used to calculate its resistance for any given
temperature point. However, when used in series with a resistance such as in a voltage divider or Wheatstone Bridge type
arrangement, the current obtained in response to a voltage applied to the divider/bridge network is linear wi th
temperature. Then, the output voltage across the resistor becomes linear with temperature.
9.2 Light Sensors
Light Sensors are used to measure the radiant energy that exists in a very narrow range of frequencies basically called
"light", and which ranges in frequency from "Infrared" to "Visible" up to "Ultraviolet" light. Light sensors are passive
devices that convert this "light energy" whether visible or in the infrared parts of the spectrum into an electrical signal
output. Light sensors are more commonly known as "Photoelectric Devices" or "Photosensors" which can be grouped into
two main categories, those which generate electricity when illuminated, such as Photovoltaics or Photoemissives etc,
and those which change their electrical properties such as Photoresistors or Photoconductors. This leads to the
following classification of devices.
•
•
•
•
•
•
•
Photo-emissive Cells - These are photo devices which release free electrons from a light sensitive
material such as caesium when struck by light.
Photo-conductive Cells - These photo devices vary their electrical resistance when subjected to light.
The most common photoconductive material is Cadmium Sulphide
Photo-voltaic Cells - These photo devices generate an e.m.f. in proportion to the radiant light energy
received. The most common photovoltaic material is Selenium.
Photo-junction Devices - These photodevices are mainly semiconductor devices such as the photodiode
or phototransistor which use light to control the flow of electrons and holes across their PN-junction.
9.2.1 The Photoconductive Cell.
Photoconductive light sensors change their physical properties when subjected to light energy. The most common type
of photoconductive device is the Photoresistor which changes its electrical resistance in response to changes in the light
intensity. Photoresistors are Semiconductor devices that use light energy to control the flow of electrons, and hence
the current flowing through them. The commonly used Photoconductive Cell is called the Light Dependant Resistor or
LDR.
9.2.2 The Light Dependant Resistor.
As its name implies, the Light Dependant Resistor is a resistive light sensor that changes its electrical resistance from
several thousand Ohms in the dark to only a few hundred Ohms when light falls upon it. The net effect is a decrease in
resistance for an increase in illumination. Materials used as the semiconductor substrate include, Lead Sulphide, (PbS)
Lead Selenide, (PbSe) Indium Antimonide, (InSb) which detect light in the INFRARED range and the most commonly
used of all is Cadmium Sulphide (Cds), as its spectral response curve closely matches that of the human eye and can
even be controlled using a simple torch as a light source. Typically it has a peak sensitivity wavelength ( λp) of about
560nm to 600nm in the visible spectral range.
104
The Light Dependant Resistor Cell
The most commonly used photoresistive light sensors is the ORP12 Cadmium Sulphide photoconductive cell. This light
depedant resistor has a spectral response of about 610nm in the yellow to orange region of light. The resistance of the
cell when unilluminated (dark resistance) is very high at about 10MΩ's which falls to about 100Ω's when fully illuminated
(lit resistance). To increase the dark resistance and therefore reduce the dark current, the resistive path forms a zigzag
pattern across the ceramic substrate. The CdS photocell is a very low cost device often used in auto dimming, darkness
or twilight detection for turning the street lights "ON" and "OFF", and for photographic exposure meter type applications.
One simple use of a Light Dependant Resistor, is as a light sensitive switch as shown below.
This basic light sensor circuit is of a relay output light activated switch.
A potential divider circuit is formed between the photoresistor, LDR
and the resistor R1. When no light is present ie in darkness, the
resistance of the LDR is very high in the Megaohms range so zero
base bias is applied to the transistor TR1 and the relay is deenergised or "OFF". As the light level increases the resistance of the
LDR starts to decrease causing the base bias voltage at V1 to rise. At
some point determined by the potential divider network formed with
resistor R1, the base bias voltage is high enough to turn "ON" the
transistor TR1 and thus activate the relay which inturn is used to
control some external circuitry. As the light level falls back to darkness
again the resistance of the LDR increases causing the base voltage of
the transistor to decrease, turning the transistor and relay "OFF" at a
105
fixed light level determined again by the potential divider network.
By replacing the fixed resistor R1 with a potentiometer VR1, the point at which the relay turns "ON" or "OFF" can be
preset to a particular light level. This type of simple circuit shown above has a fairly low sensitivity and its switching point
may not be consistent due to variations in either temperature or the supply voltage. A more sensitive precision light
activated circuit can be easily made by incorporating the LDR into a "Wheatstone Bridge" arrangement and replacing the
transistor with an Operational Amplifier as shown.
9.2.3 Light Level Sensing Circuit
In this basic circuit the Light Dependant Resistor, LDR1 and the potentiometer VR1 form one arm of a simple
Wheatstone bridge network and the two fixed resistors R1 and R2 the other arm. Both arms of the bridge form potential
divider networks whose outputs V1 and V2 are connected to the inverting and non-inverting voltage inputs respectively of
the operational amplifier. The configuration of the operational amplifier is as a Differential Amplifier or Voltage
Comparator with its output signal being the difference between the two input signals or voltages, V2 - V1. The feedback
resistor Vf can be chosen to give a voltage gain if required.
The resistor combination R1 and R2 form a fixed reference voltage input V2, set by the ratio of the two resistors and the
LDR - VR1 combination a variable voltage input V1. As with the previous circuit the output from the operational amplifier
is used to control a relay, which is protected by a free wheel diode, D1. When the light level sensed by the LDR and its
output voltage falls below the reference voltage at V2 the output from the op-amp changes activating the relay and
switching the connected load. Likewise as the light level increases the output will switch back turning "OFF" the relay.
The operation of this type of circuit can also be reversed to switch the relay "ON" when the light level exceeds the
reference voltage level and vice versa by reversing the positions of the Light Dependant Resistor LDR and the
potentiometer VR1. The potentiometer can be used to "pre-set" the switching point of the differential amplifier to any
particular light level making it ideal as a light sensor circuit.
9.3Photojunction Devices.
Photojunction Devices are basically PN-Junction light sensors or detectors made from silicon semiconductors and
which can detect both visible light and infrared light levels. This class of photoelectric light sensors incl ude the
Photodiode and the Phototransistor.
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9.3.1 The Photodiode.
The construction of the Photodiode light sensor is similar to that of a conventional PN-junction diode except that the
diodes outer casing is transparent so that light can fall upon the junction. LED's can also be used as photodiodes as they
can both emit and detect light. All PN-junctions are light sensitive and can be used in a photoconductive (PC) mode with
the PN-junction of the photodiode always "Reverse Biased" so that only the diodes leakage or dark current can flow. This
reverse bias condition causes an increase of the depletion region which is the sensitive part of the junction.
9.3.2 Photo-diode Construction and Characteristics
The photodiodes dark current (0 lux) is about 10uA for geranium and 1uA for silicon type diodes. When light falls upon the
junction more hole/electron pairs are formed and the leakage current increases. The leakage current increases as the
illumination of the junction increases. Diode current is directly proportional to light intensity. One main advantage of
photodiodes when used as light sensors is their fast response to changes in the light levels, but one disadvantage of this
type of photodevice is the relatively small current flow even when fully lit.
Photodiodes are very versatile light sensors and are commonly used in cameras, light meters, CD and DVD-ROM drives,
TV remote controls, scanners, fax machines and copiers etc, and when integrated into operational amplifier circuits as
infrared spectrum detectors for fibre optic communications, burglar alarm motion detection circuits and numerous imaging,
laser scanning and positioning systems etc.
9.4 The Phototransistor.
An alternative photojunction device to the photodiode is the Phototransistor which is basically a photodiode with
amplification and operates by exposing its base region to the light source. Phototransistor light sensors operate the same
as photodiodes except that they can provide current gain and are much more sensitive than the photodiode with currents
are 50 - 100 times greater than that of the standard photodiode.
107
Phototransistors consist mainly of a bipolar NPN Transistor with the collector-base PN-junction reverse-biased. The
phototransistor´s large base region is left electrically unconnected and uses photons of light to generate a base current
which inturn causes a collector to emitter current to flow.
9.4 .1 Photo-transistor Construction and Characteristics
In the NPN transistor the collector is biased positively with respect to the emitter so that the base/collector junction is
reverse biased. therefore, with no light on the junction normal leakage or dark current flows which is very small. When
light falls on the base more electron/hole pairs are formed in this region and the current produced by this action is
amplified by the transistor. The sensitivity of a phototransistor is a function of the DC current gain of the transistor.
Therefore, the overall sensitivity is a function of collector current and can be controlled by connecting a resistance
between the base and the emitter but for very high sensitivity optocoupler type applications, Darlington phototransistors
are generally used.
Photodarlington transistors use a second bipolar NPN transistor to
sensitivity of a photodetector are required, but its response is slower
consists of a normal phototransistor whose emitter output is coupled
Because a darlington transistor configuration gives a current gain equal
transistors, a photodarlington device produces a very sensitive detector.
provide additional amplification or when higher
than that of an ordinary NPN phototransistor. It
to the base of a larger bipolar NPN transistor.
to a product of the current gains of two individual
Typical applications of Phototransistors light sensors are in opto-isolators, slotted opto switches, light beam sensors,
fibre optics and TV type remote controls, etc. Infrared filters are sometimes required when detecting visible light.
108
Another type of photojunction semiconductor light sensor worth a mention is the Photothyristor. This is a light activated
thyristor or Silicon Controlled Rectifier, SCR that can be used as a light activated switch in a.c. applications. However
their sensitivity is usually very low compared to photodiodes or phototransistors, as to increase their sensitivity to light
they are made thinner around the gate junction which inturn limits the amount of current that they can switch. Then for
higher current applications they are used as pilot devices in opto-couplers to switch larger more conventional thyristors.
9.5 Photovoltaic Cells.
The most common type of photovoltaic light sensor is the Solar Cell. This device converts light energy directly into
electrical energy in the form of a voltage or current. Solar cells are used in many different types of applications to offer an
alternative power source from conventional batteries, such as in calculators and satellites. Photovoltaic cells are made
from single crystal silicon PN junctions, the same as photodiodes with a very large light sensitive region but are used
without the reverse bias. They have the same characteristics as photodiodes when in the dark. When illuminated the light
energy causes electrons to flow through the PN junction and an individual solar cell can generate an open circuit voltage
of about 0.58v (580mV). Solar cells have a "Positive" and a "Negative" side just like a battery.
Individual solar cells can be connected together in series to form solar panels which increases the output voltage or
connected together in parallel to increase the available current. Commercially available solar panels are rated in Watts,
which is the product of the output voltage and current (VxI) when fully lit.
9.5.1 Characteristics of a typical Photovoltaic Solar Cell.
The amount of available current from a solar cell depends upon the light intensity, the size of the cell and its efficiency
which is generally very low at around 20%. To increase the overall efficiency of the cell commercially available solar cells
use Polycrystalline Silicon and Amorphous silicon, which has no crystalline structure and can generate currents of
between 20 to 40mA per cm 2. Other materials used include Gallium Arsenide, Copper Indium Diselenide and Cadmium
Telluride. These different materials each have a different spectrum band response, and so can be "tuned" to produce an
output voltage at different wavelengths of light.
Problem 1.
109
For the circuit shown all resistors are 1k .
i.
ii.
Find the equivalent resitstance between
terminals A and B.
Between C and D.
iii.
Between B and C.
Problem 2.
For the circuit shown:
i.
Find the Thevenin equivalent between
nodes B and C. Do it once without using
superposition, and once with it.
ii.
Find the Thevenin equivalent between
nodes A and C. Use any method
Problem 3.
For the circuit shown (note this is case 3 of the
Thevenin Theorem):
i.
Find the Thevenin equivalent between nodes
A and B..
ii.
Between C and D.
Problem 4.
110
For the circuit shown:
i.
Find Rx such that the power
dissipated through it is
maximized (hint: find the
Thevenin equivalent circuit
experienced by Rx and use
maximum power transfer)
ii.
With the value of Rx chosen,
find the power dissipated in Rx,
and the total power dissipated in
the other resistors.
Problem 5.
For each circuit find the voltage at the output
terminals if:
i.
ii.
the input has been zero volts for a long time,
and then goes to 1 volt.
the input has been 1 volt for a long time, and
then goes to -1 volt.
The resistor has value=R, and the capacitor has
value=C.
Problem 6.
For the two circuits of the previous problem, with R=1k , C=1uF:
i.
ii.
iii.
Construct the Bode plots.
From the Bode plots determine the output voltage if the input is 10sin(100t).
Determine the output voltage if the input is 10sin(10000t).
iv.
What type of filtering operation does each circuit perform?
Problem 7.
111
For the circuit shown find vx(t) if:
i.
ii.
the switch has been closed for a long time,
and is opened at t=0.
the switch has been open a long time and is
closed at t=0.
R1=R2=R3=1k , C=1uF.
Problem 8.
Solve for the two cases given.
i.
ii.
The voltage across the capacitor is 1/3 V1 at
t=0 when the switch opens. Solve for the
time at which the voltage across the capacitor
is 2/3 V1.
The voltage across the capacitor is 2/3 V1 at
t=0 when the switch closes. Solve for the
time at which the voltage across the capacitor
is 1/3 V1 if at t=0:
Problem 9.
For the circuit shown
i.
ii.
iii.
The capacitor is charged to 2.2 V
when the switch is closed at t=0,.
Calculate how long it takes the voltage
across the capacitor to go from 2.2V
(at t=0) to 1.4V.
The capacitor is charged to1.4 V when
the switch is opened at t=0. Calculate
how long it takes the voltage across
the capacitor to go from 1.4V (at t=0)
to 2.2V. Since R5 is so much less than
the other resistances in the circuit,
assume it has a value of 0 for this part
of the problem.
(extra - you need not do this) Repeat
the previous part without assuming
112
R5=0( .
Problem 10.
For the circuit shown:
i.
ii.
iii.
iv.
What type of filter is this? Make your
argument without equations, based upon the
low and high frequency behavior of the
circuit elements.
Find the transfer function (Vo(s)/Vi(s))
Verify that your transfer function agrees with
part a.
Does the damping factor increase or decrease
as R increases?
Problem 11.
For the circuit shown:
i.
ii.
iii.
iv.
What type of filter is this? Make your
argument without equations, based upon
the low and high frequency behavior of
the circuit elements.
Find the transfer function (Vo(s)/Vi(s))
Verify that your transfer function agrees
with part a.
Does the damping factor increase or
decrease as R increases?
Problem 12.
The circuit shown is a Wheatstone bridge, and is often
use to measure small changes in resistance (denoted by
the lowercase "r").
i.
ii.
Show that when r=0, Vo=0.
Derive an expression for Vo in terms of Vi, R,
and r.
iii.
If r<<R, find a linear approximation for Vo in
terms of Vi , R and r.
113
Problem 13.
For the circuit shown:
i.
Find the voltage between B and C using
superposition.
ii.
Find the voltage between A and C.
Problem 14.
The resistor network shown
repeats forever to the right.
What is the equivalent seen
between A and B? All
resistors are equal with
resistance R.
Hint: The resistance to the right
of A/B is the same as the
resistance to the right of C/D.
114