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Calculating Resistance
Area,A
l
lR
A
When conductor has uniformcross-section
6
6
1.67 10
2.70 10
cu
al
cm
cm
Temperature Coefficient of Resistance
Metallic conductors have a linear increase of resistance with increased temperature.
To is the reference temperature (usually 20oC) and Ro is the resistance at the reference temperature. is the temperature
coefficient of resistance for the material. At 20oC, some values for are:
Material Alpha @ 20oC
Aluminum 0.004308
Copper 0.004041
( ) 1o oR T R T T
Resistors in Series
Vs
R1
R2
+ V1 -Is
I1
I2
+V2
-
By KCL: Is = I1= I2
By Ohm’s Law: V1 = R1·I1 and V2 = R2·I2
Combine: Vs = R1I1 + R2I2 = (R1 + R2) Is = ReqIs
In General: Req = R1 + R2 +···+ Rn
+Vs
-Vs Req
Is
Resistors in Parallel (1/2)
Is R1 R2
+V1 -
I1 I2+V2
-
+Vs -
+Vs
-Is Req
11
1
VI
R
By KVL: Vs = V1 = V2 By KCL: Is = I1 + I2
By Ohm’s Law: and
Combine:
22
2
VI
R
1 2
1 2 1 2
1 1 ss s
eq
VV VI V
R R R R R
Resistors in Parallel (2/2)
1 2
1 2eq
R RR
R R
1 2
1 1 1 1
eq nR R R R
For two resistors:
For many resistors:
In terms of conductance:
1 2eq nG G G G
Voltage Divider Circuit
Vs
R1
R2
+ V1 -
+V2
-Measure
V2
I
1 2sV I R R 1 2
sVIR R
2
2 2 21 2 1 2
ss
V RV I R R V
R R R R
Voltage Divider Equations
2
1 2o s
RV V
R R
2
1 2o s
RV V
R R
Unloaded:
Loaded:
If RL >> R2:
2
21 21
o s
L
RV V
RR R
R
Current Divider Circuit
Is G1 G2
+vo
-
i1 i2
1 2
1 2 1 2
so
Ii iv
G G G G
2 2
21 2
1 2
1
1 1s s
G Ri I I
G GR R
If there are onlytwo paths:
In general:
2 1 2 12
1 2 1 21 2
1
1 1s s
R R R Ri I I
R R R RR R
1 2
nn s
n
Gi I
G G G
D’Arsonval Meter Movement• Permanent Magnet Frame
• Torque on rotor proportional to coil current
• Restraint spring opposes electric torque
• Angular deflection of indicator proportional to rotor coil current
S N
D’Arsonval Voltmeter• Small voltage rating on movement (~50 mV)• Small current rating on movement (~1 mA)
• Must use voltage dropping resistor, Rv
Rv
+Vd'A
-
+ VRv -+Vx-
Id'A
Example: 1 Volt F.S. Voltmeter
Note: d’Arsonval movement has resistance of 50
Scale chosen for 1.0 volt full deflection.
950
+50 mV
-
+ 0.95 V -+1.0 V
-
1 mA
Example: 10V F.S. Voltmeter
Scale chosen for 10 volts full deflection.
9950
+50 mV
-
+ 9.95 V -+10 V
-
1 mA
D’Arsonval Ammeter
• Small voltage rating on movement (~50 mV)
• Small current rating on movement (~1 mA)
• Must use current bypass conductor, Ga
Ga
+Vd'A
-
IGa Id'AIx
Example: 1 Amp F.S. Ammeter
Note: d’Arsonval movement has conductance of 0.02 S
Scale chosen for 1.0 amp full deflection.
Ga = 19.98 S has ~50.050 m resistance.
19.98 S
+50 mV
-
999 mA 1 mA1.0 A
Example: 10 Amp F.S. Ammeter
Scale chosen for 10 amp full deflection.
Ga = 199.98 S has ~5.0005 m resistance.
199.98 S
+50 mV
-
9.999 A 1 mA10 A
Measurement Errors
• Inherent Instrument Error
• Poor Calibration
• Improper Use of Instrument
• Application of Instrument Changes What was to be Measured– Ideal Voltmeters have Infinite Resistance– Ideal Ammeters have Zero Resistance
Example: Voltage Measurement
True Voltage:
(If voltmeter removed)
10045 9
500oV V V
45 V
400
100 +Vo-
10 kvolt-
meter
Example: Voltage Measurement
Measured Voltage:
10045 8.9286
100400 1 100
10
oV V
k
8.9286% 1 100% 0.794%
9.0
VError
V
Another Voltage Measurement (1/2)
True Voltage:
(If voltmeter removed)
1045 9
50o
kV V V
k
45 V
40 k
10 k+Vo-
10 kvolt-
meter
Another Voltage Measurement (2/2)
Measured Voltage:
1045 5.0
1040 1 10
10
o
kV V V
kk k
k
5.0% 1 100% 44.44%
9.0
VError
V
Example: Current Measurement (1/2)
True Current:
(If ammeter replaced by short circuit)
255 1.0
125oI A A
5A
100
25 50 mAmmeter
Io
Example: Current Measurement (2/2)
Measured Current:
255 0.9996
125.05oI A A
0.9996% 1 100% .04%
1.0
AError
A
Another Current Measurement (1/2)
True Current:
(If ammeter replaced by short circuit)
255 1.0
125o
mI A A
m
5A
100 m
25 m 50 mAmmeter
Io
Another Current Measurement (2/2)
Measured Current:
255 0.7143
175o
mI A A
m
0.7143% 1 100% 28.57%
1.0
AError
A
Measuring Resistance
• Indirect– Measure Voltage across Resistor– Measure Current through Resistor– Calculate Resistance (Inaccurate)
• d’Arsonval Ohmmeter– Very Simple– Inaccurate
• Wheatstone Bridge (Most Accurate)
Ohmmeter Example
10 mA Full Scale (Outer Numbers)
Rb+Radj+Rd’A=150 Vb=1.5 V
Inner (Nonlinear) Scale in Ohms
5
1002.5
7.5
0
50150
450
8
Wheatstone Bridge
Vg
Rg R1R2
R3 Rx
a b
c
d
+ Vab -
I1 I2
I3 Ix
Iab
Vab= 0 and Iab= 0
Vad = Vbd
I1 = I3 I2 = Ix
R1I1=R2I2
R3I3=RxIx
2 3
1x
R RR
R