Download ELECTRICITY II

Classification of Materials from the Standpoint of Electrical Conductivity
A. Conductors – materials that have a low value of resistivity that allowing them to easily pass
an electrical current due to their being plenty of free electrons floating about within their
basic atomic structure. Examples of good conductors are generally metals because these
materials have very few electrons in their outer valence shell, resulting in them being easily
knocked out of the atom’s orbit. This allows them to flow freely through the material until
they join up with other atoms thereby creating electric current.
B. Insulators – the exact opposite of conductors. They are made of generally non-metals, that
have very few or no “free electrons”. 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
insulating properties. Insulators also have a very high resistance, millions of ohms per
meter and are generally not affected by normal temperature changes.
C. Semiconductor – have electrical properties somewhere in the middle between those of
conductor and insulator. Examples are Si, Ge, Ga, As. The most commonly used
semiconductor material by far is silicon. The significant advantage of silicon is that it is
less temperature sensitive which became an important requirement for any electronic
device to achieve high levels of reliability.
Voltage, Current, Resistance and Conductance
I.
Voltage
a. The potential energy of an electrical supply stored in the form of an electric charge.
b. It can be thought as a force that pushes the electrons through a conductor.
c. As energy, it has the ability to do work and this potential energy can be described
as the work required in joules to move the electrons in the form of an electric current
around a circuit from one point of node o another.
The difference in the voltage between the two nodes in a circuit is known as potential
difference, sometimes called as Voltage Drop.
The constant voltage source is called a DC voltage. A voltage that varies periodically with
time is called AC voltage. Voltage is measured in volts, with one volt being defined as the
electrical pressure require to force an electric current of one ampere through a resistance
of one Ohm. Voltage can be neither positive or negative.
Voltage Symbol
Single Cell
II.
Multiple Cell
DC Voltage Source
AC Voltage Source
Electrical Current
a. Movement or flow of the electrical charge.
b. Measured in Amperes, “I”, for intensity.
c. It is the continuous and the uniform flow (drift) of electrons around a circuit that
are being pushed by voltage.
Conventional Current Flow
Conventionally, this is the flow of positive charge around the circuit. The diagram
shows the movement of positive charge which close to positive terminal of the battery,
through the circuit and returns to the negative terminal of the battery. Conventional
current flow is the opposite in the direction to the flow of electrons.
Electron Flow
The flow of electrons around the circuit is opposite to the direction of the
conventional current flow. The current flowing in a circuit is composed of electrons
that flow from the negative pole to the positive pole. The direction of current flow does
not affect what the current does to the circuit.
III.
Resistance
a. The ability to resist or prevent the flow of current through it making it necessary to
apply a bigger voltage to the circuit.
b. The opposition to the flow of current.
c. Resistance can never be negative.
d. Resistance is measured in Ohms, “Ω”
Since electrical conductivity varies with different materials, it is therefore proper and
convenient to assume that all substances possess a reciprocal property, a tendency to
oppose a current. This is called electrical resistance and the object possessing the property
is designed as a resistor.
Laws on Resistance
1.
2.
3.
4.
It varies directly with the length of the conductor.
It varies inversely with the cross-sectional area of the conductor.
It depends upon the material used.
It depends upon the temperature to which the material is being measured.
𝐿
Since: 𝑅 ∝ 𝐴 : 𝑅 = 𝑘𝐿/𝐴
Where: 𝑘 = 𝜌, 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑜𝑓 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙𝑖𝑡𝑦
Therefore:
𝑅=
Where:
𝜌𝐿
𝐴
R = resistance
𝜌 = resistivity, Ω-CM/ft
L = length of conductor
A = Cross-sectional Area: measured in circular mil
Circular Mil, CM – unit of cross section of small diameter conductor whose
diameter is equal to one mil.
1 mil = 10-3 inch
Square Mil – unit of cross section of small diameter conductor whose side is equal
to one mil.
Specific Resistance at 20oC
Material
Silver
Copper, annealed
Copper, hard drawn
Gold, Pure
Aluminum
Magnesium
Tungsten
Zinc
Platinum
Iron, Cast
Iron, commercial
Lead
Mercury
Brass
German Silver
Manganin
Lucero
Advance
Constatan
Excello
Nichrome
Nichrome II
Chromel
Element/Alloy
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Alloy
Alloy
Alloy
Alloy
Alloy
Alloy
Alloy
Alloy
Alloy
Alloy
Example:
1. Calculate the resistances of the following conductors at 20oC.
a. Material – Copper, L = 1000 ft, CM = 3200
b. Material – Aluminum, L = 4 miles, dia = 162 mils
Ω-CM/ft
9.9
10.37
10.65
14
17
28
33
36
47
54
75
132
577
42
199
265
280
294
302
550
600
660
625-655
2. A kilometer of wire having a diameter of 11.5 mm and a resistance of 0.035 ohm is drawn
so that its diameter is 5 mm. What does its resistance become?
3. An undetermined number of feet of wire have been used from the carton. Find the length
of the copper wire if it has a diameter of 1/16 inch and a resistance of 0.5 ohm.
Temperature – Resistance Effect
Resistance of an electric conductor depends among the other things, upon the
temperature. Experiment has demonstrated that the resistance of all wire, generally used in
practice in electrical systems increases as the temperature raised, moreover, within the
usual operating range, the resistance varies linearly with the temperature change.
Absolute Zero Temp – actual value of the temperature when the resistance is 0.
Inferred Zero Temp – theoretical value of the temperature of the material when its
resistance is 0.
𝑅2 = 𝑅1 [1 + 𝛼(𝑇2 − 𝑇1 )]
Where:
α – temperature coefficient of the resistance of the conductor
Example:
1. A coil of copper wire has a resistance of 20 ohms at 18oC. If the temperature coefficient
resistance is 0.004/oC, determine the resistance of the coil when the temperature rises to
98oC.
2. The tungsten filament of incandescent lamp has a resistance of 9.8 ohms at 20oC and a
resistance of 132 ohms at normal operating temperature. Calculate the temperature of the
heated element.
3. The temperature coefficient of copper is 0.00427/oC. The resistance of the coil is 8.82 ohms
at 24oC. Determine its resistance at 0oC and 42oC.
IV.
Conductance
a. The inverse of resistance
b. The greater the resistance, the lesser the conductance
c. The total conductance is equal to the sum of the individual conductance in a
parallel circuit.
Standard Electrical Units
Parameter
Symbol
Measuring Unit
Description
Voltage
Volt
V or E
Current
Ampere
I
Resistance
Ohm
R or Ω
Unit of DC Resistance
Conductance
Siemen
G or ℧
G = I/R
Capacitance
Farad
C
C = Q/V
Charge
Coulomb
Q
Q=CxV
Inductance
Henry
L or H
VL = -L(di/dt)
Power
Watts
W
P=VxI
Impedance
Ohm
Z
Z2 = R2 + X2
Frequency
Hertz
Hz
F = 1/T
Unit of Electrical Potential
V=IxR
Unit of Electrical Current
I = V/R
R = V/I
Electric Charge and Electric Current
The magnitude of a charge is commonly given in coulombs and for each negatively charged
electron is 1.602 x 10-19 coulomb. Putting it in another way, it would be necessary to have a
concentration of 6.28 x 1018 electrons before a charge of one coulomb is accumulated.
Furthermore, when one coulomb of current is said to be in ampere. Stating the foregoing in the
equation form,
𝐼=
𝑄
𝑡
If the current in amperes is constant, charge is transferred in a constant rate. For a non –
uniform current, the transferred charge will vary with the current changes.
Example:
1. The current in a conductor changes uniformly from zero to 2 amperes in 3 sec then remains
steady at 2 amperes for 6 sec and then drops uniformly to 1.5 amperes in 8 sec. Calculate
the total amount of charge transferred in the elapsed time of 17 sec.
2. The charge flowing through a copper wire between two batteries terminal is 0.6 C every
64ms. Determine the current in amperes.
3. Determine the time required for 4 x 1016 electrons to pass through a copper wire between
two batteries if the current is 5 mA.
Ohm’s Law and Electric Circuit
Ohm’s Law
It states that the current through a conductor between two points is directly proportional to
the potential difference across the two points and inversely proportional to the resistance between
them.
𝐼 = 𝑉/𝑅
Where:
E – impressed voltage (V)
I – circuit current (A)
R – circuit resistance, (Ω)
The resistance of most devices is constant over a large range of values of current and
voltage. When a resistor is used under these conditions, the resistor is referred to as an ohmic
device or ohmic resistor because a single value for the resistance suffices to describe the behavior
of the device over the range. When sufficiently high voltages are applied to the resistor, forcing a
high current through it, the device is no longer ohmic because its resistance, when measured under
stressed conditions is different from the value measured under normal conditions.
Ohm’s Law in the form above is an extremely useful equation in the field of engineering
because it describes how the voltage, current and resistance are interrelated on a “macroscopic”
level, that is, commonly, as circuit elements in an electrical circuit.
Electric Circuit
An electric circuit in its simplest form consists of – an electric source, a load and connecting
wires. Other elements maybe added: devices that perform control and regulating functions,
measuring instruments and protective devices.
Circuit – an interconnection of simple electrical device in which there is at least closed path
in which the current may flow.
Electrical Energy, Power and Heat
Power
a. measures the rate at which the energy is transformed, or the time rate of doing work.
b. The work done when one coulomb of electricity is moved through a potential
difference of one volt in one second.
Electromotive Force (EMF) – characteristics of a device or machine that tends to create
electron flow.
Potential Difference – when an emf is applied to the ends of conductor, it is proper to refer
to the existence of a potential difference between such ends.
Potential Drop – since increment of emf is required for successive increments of conductor,
it is customary to regard a potential difference as a drop in potential along the length of the
conductor.
Electrical Energy
If a charge Q flows between two points at a potential difference V, the change in potential
energy is given by:
𝑊 = 𝑄𝑉
Derived:
𝑊 = 𝐼 2 𝑅𝑡
𝑊 = 𝑉𝐼𝑡
Where:
𝑊=
V = Volts
R=Ω
W = Joule
I = Ampere
Q = Coulomb
t = seconds
𝑊=
Electrical Power (P)
𝑉2
𝑅
𝑡
𝑃
𝑡
Derived:
𝑃=
Where:
𝑄𝑉
𝑃 = 𝑉𝐼
𝑡
𝑃 = 𝐼2𝑅
𝑃=
𝑉2
𝑅
P = Watt, J/s
1kW = 1.34 hp
1 hp = 746 watts
Cost of Electrical Energy = W x Unit Cost
Joule’s Law
𝐻=𝑊
Heat equivalent of Electrical Energy:
𝐻=
𝑊
𝐽
;
𝐻 = 0.239
J = mechanical equivalent of heat (4.184 J/cal)
𝑐𝑎𝑙
𝑗
(𝑊)
Simple Electrical Circuit
1. Series Circuit
a. A single path for electric current through all of its components.
b. Sometimes called as current – coupled or daisy chain coupled.
c. The current in a series goes through every component in the circuit.
Laws governing Series circuit
1. The total resistance is equal to the sum of resistance in each branch.
𝑅𝑡 = 𝑅1 + 𝑅2 + 𝑅3 + . ..
2. The total current is equal to the current passing through each branch
𝐼𝑡 = 𝐼1 = 𝐼2 = 𝐼3 = ⋯
3. The total voltage is equal to the voltage drop across each branch
𝑉𝑇 = 𝑉1 + 𝑉2 + 𝑉3 + ⋯
4. Power Calculation
𝑃𝑇 = 𝑃1 + 𝑃2 + 𝑃3 + ⋯
Voltage Divider
𝑉1 =
𝐸𝑅1
𝑅𝑡
2. Parallel Circuit
a. A different path for current through each of its components.
b. Two or more components have the same potential difference
c. The potential differences across the components are the same in magnitude and
they also have identical polarities
d. The total current is in accordance with the Kirchhoff’s Current Law
Laws governing
1. The total resistance is equal to the sum of the resistance in each branch
1
1
1
=
+
+⋯
𝑅𝑇 𝑅1 𝑅2
2. The total current is equal to the sum of the current passing through each branch
𝐼𝑡 = 𝐼1 + 𝐼2 + 𝐼3 + ⋯
3. The total voltage is equal to the sum of the voltage drop across each branch
𝑉𝑇 = 𝑉1 = 𝑉2 = 𝑉3 = ⋯
4. Power Calculation
𝑃𝑇 = 𝑃1 + 𝑃2 + 𝑃3 + ⋯
Current Divider
𝐼1 =
3. Series – Parallel Circuit
𝐼𝑇 𝑅𝑇
𝑅1
4. Parallel – Series Circuit
Example:
1.
2.
3. In this circuit, three resistors receive the same amount of voltage (24 volts) from a single
source. Calculate the amount of current “drawn” by each resistor, as well as the amount of
power dissipated by each resistor:
4.
5. From the circuits shown below, determine the equivalent resistance at terminal AB
6. Find the equivalent RAB
Delta – Wye and Wye – Delta Transformation
To convert Delta to Wye
𝐵𝐴
𝑅3 = 𝐴+𝐵+𝐶
𝐴𝐶
𝑅2 = 𝐴+𝐵+𝐶
𝐵𝐶
𝑅1 = 𝐴+𝐵+𝐶
To Convert Wye to Delta
𝑅𝐴 =
12+23+13
Example:
1. Find RAB:
𝑅1
𝑅𝐵 =
12+23+13
𝑅2
𝑅𝐶 =
12+23+13
𝑅3
2. Find RAB:
Cells – a combination of solid materials, usually metals called electrodes, immersed in a chemical
solution called an electrolyte, that is held by a single container; converts chemical energy to
electrical energy
Three Major Components of Cell
1. Anode – the reducing or fuel electrode which gives up electrons to the external circuit and
is oxidized during the reaction
2. Cathode – the oxidizing electrode that accepts electrons
3. Electrolyte – ionic conductor which provides the medium for transfer of charge
Kinds of Cell:
1. Primary Cell
a. Most active in converting chemical energy to electrical energy when they are
assembled.
b. Lose activity in service and with age, discarded when exhausted.
c. Undergoes irreversible chemical reaction
2. Secondary Cell
a. Undergoes reversible reaction
b. May therefore be charged
A. Wet cell – has liquid electrolyte. Other names are flooded cell since liquid covers all
internal parts, or vented cell since gases are produced during operation.
B. Dry Cell – has the electrolyte immobilized as a paste with only enough moisture in the
paste to allow current to flow. The makeup of a standard dry cell is a zinc anode
(negative pole), usually in the form of cylindrical pot and carbon cathode in the form
of central rod. The electrolyte is ammonium chloride in the form of a paste next to zinc
anode.
Classification of DC Source in terms of their energy source
1.
2.
3.
4.
5.
6.
Chemical Source
Solar and Photovoltaic Cells
Thermoelectric Generation
Piezo – Electric Generation
Electromagnetic Generation
Electrical Conversion
Combination of Cells
1. Series
Where:
e = emf
r = internal resistance
rt = total internal resistance
Rt = total external resistance
E = total emf
a. 𝐸 𝑜𝑟 𝑉𝑡 = 𝑛𝑒
b. 𝑟𝑡 = 𝑛𝑟
𝑛𝑒
c. 𝐼𝑡 = 𝑛𝑟+𝑅
𝑒𝑥𝑡
It = total current in the circuit
n = no. of cells
M = no. of rows in parallel
2. Parallel
a. 𝐸 𝑜𝑟 𝑉𝑡 = 𝑒
𝑟
b. 𝑟𝑡 = 𝑛
c. 𝐼𝑡 =
𝑛𝑒
𝑟
+𝑅𝑒𝑥𝑡
𝑛
3. Series – Parallel
a. 𝐸 𝑜𝑟 𝑉𝑡 = 𝑛𝑒
𝑟
b. 𝑟𝑡 = 𝑀𝑛
c. 𝐼𝑡 =
𝑛𝑒
𝑟
+𝑅𝑒𝑥𝑡
𝑀𝑛
Network Laws and Theorems
Elements of a network:
1. Branch – any portion of a circuit with two terminals connected to it. A branch may
consist of one or more circuit elements.
2. Node – the junction consists of 2 or more branches. A supernode is obtained by defining
a region that encloses more than one node.
3. Loop – any closed connection of branches.
4. Mesh – a loop that does not contain other loops.
Kirchhoff’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.
Kirchhoff’s Voltage Law
States that “in any closed loop network, the total voltage around the loop is equal to the
sum of all voltage drops within the same loop.” In other words, the algebraic sum of all the voltage
drop within the loop must be equal to zero.
Example:
1. Find the current IA, IB, IL.
2. Two resistors RA and RB are connected parallel across 120 V source. What voltage exists
between the midpoint of Ra and a point that is one-third from either end of Rb.
3. For the circuit shown, find the current II and the voltage V.
Problem Set:
1. Determine the electrical force of attraction between two balloons with separate charges of
+3.5 x 10-8 C and -2.9 x 10-8 C when separated a distance of 0.65 m.
2. Determine the electrical force of attraction between two balloons that are charged with the
opposite type of charge but the same quantity of charge. The charge on the balloons is 6.0
x 10-7 C and they are separated by a distance of 0.50 m.
3. Joann has rubbed a balloon with wool to give it a charge of -1.0 x 10-6 C. She then acquires
a plastic golf tube with a charge of +4.0 x 10-6 C localized at a given position. She holds
the location of charge on the plastic golf tube a distance of 50.0 cm above the balloon.
Determine the electrical force of attraction between the golf tube and the balloon.
4. A balloon with a charge of 4.0 µC is held a distance of 0.70 m from a second balloon
having the same charge. Calculate the magnitude of the repulsive force.
5. At what distance of separation must two 1.00-microCoulomb charges be positioned in
order for the repulsive force between them to be equivalent to the weight (on Earth) of a
1.00-kg mass?
6. How many calories per second are generated in an electric heater that has a resistance of
20 ohms and uses 5A?
7. A 60 W lamp at 15 V and a 100 W at 30 V are connected in series. If the voltage applied
to the connection is 50, determine the power of each lamp.
8. The total power used by three equal resistors connected in series is 15 W when a constant
voltage is applied to the connection. If the resistances are connected in parallel and the
same voltages applied, what will be the total power?
9. Determine the equivalent resistance between c and d for the combination shown.
10. Find RAB:
11. RAB:
12. Find RAB: