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Electrical, Electronics
Engineering Department
Solve problems
in
multiple path d.c.
circuitsUEUNEEE004A
Revision 1 2 3 4 5 6Date 08/2005 03/2007 01/09
Contact IH IH DK
Chisholm Institute
Stud Road
Dandenong 3175
Tel: +61 3 92124526
Fax: +61 3 92124999
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Required Skills and Knowledge
Duration 40 hours
E2.8.2.1 Direct current circuit principles
Evidence shall show an understanding of electrical principles to an extent indicated by
the following aspects:
a) Factors affecting resistance encompassing:
The factors of length, cross-sectional area and material effect the resistance of
conductors.
effects of temperature change on the resistance of various conducting materials.
the resistance of a conductor from factors such as conductor length, cross-sectional
area, resistivity and changes in temperature.
b) Resistors encompassing:
features of fixed and variable resistor types and typical applications.
characteristics of temperature, voltage and light dependent resistors and typical
applications of each.
specifying a resistor for a particular application.
resistance of a colour coded resistor from colour code table and confirm the value by
measurement.
c) Series circuits encompassing:
setting up and connecting a single-source series dc circuit.
Measurement of resistance, voltage and current values in a single source series
circuit.
the voltage, current, resistances or power dissipated from measured or given values
of any two of these quantities.
relationship between the voltage drops around a circuit and the applied voltage.
relationship between voltage drops and resistance in a simple voltage dividernetwork.
output voltage and current levels of connecting cells in series.
d) Parallel circuits encompassing:
setting up and connecting a single-source parallel circuit.
Measurement of resistance, voltage and current values in a single-source parallel
circuit.
the voltage, current, resistance or power dissipated from measured or given values of
any of these quantities.
relationship between currents entering a junction and currents leaving a junction.
relationship between branch currents and resistances in a two branch current divider
network. voltage and current levels of connecting cells in parallel.
e) Series/parallel circuits encompassing:
setting up and connecting a single-source series / parallel circuit.
Measurement of resistance, voltage and current values in a single-source
series/parallel circuit.
the voltage, current, resistances or power dissipated from measured or given values
of any two of these quantities.
relationship between voltages, currents and resistances in a bridge network.
voltage and current levels of connecting cells in series parallel.
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f) Measurement of electrical quantities encompassing:
hazards involved in using electrical instruments and the safety control measures that
should be taken.
operating characteristics of analogue and digital meters.
selecting an appropriate meter in terms of units to be measured, range, loading effect
and accuracy for a given application. measuring resistance using direct, volt-ammeter and bridge methods.
instruments used in the field to measure voltage, current, resistance and insulation
resistance and the typical circumstances in which they are used.
g) Capacitance encompassing:
definition of capacitance and explain how a capacitor is charged.
the units by which capacitance is measured.
relationship between capacitance, voltage and charge.
Behaviour of a series d.c. circuit containing resistance and capacitance components.
h) Capacitors encompassing: hazards involved in working with capacitance effects and the safety control
measures that should be taken.
factors which determine the capacitance of a capacitor and explain how these factors
are present in all circuits to some extent.
effects of capacitors connected in parallel by calculating their equivalent
capacitance.
effects on the total capacitance of capacitors connected in series.
common faults in capacitors.
testing of capacitors to determine serviceabity.
Useful references include:
Jenneson, J.R.,Electrical Principles for the Electrical Trades. Prentice Hall, Sydney,
5th Ed McGraw Hill, Sydney.
Phillips, P.,Electrical Principles. Prentice Hall, Sydney.
Van den Bergen, B.,Mathematics for the Electrical Trades. TAFE Publications,
RMIT Melbourne.
Pethebridge, K., Neeson, I.,Electrical Wiring Practice. McGraw Hill, Sydney.
Batty, I. 1996,Electrical Principles. Prentice Hall, Sydney.
Occupational Health & Safety Requirements
A safe and healthy environment will be provided for students and teachers as well as
safety procedure with regard to learning and teaching activity.
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Table of ContentsReview Questions ........................................................................................................7Resistors and Resistance ..........................................................................................11
Variable resistor......................................................................................................11Temperature-dependent resistors ..........................................................................12
Light-dependent resistors.......................................................................................14Voltage-dependent resistors...................................................................................15Resistor colour code...............................................................................................16Resistivity ...............................................................................................................18Summary................................................................................................................24Unit test ..................................................................................................................25Review questions ...................................................................................................27
Series Resistive Circuits ............................................................................................29Resistance in a series circuit..................................................................................29Unit test ..................................................................................................................35Review questions ...................................................................................................36
Parallel Resistive Circuits...........................................................................................43Current in a parallel circuit......................................................................................48Power in a parallel circuit........................................................................................49Voltage in a parallel circuit......................................................................................50Summary of characteristics of parallel circuits .......................................................51Unit test ..................................................................................................................52Review questions ...................................................................................................53
Combined SeriesParallel Resistive Circuits .............................................................60Unit test ..................................................................................................................70Review questions ...................................................................................................74
Electrical Measuring Instruments...............................................................................81Introduction.............................................................................................................81
Moving coil meter ...................................................................................................81Moving iron meter...................................................................................................85Reading a meter error of parallax........................................................................88Extending the range of voltmeters..........................................................................88Extending the range of ammeters ..........................................................................90Non contact testing.................................................................................................91Dynamometer movement instruments....................................................................92Care, selection and protection of instruments ........................................................93
The multimeter......................................................................................................100Self help questions...............................................................................................105Digital meters........................................................................................................106Review questions .................................................................................................107
Capacitors and Capacitance....................................................................................112Electrostatics and Capacitance................................................................................112
Definitions.............................................................................................................112Electrostatics ........................................................................................................113Electrostatic charge and discharge ......................................................................114Calculation of capacitance....................................................................................115Commercially available capacitors .......................................................................117
Calculation of Measurement of Capacitance Networks ...........................................124Capacitance in series circuits...............................................................................124Capacitance in parallel circuits.............................................................................127Capacitance measurement...................................................................................128Unit test ................................................................................................................133
Review questions .................................................................................................136Chapter review questions.....................................................................................139
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Review QuestionsThe intention of these review questions is to refresh your memory of work covered in
EUENEEE003A Electrical Fundamentals. Complete these now before starting the
section on resistors and resistance.
For each question circle the response you consider best fits the sentence.
1. The unit of electrical power is the:a. Volt.
b. Ampere.c. Ohm.d. Watt.
2. Power is directly related to:
a. How quickly a body is accelerated.b. How heavy a body is.c. The distance a force moves a body.d. The rate at which work is done on the body.
3. The ability to do work is called:a. Energy
b. Powerc. Velocityd. Acceleration
4. The current in a circuit which is consuming power can be calculated by usingthe formula:
a. 2RPI=
b. RPI=
c. PVI 2=
d. RPI=
5. Power in a DC circuit can be determined by combining the readings from twoseparate instruments. These are the:
a. Voltmeter and wattmeter
b. Ammeter and wattmeterc. Voltmeter and ammeterd. Ohmmeter and wattmeter
6. The voltage coil of a wattmeter is connected in parallel with the circuitresistance and it has a resistance that is:
a. Lowb. Highc. Negligibled. Variable
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7. The current coil of a wattmeter is connected in series with the circuitresistance and it has a resistance that:
a. Is lowb. Is highc. Equals zero
d. Is variable
8. Assuming the voltage remains constant, if the resistance of a circuit isdoubled, the power will:
a. Doubleb. Decrease by four timesc. Halved. Increase by four times
9. If the current flowing through a resistor falls to half its original value, thepower will:
a. Doubleb. Decrease by four timesc. Halved. Increase by four times
10.If the voltage applied to a resistor is halved the power will:a. Double
b. Increase by four timesc. Halved. Decrease by four times
11.Calculate the power being consumed by an electrical appliance that is drawing5 A from a 240 V supply.
12.An electric toaster element has a voltage rating of 240 V and a power rating of550 W. Calculate the:
a. Resistance, of the element.b. The circuit current when the rated voltage is applied to the element.
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13.Calculate the power consumed by a 50 ohm resistor that has 5 amperesflowing through it.
14.A lamp with a resistance when hot of 960 ohms is connected to the 240 Vmains. Determine the power being consumed by the lamp.
15.Calculate the power dissipated by a coil with a resistance of 8 ohms that ispassing 15 amperes. Express your answer in kW.
16.Calculate the maximum working current of an appliance rated at 2.4 kW withan operating voltage of 240 V.
17.Calculate the resistance of a radiator element that consumes 1000 W whenconnected to the 240 V mains.
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18.The power circuit of a domestic installation consists of three power outlets,each rated at 1000 W for a supply voltage of 250 V. Calculate the maximum
circuit current.
19.A length of single core copper cable has a resistance of 0.02 ohm per metre.Calculate the power loss in the cable if it is supplying 10 amperes to a load
located 50 metres from the source of supply.
20.For the circuit of Fig.1:a. Complete the circuit so that the ammeter indicates the circuit current, the
voltmeter indicates the supply voltage and the wattmeter indicates the
circuit power.
b. Determine the ammeter, voltmeter and wattmeter readings.c. Indicate with an arrow the direction of the current (use conventional
current flow). Label the arrow as I (for current) and also identify the
polarity of the battery terminals with a + and sign.
Figure 1
Load
A
+
V
+
+
W
LM
V2V1
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Resistors and ResistanceIn Applied Electricity 1, you learned that temperature can affect the resistance of a
conductor. Sometimes this effect is useful. Special types of resistors that change
resistance due to an external influence, such as temperature or light, have many uses.
Sometimes the change in resistance is linear, which means the resistance change
exactly follows the change of the external influence. For example, the resistance of a
metal conductor increases in a linear manner with temperature (for temperatures
above 0C). If the change was non-linear, the resistance change between; say 0C to
100C would be different to the change between 100C and 200C, even though the
temperature change in both cases is 100C.
The most common used types of resistors that change their value due to an external
influence are the variable resistor, the temperature-dependent resistor, the light-
dependent resistor and the voltage-dependent resistor.
Variable resistor
This type has a wiper that is moved along a resistive track. The resistance of a
variable resistor is therefore changed mechanically by positioning the wiper. If the
change in resistance is directly proportional to how much the wiper is moved, the
resistance element is linear.
Some types of variable resistors have a non-linear resistance element. The effect of a
linear and a non-linear resistance element is shown in Figure 2. As you can see, the
minimum and maximum resistance is the same in both cases. However, when thewiper is moved half-way, the linear resistance element measures 5 ohms, which is half
its maximum resistance, while the non-linear element is about one-tenth at 1 ohm. The
wiper would need to move almost three-quarters of its full travel to give a resistance
reading of 5 ohms.
Figure 2 Linear and non-linear variable resistors
Minimum Half-way Maximum
Linear
Non-linear
0 ohm 5 ohm 10 ohm
0 ohm1 ohm 10 ohm
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The most common use for a non-linear variable resistor is the volume control in a
radio or an amplifier. Because human hearing is logarithmic, the resistance element in
a volume control is also logarithmic to make the change in sound level appear to
follow the change in the position of the control.
Because our hearing is logarithmic, we can hear the gentlest whisper yet still bear thesound of a jet plane at take-off. However, the difference in the power of these two
sounds is immense, far more than it seems to our ears.
Temperature-dependent resistors
Resistors that change their resistance value with a change in temperature have a
number of applications. For example, they are often used as the sensor in a
temperature measuring system, an over-temperature protection system or a
temperature control system.
A temperature-dependent resistor can be made by winding copper, nickel or platinum
wire around a ceramic former. The resistive element is placed inside a protective
sheath. The whole assembly can then be used as a probe in a temperature measuring,
protection or control system. The change in resistance for each degree of temperature
change is relatively small, although it will be linear. For example, if the resistance is
100 ohm at 0C, it will be about 140 ohms at 100C.
Another very commonly used temperature dependent resistor is the thermistor
(thermal resistor). These devices are made with a semiconductor material and have
either a positive temperature coefficient (PTC) or a negative temperature coefficient
(NTC). They give a large change in resistance over their operating range, althoughthey are limited to uses where the temperature doesnt exceed a few hundred degrees
Celsius.
Thermistors are made in many styles. Those shown in Figure 4 are the evacuated glass
bulb (no air inside) and the disc type. Both come in various sizes. In the glass bulb
type, the resistance element is in a vacuum. As a vacuum cannot conduct heat, this
type of thermistor doesnt change resistance with external temperature variations.
Instead, it changes resistance by the heating effect of the current passing through the
element. Currents less than a few milliamps will cause a resistance change, and this
type of thermistor is mainly used in electronic circuits.
Thermistors are often used to protect an electric motor against overheating. In small
electric motors, a PTC thermistor is embedded inside the windings and connected so
the motor current passes through the thermistor. If the temperature of the windings
rises above a certain value, the resistance of the thermistor will quickly increase (from
a low resistance to several thousand ohms), reducing the motor current to almost zero.
In other words, the thermistor behaves almost like a switch that turns off when the
temperature is too high. When the motor cools down, the resistance of the thermistor
drops, letting the motor run. In larger motors, the thermistor is connected to operate a
switch that cuts off power to the motor.
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Light-dependent resistors
Light-sensitive devices have a number of uses in electrical and electronic circuits. The
light-dependent resistor (LDR) consists of a thin ceramic disc sintered with cadmium-
sulphide. Sintering is the process of using heat to combine two materials. Cadmium-
sulphide is a photo-conductive material, which means its resistance is affected bylight. A vacuum-deposited metallic grid is applied to the surface of the disc and the
whole assembly is then covered in clear plastic.
The resistance of LDRs in complete darkness is over 10 million ohms (10 M),
which falls to less than 100 ohms in normal daylight. They are used in light meters
and automatic exposure controls in cameras.
LDRs are also used as a light sensor in automatically controlled lighting installations,
where the increase in resistance of the LDR is used to switch on the lights at sunset.
The LDR is connected to operate a relay, which in turn switches the lights. When its
daylight, the resistance of the LDR falls, switching off the relay that controls the
lights.
The LDR has a time lag between changes in light to a change in resistance. It takes
about 10 milliseconds for an LDR to respond to a change from total darkness to
daylight. In this case, the resistance of the LDR drops. However, it takes over one
second for the resistance to increase when the light is removed. Therefore, the LDR is
not used to detect fast changes of light. This is useful, as flashes of lightning in the
night are too fast to cause the LDR to turn off the lights.
The construction, symbol and response curve of an LDR is shown in Figure 6. Atypical LDR is about the size of a five cent piece. As the response curve shows, the
LDR is a non-linear resistor.
Figure 6 The LDR symbol, construct ion and response curve
Resistance - ohms
Response curve
10 M
100
Dark Daylight
Light intensity
Symbol
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Voltage-dependent resistors
A voltage-dependent resistor (VDR), as the name suggests, is a component in which
the voltage across the device affects its resistance. VDRs are usually made from
silicon carbide, which is mixed with a ceramic material formed into either a disc or a
rod. The assembly is fired at a high temperature and covered with a protectiveinsulation.
The main use of VDRs (or varistor) is to protect a circuit from a voltage surge. In
principle, when the voltage across a VDR goes over a certain value, the resistance of
the VDR will quickly drop to a very low value. If the voltage surge lasts long enough,
the large current flowing in the VDR will blow the circuit fuse (or trip a circuit
breaker) and isolate the circuit from the supply. If the surge is very brief, the energy
contained in the surge will be dissipated by the VDR without blowing the fuse.
A VDR is therefore given two ratings: the break-over voltage and the energy (in
joules) it can dissipate. Voltage ratings vary from 5V to several hundred volts. A
typical voltage rating for a VDR connected across the 230V AC mains supply is
275V. A VDR is connected across (or in parallel with) the circuit it is protecting as
illustrated in Figure 7. Notice that the VDR is connected after the fuse.
Figure 7 A VDR is connected so that it will cause the fuse to blow if the voltage to thecircuit is too high
VDRs are also used as surge diverters in overhead power supply lines by connecting
them from each line to a metal stake driven into the ground. If the lines are struck by
lighting, the VDRs divert the voltage surge to the ground, minimising or preventing
damage to equipment connected to the lines. The shape and size of a VDR determines
the amount of energy it can dissipate. Large VDRs are made by combining a number
of smaller ones.VDRs used as voltage protection devices with low power electrical or electronic
equipment are about 2.5 cm in diameter, and look like those shown in Figure 8. Surge
diverters used to protect large electric motors or mains supply lines are much larger as
they need to be able to dissipate lots of energy.
Construction Symbol
Figure 8 Typical construction and symbol of VDR
VDR230 V
mains
fuse
V
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Although not shown, the resistance of a VDR is non-linear. The usual way of showing
the electrical behaviour of a VDR is with a graph that plots current against voltage,
rather than showing resistance versus voltage. The response time of a VDR is very
fast an important characteristic if it is to give enough protection against a voltage
surge caused by a lighting strike.
You can buy 230V plugs fitted with VDRs. These plug into a power point and
protect an appliance connected to the power point from a voltage surge these are
especially useful if a computer is connected to the power point.
Resistor colour code
Fixed value resistors are made in size to suit the power rating of the resistor. Those
shown in Figure 9 are typical low power resistors with a power rating on one watt or
less. Because these resistors are so small, it is not possible to print their resistance
value on the body of the resistor.
Rating Resistor size
0.25 Watt
0.5 Watt
1 Watt
Figure 9 Lowpower resistors; their value indicated with a colour code
Instead, a series of coloured bands are painted around the body of the resistor, as
shown on the resistors. Each colour represents a number and the resistance value is
therefore determined by reading the colour code. Larger resistors have their
resistance value marked with numbers. There are two types of resistors that use colour
codes: general purpose resistors and precision resistors. The difference between these
two resistors is their tolerance value.
Tolerance refers to how close the actual value of the resistor is to its marked value.
For example, a resistor might be coded as having a resistance of 1000 ohms with a
tolerance of 10%. This means the maker guarantees the actual value wont be more
than 10% higher or lower than 1000 ohms. As 10% of 1000 is 100, the actual value ofthe resistor could be anywhere between 900 ohms (1000 100) to 1100 ohms (1000 +
100).
General purpose resistors are those with a tolerance of 5% or more. Precision
resistors have a tolerance of less than 5%, usually 2% or 1%. General purpose
resistors have their value indicated with four bands, and precision resistors have five
bands. The last band is always the tolerance band. The two types of resistors and what
each band represents are shown in Figure 10.
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General purpose fourband colour code
Precision fiveband colour code
Figure 9 Colour bands for four and fi ve band resistors
The colours used in the resistor colour code are shown in Table 1. (You dont need to
remember this table, but you should know how to use it).
Colour
Significant
figures Multiplier Tolerance
Black 0 1 (100)
Brown 1 10 (101) 1%
Red 2 100 (102) 2%
Orange 3 1 000 (103)
Yellow 4 10 000 (104)
Green 5 100 000 (105)
Blue 6 1 000 000 (106)Violet 7
Grey 8
White 9
Gold 0.1 (101) 5%
Silver 0.01 (102) 10%
None 20%
Table 1 Resistor colour code
1ST significant
figure
2ND significant
figureMultiplier
Tolerance
1ST significant
figure
2ND significant
figureMultiplier
Tolerance
3RD significant
figure
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The following examples show how to use the colour code. The first two bands (or
three for precision resistors) are given a number, the next band is the multiplier and
the last (on the right) is the tolerance band.
= 120,000 ohm or 120 k 5% = 10 ohm 10%
= 4.7 ohm 5% = 1,200 ohm or 1.2 k 2%
Figure 11 Using the resistor colour code
Resistivity
Because each different type of conductor has a different atomic construction, each
conductor has a different electrical resistance. Previously (NUE052 Applied
Electricity 1), it was stated that lR and AR 1 .
Combining both of these gives:
A
lR
Length and cross-sectional area are the standard physical sizes for comparison
purposes and each is applied to the various types of materials in order to compare
their resistances. The unit for length is the metre (m) and for area the square metre
(m2). This means that theoretically each type of material is made up into a block 1 mlong with a cross-sectional area of 1 m2. the resistance is then measured along its
length at a specified temperature and the value becomes the reference or standard. The
formula AR 1 is thus expressed in the form:
A
lR
=
where (pronounced rho) = resistivity.
The resistivity for a material is defined as the resistance between the oppositefaces of a 1 metre cube at a specified temperature.
brown red yellow gold brown black black silver
1 2 x 10,000 5% 1 0 x 1 10%
yellow violet gold gold brown red black brown
4 7 x 0.1 5% 1 2 0 x 10
red
2%
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In practice a block of material 1 m x 1 m x 1 m is cumbersome and expensive so a
smaller sample of material is used and the resistance value obtained adjusted
mathematically to the base size.
Knowing the resistivity of any material, the resistance of any conductor can becalculated, due allowances being made for temperature differences where necessary.
In Table 2, some electrical materials are listed, together with their resistivity value.
The resistivity values given in the table are the resistance values between opposite
faces of a 1 m x 1 m x 1 m cube at 20oC. the units are given in ohmmetres rather
than as ohms/m3 because by transposition, the formula AlR = can be written as
( ) lRA= .
Using units (R = , A = m2, l = m) this becomes:
mm
mxmx=
=
Conductor Resistivity () at20
oC in ohm
metres
aluminium 2.83 x 108
copper 1.725 x 108
gold 2.32 x 108
lead 2.04 x 108
platinum 10.09 x 108
silver 1.62 x 108
steel 16.6 x 108
Pure metals used
for conductors
German silver 33 x 108
advance 49 x 108
manganin 48 x 108
Nichrome 112 x 108
constantan 47 x 108
Alloys used as
resistance wire
carbon 5 x 105
germanium 5.5 x 101
silicon 5.5 x 102
Semiconductors
paper 1 x 1010
mica 2 x 1014
Teflon 1 x 1015
porcelain 1 x 1016
glass 8 x 1016
Insulators
Table 2 Resistivity of selected materials
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Self help questions
1. Which of the materials in Table 2 has the least resistance?
2. Which of the materials in Table 2 has themost resistance?
As you can see from Table 2, copper will have less resistance than steel. This explains
why the most common material used for wire in electronics is copper. On the other
hand, carbon is commonly used in the manufacture of resistors, Insulators, which have
a very high resistance prevent the flow of electricity.
3. List five items from Table 2 that will make good insulators. Insulators arematerials such as plastic, ceramic and mica. Each has its own use, determined
by its characteristics. Engineers will select the best material for a particular
job.
1.
2.
3.
4.
5.
Conductors Insulators Semi-conductors
Aluminium Ceramics Germanium
Brass Glass Silicon
Copper Mica Carbon
Gold Neon
Silver Plastic
Solder Wood
Steel Paper
Distilled WaterTeflon
Fibreglass
Table 3 Conductors, insu lators and semi-conductors
Note: Semi-conductors have resistance somewhere between a conductor and an
insulator. The makers of transistors and integrated circuits use germanium and
silicon. You will learn more about these materials and their use in other
modules in the course.
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Resistivity is abbreviated by the Greek letter rho (p). The greater the value of a
materials resistivity, the greater the amount of resistance offered by that material to
current flow.
The formula for the amount of resistance offered by a piece of material is:
A
lR
=
Where: R is the resistance in ohms
lis the length of the material in metres
A is the cross-sectional area in metres squared (m2)
is the resistivity of the material
Example
What is the resistance of a 2 m length of copper wire that has a cross-sectional area of
0.5 x 106 m2?
=
=
=
10xx
6-
0688.0
5.0
21072.1 8
A
lR
Notice the resistance is very small; this is what is required for a conductor
that connects component together. The connecting wires in a circuit should
have as little resistance as possible.
4. What is the resistance of a 2rn length of copper wire that has a cross sectionalarea of 1.0 x 10 m2?
5. What is the resistance of a block of germanium that is 0.4 cm long and is1.2 cm wide and 1.25 cm high? Note: The cross-sectional area A = width x
height
You can see from the formula and from Figure 12 that as cross-sectional area (A)increases the resistance of the material decreases. If both wires are equal in length, the
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resistance of the wire in Figure 12 (b) is half that of the wire in Figure 12 (a) because
its cross-sectional area has doubled.
(a) = xR
(b) = 2
xR
Figure 12 Relationship of resistance to area
Why then arent we surrounded by very thick copper wires, for telephone and power
distribution to our homes? The answer is simple; there are other factors to think about
when selecting materials and sizes for a particular project. Things such as cost,weight, availability, strength and many more all have to be considered.
6. What is the resistance of a 100 m length of copper wire that has a cross-sectional resistance area of 0.25 x 106 m2?
7. What is the resistance of a 100 m length of gold wire that has a cross-sectionalarea of 0.25 x 106 m2?
8. What is the resistance of a 100 m length of tungsten wire that has a cross-sectional area of 0.25 x 106 m2?
The copper wire is the one with the least resistance of the three in these Self Help
Questions. There are specialist applications however, where the tungsten or the gold
wires would be preferred. You will learn about these applications later.
What happens to the resistance of the wire if we decrease the length of the wire?
Well use similar wire to Self Help Questions 6, 7 and 8.
Copper wire A = 0.5 mm2
Copper wire A = 1.0 mm2
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9. What is the resistance of a 50 m length of copper wire that has a cross-sectional area of 0.25 x 106 m2?
10.What is the resistance of a 200 m length of copper wire that has a cross-sectional area of 0.25 x 106 m2?
It can be seen from Figure 13, if the length of wire is halved, the resistance of each
piece becomes half that of the original length, provided everything else remains
constant.
(a) = 2xR
(b) = xR
Figure 12 Relationship of resistance to area
If the wire is replaced with a wire having twice the length, then the resistance will be
doubled. This again assumes that all other characteristics remain the same.
Resistance is directly proportional to length. That is, any increase (or decrease) in
length will give an increase (or decrease) in resistance by the same amount. You
should also have learnt that resistance is inversely proportional to area. That is, any
increase (or decrease) in area will give a decrease (or increase) in resistance by the
same amount.
Copper wire l= 50 m
Copper wire l= 100 m
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Summary
Resistance is the opposition that an electrical circuit offers to current flow.
As circuit resistance increases, circuit current decreases. As circuit resistance decreases, circuit current increases.
There is no current flow if a circuit is open circuit.
Materials with a valence number of 1 are good conductors of electricity.
Materials with a valence number of 8 are poor conductors of electricity.
Materials with a valence number of 4 are semi-conductors.
Resistivity is the specific resistance rating of a material. It shows a materialsability to oppose current.
Resistance is directly proportional to the length of material, and inverselyproportional to the cross-sectional area of material.
A
lR
=
SI units used to express common electronic units.
Engineering notation is a short hand method of writing large and smallnumbers.
Prefixes are used to express multiples and sub-multiples of numbers.
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Unit test
1. List three quantities that are common to all electrical/electronic circuits.
a.
b.
c.
2. What is the name given to the quantity that opposes current flow?
3. A circuit having a greater resistance will offer less opposition to the flow ofcurrent.
True / False
4. State the three factors that determine the resistance of a material.5.
a.
b.
c.
Question 5 to 10 related to the following table:
Table 1 Resistivity of Common Electrical Materials
Material Resistivity (in -m) at
20C
Aluminium 2.83 x 108
Carbon 5000 x 108
Copper 1.725 x 108
Gold 2.32 x 108
Nichrome 112 x 108
Silver 1.62 x 108
Steel 16.6 x 108
6. Which of the following would be preferred as resistance wire?a. Gold
b. Nichromec. Silverd. Copper
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7. Which of the following is the most commonly used for electrical wire?a. Gold
b. Nichromec. Silver
d. Copper
8. Which of the following materials is commonly used to make resistors?a. Carbon
b. Aluminiumc. Copperd. Gold
9. The resistivity of steel is 16.6 x 108m. What will be the resistance of around steel bar that is 25 cm long and has a cross-sectional area of
2 x 106 m2?
10.A nichrome wire resistor of 25 ohms is needed. What length of wire, having across- sectional area of 2.5 x 107 m2 would be needed?
L = ______________
11.A 2.6m length of copper wire has a resistance of 0.0598 ohms. What is thecross-sectional area of the wire?
A = _____________
12.A length of gold wire is cut in half. What happens to its resistive value?
13.A copper conductor is replaced by another copper conductor that has greatercross-sectional area than the original. What change will have occurred in
resistance of the conductor?
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Review questions
1. If the length of a conductor is increased, the resistance of the conductor
2. A conductor with a cross-sectional area of 2 mm2 has a ________________resistance than a conductor with a cross-sectional area of 4 mm2.
3. As the temperature of a copper conductor rises, the resistance of the conductor
4. An aluminium conductor has _________________resistance than a copperconductor of the same dimensions.
5. A positive temperature coefficient of resistance means the resistance of thematerial ______________________if the temperature increases.
6. Most semiconductor materials have a _____________________ temperaturecoefficient of resistance.
7. A thermistor with a negative temperature coefficient (NTC) will_____________ in resistance if the temperature rises.
8. A varistor (or VDR) is a component whose resistance changes with____________
9. The resistance of an LDR _________________________ as the light intensity
increases.
10.A VDR is used to protect electrical and electronic equipment against damagecaused by
11.A 100 ohm variable resistor that has a resistance of 15 ohms when the wiper ismoved half-way has a _______________ resistance element.
12.A 470 ohm resistor with a tolerance of 10% has a colour code (left to right) of:
14.The resistor shown below has a resistance of __________ohms and a toleranceof ___________ %.
15.A 680 ohm, 10% resistor could have a resistance as high as ________ ohms.
16.A four band resistor with all bands coloured red, except the tolerance bandwhich is gold, has a resistance of _______ohms.
orange white red gold
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17.The five band resistor shown below has a resistance of _______ohms and atolerance of ______%.
18.A wire-wound resistor has a higher rating than a carbon-film resistor.
19.Resistor values are usually limited to the ___________________ range ofvalues.
20.Before a resistance value is measured with an analogue ohmmeter, the meterprobes should be connected together and the meter pointer adjusted with the
OHMS ADJ to read __________ohms.
21.When measuring resistance with an ohmmeter, its important not to______________ both probes at the same time.
blue red black orange red
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Series Resistive CircuitsIn a series-connected circuit there is only one path for the current flow when from the
higher potential terminal to the lower potential terminal. Any one piece of apparatus
will never have more than one conductor connected to each connecting terminal.
Resistance in a series circuit
When an electrical circuit is connected so that there is only one path through which
current can flow the circuit is said to be connected in series.
Figure 1
The value of the resistance can be shown in a number of ways:
Component identification code and value shown together
on the diagram.
Component identification code on the diagram and a table
of components and their values listed elsewhere either on
the diagram or in a separate document.
Current drawing standards state that the component identification and value should be
located alongside the related drawing symbol; however, older drawings may show
either the component identification or the component value inside the related symbol.
R1 R2
R3
R5 R4
AppliedVoltage
R2
100 k
R2
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Calculations
The total resistance to the flow of current in a series circuit can be obtained by
adding together the value of each individual resistance.
Figure 2
Therefore the formula for calculating total resistance in a series circuit can be written
thus:
Rtotal = R1 + R2 + R3 + R4 + R5
In the above example:
ohms
R
RRRRRR
T
T
16
64123
54321
=
++++=
++++=
Current in a series circui t.
The current is a series circuit has only one path through which to flow. Therefore the
value of the current must be the same through each component in the circuit.
Ammeter measurements taken at any point around a series circuit will show the same
value of current flowing at all points.
Figure 3
R1 R2
R3
R5 R4
Applied
Voltage
3 2
1
6 4
R1 R2
R3
R5 R4
Applied
Voltage
3 2
1
6 4
2 A
2 A
2 A
2 A2 A
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Voltage across a series circu it
The value of the applied voltage shown in Figure 3 is the value of supply voltage
applied across the whole circuit to make a current of 2A flow in that circuit. The value
of the applied voltage will decrease as the current is forced through the various
resistances. This loss of voltage is known as the voltage drop.
Figure 4
By connecting a voltmeter across each resistance as shown in Figure 5 it will be seen
that the voltage is different in each case because of the drop in voltage over the
different values of resistors. When the five voltage readings are added together the
sum will be found to equal the applied voltage. Hence the total voltage applied is
equal to the sum of voltage drops. (May be expressed as Vd), i.e.
VTotal = V1 + V2 + V3 + V4 + V5
This is often expressed as Kirchhoffs Voltage Law. It is possible to calculate the
voltage drop across each resistance, if the value of current and resistance are known,
by using Ohms Law.
N.B. A neat methodical approach is a must when calculating unknown values using
Ohms Law.
Example: A circuit consisting of 5 resistances each; 3 ohm, 2 ohm, 1 ohm, 4 ohm
and 6 ohm respectively are connected in series to a 32V supply
voltage.Calculate:
a. The total resistanceb. The total current flowing through the circuitc. The value of the voltage drops across each resistance.
N.B. When attempting a problem involving Ohms Law the following methodical
approach should be adopted.
1. Read the question.2. Make sure you understand the question.3. Draw the circuit and mark in known details.
4. Calculate the required values.5. If necessary redraw the circuit with calculated values.
R1 R2
R3
R5 R4
Applied
Voltage
3 2
1
6 4
VT
VR1 VR2
VR3
VR5 VR4
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Figure 5
a. Total resistance:
ohm
RRRRRRT
16
64123
54321
=
++++=
++++=
b. The total current flowing through the circuit:
A
R
VI
T
TT
2
16
32
=
=
=
c. The value of the voltage drops across each resistance:
Using Ohms law:R
VI=
Transpose to make V the subject (that is on the top line on its own)
Multiply both sides by R RR
VRI xx =
Cancel R on the right-hand side: VRI =x
For convenience, the formula may be written as:
IRV=
VR1:
R1 R2
R3
R5 R4
32 V
3 2
1
6 4
( )( )V
IRV
6
32
=
=
=
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VR2:
VR3: ( )( ) VV 212 ==
VR4: ( )( ) VV 842 ==
VR5: ( )( ) VV 1262 ==
VT:
Power in a series circuit
Figure 6
From your previous Ohms Law work, you discovered that the power consumed in a
circuit could be easily calculated using the formula:
R
VP
RIP
VIP
2
2
=
=
=
In a series circuit such as that in Figure 6, all the values of V, I and R are known; and
so you can easily determine the power consumed by each resistor.
In a series circuit the sum of all the power values is equal to the total power consumed
by the circuit.
54321 RRRRRT PPPPPP ++++=
R1 R2
R3
R5 R4
32 V
3 2
1
6 4
2 A
2 A
2 A
2 A
2 A
2 A
( )( )V
IRV
4
22
=
=
=
V
VVVVVV RRRRRT
32
128246
54321
=
++++=
++++=
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Now calculate the power for each resistor and the total circuit power:
Example
( ) ( )( )( )W
RIPR
12
34
322
21
=
=
=
=
Now calculate the power in each of the remaining resistors, R2, R3, R4, and R5 and
then obtain the total power used by the circuit.
PR2 = ________ PR3 = _________ PR4 = ________ PR5 = _________
PT = __________________________
If any of the resistors was to be shorted out, then the current flowing in the circuit
would increase due to the reduced resistance. This extra current flowing in a series
circuit will cause the power being consumed by the remaining resistors to increase
also. This could pose a danger to the circuit because the power rating of the resistors
left in the circuit may be exceeded, causing them to burn out.
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Unit test
1. Calculate the total resistance of three (3) resistances, of 2.5 ohms; 4.3 ohm and3.2 ohm respectively, connected in series.
2. If a supply voltage of 40 volts were applied to the above circuit, what would
be the total current flow through the circuit?
3. An electrical appliance draws a current of 8 amps from a 240 volt supply.What is the resistance of the appliance?
4. The current in a series resistive circuit:a. Changes at each resistance
b. Remains the same throughout the circuitc. Drops in value around the circuitd. Is inversely proportional to the voltage.
5. Voltage drops and applied voltage in a series circuit have the same value.True / False
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Review questions
For each question, circle the response you consider best answers the question.
1. A series circuit has:
a. One current path.b. One component.c. Many current paths.d. Nothing to do with current paths.
2. Current in a series circuit is:a. Always zero.
b. Always the same for every component.c. Different for components of differing resistances.d. Different for components of equivalent resistance.
3. If the voltage supply to a series circuit decreased by half and the circuitresistance remains the same, the total circuit current will:
a. Decrease to zero.b. Decrease by half.c. Double.d. Remain the same.
4. The total resistance of a series circuit s equal to the:a. Sum of the circuit resistances.
b. Reciprocal of sum of the circuit resistances.
c. Largest resistance value.d. Average of the resistance values.
5. In a series circuit containing ten lamps, the equivalent resistance of the circuit,if one lamp is burnt out will be:
a. Zero.b. Infinity.c. Nine times the resistance of one lamp.d. The same as it was before the lamp burnt out.
6. The voltage that will be indicated by a voltmeter across an open-circuit
component is a series circuit will be:a. Zero volts.
b. A voltage of reversed polarity.c. The applied voltage.d. Impossible to measure.
7. In a series circuit containing two lamps, lamp 1 is on and lamp 2 is off. Thefault is:
a. Lamp 1 is a short-circuit.b. Lamp 2 is a short-circuit.c. Lamp 1 is an open-circuit.
d. Lamp 2 is an open-circuit.
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8. In a series circuit operating from a 6V supply, two lamps are connected inseries, and both lamps are off. If a voltage of 6V is measured across lamp 1
and CV is measured across lamp 2, the fault is:
a. Lamp 1 is a short-circuit.b. Lamp 2 is a short-circuit.
c. Lamp 1 is an open-circuit.d. Lamp 2 is an open-circuit.
9. In a series circuit, the applied voltage is equal to the:a. Equivalent resistance multiplied by the circuit current.
b. Difference of all the individual voltage drops.c. Square of the current multiplied by the equivalent resistance.d. Equivalent resistance divided by the circuit current.
10.The voltage drop across each resistor in a series circuit is:a. Always the same.
b. Inversely proportional to the total circuit current.c. Inversely proportional to the supply voltage.d. Proportional to the resistance of each resistor.
For the following questions, include a circuit diagram in your answer. All answers
should be expressed using the correct units.
11.A circuit contains two resistors, R1 and R2 connected in series. If R1 equals30 ohms, R2 equals 500 ohms and the applied voltage is 200V, calculate the
voltage drop across R1.
12.Three lamps are connected in series across a 100V supply. If the circuitcurrent is 2A and the resistance of two of the lamps totals 32 ohms, calculate
the resistance of the third lamp.
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13.Fourteen lamps used for Christmas decorations, each rated at 4W, 20V areconnected in series across a 280V supply. Calculate:
a. The resistance of each lamp.
b. The total resistance of the circuit.c. The total circuit current drawn from the supply.d. The current flowing in each lamp.
14.Three resisters, R1, R2 and R3 are connected in series with a battery. If theresistor values are R1 = 1.2k ohm, R2 = 850 ohm, R3 = 350 ohm and the
voltage drop across R2 is 16V, calculate:
a. The voltage drops across R1 and R3.b. The battery voltage.
15.For the circuit of figure 1 determine the voltage between points:
a. A and Bb. B and Cc. A and Cd. D and Ee. B and Ef. E and Fg. D and Fh. C and F
i. A and Ej. A and F
16.For the circuit of figure2 calculate:a. The total circuit resistance.
b. The voltage drop across each resistor.c. The supply voltage VT.d. The power dissipated by each resistor.e. The total power dissipated by the circuit.f. The value of the circuit current if resistor R2
is short circuited.
A B C D E F
240 V
coil
Figure 1
VT
R1 2k
R2
6k
R3 7k
IT = 1 mA
Figure 2
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17.For the circuit of figure 3, calculate:a. The circuit current.
b. The voltage drop across R1.c. The value of R1.
d. The voltage drop across R3.e. The power dissipated by each resistor.f. The total power dissipated by the circuit.
18.For figure 4:a. Complete the circuit so that:
The three resistors are connected in series.
The ammeter indicates the circuit current.
The voltmeter indicates the voltage across R2.
The wattmeter indicates in circuit power. The switch controls the circuit current.
b. Determine the ammeter, voltmeter and wattmeter readings.c. Indicate with an arrow the direction of the current (use conventional
current flow).
Figure 4
19.The current in a circuit containing three series connected resistors is:a. The sum of the currents in each resistor.
b. Proportional to the total resistance of the circuit.c. Inversely proportional to the voltage applied to the circuit.d. The same in all parts of the circuit.
R1 R2 R3B CA D
1k 2 k
40 V
60 V
Figure 3
R2
1 kA
+
V
+
+
W
LM
V2V1
R1
470
R1
2.2 k
100 V
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20.If the resistance in a series circuit is doubled and the supply voltage is doubledthe circuit current will:
a. Double.b. Remain the same.
c. Increase by four times.d. Decrease by four times.
21.If five lamps are connected in series, and the third lamp becomes open-circuit:a. Lamps one and two go out and lamps four and five remain on.
b. All lamps except the third lamp remain on.c. All lamps go out.d. The fuse protecting the circuit will blow.
22.If a current of 12A flows in a circuit containing three series connected resistorseach of the same value and two of the resistors become a short-circuit, the
current will equal:a. 36A.
b. 4A.c. 12A.d. zero.
23.Determine the equivalent resistance for the circuit of Figure 5.
Figure 5
24.For the circuit of figure 6 calculate the applied voltage.
Figure 6
Req
R1
R268
R3
47
100
R1
R2VT = ?VT
25V
15V
Iconventional
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25.The circuit of figure 7 has a total current, IT, of 250 mA. Determine thecurrents indicated by the three ammeters.
Figure 7
26.Redraw the circuit of figure 8 to include the following:
a. An ammeter to measure circuit current.b. A voltmeter to measure the applied voltage.c. A voltmeter to measure the voltage drop across resistor R3.d. An ammeter to measure the current through resistor R2.
Figure 8
27.For the circuit of figure 9, given that V1 = 8.4 V, V2 = 12.3 V and V3 = 5.85 V,calculate the:
a. Applied voltage.b. Equivalent resistance of the circuit.c. Circuit current.
Figure 9
A1
A2
A3
IT
Lamp 1
Lamp 2
Lamp 3
+
R1
R2
820
1200
VT = 20 V
470
R3
R1
R2
82 VT = ?
V3
V1
V2
56
R3
39
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28.Determine the value of the unknown resistance R2 in the circuit of figure 10.
Figure 10
29.Calculate the voltage drop across resistor R1 in the circuit of figure 11.
Figure 11
30.For the circuit of figure 12, determine the voltmeter reading for the followingswitch conditions:
a. Both switches A and B open.b. Both switches A and B closed.c. Switch A open and switch B closed.d. Switch A closed and switch B open.
Figure 12
Req = 445
R1
R2
R3
25
150
R1
R2VT = 30 VVT
V1
17V
IT
V
32 V lamp
BA
VT = 32 V
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Parallel Resistive CircuitsWhen an electrical circuit is connected so that there are two or more paths through
which current can flow the circuit is said to be connected in parallel.
Figure 1
This form of parallel connection is the method generally used in electrical wiring
installations in homes, shops etc. The reason for this will become more evident as the
module progresses.
Calculations
If a 6 ohm resistance is connected across a 12 V supply a current of 2 amps will flow.
A
R
VI
2
6
12
=
=
=
However, if a second 6 ohm resistance is connected into the circuit in parallel the
current through this resistance must also be 2 amps as the applied voltage is the same,as shown in Figure 2.
Figure 2
R1 R2 R3
I1 I2 I3
IT I2,3 I3
IT I2,3 I3
R
6 12 V
2 A
2 A
R
6 12 V
2 A2 A
R
6
IT
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The total current leaving the source divides at point A and a portion of the current
flows through each resistance. The currents rejoin at Point B and return to the source
of supply.
If R2 takes a similar current to R1, then the total current in figure 2 is higher than the
total current in figure 1. If the total current increases whilst the supply voltage remains
constant then the only other factor which could cause an increase in current would befor the total resistance to decrease.
The total value of resistance can be found by calculation using Ohms Law.
=
=
=
=
+=
+=
+=
==
+=
+=
==
3
4
12
4
22
6
12
6
12
&
21
21
2
2
1
1
21
2
22
1
11
T
T
T
T
T
I
VR
AI
R
V
R
VI
VVV
R
V
R
V
III
R
VI
R
VI
circuitparallelaforHowever
An alternative and more direct method of finding the total resistance of a parallel
circuit is to use the conductance of the branches (Remember! Conductance is the
opposite of resistance and is expressed as the reciprocal of the resistance).
ConductanceR
G1
=
Referring to Figure 2:
21
21
21
111
1
11
RRR
RG
RR
GGG
T
T
T
T
+=
=
+=
+=
So
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For the circuit in figure 2:
==
+=
+=
+=
32
6
6
111
6
1
6
1
111
21
T
T
T
R
R
RRR
sidesbothInvert
(LCD)rdenominatocommonlowesttheSelecting
This is the most common formula, but, when using a modern calculator, it may not be
necessary to use the method of determining the Lowest Common Denominator.
Obtaining the inverse of each parallel resistance using the inverse function button on
the calculator (x1) and adding the inverse of each parallel leg of resistance in the
circuit and finish the calculation by taking the inverse of the answer will provide thetotal resistance of the circuit.
For the circuit in figure 2, typical steps using a scientific calculator would be:
Step 6 x1 + 6 x1 =
Display 6 61 61 + 61 + 6 61 + 61 0.33333333
Depending on the brand of calculator, the inverse function may be a direct keypress or
a shiftkeypress.
When the inverse of each parallel resistance has been summed, obtain the inverse of
the answer and the result is the total resistance for the parallel circuit.
Step 6 x1 + 6 x1 = x1 =
Display 6 61 61 + 61 + 6 61 + 61 0.33333333 0.33333333 3
Where a circuit contains only two resistances in parallel, the following formula can be
applied:
21
21
12
21
21
12
21
12
21
21
21
21
1
1
1
11
1
111
RRRRR
RRRRR
RR
RRR
RR
RR
RR
RR
R
RR
RRR
TT
T
T
T
+=
+=
+=
+=
+
=
+=
:asittenusually wr
:givesequationtheofsidesbothInverting
:issidehand-lefttheforrdenominatocommonlowestThe
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Example
Determine the total resistance between terminals A and B
Solution:
=
=
=
+=
+=
60
66666667.0
1
66666667.01
100
1
150
1
111
21
T
T
T
R
R
RRR
Did you follow the work above?
It can be simplified using the calculator and the following button sequence will apply
to most calculators.
Calculator button sequence to calculate parallel resistance
R1
150
B
R2
100
A
1
5
0
x1
+
1
0
0
x1
=
x1
=
150
150
1
+
100
100
1
=
100
1
150
1+
=
1001
1501
1
+
60
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Example
Determine the resistance of a 10 and a 10 k connected in parallel.
Example
Determine the resistance of a 10 k and a 10 k in parallel.
1
0
EXP
3
x1
+
1
0
x1
=
x1
=
10
+
1010
1
=
10000
1
=
10
1
10000
1
1
+
9.9900099
10,000
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It should be noted that the value of the equivalent or total resistance in a parallel
circuit is always LESS than the value of the smallest resistance in the circuit.
Figure 3
Current in a parallel circuit
As previously mentioned when referring to Figure 2, the current divides and portionsof the current flow through the various resistances and then rejoin to return to the
source of supply. It can therefore be said that the total current is the sum of individual
currents around the circuit.
This can be written as
nT IIII +++= ....21
Figure 4
A
R
VI
2
6
12
1
1
=
=
=
A
R
VI
2
6
12
2
2
=
=
=
A
R
VI
T
T
4
3
12
=
=
=
OR
A
IIIT
4
22
21
=
+=
+=
R
3 R
6
R
6
R1
6 12 V
I2I1
R2
6
IT
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Voltage in a parallel circui t
As all the Branch Circuits (Resistances) are connected to the same point the voltage,
or volt drop, across each branch is the same.
Figure 5
This might also be drawn as:
Figure 6
Or:
Figure 7
These three circuits are the same and it can be seen from them that the supply voltage
is applied across each of the three resistances from one point.
So it can be said that the value of the supply voltage is also the value of each
individual volt drop.
Or 321 RRRT VVVV === etc
R1
B
R2
A
R3V
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Summary of characteristics of parallel ci rcuits
1. The total resistance of a parallel circuit is always less than the lowest valueresistance in the circuit.
2. The total current in the circuit is the sum of all the Branch Currents.
3. The voltage is the same in all parts of the circuit.
In symbolic representation:
nT RRRR
1...
111
21
+++=
nT IIII +++= ...21
RnRRR VVVVV ===== ...321
nT PPPP +++= ...21
Also, TT IVP x=
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Unit test
1. Calculate the total resistance of three (3) resistances of 4 ohm; 6 ohm and 5ohm respectively connected in parallel.
2. Calculate the total current flow in the above circuit when a voltage of 12 voltsis applied.
3. Current in a parallel resistive circuit remains the same in all parts of a circuit.
True / False
4. The voltage in a parallel resistive circuit:a. Changes at each resistance;
b. Remains the same in all parts of the circuit;c. Drops in value at each branch;d. Is directly proportional to the resistance.
5. A 25V heater has two elements, one of which is missing. If the resistance ofthe remaining element is measured to be 50 ohm, what must the resistance of a
second element be when connected in parallel to maintain a current of 1 amp.
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Review questions
For each question, circle the response you consider best answers the question.
1. A parallel circuit is different to a series circuit in that it has:
a. Fewer current paths.b. A single current path.c. More than one current path.d. No current paths.
2. Components that are connected in parallel form:a. Several branches for current flow.
b. A single path for the current.c. An open circuit.d. A voltage divider.
3. The total resistance in a parallel circuit is:a. Less than the smallest resistance.
b. Equal to the average resistance.c. Equal to the sum of the individual resistances.d. Greater than the largest resistance.
4. The largest resistance in a parallel circuit will always have the:a. Highest voltage drop across it.
b. Highest current flowing through it.c. Smallest voltage drop across it.
d. Smallest current flowing through it.
5. If an open-circuit occurs in a parallel circuit, the total resistance will:a. Increase.
b. Remain the same.c. Decrease.d. Be unpredictable.
6. In a parallel circuit containing two lamps, if lamp 1 is open-circuit:a. Both lamps will be off.
b. Lamp 1 will be on and lamp 2 off.
c. Both lamps will be on.d. Lamp 1 will be off and lamp 2 on.
7. In a circuit containing two resistors connected in parallel, if resistor R2 isconducting excessive current:
a. Resistor R1 is open-circuit.b. Resistor R2 has a much reduced resistance.c. Resistor R2 is open-circuit.d. Resistor R1 has a much reduced resistance.
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8. The lowest value of individual resistance in a parallel combination of resistorsis always:
a. Equal to the equivalent resistance of the combination.b. Less than the equivalent resistance of the combination.c. Dependent on the voltage and current for its resistance.
d. Greater than the equivalent resistance of the combination.
9. In a parallel circuit containing two resistors, if one resistor is dissipating 6 Wof power and the other is dissipating 10 W of power, the total power
dissipation is:
a. 3.75W.b. 4W.c. 16W.d. 60W.
10.Two resistors are connected in parallel, in which resistor R1 has twice the
resistance of R2. The current taken by resistor R2 is:a. Two thirds of the total supply current.
b. Twice that taken by resistor R1.c. One third of the total supply current.d. One half of the total supply current.
For the following questions, include a circuit diagram in your answer. All answers
should be expressed using the correct units.
11.Three resistors of 20, 40 and 110 ohms are connected parallel across a 200 DCsupply. Calculate the:
a. Equivalent resistance of the circuit.b. Current flowing in each resistor.c. Total current taken from the supply.d. Total power dissipated by the circuit.
12.A 4 ohm and a 6 ohm resistor are connected in parallel across a 60 V DCsupply. Calculate the:
a. Equivalent resistance of the circuit.
b. Total current taken from the supply.c. Current flowing in the 6 ohm resistor.d. Power dissipated by the 4 ohm resistor.
13.If a 100 ohm resistor has 20 mA of current flowing through it, calculate thecurrent that will flow in a 60 ohm resistor connected in parallel with the 100
ohm resistor
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14.A parallel circuit containing three resistors of 1, 2 and 4 ohms respectively hasa total circuit current of IT = 5.6 A. Calculate the current flowing in each
resistor.
15.Calculate the value of the resistor that will give a total resistance of 4 ohms if
it is connected in parallel with a 12 ohm resistor.
16.Complete Table 1 below for the five conditions listed for the circuit of Fig. 1.
Figure 1
Circuit
condition VT IT
RT
(ohms) PT
R1
(ohms)
R2
(ohms) I1 I2 P1 P2
1 10 V 4 6
2 20 V 200 mA 5 mA
3 100 V 12 M 20 M
4 3.8 1.58 A 1.9 A
5 5 W 41 1.5 W
Table 1
R1VT
I2I1
R2
IT
+
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17.For the circuit of Figure 2, determine the voltage between points:
a. A and Cb. A and B
c. C and Bd. A and De. D and Bf. A and Eg. E and Bh. + and
18.For figure 3:
a. Complete the circuit so that: The three resistors are connected in parallel. The ammeter indicates the circuit current. The voltmeter indicates the voltage across the resistors.
The wattmeter indicates the circuit power.
The switch controls the circuit current.b. Determine the ammeter, voltmeter and wattmeter readings.c. Indicate with an arrow the direction of the current (use conventional
current flow).
d. Calculate the meter readings if a fourth resistor, R4, with a value of 100
ohms was connected in parallel with the other three resistors
Figure 3
A
B
C D E12 V
+
Figure 2
R2
470
A+
V
+
+
W
LM
V2V1
R1
220
12 V
R1
120
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19.In a parallel circuit, all components are directly connected to the
20.The total current in a parallel circuit equals the _____________of the branchcurrents.
21.The voltage across each component in a parallel circuit equals the____________
22.If theres an open-circuit in a component in a parallel circuit the total currenttaken by the circuit will ________________
23.The total resistance of a parallel circuit is always less than the__________________ individual resistance value in the circuit.
24.The current in resistor R2 in the diagram on
the right is ________mA.
25.The total resistance of the circuit on the rightis _______.
26.The total current taken by the circuit on theright is ____A.
27.Calculate quantities indicated:a. R1 ________
b. lR1 ________
c. R2 ________
d. IR2 ________
e. lT ________
28.Calculate quantities indicated:a. lR1 ________
b. R2 ________
c. lR2 ________
d. lT ________
R1
100V = 30 V R2
900
+
R1
10 V = 50 V R2
10
IT
IR1 IR2
R120 V = 100 V R220
IT
IR1 IR2
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29.Calculate quantities indicateda. lR1 ________
b. lR2 ________
c. lR3 ________
d. lT ________
e. R1 ________
f. R2 ________
g. R3 ________
For each question, circle the response you consider best answers the question.
30.For a parallel circuit containing two resistors:a. VT = V1 V2b. IT = I1 + I2c. RT = R1 R2.d. VT = 1/V1 1/V2
31.A current of 10 A enters a junction that divides into two branches, with onebranch taking 2 A. The other branch takes:
a. 0.5 Ab. 5 Ac. 8 A
d. 12 A
32.The unknown current in this diagram equals:a. 50 mA
b. 230 mAc. 320 mAd. 520 mA
33.In a parallel circuit the supply current equals the:a. Total power multiplied by the supply voltage.
b. Sum of the branch currents.c. Supply voltage divided by the resistance of any one branch.d. Ratio of the branch currents
34.For the circuit shown, calculate the:a. Voltage across each resistor.
b. Equivalent circuit resistance.c. Current taken from the supply.d. Current in the R3.
R1
5 V = 20 V R3
10
IT
IR1 IR3
R2
5
IR2
20 mA
500 mA
200 mA
? mA
R1
25V = 300 V R3
100
IT
IR1 = 12 A IR3
R2
60
IR1 = 5 A
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35.A 6 ohm and a 12 ohm resistor are connected in parallel across a 100 Vsupply. Calculate the:
a. Equivalent resistance of the circuit.b. Current taken from the supply.
c. Current in the 12 ohm resistor.
36.Determine the value of a parallel connected resistor required to reduce theresistance of a circuit from 10 ohms to 6 ohms.
37.Two resistors with the values of 4.7 k and 12 are connected in parallel.Calculate the:
a. Equivalent resistance of the combination.b. The current taken from a 9 V supply.
38.Draw a circuit diagram that shows three resistors connected in parallel across a60 V DC supply. Include an ammeter connected to measure the total circuit
current. Given that the resistor values are R1 = 6 ohms, R2 = 8 ohms andR3 = 24 ohms, calculate the reading the ammeter should indicate.
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Combined SeriesParallel Resistive CircuitsWhen three or more resistances are connected partly in series and partly in parallel the
circuit is said to be aseries/parallelconnection.
The combined series/parallel circuit is commonly found in practice. It may consist of
parallel loads, with the series resistance being that of the supply lines, or it may
consist of far more complicated circuits. In analysing combined circuits it is important
to consider the components as separate parts of the whole and apply only those values
to them that apply according to Ohms law. The circuit rules are those of the normal
series or parallel circuits.
Figure 1
Figure 1 shows the first of two basic series/parallel arrangements: a series resistance
connected to a parallel combination:
Figure 2
Figure 2 depicts the second basic arrangement: a series resistance connected in
parallel with a series combination.
With series/parallel networks no additional formulae are required. However, great
care should be taken when making calculations. The combined circuit must be divided
into simple series and parallel arrangements. Each part is calculated separately and
then combined to obtain the required answer.
R1
R2
R3
R1 R2
R3
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Example 1
Figure 3
Calculate the total or equivalent resistance (Req) of the above series parallel circuit.
Step 1 Determine the equivalent resistance of the parallel arrangement.
( )( )
=
+=
+=
2
44
44
32
32
T
T
R
RR
RRR
Step 2 Re-draw the circuit using the equivalent resistance.
The circuit is now a simple series circuit. Calculate the total resistance.
=
+=
10
28eqR
This means that a circuit with a single 10 ohm resistance is equivalent to the
series/parallel combination in Figure 3.
Figure 3
R2
R3
R1
8
4
4
Req
R2 // R3
R1
8
2
Req
Req
10
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A straight mathematical solution can be calculated for the problem but care must be
taken to avoid error, e.g.
( )( )
=
+=
+=
++=
++=
+=
+=
parallelseries
10
28
8
168
44
448
//
32
321
321
eq
eq
R
RR
RRR
RRR
R
Example 2
Figure 4
Calculate the equivalent resistance of the circuit shown in Figure 4.
Again: Determine the equivalent resistance of R1 and R2 in series.
=
+=
+=
16
610
21
series
series
R
RRR
Redraw the circuit using Rseries:
The circuit is now a simple parallel circuit and Req can be calculated using Ohms lawfor resistances connected in parallel.
R1 R2
R3
10 6
4
Rseries
R3
16
4
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The calculation required is:
( )( )
=
=
+=
+=
2.3
20
64
416
416
3
3
eq
series
serieseq
R
RR
RRR
A 3.2 ohm resistance is equivalent in value to the series/parallel network.
The total resistance of more complex series/parallel circuits is calculated using exactly
the same methods except that more steps will be needed. It is important that each step
is calculated carefully and that at each stage the circuit is redrawn with the newly
calculated values. (This may be dispensed with later as you become more competent
at solving series/parallel problems).
Example 3
Some circuits appear quite daunting at first sight, but by following the procedures
already outlined, even the most complex circuits can be reduced step-by-step.
In above circuit, we start by reducing the series/parallel combination of R2, R4 and
R5 to a single equivalent resistance:
Req
3.2
R1
6
R2
3
R3
6
R4
4
R5
12
R6
3
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( )( )
=+=
+=+
+=
+=
633
16
483
124
1243
//
1
5421
eq
eq
R
RRRR
Redrawing the circuit and replacing R2, R4 and R5 with their equivalent resistance
gives:
Examining the revised circuit, the next combination to reduce to a single equivalent
resistance is R1 in series with R3 // Req1.
The equivalent resistance for this network will be:
( )( )
=
+=
+=
++=
+=
9
36
12
366
66
666
// 1312
eq
eqeq
R
RRRR
Redrawing the circuit once again using the equivalent resistance for R1 in series with
R3 // Req1 shows:
The final calculation can now be performed for R6 // Req2:
R1
6
Req1
6
R3
6
R6
3
Req2
9
R6
3
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( )( )
=+
=
+=