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1
E. ELECTRICITY AND MAGNETISM
Chapter 15 Direct 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 variable
voltage (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 bridge 4
15.1 Internal resistance of sources
5
Electric Current A 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 (emf EMF = 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
qWne
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 charges on the battery terminals, an electric potential difference exists between them. The maximum potential difference is called the electromotive force* (emf) of the battery. The electric potential difference is also known as the voltage, V. The SI unit for voltage is the volt, after Alessandro Volta (1745-1827) who invented the electric battery. 1 volt = 1 J/C.
9
Electromotive Force (emf) Electromotive force (emf) is the potential difference that appear between the terminals of a battery when no current is present. A source of emf is a device that converts chemical, mechanical or other forms of energy into the electric energy necessary to maintain a continuous flow of electric charge. In electric circuit, source of emf is usually represented by and it is measured in volts.
10
Electromotive Force (emf) A source of emf will maintain a potential difference and supply current to an external circuit .Example: 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 case there is no potential difference to drive the charge. But if the emf in nonzero, a current is present when the terminals are connected. The greater the emf, the greater the current in the circuit.
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
Zn Cu The plates react with the acid Zn plate becomes neg. charged, Cu +. Thus a potential difference exists so electrons can flow from -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 be recharged (usually by pushing current through it in the wrong direction) (e.g. car battery)
16
Example When you press one of the buttons on a pocket calculator, the battery provides a current of 300 µA for 10 ms.
How much charge flows during that time? How many electrons flow in that time?
1319 1090.1
1060.10.3
electron 1on charge10msin charge totalelectrons ofNumber
00.310300
CC
CmsAtIq
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 relationship I = I2 R + I2 r
When R >> r, most of the power delivered by the battery is transferred to the load resistor
21
Internal resistance In 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 the cell to the flow of electric current through it is called the internal resistance of the cell. Factors affecting Internal Resistance of a cell:
Larger the separation between the electrodes of the cell, more the length of the electrolyte through which current has to flow and consequently a higher value of internal resistance. 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.
23
E = V + v
= IR + Ir = I (R + r)
I = E / (R + r) This relation is called circuit equation.
R
r E
I I
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
r E
I I
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:
25
Determination of Internal Resistance of a cell by voltmeter method:
r
K R.B (R)
V +
r
I I 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 Battery Battery 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 greater voltage 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 a resistance. The resistance, r is the internal resistance of the battery. If the current is connected to an external resistance, R (load resistance), the circuit can be drawn as shown in the given figure.
28
Internal Resistance of a Battery The current through the circuit depends on the emf and the total resistance. The potential difference across the terminals of the battery is called the terminal potential difference (TPD). It is the emf reduced by the voltage drop across the internal resistance.
rRI
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.
rRR
rRrIrTPD
emf votageterminalemf voltageterminal
RR
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. 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?
VrR
RTPD
AV
Ir
23.90.12310
10
0.30.40.12
32
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 is transversed in the direction of the emf (from to +), the change in the electric
In d, the source of emf is transversed in the direction opposite of the emf (from + to -), the change in the electric potential is -
40
Rules Use 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
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 Rules
Draw the circuit diagram and assign labels and symbols 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 needed to solve for the unknowns Solve the equations simultaneously for the unknown quantities Check 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
0I
0V
conservation of charge: junction rule, valid at any junction Junction (Node) Rule: At any junction point, the sum of all currents entering the junction must equal the sum of the currents leaving the junction.
conservation of energy:
loop rule, valid for any loop Loop Rule: The some of the changes in potential around any closed path of a circuit must 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 junction 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
Rules Some circuits cannot be broken down into series and parallel connections.
47 213 III
I1
I2 I3
11 II
213 III
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 to conservation of energy.
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 the Junction Rule
I
1 2 31 2 3 1 2 3 eq
V V V 1 1 1 VI I I I VR 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 loop from a to b In (a), the resistor is traversed in the direction of the current, the potential across the resistor is IR In (b), the resistor is traversed in the direction opposite of the current, the potential across the resistor is is + IR
55
Loop Rule In (c), the source of emf is traversed in the direction of the emf (from to +), and the change in the electric potential is + In (d), the source of emf is traversed in the direction opposite of the emf (from + to -), and the change in the electric potential is -
56
Example Problem 1
1R 1690
3R 1000
4R 3000
Given:
Find: current in each resistor
V = 3 Volts
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 produces an output voltage that is a fraction of the supply voltage V. This is done by connecting two resistors in series as shown. Since the current flowing through each resistor is the same, thus
V
1V
1RI
2V
2RI
21eff RRReffRVI and
21 RRVI
62
Therefore, the potential difference (voltage) across R1 is given by Similarly, Resistance R1 and R2 can be replaced by a uniform homogeneous wire as shown.
11 IRV VRR
RV21
11
VRR
RV21
22
V
I2l1l
IBA C
2V1V
63
The total resistance, RAB in the wire is Since the current flowing through the wire is the same, thus
ARCBACAB RRR
AAR 21
AB
and
ABRVI
21 llA
VI
21AB llA
R
64
Therefore, the potential difference (voltage) across the wire with length l1 is given by Similarly,
A C1 IRVAll
A
VV 1
21
1
Vll
lV21
11
Vll
lV21
22
lVA
IIRV
65
0004
V 21
0008
ou tV
V 0.4outV
V 4.2outV
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 max input voltage when the contact is at B
A
B
Vout
68
Potential Divider Fixed Valued Potential Divider
21 RRV
R1
V
R2 V2
V1
I I
Effective resistance = R1 + R2
Current through each resistor =
Voltage across R1 , V1 = R1I = 121
RRR
V
Voltage across R2 , V2 = R2I = 221
RRR
V
69
Potential Divider Variable 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
+12 V
0 V
R2
R1
+12 V
0 V
V
I
I
Potential Divider
71
R2
R1
+12 V
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
73
Usage of Potential Divider In reality, series circuits are used as potential dividers
to control a device automatically. Eg.: to turn on an electric heater automatically in an
incubator. 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 input voltage is shared will also be affected. As a result, the output voltage will vary depending on
the environment. This can then control the device by switching it off (of the voltage to it is too low), or on.
74
Control the temperature in an incubator Consider a potential divider which uses a fixed resistor in series with a thermistor. Remember that the resistance of the thermistor falls with increasing temperature. As the temperature of the incubator drops, the resistance of the thermistor will increase. A larger portion of the input voltage will then be used across it. Place an electric heater across the thermistor. The heater will come on when the
voltage to it is high enough, i.e when the temperature has dropped sufficiently. Choosing different values for the fixed resistor will allow the heater to come on at different temperatures.
R
T
Vout to heater
75
15.3.1 Shunt and multiplier
76
The Measurement of Current and Voltage
A dc galvanometer. The coil of wire and pointer rotate when there 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 multiplier The 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 total current 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 µA, 5000 meter movement, multiplier resistors, and one range switch
81
Multi-range ammeter typical of those found in many VOMs. The meter is a 50 µA full-scale, 5000 movement
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 the galvanometer deflection to the voltage producing this deflection
Amm
IdS I
mVmm
VdSV
AmmS
IdS IR
Cmm
Qd
S mQ
83
Megohm sensitivity may be defined as the number of megohms required in series with the (CDRX shunted) galvanometer to produce one scale division deflection when 1 V is applied to the circuit Ballistic sensitivity and is defined as the ratio of the maximum deflection, dm, of a galvanometer to the quantity Q of electric charge in a single pulse which produces this deflection.
Galvanometer Sensitivity
Amm
IdS I
mVmm
VdSV
AmmS
IdS IR
Cmm
Qd
S mQ
84
DC Ammeters Shunt Resistor
m
mms II
RIR
+
-
I Is
Rs Rm Movement
Im
85
Ayrton Shunt Schematic diagram of a simple multirange ammeter -------- Universal or Ayrton shunt --
Ra
+
- S
Rb Rc Rd
+
-
Rc
Rb
Ra
5A 5A
1A
10A
86
DC Voltmeters
Basic dc voltmeter circuit -- Multirange voltmeter ------- Voltmeter sensitivity :
mmm
mms R
IV
IRIV
R
VIS
fsd
1
R1
+
-
R2
R3
R4
Im
V1
V2
V3
V4
+
-
Im
v
Multiplier
Rm
Rs
87
Voltmeter-Ammeter Method A popular type of resistance measurement Effect of voltmeter and ammeter positions in voltmeter-ammeter measurements --
+
-
V
A
Rx Vx Load V
I Ix
+
-
V
A
Rx Vx Load V
I Ix
88
Voltmeter-Ammeter Method
Effect of the voltmeter position in a voltmeter-ammeter measurements
+
- V
A
Rx Load
V
2
1
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 resistance of the order of 60 W - that could significantly disturb (reduce) a current measurement. Built to have full scale for small current ~ 1 mA or less. Must therefore be mounted in parallel with a small resistor 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 the galvanometer. 0.001 60 1.999
0.03002p
p
A A RR
Rp is rather small! The equivalent resistance of the circuit is also small! 92
Galvanometer used as Voltmeter Finite internal resistance of a galvanometer must also 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 disturbing circuit when measured in parallel.
Galvanometer 60 Rs
93
Galvanometer 60 Rs
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
99940p
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 Rs form 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
Rsh
Rm
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
Rsh
Rm
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 factor n=I/Im
I = nIm ---(b)
Substitute b to a
Rsh = ImRm/(nIm Im)
Rsh= Rm/(n-1) -----(c)
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
Rm
Rc RaRb
Rsh
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
1nRR m
sh ----(c)
99
Rm
Rc RaRb
Rsh
I3 I1
+ -
I2
I
I - Im
Im
B
Middle sensitive range
macb RRRR VV
(Rb + Rc )(I2 -Im) = Im(Ra +Rm)
Since,
Ra = Rsh (Rb + Rc),
yield,
I2 (Rb + Rc ) Im(Rb+Rc) = Im [Rsh (Rb + Rc ) + Rm]
2
)(I
RRIRR mshmcb ----(d)
At point B, (Rb+Rc)||(Ra+Rm)
100
Rm
Rc RaRb
Rsh
I3 I1
+ -
I2
I
I - Im
Im
C
At point C, Rc||(Ra+Rb+Rm)
3
)(IRRIR mshm
c----(e)
mbac RRRR VV
(I3-Im)Rc = Im(Ra+Rb+Rm)
I3Rc = Im(Ra+Rb+Rc+Rm)
I3Rc = (Rsh+Rm)
101
Substitute eqn (d) into eqn (e), yields
32
11)(II
RRIR mshmb----(f)
Ra = Rsh (Rb+Rc) ----(g)
102
Rm
Rc RaRb
Rsh
I3 I1
+ -
I2
I
I - Im
Im
Example 2: The Aryton Shunt Calculate 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
R1
Ie
X
E
Y
Rm
R1
Im
X
E
Y
1REIe
mm RR
EI1
me
m
RRR
II
1
1
Ammeter Insertion Effect
%100e
me
IIIrrorInsertionE
%100e
me
IIIrrorInsertionE
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.
Rm
R1
Im
X
E
Y
1k
3V
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 PU
RPO
SE
106
Rm
Rs Im+
fsI1ySensitivit ( /V)
voltohms
ohmsvolt
1amperes
1 y SensitivitIfs= Im = full scale deflection current
Rs + Rm= (S x Vrange) It is desirable to make R(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( RRmV
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 Voltmeter
Calculate 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
RA
RBRm
Rs ImRT = Rs +Rm
VRB Req = RB //RT
Ifs= Im
Rs= (S x Vrange) - Rm Total voltmeter resistance, RT
RT = Rs + Rm = S x Vrange
SRRV ms
range
Vrange = ( Rs + Rm) Im
111
Calculation: 1) 2) 3) 4) 5)
xERR
RVBA
BRB
RT = Rs + Rm = S x Vrange
Req = RB // RT
Without volt-meter
With volt-meter xERR
RV
Aeq
eqmRB
Insertion error %100x
VVV
RB
mRBRB
(expected value)
(measured value)
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Example 6: Voltmeter Loading Effect
E
RA
RBRm
Rs ImRT = Rs +Rm
VRB Req = RB //RT
A volt meter (0-10V) that has an internal resistance of 78 is used to measure the voltage across resistor RB. Determine the percentage of error of the reading due to voltmeter insertion. Let E = 4V, RA=RB = 1k , S = 1k /V
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DC Ohmmeter Basic Ohmmeter circuit
Ifs
Rm0.9Rz0.1Rz
Rz
EX Y
Rx
Variable portion
Fixed portion
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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.
mzfs RR
EI w/o Rx
xmz RRREI with Rx
I < Ifs
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Relationship between full-scale deflection to the
value of Rx is :
xmz
mz
fs RRRRR
IIP
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
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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%) .
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Solution Ex:7
0 0%
20%
40% 50%
75%
100%
12k
4.5k 3k
1k
Ohm
Full scale percentage
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IfsRz - fixed resistance &zeroing potentiometer
E X Y
R x 1
R x 10
R x 100
R1
Rm
R2
R3
Multiple-range Ohmmeter The 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
121
0 l
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 a
voltmeter. It is because the readings of the potentiometer are measured from zero to 100 cm. A large scale gives a more accurate reading.
Potentiometer can be used to measure emf of an unknown cell, measure the internal resistance of a cell, measure current measure thermoelectric emf calibrate a voltmeter, compare resistances
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Potentiometer If the galvanometer shows defection in one direction
only, it may be due to The connections of the terminals of the cells are wrong. The positive terminal of the cell must be connected to the positive terminal of another cell. The emf of the unknown cell is more then the emf of the cell connected across the wire of the potentiometer, AB. The connections are not tight and the current does not flow in certain part of the circuit.
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+
E1
E2
+
R.B G
J1 l1
J2 l2 E
A
K
A
B Rh +
I
100
200
300 400
0
E1 = VAJ1 = I l1 /A
E2 = VAJ2 = I l2 /A
E1 / E2 = l1 /l2
Potentiometer: The balance
point is obtained for the cell when the potential at a point on the potentiometer wire is equal and opposite to the emf of the cell.
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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 can be regarded as an ideal voltmeter.
127
15.4.2 Wheatstone bridge
128
Wheatstone bridge How does it work?
If the galvanometer reading is zero,
VA = VC VAB = VCB and VAD = V CD
P and R carry the same current, I1 and X and Q carry the same current I2.
I1P=I2Q and I1R = I2X Dividing the equations, Then
P QR 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
lXR l
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 null current.
v
v
RRRR
IRIRRIIR
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:
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Wheatstone bridge
A device for measuring the value of an unknown resistance The values of the resistances are varied until no current flows through 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 A and D (VAB=VAD) Similarly VBC = VDC
I1R1 = I2R3
I1R2 = I2R4
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Wheatstone bridge Thus
R1 / R2 = R3 / R4 Thus, if three of the resistors are known, you can calculate the value of the last. Experimentally a resistor is placed in series with the galvanometer to protect it from too much current. This resistor is then removed when the aprox. balance point is found
135
A B
R.B (R) X
G
J
K E
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 wire L1 L2
138
We know from the wheatstone bridge circuit R1 / R2 = R3 / R4 In this case R3 and R4 are wires of uniform cross section (A) and the same material ( 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 the wheatstone bridge starts balanced. If the temperature of one of the resistors changes then its resistance will change, the bridge will no longer be balanced and so current flows through the galvanometer.
140
Uses of wheatstone bridge circuits
The size and direction of the current indicate the size and direction of the temperature change, and so can be used to control a heater and bring the temp. back to its original value
141
Uses of wheatstone bridge circuits
Fail-safe device if the pilot light in a gas boiler goes out, you need the gas to shut off automatically.
142
Uses of wheatstone bridge circuits
A thermistor placed near the flame is used as one resistor in a wheatstone bridge. If the flame goes out the resistance increases, unbalances the bridge and current flows in the galvanometer. This current can be 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 small amounts of current) is inserted across terminals b and d to indicate the condition of balance.
144
MEASUREMENT TECHNIQUES FOR RESISTANCE WHEATSTONE BRIDGE
When the bridge is properly balanced, no difference in potential exists across terminals b and d; when switch S2 is closed, the galvanometer reading is zero.
145
MEASUREMENT TECHNIQUES FOR RESISTANCE WHEATSTONE BRIDGE
During balance state, I1 follows a-b-c path and I2 follows a-d-c- path. Thus,
3211
31
RIRIEE
146
MEASUREMENT TECHNIQUES FOR RESISTANCE WHEATSTONE BRIDGE
Similarly
x
x
RIRIEE
221
2
147
MEASUREMENT TECHNIQUES FOR RESISTANCE WHEATSTONE BRIDGE
Results
xRIRIRIRI
221
3211
148
MEASUREMENT TECHNIQUES FOR RESISTANCE WHEATSTONE BRIDGE
Divide between these two
xRIRI
RIRI
2
32
21
11
Simplify
xRR
RR 3
2
1
Summary: Direct Current Circuits
Internal resistance
E = I(R+r)
Law Junction Rule: I = 0 Loop Rule: (IR) = E
Potential Divider V = R1V0/(R1 + R2)
Potentiometer VAB l
Wheatstone Bridge R/S = P/Q
149