Chapter15 Direct Current Circuits
Chapter15 Direct Current Circuits
Chapter15 Direct Current Circuits
Chapter 15
Direct current circuits
1 2
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 variable 15.1 Internal resistance
voltage
(d) explain the uses of shunts and multipliers
of sources
(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 bridge
3 4
Wne
q
5 6
7 8
Electromotive Force (emf) Electromotive Force (emf)
Electromotive force (emf) is the potential difference A source of emf will maintain a potential difference
that appear between the terminals of a battery when and supply current to an external circuit .Example:
no current is present. batteries, solar cells, generators etc.
If the emf of a battery is zero, there is no current
when a wire is connected across its terminals. In this
A source of emf is a device that converts chemical, case there is no potential difference to drive the
mechanical or other forms of energy into the electric charge.
energy necessary to maintain a continuous flow of But if the emf in nonzero, a current is present when
electric charge. the terminals are connected.
The greater the emf, the greater the current in the
In electric circuit, source of emf is usually represented circuit.
by and it is measured in volts.
9 10
Primary cell
Battery of
cells
11 12
IR + Ir
17 18
Internal Resistance and emf, cont Internal Resistance and emf, final
When R >> r, r can be ignored
current is zero Generally assumed in problems
Also called the open-circuit voltage Power relationship
R is called the load resistance I = I2 R + I2 r
The current depends on both the resistance When R >> r, most of the power
external to the battery and the internal delivered by the battery is transferred to
resistance the load resistor
19 20
R
Internal resistance Internal Resistance of a cell:
r
+ -
In reality, batteries and generators also add The opposition offered by the electrolyte of the
cell to the flow of electric current through it is
some resistance to a circuit. This resistance is called the internal resistance of the cell.
called the internal resistance of the battery. Factors affecting Internal Resistance of a cell:
When an external resistance R is connected to Larger the separation between the electrodes
of the cell, more the length of the electrolyte
the battery, the resistance is connected in through which current has to flow and
series with the internal resistance. This internal consequently a higher value of internal
resistance causes the voltage between the resistance.
terminals to drop below the emf. Greater the conductivity of the electrolyte,
lesser is the internal resistance of the cell. i.e.
internal resistance depends on the nature of
the electrolyte.
21 22
27 28
Example
A transistor radio battery has an emf of 12.0 V. A
current of 4.0 A passes through a wire which is
connected directly across the battery terminals. What
is the internal resistance of the battery ? What is the
TPD across a 10 load?
12.0V
r 3.0
I 4.0 A
R 10
TPD 12.0 9.23V
R r 10 3
31 32
Gustav Kirchhoff
1824 1887 There are ways in which resistors can be
Invented spectroscopy connected so that the circuits formed cannot be
with Robert Bunsen reduced to a single equivalent resistor
Formulated rules about
radiation
instead
33 34
35 36
37 38
39 40
Problem-Solving Strategy
Rules
The loop rule can be used as often as needed Draw the circuit diagram and assign labels and
so long as a new circuit element (resistor or symbols to all known and unknown quantities
battery) or a new current appears in each new Assign directions to the currents.
equation Apply the junction rule to any junction in the circuit
Apply the loop rule to as many loops as are needed
You need as many independent equations as to solve for the unknowns
you have unknowns Solve the equations simultaneously for the unknown
quantities
Check your answers
41 42
junction
Rules
A junction is a point in a circuit where a number Some circuits cannot be broken down into
of wires are connected together. series and parallel connections.
45 46
I3 I1 I 2
47 48
Analogy Voltage and GPE
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.
Pump
49 50
I
I1 I2 I3
Apply the
+
Apply the Loop Rule Vac Vab Vbc 0 Junction Rule
V
+
Vac V V V 1 1 1 V
I I1 I 2 I3 V
Vac Vab Vbc IR1 IR 2 I R1 R 2 IR eq R1 R2 R3 R1 R2 R3 R eq
1 1 1 1
R eq R1 R 2 ..... ....
R eq R1 R2 R3
51 52
53 54
Given: 5 10
57 58
R2
59 60
The total resistance, RAB in the wire is Therefore, the potential difference (voltage) across
the wire with length l1 is given by
R AB R AC R CB and R
A V 1
1 2
V1 IR A C V1
RA B l1 l2 A
A A A
RA B l1 l 2 l1
A V1 V
l1 l2
Since the current flowing through the wire is the
same, thus V l2
I Similarly, V2 V
RAB l1 l2
V
I V IR I
l1 l2 A
A
V l
63 64
Example 21.16 : More Potential Divider : Variable Resistor
For the circuit below, used as a potential divider (potentiometer)
a. calculate the output voltage.
b. If a voltmeter of resistance 4000 is connected across circuit symbol
the output, determine the reading of the voltmeter.
8 000
V o ut 4 .0 V
12 V
V o ut 2 .4 V
4 000 Vou t
65 66
B V
Voltage across R2 , V2 = R2I = R2
R1 R2
67 68
69 70
+12V
Just like in our pencil the voltage V1
will distribute itself proportional V2 = V1 * R2 / (R1+R2)
R1 R1
to the resistance.
I I (We can prove this from Ohms
E.G if R1 is twice R2 then 1/3 of law)
V the voltage will be across R2. V2
I I I = V1/(R1+R2)
So V will be 4 volts.
R2 R2
I = V2/R2
0V 0V
71 72
Usage of Potential Divider Control the temperature in an incubator
Consider a potential divider which R
In reality, series circuits are used as potential dividers uses a fixed resistor in series with a
thermistor.
to control a device automatically. Remember that the resistance of the
Eg.: to turn on an electric heater automatically in an thermistor falls with increasing
temperature.
incubator. As the temperature of the incubator T
drops, the resistance of the
setting up the circuit with a component that is affected thermistor will increase. A larger
by heat (thermistors) or light (LDRs). portion of the input voltage will then
be used across it.
The total resistance of the circuit will depend on Place an electric heater across the
some environmental factor, and the way the input thermistor.
The heater will come on when the
voltage is shared will also be affected. voltage to it is high enough, i.e when
As a result, the output voltage will vary depending on the temperature has dropped
sufficiently.
the environment. This can then control the device by Choosing different values for the fixed
switching it off (of the voltage to it is too low), or on. resistor will allow the heater to come Vout to
on at different temperatures.
73 heater 74
A dc galvanometer.
The coil of wire and
15.3.1 Shunt and pointer rotate when
there is a current in
multiplier the wire.
75 76
77 78
79 80
Multi-range ammeter Galvanometer Sensitivity
Multi-range Current sensitivity may be
d mm
ammeter typical of defined as a ratio of the SI
I A
those found in deflection of the galvanometer to
d mm
the current producing this SV
many VOMs. The V mV
deflection
meter is a 50 µA d mm
Voltage sensitivity may be SR SI
full-scale, 5000 I A
defined as the ratio of the
movement galvanometer deflection to the
d m mm
voltage producing this deflection SQ
Q C
81 82
Ayrton Shunt +
DC Voltmeters
Multiplier
Ra Rb Rc Rd Basic dc voltmeter +
Schematic diagram Im
of a simple circuit -- Rs
S
multirange ammeter V I m Rm V Rm
- Rs Rm v
-------- Im Im
-
1A
Rc Multirange V1
voltmeter ------- R1
Universal or Ayrton 5A Rb + V2
shunt -- Voltmeter R2 Im
10A sensitivity : V3
Ra R3
1 +
- V4
S R4
I fsd V -
85 86
-
87 88
Galvanometer/Applications Galvanometer used as Ammeter
Typical galvanometer have an internal resistance
of the order of 60 W - that could significantly
Device used in the disturb (reduce) a current measurement.
Scale
construction of ammeters Built to have full scale for small current ~ 1 mA or
and voltmeters. less.
Current loop
Must therefore be mounted in parallel with a small
or coil resistor or shunt resistor.
Galvanometer
60
Magnet
R
p
Spring
89 90
Galvanometer
60
Galvanometer used as Voltmeter
Finite internal resistance of a galvanometer must
also addressed if one wishes to use it as voltmeter.
Rp
Must mounted a large resistor in series to limit the
current going though the voltmeter to 1 mA.
to an ammeter that can measure up to 2 A current. Must also have a large resistance to avoid disturbing
circuit when measured in parallel.
Rp must be selected such that when 2 A passes
through the ammeter, only 0.001 A goes through the Rs Galvanometer
60
galvanometer. 0.001A 60 1.999 A R p
Rp 0.03002
Rp is rather small!
The equivalent resistance of the circuit is also
small! 91 92
Rs Galvanometer
60
Ammeter, Voltmeter and Ohmmeter?
DC Ammeter : The shunting resistor Rsh
movement form a current divider
DC Voltmeter : Series resistor R s
form a voltage divider.
Maximum voltage across galvanometer: Ohmmeter : Measures the current to find the resistance
Vmax 0.001A 60 0.06V Rs
Rsh
Suppose one wish to have a voltmeter that can
measure voltage difference up to 100 V:
100V 0.001A Rp 60
Rs
DC Ammeter
Vm = ImRm Vsh = IshRsh
m || shunt resistor, Rsh d'Arsonval movement Vsh = Vm
Im + - IshRsh = ImRm
coil by shunting some of it through Rsh Rm Rsh = I mRm / Ish ( ) ----(a)
Rsh = resistance of the shunt Rsh I = Ish + Im Ish = I Im
Ish
d'Arsonval movement Rm = internal resistance of the Therefore, Rsh = ImRm/(I Im)
Im + - meter movements (movable coil)
Purpose I >> n Im , n = multiplying factor
Rm
n=I/Im
Rsh Ish = shunt current I
Ish I = nIm ---(b)
Im = full scale deflection current Ammeter terminal
Substitute b to a
of the meter movement
I Rsh = I mRm/(nIm Im)
I = full-scale deflection current
Ammeter terminal
for the ammeter Rsh= Rm/(n-1) -----(c)
| | = Parallel symbol
95 96
Example 1: DC Ammeter The Aryton Shunt
A 100uA meter movement with an internal Rm
- 100 mA R sh
Rc Rb Ra
ammeter . Find the value of the required shunt
resistance.
5A
Solution: 10A 1A
Im Im
Rm At point B, (Rb+Rc)||(Ra +Rm) Rm At point C, Rc||(Ra+Rb+Rm)
Rsh
Rsh
Rc Rb
B Ra VRb Rc VRa Rm Rc Rb Ra VRc VRa
C Rb Rm
I - Im
I - Im
I2 (Rb + Rc )(I2 -Im) = Im(Ra +Rm)
I2 (I3-Im)Rc = Im(Ra+Rb+Rm)
Since,
I3 I1 I3Rc = Im(Ra +Rb+Rc+Rm)
Ra = Rsh (Rb + Rc), I3 I1
I3Rc = (Rsh +Rm)
yield,
I
+ - I2 (Rb + Rc ) Im(Rb+Rc ) = Im [Rsh (Rb + Rc ) + Rm]
Middle
I I m ( Rsh Rm ) ----(e)
+ - Rc
sensitive I m ( Rsh Rm ) I3
range Rb Rc ----(d)
I2
99 100
1 1 Im
Rb I m (Rsh Rm ) ----(f) Rm
I2 I3 Rc
Rs h
Rb Ra
I - Im
I3 I1
I
+ -
101 102
InsertionError
Ie Im
100%
Y
R1
E Y Ie
E X
Ie Im
R1 R1 Rm 1k Im
Im R1 3V E Rm
Ie R1 Rm
Ie I m
InsertionError 100%
Ie Y
103 104
DC Voltmeter
DMM become VOLTMETER multiplier Rs in Rs Im
+
series with the meter movement.
Rm
To extend the
voltage range
PURPOSE
1 Unit derivation:
To limit current through the DMM to Sensitivit y ( /V)
a maximum full-scale deflection
I fs 1 1 ohms
Sensitivit y
current Ifs= Im = full scale deflection current amp eres volt volt
ohms
Rs + Rm= (S x Vrange)
It is desirable to make
R(voltmeter) >>R ( circuit)
105 106
V RB RB
Rm Req = RB //RT
3V
10V
Ifs= Im Rs Rm
30V Vrange
S
Rs= (S x Vrange) - Rm
Total voltmeter resistance, RT
Figure (a) Figure (b)
Vrange = ( Rs + Rm) Im RT = Rs + Rm = S x Vrange
109 110
RB
3) Without volt-meter VRB xE VRB RB
Rm R eq = R B //R T
(expected value) RA RB
E
I with Rx
Rz Rm Rx
Rx
113 114
115 116
75%
20% R1
Rx1
0
0% 100% R x 10
R2
Full scale
R3
percentage R x 100
E
X Y
117 118
I
+
Potentiometer: E A
V
0 cm J 100
A 200
+
Principle: Rh
300
B 400
V=IR
15.4.1 Potentiometer = I l/A K
119 120
Potentiometer: Potentiometer
V l The potentiometer has a better accuracy then a
V voltmeter.
V /l is a constant. It is because the readings of the potentiometer are
measured from zero to 100 cm. A large scale gives a
The potential difference more accurate reading.
0 l
across any length of a Potentiometer can be used to
wire of uniform cross- measure emf of an unknown cell,
measure the internal resistance of a cell,
section and uniform
measure current
composition is measure thermoelectric emf
proportional to its length calibrate a voltmeter,
when a constant current compare resistances
flows through it.
121 123
Potentiometer
Potentiometer: E1
R.B
If the galvanometer shows defection in one direction The balance I + G
+
only, it may be due to point is E2
The connections of the terminals of the cells are obtained for E A
0 l2 J2 100
wrong. The positive terminal of the cell must be the cell when l1 J1
connected to the positive terminal of another cell. the potential at A +
200
The emf of the unknown cell is more then the emf of a point on the Rh B 400
300
Potentiometer:
Note:
The balance point will not be obtained on the 15.4.2 Wheatstone
potentiometer wire if the fall of potential along
the potentiometer wire is less than the emf of the bridge
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 can
be regarded as an ideal voltmeter.
126 127
R
R1
Rv By manipulating the above equations, we get P R
R2 Q
130 S 131
132 133
R.B (R)
Wheatstone bridge Metre Bridge: X
Therefore, X = R (100 l) l
134 135
Metre bridge
This uses the same logic as the wheatstone
bridge, but two of the resistors are replaced by
R2
a length of wire. A sliding contact divides the R1
136 137
We know from the wheatstone bridge Uses of wheatstone bridge circuits
circuit R1 / R2 = R3 / R4
Temperature control in this case the
In this case R3 and R4 are wires of wheatstone bridge starts balanced. If the
uniform cross section (A) and the same temperature of one of the resistors changes
material ( is the same) then its resistance will change, the bridge will
no longer be balanced and so current flows
Thus R3 =constant L1 through the galvanometer.
R4 = constant L2
R1 / R2 =L1 / L2
138 139
140 141
142 143
144 145
MEASUREMENT TECHNIQUES FOR MEASUREMENT TECHNIQUES FOR
RESISTANCE WHEATSTONE BRIDGE RESISTANCE WHEATSTONE BRIDGE
Similarly Results
E2 Ex I1 R1 I 2 R3
I1 R2 I 2 Rx I1 R2 I 2 Rx
146 147
Internal
E = I(R+r)
Divide between these two resistance
I1 R1 I 2 R3 Junction Rule: I = 0
Law Loop Rule: (IR) = E
I1 R2 I 2 Rx Potential
V = R1V0/(R1 + R2)
Divider
Simplify Potentiometer VAB l
R1 R3 Wheatstone
Bridge
R/S = P/Q
R2 Rx
148 149