Download Electronic Circuits

ENES100-0702
Professor R. J. Phaneuf
Fall 2002
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Electronic Circuits
One of the specifications of your project is that it must contain both a mechanical
and an electronic alarm which goes off when the volume of water in your catch basin
reaches 90% of the 5.0 gallons your pump must lift during the testing procedure. This
brings up the topic of electronic circuitry.
An electron placed in a region in which its potential energy varies with position,
due to the existence of an electrostatic field, will experience a force which accelerates it
in the direction of decreasing potential energy U(r). By convention the electrostatic
potential V(r) is the ratio of the potential energy to the charge of the electron (or whatever
“test charge” is placed in the region of varying potential energy):
V(r) = U(r)/q.
(1)
Just as is the case for potential energy, it is not possible to define a unique zero level for
potential. We are thus usually concerned with differences in potential (the terms
“electromotive force”, EMF and “voltage” are also used), V.
An electron moving within a material medium such as a wire, rather than in free
space, does not accelerate without limit. Thus Newton’s second law gives the
instantaneous, but not the average behavior. Electrons collide with or “scatter from”
obstacles within the material, and thus move in a “start-stop” manner which on
average corresponds to a so-called drift velocity, vd. The obstacles are imperfections in
the crystalline order of the material, including impurity atoms, missing atoms, the
boundaries between crystalline grains, and even the disorder associated with the
vibrations of the atoms within the material. Where then does the energy from the
change in potential go, if the the average kinetic energy remains the same? It is
transferred to the material in the form of heat.
The product of the charge of the electron, q, the number of mobile electrons per unit
length within the material, NL, and the drift velocity, vd yields a quantity known as the
current:
I = qNLvd
(2).
The current corresponds to the amount of charge per unit time passing though a plane
drawn though the material and oriented perpendicular to the direction of the drift
velocity. The unit of current in the SI system is the ampere, or “amp”. One ampere is
equal to 1 coulomb of charge per second.
V2
V1
I
A
l
You will also see a related quantity expressed in terms of the volume density of
mobile electrons Nv, yielding the current crossing a given area, or current density J
J = I/A = qNvvd
(3).
The unit of current density in the SI system is the amp/m2. In most problems in
electronic circuits it is assumed that the cross sectional area A of the material is constant,
and I is used rather than J.
Resistors
Experimentally it is found that the current in simple materials is proportional to
the potential difference placed across them, which is known as Ohm’s law:
I = (1/R)V
(4a),
V = IR
(4b).
or
The quantity R, is the resistance of the piece of material through which current flows. It
is measured in ohms, denoted by the letter . One ohm is equal to one volt per ampere.
Experimentally, it is found that the resistance of a piece of material increases linearly
with the length of material that the potential difference is placed across, and inversely
with its cross sectional area:
R =  L/A
(5),
where  is known as the resistivity of the material, a quantity which varies from material
to material, and with temperature, but is independent of the dimensions of the piece of
material and the potential difference in most cases.
Electrical resistivities vary over an enormous range, with pure metals such as
copper and silver having room temperature values of 1.6-1.7 x 10-6 ohm-cm, and ceramic
materials having values as high as 1020 ohm-cm. Materials with very low resistivity are
referred to as conductors, while those with very high resistivities are called insulators. A
third class of materials, which includes silicon, shows moderate resistivities when very
pure, but much lower resisivities when “doped” with very small concentrations of certain
impurities. These are referred to as semiconductors. Finally, although they are not
generally used in standard circuit components, there exists a fourth class of materials,
called superconductors, whose resistivity drops to zero below a material-dependent
temperature. Values for room temperature-resistivities of common materials are listed
below [1-3]:
Material
Resistivity at 25°C (ohm-cm)
Copper
Silver
Aluminum
Gold
Brass (70%Cu/30%Zn)
Stainless Steel (321)
1.7x10-6
1.6x10-6
2.7x10-6
2.2x10-6
6.2x10-6
11 x10-6
Germanium
Silicon
Gallium Arsenide
47
2.3x105
1x108
Glass (silica)
Pyrex 7052
Aluminum Oxide
Porcelain
4x109-3x1010
25x106
1017 - 1020
1012-1014
One of the most commonly used circuit elements is a so-called resistor. In
practical circuits, resistors are generally chosen to have resistance much higher than that
of the wires and electrical contacts of the circuit. The purpose of a resistor is to allow a
well defined current to be set within a circuit by application of a potential difference
across it. Commercial resistors are fabricated either from very long wires wound around
a spool, or more commonly from amorphous carbon.
Power dissipated in a resistor; Joule heating
In the absence of resistance, the source of potential difference would be supplying power
at a rate equal to the change in potential energy of one electron times the number of
electrons flowing through the material per unit time. This is simply:
Pin  U
n
I
 qV  IV
t
q
(6)
By the conservation of energy, an equal amount of power must be dissipated in the form
of heat:
Pdiss  Pin  IV  I ( IR )  I 2 R
(7)
The power loss to dissipation within a material is known as Joule heating. It has a very
practical application, that of radiant heat and light sources, such as incandescent lamps.
The temperature of a resistor will continue to increase until the rate at which power is
radiated away, lost to convection, and lost to conduction balances the Joule heating
power input. In a lamp, all of the air is removed from the glass envelope so convection
does not contribute. At high temperatures the radiation loss is much greater than the
conduction loss, and is given by the Stefan-Boltzmann equation:
P = ST4
(8)
where S is the surface area of the resistor,  is the Stefan-Boltzmann constant = 5.670 x
10-8 J / (m-s-K4), and  is a material dependent coefficient called the emissivity. A lamp
filament will emit light in a spectrum with the highest intensity occurring at a wavelength
which decreases with temperature. The temperature dependence is given (approximately)
by Wien’s law:
MAX =(2.898 x 10-3 m- K) / T
(9)
Typically tungsten filament lamps operate at a temperature near 2500 K, which gives a
peak in the emission spectrum in the near infrared, but also a substantial amount of light
in the visible part of the spectrum. The high intensity causes the light to appear nearly
white to the eye.
Running a lamp at temperatures much higher than this results in a lifetime which
decreases exponentially with temperature, due to rapid evaporation of tungsten. From
equations (7) and (8), the potential “dropped” across the filament must be limited to
avoid over heating. Since batteries provide a fixed voltage, we next consider how to
control how much of the battery potential is across the lamp filament.
Combining Resistances; Voltage and Current Dividers
By examining the form of equation (5) we can predict how resistances combine.
Let’s say we had two wires of the same area joined end to end. If the joint were well
done the overall length would simply be the sum of the individual lengths, and from
equation (5) the overall resistance would simply be the sum of the individual resistances.
This turns out to be quite general, even if the cross sectional areas of the two wires are
not equal, and even if they have different resistivities. Two or more resistors in series
have a resistance equal to the sum of the individual resistances:
Rseries =  Ri
(10),
i
Similarly, again consider joining two wires of equal length and cross sectional area at
both ends. The overall cross sectional area is the sum of the individual areas. If we take
the reciprocal of equation (5) we can see that the reciprocal resistance of the combined
wires is simply the sum of the reciprocal resistances of the individual resistances. Again,
this is quite general, even if the lengths, areas and resistivities of the individual wires is
not the same. For a combination of resistances in parallel the overall reciprocal
resistance is the sum of the individual reciprocal resistances:
1
R parallel
 i
1
,
Ri
(11)
or, equivalently,
R parallel 
1
1
i R
i
(12)
Very simple circuits can be built up from configurations containing only resistors, and
sources of potential difference (batteries, fuel cells, power supplies). By combining (5)
and Ohms law (4b), you can see how a series of resistors allows a voltage to be
“divided”:
V = IR = I(R1 + R2) = IR1 + IR2 = V1 + V2
(13),
While the reciprocal of (6) and Ohm’s law (4a) shows how a current can be divided by a
parallel resistor pair:
I = V/R = V(1/R1 + 1/R2) = V/R1 + V/R2 = I1 + I2
(14).
Kirchoff’s laws
We can calculate the potential differences between different points in a more complex
circuit as well as the currents flowing through different branches using two equations
known as Kirchoff’s laws. The first law states that the energy of an electron is conserved
in moving completely through any continuous circuit loop. The second states that, since
charge does not build up at points of intersection, or “nodes”, between circuit branches,
the total current into a node must equal the total current out.
Using Kirchoff’s laws to analyze a circuit is done by dividing it up into loops,
algebraically defining currents in each loop, and using these currents and ohm’s law to
sum potential drops. The equation for potential drops in each loop, plus those expressing
current conservation at nodes are solved simultaneously. Things to note:
(1) The convention is that since the voltage drop is IR, traversing a resistor
parallel to the current is a positive drop, traversing opposite the current is a
negative drop.
(2) Traversing a voltage source from the negative terminal to the positive terminal
is a potential gain, thus constitutes a negative potential drop.
(3) The total number of potential drop equations is equal to the total number of
loops.
(4) The total number of current equations is the number of nodes minus one.
(5) Choosing loops and assuming current directions can be done in more than one
way, but the results will be the same.
Example:
Consider the circuit shown in Fig. 1. The sawtooth symbols denote resistors, the parallel
lines denote a voltage source (for example, a battery) with the convention being the
longer line is the positive terminal.
Fig. 1. Schematic for example of Kirchoff’s laws.
We start by choosing the first loop to include the voltage source V, resistor R1 and R2
with the current through R1 equal to I1, assumed to go from left to right and the current
through R2 equal to I2, assumed to go from top to bottom. The potential drop equation
for this loop, starting from the positive battery terminal is:
I1 R1 + I2R2 –V =0
(e1-1)
Next we choose a loop including R2 and R3, and assume that the current passing through
R3 is I3, moving from top to bottom. Moving from the top node, clockwise gives a
second equation:
I3R3 – I2R2 = 0,
(e1-2)
where the second drop is negative, since we mathematically complete the loop against the
assumed direction of current flow through R2.
Finally, we write Kirchoff’s second law for the top node:
I1 – I2 – I3 =0,
(e1-3)
Where I1 is positive, since we assumed it moves into the top node, but I2 and I3 are
negative since they flow out of the node.
Combining (e1-1) and (e1-3):
(I2 + I3)R1 + I2R2 = V
(e1-4),
and combining this with (e1-2) and rearranging gives:
I2 = VR3/(R1R2+R1R3+R2R3)
(e1-5)
Combining this with (e1-2):
I3= VR2/(R1R2+R1R3+R2R3)
(e1-6),
and finally combining with (e1-3):
I1=I2+I3 = V(R2+R3) /(R1R2+R1R3+R2R3) = V/[R1 + R2R3/(R2+R3)]
(e1-7).
Capacitors
The second ubiquitous element in electrical circuits consists of a pair of conducting plates
separated by an insulating gap. Applying a potential difference across the plates causes
equal and opposite charge to accumulate on each plate, with the charge proportional to
the potential difference:
Q = CV
(15),
With the constant of proportionality referred to “capacitance”, measured in units of farads
(F). The capacitance is found to be proportional to the area of each plate, and inversely
proportional to the thickness of the gap between them:
C = A/d
(16),
With  a material constant for the insulator placed between the plates, known as the
dielectric constant. The value of  for vacuum is 0 = 8.85x10-12 F/m, as is that of air.
Physically, an insulator polarizes when a potential difference is placed across it. The
dielectric constant is a measure of this polarization. Values for several common
insulating materials are given below [1]:
d
Q+
+
+
+
- +
-
+
+
+
- Q- A -
Material
Dielectric Constant/0
Air, Vacuum
Paper
Plastic
Glass
Mica
Aluminum Oxide
Tantalum Oxide
Ceramic
1.00
2.0 – 6.0
2.1 – 6.0
4.8-8.0
5.4-8.7
8.4
26
12-400,000
Although the presence of an insulating gap prevents finite current from flowing from
plate to plate, the initial accumulation of charge on the plates takes a finite amount of
time if the circuit also contains resistance, R, in series with the capacitance. In fact,
starting from 0, the charge builds up to its steady state value Q0 exponentially, with a
time constant given by the product of R times C:
Q = Q0[1- exp(-t/RC)]
(17).
In practice, capacitors are used in timing circuits, and in circuits where the applied
voltage is not constant, but instead varies with time.
Inductors
When current passes through a wire it generates a magnetic field. If the current is timevarying, the field will vary with time as well. If the wire is wound as a coil, it is found
that a time varying potential is generated across it in response, which is proportional to
the time rate of change of the current, and is in a sense so as to oppose the change in
current:
V   L
I
,
t
(18)
where the constant of proportionality L, is called the inductance of the coil, and is
measured in Henry’s. The inductance of a coil is given by:
A
,
l
L  0 N 2
(19)
where  0  4 (10 7 ) Newtons/Ampere2 is the so-called permittivity of free space, N is
the number of windings in the coil, A is the cross-sectional area of the coil, and l is its
length.
Putting an inductor in series with a capacitor results in an oscillator, in which initially the
plates of the capacitor begin to charge, resulting in a time varying current which
generates a potential difference which attempts to charge the plates in the opposite sense.
Tha charge on the capacitor, and current through the inductor oscillate back and forth at
a frequency given by

2
LC
,
(20)
where v is measured in cycles per second or Hertz, C is measured in Farads, and L is
measured in Henries. If a speaker is placed in the circuit, it will generate sound at this
frequency. For an audible sound  must be in the range of 30-20,000 Hz.
Since inductors always have resistance, which dissipates power a source of power must
also be included in the circuit, or the sound will damp out with time. This is often done
by using “active” circuit elements, such as transistors
Transistors
Resistors capacitors and inductors are referred to as passive devices. The current
through a resistor or charge across a capacitor are linear with applied voltage. The
voltage generated across an inductor is linear in the time rate of change of the current.
An entirely different class of devices exist with current which is non linear with applied
voltage. These devices are based upon combinations of semiconductor materials (usually
silicon) with different types of impurity (usually boron in “p-type’ regions and
phosphorous in “n-type’ regions). These devices allow amplification, which is the
control of a large signal using a smaller control signal. They also allow switching
between states with a large “saturation” current and zero current, and thus for the
fabrication of digital electronic circuits which have two states: “on” and “off”. They are
called transistors, and come in two main varieties: hetero bipolar transistors (HBT’s) in
which a small control current between two terminals (the “base” and “emitter”) controls
a much larger current across the entire device (from “emitter” to “collector”), and field
effect transistors (FET’s) in which a small potential placed on a control terminal (the
“gate”) controls a current across the device (from “source” to “drain”). The discussion of
the physical principals behind transistor action is beyond the scope of this course, but the
interested reader can find a good introduction in the book by Kano [4], or the book by
Horowitz and Hill [5]. The book by Horowitz and Hill also discusses transistor-based LC
oscillators, which might be used to drive a speaker.
Amplifiers
Almost all electronic circuits are built up of combinations of transistors, resistors
and capacitors. Among the most important circuits are so-called “operational
amplifiers”, which come commercially packaged on a single semiconductor chip. We
will not go into detail about the design of these devices, but again refer the interested
reader to a good reference, the book by Horowitz and Hill [5]. Here we merely treat opamps as “black boxes”, with two important properties:
#1 the output of the opamp will adjust itself so as to make the difference in
potential at its two inputs zero,
#2 the inputs of an op amp draw essentially zero current.
The most common configuration for an op-amp is shown in Fig. 2, an inverting
amplifier. Using Kirchoff’s laws, and property #2, if the + terminal (the “noninverting
terminal”) is held at “ground” potential (the reference for the voltage source, then the “-“
(“inverting”) terminal is also at zero, and
Vout/R2 = - Vin/R1.
(21)
R2 is referred to as the feedback resistor, while R1 is referred to a the
programming resistor.
Note that the maximum value of Vout is limited by the supply voltages, typically
+/-15 V or +/- 5 V, depending on the design of the opamp. Thus, if R2/R1 is too large
(say R2 is infinite, i.e. missing) then a finite input voltage across the terminals will cause
the output to jump to the maximum in the opposite polarity. This is the principle of a
comparator, which switches to one extreme output or the opposite depending on
whether the voltage at the + or – terminal is larger.
Fig. 2. Schematic of Opamp used as an inverting amplifier.
References
[1] “The Electrical Engineering Handbook”, Ed. R. C. Dorf, (CRC Press, Boca Raton,
1993).
[2] “Physics of Semiconductor Devices”, 2nd Ed., S. M. Sze, ((Wiley, New York, 1981).
[3] “Handbook of Chemistry and Physics”, 53rd Ed., R. C. Weast, Ed., (CRC Press,
Cleveland, 1972-1973).
[4] Semiconductor Devices”, K. Kano, (Prentice-Hall, Upper Saddle River, 1998).
[5] “The Art of Electronic”, P. Horowitz and W. Hill, (Cambridge, New York, 1980.
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