Solve Problems in Multiple Path DC circuits

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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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Solve problems in multiple path d.c. circuits_UEUNEEE004A

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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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27

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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34

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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36

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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39

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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40

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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41

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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44

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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45

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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46

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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47

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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48

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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50

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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51

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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52

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


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 ________


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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( )( )

=+

=

+=

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Solve Problems in Multiple Path DC circuits