Chapter15 Direct Current Circuitssssssss

8/19/2019 Chapter15 Direct Current Circuitssssssss

1/19

1

E. ELECTRICITY

AND MAGNETISM

Chapter 15Direct current circuits

2

Outline

15.1 Internal resistance

15.3

15.4 Potential divider

15.5 Potentiometer & Wheatstone bridge

3

Objectives(a) explain the effects of internal resistance on the

terminal potential difference of a battery in a circuit

(b) state and apply

(c) explain a potential divider as a source of variablevoltage

(d) explain the uses of shunts and multipliers

(e) explain the working principles of a potentiometer , and

its uses

(f) explain the working principles of a Wheatstone

bridge, and its uses

(g) solve problems involving potentiometer and

Wheatstone bridge4

15.1 Internal resistanceof sources

5

Electric CurrentA battery that is disconnected from any circuit

has an electric potential difference between its

terminals that is called the electromotive force or

emf :

Remember despite its name, the emf is an

electric potential, not a force.

6

Electromotive Force (emfEMF = the voltage between two ends of a

circuit when no current is flowing in the circuit

An emf (electromotive force) is the work per

unit charge done by the source of emf in

moving the charge around a closed loop

q

W ne

7

Electromotive Force (emf)

The energy needed to run electrical devices

comes from batteries.

Within a battery, a chemical reaction occurs

that transfers electrons from one terminal

(leaving it positively charged) to another

terminal (leaving it negatively charged).

8

Electromotive Force (emf)

Because of the positive and negative chargeson the battery terminals, an electric potential difference exists between them. The maximumpotential difference is called the electromotiveforce* (emf) of the battery.

The electric potential difference is also knownas the voltage, V.

The SI unit for voltage is the volt, after Alessandro Volta (1745-1827) who invented theelectric battery. 1 volt = 1 J/C.


2/19

9

Electromotive Force (emf)Electromotive force (emf) is the potential differencethat appear between the terminals of a battery whenno current is present.

A source of emf is a device that converts chemical,mechanical or other forms of energy into the electricenergy necessary to maintain a continuous flow ofelectric charge.

In electric circuit, source of emf is usually representedby and it is measured in volts.

10

Electromotive Force (emf) A source of emf will maintain a potential differenceand supply current to an external circuit .Example:batteries, solar cells, generators etc.

If the emf of a battery is zero, there is no currentwhen a wire is connected across its terminals. In thiscase there is no potential difference to drive thecharge.

But if the emf in nonzero, a current is present whenthe terminals are connected.

The greater the emf, the greater the current in thecircuit.

11

Sources of emf

The source that maintains the current in a closed

circuit is called a source of emf

Any devices that increase the potential energy of

charges circulating in circuits are sources of emf

Examples include batteries and generators

SI units are Volts

The emf is the work done per unit charge

12

Sources of EMF

1. Electric cells-convert chemical energy to

electrical energy

Consists of 2 different metals (the electrodes)immersed in a substance called an

electrolyte.

A battery consists of a no. of cells connected

together (a car battery = 6 2V cells in series)

Battery of

cells

Primary cell

13

The electro motive force is the maximum potential

difference between the two electrodes of the cell when

no current is drawn from the cell. Comparison of EMF and P.D:

EMF Potential Difference

1 EMF is the maximum potential

difference between the two

electrodes of the cell when no

current is drawn from the cell i.e.

when the circuit is open.

P.D is the difference of potentials

between any two points in a

closed circuit.

2 It is independent of the resistance

of the circuit.

It is proportional to the resistance

between the given points.

3

source of emf.

It is measured between any two

points of the circuit.

4 It is greater than the potential

difference between any two points

in a circuit.

However, p.d. is greater than emf

when the cell is being charged.

Sources of emf:

14

Simple Cell

Cu plate and Zn Plate in a beaker of

dilute sulphuric acid

Dilute

sulphuric

acid

ZnCuThe plates react withthe acid Zn platebecomes neg. charged,Cu +. Thus a potentialdifference exists soelectrons can flowfrom -ve to +ve plate

15

Primary and Secondary cells

Primary cell = cell which cannot be

recharged-once the chemicals are used

up it must be discarded (e.g. dry

battery)

Secondary cell = cell which can berecharged (usually by pushing current

through it in the wrong direction) (e.g.

car battery)

16

ExampleWhen you press one of the buttons on a pocketcalculator, the battery provides a current of 300 µAfor 10 ms.

How much charge flows during that time?

How many electrons flow in that time?

13

191090.1

1060.1

0.3

electron1oncharge

10msinchargetotalelectronsof Number

00.310300

C

C

C ms At I q


3/19

17

emf and Internal Resistance

A real battery has some

internal resistance

Therefore, the terminal

voltage is not equal to

the emf

18

More About Internal Resistance

The schematic shows

the internal resistance, r

The terminal voltage is

b-Va

Ir

IR + Ir

19

Internal Resistance and emf, cont

current is zero

Also called the open-circuit voltage R is called the load resistance

The current depends on both the resistance

external to the battery and the internal

resistance

20

Internal Resistance and emf, final

When R >> r, r can be ignored

Generally assumed in problems

Power relationshipI = I2 R + I2 r

When R >> r, most of the powerdelivered by the battery is transferred tothe load resistor

21

Internal resistanceIn reality, batteries and generators also add

some resistance to a circuit. This resistance is

called the internal resistance of the battery.

When an external resistance R is connected to

the battery, the resistance is connected in

series with the internal resistance. This internal

resistance causes the voltage between the

terminals to drop below the emf.

R

r+ -

22

Internal Resistance of a cell:The opposition offered by the electrolyte of thecell to the flow of electric current through it iscalled the internal resistance of the cell.Factors affecting Internal Resistance of a cell:

Larger the separation between the electrodesof the cell, more the length of the electrolytethrough which current has to flow andconsequently a higher value of internalresistance.

Greater the conductivity of the electrolyte,lesser is the internal resistance of the cell. i.e.

internal resistance depends on the nature ofthe electrolyte.

23

E = V + v= IR + Ir

= I (R + r)

I = E / (R + r)

This relation is called circuit

equation.

R

rE

II

V

v

Internal Resistance of a cell:

The internal resistance of a cell is inversely

proportional to the common area of the electrodes

dipping in the electrolyte.

The internal resistance of a cell depends on the

nature of the electrodes.

24

R

rE

II

V

v

E = V + v

= V + Ir

Ir = E - V

Dividing by IR = V,

Ir E V

=

IR V

E

r = ( - 1) R

V

Internal Resistance of a cell in terms of

E,V and R:


4/19

25

Determination of Internal Resistance of a cell by

voltmeter method:

r

K

R.B (R)

V+

r

II

R.B (R)

K

V+

Open circuit (No current

is drawn)

EMF (E) is measured

Closed circuit (Current is

drawn)

Potential Difference (V) is

measured

Internal Resistance of a cell in terms of

E,V and R:

26

Internal Resistance in a BatteryBattery is a device that maintain a fixed electrical

potential difference between two points.

However, when a real battery is used to provide

electrical energy, the external voltage across the

terminals is less than the emf.

This reduction in voltage is due to the potential drop

occurring across the internal resistance of the battery

itself.

27

Internal Resistance of a Battery As more current is drawn from a battery, a greatervoltage drop occurs across its external resistance.

This effect is most easily visualized by considering a

real battery to consists of an ideal emf in series with aresistance.

The resistance, r is the internal resistance of thebattery.

If the current is connected to an external resistance,R (load resistance), the circuit can be drawn asshown in the given figure.

28

Internal Resistance of a Battery

The current through the circuit depends on theemf and the total resistance.

The potential difference across the terminals ofthe battery is called the terminal potentialdifference (TPD).

It is the emf reduced by the voltage drop acrossthe internal resistance.

r R I

29

Internal Resistance of a Battery

TPD has a value of

According to this equation, when the load resistance

R is small, the terminal voltace is applicably less than

the emf.

When the resistance is larger than the internal

resistance, the terminal voltage is approximately

equals the emf.

r R

R

r R

r Ir TPD

emf votageterminal

emf voltageterminal

R

R

30

EMF AND TERMINAL VOLTAGE All sources of emf have what is known as INTERNAL

RESISTANCE (r) to the flow of electric current. The internal

resistance of a fresh battery is usually small but increases with

use. Thus the voltage across the terminals of a battery is less

than the emf of the battery.

The TERMINALVOLTAGE (V) is given by the equation V

= - Ir, where represents the emf of the source of

potential in volts, I the current leaving the source of emf in

amperes and r the internal resistance in ohms.

The internal resistance of the source of emf is always

considered to be in a series with the external resistance present

in the electric circuit.

31

Example A transistor radio battery has an emf of 12.0 V. Acurrent of 4.0 A passes through a wire which isconnected directly across the battery terminals. Whatis the internal resistance of the battery ? What is theTPD across a 10 load?

V r R

RTPD

A

V

I r

23.90.12310

10

0.30.40.12

32


5/19

33

Gustav Kirchhoff

1824 1887

Invented spectroscopy

with Robert Bunsen

Formulated rules about

radiation

34

There are ways in which resistors can be

connected so that the circuits formed cannot be

reduced to a single equivalent resistor

instead

35

Junction Rule

The sum of the currents entering any junction must

equal the sum of the currents leaving that junction

A statement of Conservation of Charge

Loop Rule

The sum of the potential differences across all the

elements around any closed circuit loop must be

zero

A statement of Conservation of Energy

36

More About the Junction Rule

I1 = I2 + I3

From Conservation of

Charge

Diagram b shows a

mechanical analog

37

Assign symbols and directions to the currents in all

branches of the circuit

If a direction is chosen incorrectly, the resulting

answer will be negative, but the magnitude will be

correct

When applying the loop rule, choose a direction for

transversing the loop

Record voltage drops and rises as they occur

38

More About the Loop Rule

Traveling around the loop from

a to b

In a, the resistor is transversed

in the direction of the current,

the potential across the

resistor is IR

In b, the resistor is transversed

in the direction opposite of the

current, the potential across

the resistor is +IR

39

Loop Rule, final

In c, the source of emf istransversed in the directionof the emf (from to +), thechange in the electric

In d, the source of emf is

transversed in the directionopposite of the emf (from +to -), the change in theelectric potential is -

40

RulesUse the junction rule as often as needed, so long as,

each time you write an equation, you include in it a

current that has not been used in a previous junction

rule equation

In general, the number of times the junction rule

can be used is one fewer than the number of

junction points in the circuit


6/19

41

The loop rule can be used as often as needed

so long as a new circuit element (resistor or

battery) or a new current appears in each new

equation

You need as many independent equations as

you have unknowns

42

Problem-Solving Strategy

RulesDraw the circuit diagram and assign labels andsymbols to all known and unknown quantities

Assign directions to the currents.

Apply the junction rule to any junction in the circuit

Apply the loop rule to as many loops as are neededto solve for the unknowns

Solve the equations simultaneously for the unknown

quantitiesCheck your answers

43

The sum of the currents entering a junction =

sum of the currents leaving the junction

The emf across the circuit

= sum of the emfs across the individual parts of

the circuit

44

0 I

0V

conservation of charge:

junction rule, valid at any junction

Junction (Node) Rule: At any junction point, thesum of all currents entering the junction mustequal the sum of the currents leaving the

junction.

conservation of energy:

loop rule, valid for any loop

Loop Rule: The some of the changes inpotential around any closed path of a circuitmust be zero.

45

A junction is a point in a circuit where a number

of wires are connected together.

Junction rule: The total current directed into a

unction must equal the total current directed

out of the junction.

Loop rule: Around any closed circuit loop, the

sum of potential drops equals the sum of the

potential rises.

junction

46

RulesSome circuits cannot be broken down into

series and parallel connections.

47213 I I I

I1

I2

I3

11 I I

213 I I I

Junction rule: The sum of currents entering a junction equals the sum of the currents leaving it.

48

Loop rule: The sum of the

changes in potential

around a closed loop is

zero.

Loop rule: This is

equivalent toconservation of energy.


7/19

49

Analogy Voltage and GPE

Pump

50

Label each current.

Identify unknowns.

Apply junction and loop rules; you will need as

many independent equations as there are

unknowns.

Solve the equations, being careful with signs.

51

Series Circuit

+acV

ac ab bc 1 2 1 2 eqV V V IR IR I R R IR

Apply the Loop Rule

eq 1 2R R R .....

ac ab bcV V V 0

52

Parallel Circuits

+

V

1I 2I 3I

Apply theJunction Rule

I

1 2 3

1 2 3 1 2 3 eq

V V V 1 1 1 VI I I I V

R R R R R R R

eq 1 2 3

1 1 1 1....

R R R R

53

Rule Set Problem Solving Strategy A resistor transversed in the direction of assumed

current is a negative voltage (potential drop)

A resistors transversed in the opposite direction of

assumed current is a positive voltage (potential rise)

A battery transversed from to + is a positive voltage.

A battery transversed from + to - is a negative voltage.

Both the loop rule and junction rule are normally

required to solve problems.

54

Loop Rule

Traveling around the loopfrom a to b

In (a), the resistor is traversedin the direction of the current,the potential across theresistor is IR

In (b), the resistor is traversedin the direction opposite of thecurrent, the potential acrossthe resistor is is + IR

55

Loop Rule

In (c), the source of emf istraversed in the directionof the emf (from to +),and the change in theelectric potential is +

In (d), the source of emf is

traversed in the directionopposite of the emf (from+ to -), and the change inthe electric potential is -

56

Example Problem 1

1R 1690

3

R 1000

4R 3000

Given:

Find: current in each resistor

V = 3 Volts


8/19

57

Example Problem 2

10V 20V

5 10

20

Given:

Find: current in the 20 resistor

58

15.3 Potential Divider

59

Potential divider circuit

If two or more resistors are connected in series the

total potential difference is divided between the

resistors.

The bigger the resistor the bigger the potential across

it (if one resistor is much bigger than the other

effectively all the p.d. is across the big resistor)

60

Potential divider circuit

Such a system of resistors is known as a

potential divider circuit-used when a smaller

p.d. is required than the supply

R2

R1 Vout

The value of Vout

depends on R1

and R2

61

Potential divider A potential divider producesan output voltage that is afraction of the supplyvoltage V . This is done byconnecting two resistors inseries as shown.

Since the current flowing through each resistor is the same, thus

V

1V

1 R I

2V

2 R I

21eff R R R

eff R

V I and

21 R R

V I

62

Therefore, the potential difference (voltage) across R1 is given by

Similarly,

Resistance R1 and R2 can be replaced by a uniformhomogeneous wire as shown.

11 IRV V

R R

RV

21

11

V R R

RV

21

22

V

2l 1l

I

BA C

2V 1V

63

The total resistance, RAB in the wire is

Since the current flowing through the wire is thesame, thus

A RCBACAB R R R

A A R

21AB

and

AB R

V I

21 l l A

V I

21AB l l A

R

64

Therefore, the potential difference (voltage) across

the wire with length l 1 is given by

Similarly,

AC1 IRV Al l

A

V V 1

21

1

V l l

l V

21

11

V l l

l V

21

22

l V

A I IRV


9/19

65

0004

V21

0008

outV

V0.4ou tV

V4.2ou tV

Example 21.16 :

For the circuit below,

a. calculate the output voltage.

b. If a voltmeter of resistance 4000 is connected across

the output, determine the reading of the voltmeter.

66

More Potential Divider : Variable Resistor

used as a potential divider (potentiometer)

circuit symbol

67

Variable potential divider circuit

Two resistors replaced by a variable

resistor. The output voltage increases from

O V when the contact is at A to the maxinput voltage when the contact is at B

A

B

Vout

68

Potential Divider

Fixed Valued Potential Divider

21 R R

V

R1

V

R2 V2

V1

II

Effective resistance = R1 + R2 Current through each resistor =

Voltage across R1 , V1 = R1I = 121

R R R

V

Voltage across R2 , V2 = R2I = 221

R R R

V

69

Potential DividerVariable Potential Divider (potentiometer)

E R V

X

Z

Y

When the slider Y is at point Z

the voltmeter reads the

voltage of the supply

When the slider Y is at point X

the voltmeter reads the zero

voltage

70

Pencil

+12V

0 V

R2

R1

+12V

0 V

V

I

I

Potential Divider

71

R2

R1

+12V

0 V

V

I

I

Just like in our pencil the voltage

will distribute itself proportional

to the resistance.

E.G if R1 is twice R2 then 1/3 of

the voltage will be across R2.

So V will be 4 volts.

Potential Divider

72

R2

R1

V1

0 V

V2

I

I

V2 = V1 * R2 / (R1+R2)

(We can prove this from Ohms

law)

I = V1/(R1+R2)

I = V2/R2

Potential Divider


10/19

73

Usage of Potential Divider

In reality, series circuits are used as potential dividersto control a device automatically.

Eg.: to turn on an electric heater automatically in anincubator.

setting up the circuit with a component that is affected by heat (thermistors) or light (LDRs).

The total resistance of the circuit will depend on

some environmental factor , and the way the inputvoltage is shared will also be affected.

As a result, the output voltage will vary depending onthe environment. This can then control the device byswitching it off (of the voltage to it is too low), or on.

74

Control the temperature in an incubatorConsider a potential divider whichuses a fixed resistor in series with athermistor .

Remember that the resistance of thethermistor falls with increasing temperature. As the temperature of the incubatordrops, the resistance of thethermistor will increase. A largerportion of the input voltage will thenbe used across it.Place an electric heater across the

thermistor.The heater will come on when thevoltage to it is high enough, i.e whenthe temperature has dropped sufficiently.

Choosing different values for the fixedresistor will allow the heater to comeon at different temperatures.

R

T

V out to

heater

75

15.3.1 Shunt andmultiplier

76

The Measurement of Current and Voltage

A dc galvanometer.

The coil of wire and

pointer rotate whenthere is a current in

the wire.

77

The Measurement of Current and Voltage

An ammeter must be

inserted into a circuit so

that the current passes

directly through it.

78

Shunt and multiplierThe galvanometer is a sensitive current-

reading meter.

High current may burn the wiring system of

the moving-coil inside the galvanometer.

The galvanometer can be protected by using

shunt or multiplier.

79

Shunt and multiplier

A low resistance resistor connected in parallel

to the galvanometer to divert the current is

called shunt. The actual balance point is

determined when the shunt is removed.

A high resistance resistor connected in series

with the galvanometer to reduce the totalcurrent is called multiplier. The actual balance

point is then determined when the multiplier is

short-circuited

80

Multi-range voltmeter

Simplified Volt-Ohm

Meter (VOM) voltmeter

schematic diagram. This

voltmeter uses one 50

µ A, 5000 meter

movement, multiplier

resistors, and one rangeswitch


11/19

81

Multi-range

ammeter typical of

those found in

many VOMs. The

meter is a 50 µ A

full-scale, 5000movement

Multi-range ammeter

82

Galvanometer Sensitivity

Current sensitivity may be

defined as a ratio of the

deflection of the galvanometer to

the current producing this

deflection

Voltage sensitivity may be

defined as the ratio of thegalvanometer deflection to the

voltage producing this deflection

A

mm

I

d S

I

mV

mm

V

d S V

A

mmS

I

d S I R

C

mm

Q

d S mQ

83

Megohm sensitivity may be defined

as the number of megohms required

in series with the (CDRX shunted)

galvanometer to produce one scaledivision deflection when 1 V is

applied to the circuit

Ballistic sensitivity and is defined as

the ratio of the maximum deflection,

d m, of a galvanometer to the quantity

Q of electric charge in a single pulsewhich produces this deflection.

Galvanometer Sensitivity

A

mm

I

d S

I

mV

mm

V

d

S V

A

mmS

I

d S I R

C

mm

Q

d S m

Q

84

DC Ammeters Shunt Resistor

m

mm

s I I

R I R

+

-

I I s

R sR m

Movement

I m

85

Ayrton Shunt

Schematic diagram

of a simple

multirange ammeter

--------

Universal or Ayrton

shunt --

R a

+

-

S

R b R c R d

+

-

R c

R b

R a

5A 5A

1A

10A

86

DC Voltmeters Basic dc voltmeter

circuit --

Multirange

voltmeter -------

Voltmeter

sensitivity :

m

mm

mm s R

I

V

I

R I V R

V I S

fsd

1

R 1

+

-

R 2

R 3

R 4

I m

V 1

V 2

V 3

V 4

+

-

I m

v

Multiplier

R m

R s

87

Voltmeter-Ammeter Method

A popular type of

resistance

measurement

Effect of voltmeter

and ammeter

positions in

voltmeter-ammetermeasurements --

+

-

V

A

R xV x Load V

I I x

+

-

V

A

R xV x Load V

I I x

88

Voltmeter-Ammeter Method

Effect of the voltmeter

position in a voltmeter-

ammeter

measurements+

-V

A

R x Load

V

2

1


12/19

89

Galvanometer/Applications

Device used in the

construction of ammeters

and voltmeters.

Magnet

Current loop

or coil

Spring

Scale

90

Galvanometer used as Ammeter

Typical galvanometer have an internal resistanceof the order of 60 W - that could significantlydisturb (reduce) a current measurement.

Built to have full scale for small current ~ 1 mA orless.

Must therefore be mounted in parallel with a smallresistor or shunt resistor.

Galvanometer

60

Rp

91

Galvanometer

60

Rp

to an ammeter that can measure up to 2 A current.

Rp must be selected such that when 2 A passes

through the ammeter, only 0.001 A goes through thegalvanometer. 0.001 60 1.999

0.03002 p

A A R

R

Rp is rather small!The equivalent resistance of the circuit is also

small!92

Galvanometer used as VoltmeterFinite internal resistance of a galvanometer mustalso addressed if one wishes to use it as voltmeter.

Must mounted a large resistor in series to limit the

current going though the voltmeter to 1 mA.Must also have a large resistance to avoid disturbingcircuit when measured in parallel.

Galvanometer 60Rs

93

Galvanometer 60Rs

Maximum voltage across galvanometer:

max 0.001 60 0.06V A V

Suppose one wish to have a voltmeter that can

measure voltage difference up to 100 V:

100 0.001 60

99940

p

p

V A R

R Large resistance

94

Ammeter, Voltmeter and Ohmmeter?

DC Ammeter : The shunting resistor Rsh

movement form a current divider

DC Voltmeter : Series resistor Rsform a voltage divider.

Ohmmeter : Measures the current to find the resistance

Rsh

Rs

Rs

95

DC Ammeter

m || shunt resistor, Rsh

coil by shunting some of it through Rsh

+ -

Im

Ish

I

R sh

R m

d'Arsonval movement

Ammeter terminal

Rsh = resistance of the shunt

Rm = internal resistance of the

meter movements (movable coil)

Ish = shunt current

Im = full scale deflection current

of the meter movement

I = full-scale deflection current

for the ammeter

| | = Parallel symbol 96

+ -Im

Ish

I

R sh

R m

d'Arsonval movement

Ammeter terminal

Vm = ImRm Vsh = IshRsh

Vsh = Vm

IshRsh = ImRm

Rsh = ImRm / Ish ( ) ----(a)

I = Ish + Im Ish = I Im

Therefore, Rsh = ImRm/(I Im)

Purpose I >> n Im , n = multiplying factorn=I/Im

I = nIm ---(b)

Substitute b to a

Rsh = ImRm/(nIm Im)

Rsh= Rm/(n-1) -----(c)


13/19

97

Example 1: DC Ammeter

A 100uA meter movement with an internal

- 100 mA

ammeter . Find the value of the required shunt

resistance.

Solution:

n = I/Im = 100 mA / 100 µA = 1000

Thus,

Rsh = Rm / (n

98

The Aryton Shunt

R m

R c R aR b

R sh

5A

10A 1A

+-Most sensitive range

Used in multiple range ammeter

Eliminates the possibility of the

moving coil to be in the circuit

without any shunt resistance

Rsh = Ra + Rb + Rc

1n

R R m

sh ----(c)

99

R m

R c R aR b

R sh

I3 I1

+ -

I2

I

I - Im

Im

B

Middle

sensitive

range

macb R R R R V V

(R b + R c )(I 2 -I m ) = I m(R a +R m )

Since,

R a = R sh (R b + R c ),

yield,

I 2 (R b + R c ) I m(R b+R c ) = I m [R sh (R b + R c ) + R m ]

2

(

I

R R I R R

m shm

cb ----(d)

At point B, (Rb+Rc)||(Ra+Rm)

100

R m

R c R aR bR sh

I3 I1

+ -

I2

I

I - Im

Im

C

At point C, Rc||(Ra+Rb+Rm)

3

)(

I

R R I R

m shm

c

----(e)

mbac R R R R V V

(I 3-I m )R c = I m(R a+R b+R m )

I 3R c = I m(R a+R b+R c +R m )

I 3R c = (R sh+R m)

101

Substitute eqn (d) into eqn (e), yields

32

11)(

I I R R I R

m shmb----(f)

R a = R sh (R b+R c ) ----(g)

102

R m

R c R aR b

R sh

I3 I1

+ -

I2

I

I - I m

Im

Example 2: The Aryton ShuntCalculate the value for Ra, Rb and Rc as shown, given the

value of internal resistance, Rmthe moving coil = 100 µA. The required range of current are:

I1 = 10 mA, I2 = 100 mA and I3 = 1A.

103

Connect

Ammeter

R 1

Ie

X

E

Y

R m

R 1

Im

X

E

Y

1 R

E

I e mm R R

E

I 1

me

m

R R

R

I

I

1

1

Ammeter Insertion Effect

%100e

me

I

I I rror InsertionE

%100e

me

I

I I rror InsertionE

104

Example 3: Ammeter Insertion Effects

A current meter that has an internal resistance of 78

is used to measure the current through resistor R1.Determine the percentage of error of the reading due

to ammeter insertion.

R

R 1

Im

X

E

Y

1k

3V


14/19

105

DC Voltmeter

DMM become VOLTMETER multiplier Rs in

series with the meter movement.

To extend the

voltage range

To limit current through the DMM to

a maximum full-scale deflection

current P U R

P O S E

106

R m

R s Im+

fs I

1ySensitivit ( /V)

volt

ohms

ohms

volt

1

amperes

1 ySensitivit

Ifs= Im = full scale deflection current

Rs + Rm= (S x Vrange) It is desirable to makeR(voltmeter) >>R ( circuit)

Unit derivation:

107

Example 4: DC Voltmeter

Calculate the value of the multiplier resistance

on the 50 V range of a dc voltmeter that used a

108

A commercial

version of a

multi-range

voltmeter

The multiplier resistors are

connected in series, and each

junction is connected to one of

the switch terminals. The range

of this voltmeter can be also

calculated from the equation

)Im( R RmV Where the multiplier, R, now can be

R1 or (R1 + R2) or (R1 + R2 + R3)

(Note: the largest voltage range must be

associated with the largest sum of the

multiplier resistance)

Multi-

3V

30V

10V

109

Example 5: multi-range VoltmeterCalculate the value of the multiplier resistance for

the multiple range dc voltmeter circuit shown in

Figure (a) and Figure (b), if Ifs

= 50 and Rm

= 1k

3V

30V

10V

Figure (a) Figure (b)

110

Voltmeter Loading Effect

E

R A

R BR m

Rs ImR T= R s +R m

VRB R eq= R B //R T

Ifs= Im

Rs= (S x Vrange) - Rm

Total voltmeter resistance, RT RT = Rs + Rm = S x Vrange

S

R RV m srange

Vrange = ( Rs + Rm) Im

111

Calculation:

1)

2)

3)

4)

5)

xE R R

RV

B A

B

RB

RT = Rs + Rm = S x Vrange

Req = RB // RT

Without volt-meter

With volt-meter xE R R

RV

Aeq

eqm

RB

Insertion error%100 x

V

V V

RB

m

RB RB

(expected value)

(measured value)

112

Example 6: Voltmeter Loading Effect

E

R A

R BR m

Rs ImR T = R s +R m

VRB R eq = R B //R T

A volt meter (0-10V) that has an internal

resistance of 78 is used to measure the voltage

across resistor RB. Determine the percentage oferror of the reading due to voltmeter insertion. Let

E = 4V, R A=RB = 1k , S = 1k /V


15/19

113

DC Ohmmeter

Basic Ohmmeter circuit

IfsR m0.9R z0.1R z

R z

EX Y

R x

Variable portion

Fixed portion

114

Before measuring the Rx, the zero -

calibration

Definition zero = shorting the terminal x-y & adjust Rz

to obtain the full-scale deflection on the meter

movement.

m z

fs R R

E I w/o Rx

xm z R R R

E I with Rx

I < Ifs

115

Relationship between full-scale deflection to the

value of Rx is :

xm z

m z

fs R R R

R R

I

I P

This equation is used for marking off the scale on

the meter face of the ohmmeter to indicate the

value of a resistor being measured

116

Example 7:DC Ohmmeter

A 1 mA full-scale deflection current meter movement is to be

used in an ohmmeter circuit. The meter movement has an

internal resistance, Rm, of 100 , and a 3 V battery will be used

in the circuit. If the measured resistor has resistance of 1k ,mark off the meter face for the reading (20%, 40%, 50%, 75%

and 100%) .

117

Solution Ex:7

0

0%

20%

40%50%

75%

100%

12k

4.5k 3k

1k

Ohm

Full scale

percentage

118

IfsR z - fixed resistance &

zeroing potentiometer

E X Y

R x 1

R x 1 0

Rx 100

R 1

R m

R 2

R 3

Multiple-range OhmmeterThe previous section is not capable of measuring

resistance over wide range of values.

We need to extend our discussion of ohmmeters to

include multiple-range ohmmeters

119

15.4.1 Potentiometer

120

J

V

+

K

E

A

Rh

+

cm

I

A

B

100

200

300

400

0

Potentiometer :

Principle:

V = I R

= I l/A

If the constant current flows through the

potentiometer wire of uniform cross sectional

area (A) and uniform composition of material

( ), then

V = Kl or V l


16/19

121

0l

V

V l

V /l is a constant.

The potential difference

across any length of a

wire of uniform cross-

section and uniform

composition is

proportional to its length

when a constant current

flows through it.

Potentiometer:

123

Potentiometer

The potentiometer has a better accuracy then avoltmeter.

It is because the readings of the potentiometer aremeasured from zero to 100 cm. A large scale gives amore accurate reading.

Potentiometer can be used to

measure emf of an unknown cell,

measure the internal resistance of a cell,

measure currentmeasure thermoelectric emfcalibrate a voltmeter,

compare resistances

124

Potentiometer

If the galvanometer shows defection in one directiononly, it may be due to

The connections of the terminals of the cells are

wrong. The positive terminal of the cell must beconnected to the positive terminal of another cell.

The emf of the unknown cell is more then the emf ofthe cell connected across the wire of thepotentiometer, AB.

The connections are not tight and the current does notflow in certain part of the circuit.

125

+

E1

E2

+

R.BG

J1l1

J2l

2

E

A

K

A

BRh

+

I

100

200

300

400

0

E1 = VAJ1 = I l1 /A

E2 = VAJ2 = I l2 /A

E1 / E2 = l1 /l2

Potentiometer:The balance

point isobtained for

the cell whenthe potential ata point on thepotentiometerwire is equaland oppositeto the emf ofthe cell.

126

Potentiometer :Note:

The balance point will not be obtained on the

potentiometer wire if the fall of potential along

the potentiometer wire is less than the emf of the

cell to be measured.

The working of the potentiometer is based on

null deflection method. So the resistance of the

wire becomes infinite. Thus potentiometer canbe regarded as an ideal voltmeter.

127

15.4.2 Wheatstone

bridge

128

Wheatstone bridge

How does it work?

If the galvanometer readingis zero,

V A = VCV AB = VCB and V AD = V CD

P and R carry the samecurrent, I1 and X and Q carry

the same current I2.I1P=I2Q and I1R = I2X

Dividing the equations,

Then

P Q

R X 129

Wheatstone bridge

A simple form of Wheatstone bridge is a slide

wire Bridge as shown below

At the balance point,

and thus the unknown resistance, X can be

determined.

1

2

l X

R l


17/19

130

Wheatstone bridge

It can be used to measure

resistance.

To measure the unknown

resistance R, the variable

resistance Rv is adjusted

until the galvanometer

registers zero or nullcurrent.

v

v

R R

R R

I R I R R I IR

2

1

1211

131

I1

I

Ig

I1 - Ig

I - I1

E

A

B

C

D

P Q

R S

G

I

I

I

I - I1 + Ig

Loop ABDA:

-I1.P - Ig.G + (I - I1).R = 0

Currents through the arms are

Junction Rule.

When Ig = 0, the bridge is said to balanced.

By manipulating the above equations, we get

Loop BCDB:- (I1 - Ig).Q + (I - I1 + Ig).S + Ig.G = 0

P

Q

R

S

Wheatstone Bridge:

132

Wheatstone bridge

A device for measuring

the value of an unknownresistance

The values of theresistances are varieduntil no current flowsthrough the galvanometer

R1 R2

R3 R4

A

B

C

D

133

Wheatstone bridge

At this point, the potential at B =potential at D

(since no current flows)

Thus p.d. between A and B = p.d. between Aand D (V AB=V AD)

Similarly VBC = VDC

I1R1 = I2R3

I1R2 = I2R4

134

Wheatstone bridgeThus

R1 / R2 = R3 / R4

Thus, if three of the resistors are known, youcan calculate the value of the last.

Experimentally a resistor is placed in series withthe galvanometer to protect it from too muchcurrent. This resistor is then removed when theaprox. balance point is found

135

A B

R.B (R) X

G

J

KE

cm 100 - l cm

Metre Bridge is

based on the

principle of

Wheatstone Bridge.

When the galvanometer current is made zero by adjusting

the jockey position on the metre-bridge wire for the given

values of known and unknown resistances,

R RAJ

X RJB

R AJ

X JB

R l

X 100 - l

(Since,

Resistance

length)

Therefore, X = R (100 l) l

R RAJ

X RJB

Metre Bridge:

136

Metre bridge

This uses the same logic as the wheatstone

bridge, but two of the resistors are replaced by

a length of wire. A sliding contact divides the

wire into two lengths, and so into 2 resistances.

This makes it easier to adjust the resistance

137

The position of the sliding contact varies L1

and L2

R1 R2

G

Length of wireL1

L2


18/19

138

We know from the wheatstone bridgecircuit R1 / R2 = R3 / R4

In this case R3 and R4 are wires ofuniform cross section (A) and the samematerial ( is the same)

Thus R3 =constant L1

R4 = constant L2

R1 / R2 =L1 / L2

139

Uses of wheatstone bridge circuits

Temperature control in this case thewheatstone bridge starts balanced. If thetemperature of one of the resistors changesthen its resistance will change, the bridge willno longer be balanced and so current flowsthrough the galvanometer.

140


The size and direction of the current indicate

the size and direction of the temperature

change, and so can be used to control a heaterand bring the temp. back to its original value

141


Fail-safe device if the pilot light in a gas boilergoes out, you need the gas to shut offautomatically.

142

Uses of wheatstone bridge circuits A thermistor placed near the flame is used as oneresistor in a wheatstone bridge. If the flame goes outthe resistance increases, unbalances the bridge andcurrent flows in the galvanometer. This current canbe used to cut off the fuel

143

MEASUREMENT TECHNIQUES FOR

RESISTANCE WHEATSTONE BRIDGE

Resistors R1, R2, and R3 are

precision, variable resistors.The value of Rx is an unknown

value of resistance that must be

determined.

The galvanometer (an

instrument that measures smallamounts of current) is inserted

across terminals b and d to

indicate the condition of balance.

144



When the bridge is

properly balanced, no

difference in potentialexists across

terminals b and d;when switch S2 isclosed, the

galvanometer reading

is zero.

145



During balance state, I1 follows a-b-c path and

I2 follows a-d-c- path. Thus,

3211

31

R I R I

E E


19/19

146



Similarly

x

x

R I R I

E E

221

2

147



Results

x R I R I

R I R I

221

3211

148



Divide between these two

x R I

R I

R I

R I

2

32

21

11

Simplify

x R

R

R

R 3

2

1

Summary: Direct Current Circuits

Internal

resistanceE = I(R+r)

LawJunction Rule: I = 0

Loop Rule: (IR ) = E

PotentialDivider

V = R 1V 0/(R 1 + R 2)

Potentiometer V AB l

WheatstoneBridge

R /S = P /Q

149

Documents

Chapter15 Direct Current Circuitssssssss