Department of Electrical and Electronic Engineering
Lecture on EE 3115
Electrical Measurement & Instrumentations
Contact Hours: 3, Credit: 3
Md. Alamgir Hossain
Assistant Professor
Department of Electrical and Electronic Engineering(EEE)
Khulna University of Engineering & Technology(KUET)
Khulna-9203, Bangladesh
Contents
1 Measurement of Resistance
1.1
1.2
1.3
1
Measurement of Medium Resistance . . . . . . . . . . . . . . . .
1
1.1.1
Method of Measurement of Medium Resistance . . . . . .
1
Measurement of Low Resistance . . . . . . . . . . . . . . . . . .
10
1.2.1
Method of Measurement of Low Resistance . . . . . . . .
10
Measurement of High Resistance . . . . . . . . . . . . . . . . . .
14
1.3.1
Use of Guard Circuit . . . . . . . . . . . . . . . . . . . .
14
1.3.2
Methods of Measurement of High Resistance . . . . . . .
15
1.3.3
Loss of Charge Method . . . . . . . . . . . . . . . . . . .
16
1.3.4
Methods of Measurement of Earth Resistance . . . . . . .
18
Bibliography
22
Chapter 1
Measurement of Resistance
Contents
1.1
1.2
1.3
1.1
Measurement of Medium Resistance . . . . . . . . . . . . . . . .
1
1.1.1
Method of Measurement of Medium Resistance . . . . . .
1
Measurement of Low Resistance . . . . . . . . . . . . . . . . . . .
10
1.2.1
Method of Measurement of Low Resistance . . . . . . . .
10
Measurement of High Resistance . . . . . . . . . . . . . . . . . .
14
1.3.1
Use of Guard Circuit . . . . . . . . . . . . . . . . . . . .
14
1.3.2
Methods of Measurement of High Resistance . . . . . . .
15
1.3.3
Loss of Charge Method . . . . . . . . . . . . . . . . . . .
16
1.3.4
Methods of Measurement of Earth Resistance . . . . . . .
18
Measurement of Medium Resistance
Classification of Resistances: Low resistance- ≤ 1Ω, Medium resistance- 1Ω ∼
100kΩ and High resistance- ≥ 100kΩ
1.1.1
Method of Measurement of Medium Resistance
i. Ammeter-Voltmeter method
ii. Substitution method
iii. Wheatstone bridge method and
1.1. Measurement of Medium Resistance
2
iv. Ohmmeter method
1.1.1.1
Ammeter Voltmeter (AV) Method
In AV method the measured resistance is given by Rm = V/I, where V and I is the
voltage and current reading of voltmeter and ammeter respectively. The available
connection methods are shown in figure (1.1).
A
Ra
I=IV+IR
A
IV
VA
V
V
VR
R
V
V
RV
IR
R
(a)
(b)
Figure 1.1: AV method to measure medium resistance
For figure (1.1)(a),
Rm1 =
V Va + VR IRa + IR
=
=
= Ra + R
I
I
I
(1.1)
Therefore, the true value of resistance,
R = Rm1 − Ra = Rm1 (1 − Ra /Rm1 )
(1.2)
Relative error,
Rm1 − R Ra
=
R
R
(1.3)
V
V
V
R
=
=
=
I
IV + IR V/RV + V/R 1 + R/RV
(1.4)
εa =
For figure (1.1)(b),
Rm2 =
(1.5)
Therefore, the true value of resistance
Rm2 RV
1
R=
= Rm2
RV − Rm2
1 − Rm2 /RV
!
(1.6)
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.1. Measurement of Medium Resistance
3
If Rv >> Rm2 then,
1
1 − Rm2 /RV
!−1
= (1 + Rm2 /RV ) + .....
Neglecting higher order term, the true value of resistance become,
R = Rm2 (1 + Rm2 /RV ))
(1.7)
R2
Rm2 − R
= − m2
R
RRV
(1.8)
Relative error,
εb =
Since, the value of Rm2 ≈ R, relative error
εb = −
R
RV
(1.9)
The magnitude of error in both cases will be same if
Ra
R
=
R
RV
p
R =
Ra RV
(1.10)
If test resistance is greater than that of calculated from equation (1.10), then figure
(1.1)(a) is used, otherwise figure (1.1)(b).
1.1.1.2
Wheatstone Bridge Method
The method of measuring medium resistance by using Wheatstone bridge is shown
in figure (1.2). The bridge is said to be balanced if there is no current flow through
the galvanometer or the potential difference across the galvanometer is zero.
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.1. Measurement of Medium Resistance
4
b
P
Q
I3
I1
a
c
G
I4
I2
R
S
d
E
Figure 1.2: Wheatstone bridge
At balance condition,
I1 P = I2 R
(1.11)
The galvanometer current will be zero if
E
P+Q
E
I2 = I4 =
R+S
I1 = I3 =
(1.12)
(1.13)
Combining equations (1.11), (1.12) and (1.13) result
P
R
=
P+Q
R+S
P
R =
S
Q
(1.14)
In equation (1.14), P and Q are known standard resistors and by varying standard
variable resistor S the null point of the galvanometer is obtained and the value of
unknown resistor R is calculated.
1.1.1.3
Sensitivity of Wheatstone Bridge
It is frequently desirable to the know the galvanometer response to be expected in a
bridge which is slightly unbalanced so that a current flows through the galvanomeMd. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.1. Measurement of Medium Resistance
5
ter. This may be used in order to
• select a galvanometer with which a given unbalance may be observed in a
specified bridge arrangement,
• determine the minimum unbalance which can be observed with a given galvanometer in a specified bridge arrangement and
• determine the deflection to be expected for a given unbalance.
The sensitivity to unbalance can be measured by solving bridge circuit for a small
unbalance. Suppose, the resistance R is replaced by R + ∆R creating a small unbalance. This will create an emf e across the galvanometer branch. The voltage drop
between points a and b as well as between a and d are given by
EP
P+Q
E(R + ∆R)
= I2 (R∆R) =
R + ∆R + S
Eab = I1 P =
Ead
Therefore, the voltage difference between points d and b is
"
#
R + ∆R
P
e=E
−
R + ∆R + S P + Q
(1.15)
(1.16)
(1.17)
since,
P
R
=
P+Q R+S
#
R
ES ∆R
R + ∆R
=E
−
=
2
R + ∆R + S r + S
(R + S ) + ∆R(R + S )
"
e = Ead − Eab
(1.18)
as
∆R(R + S ) << (R + S )2
e≈
ES ∆R
(R + S )2
(1.19)
Let, S v be the voltage sensitivity of a galvanometer.
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.1. Measurement of Medium Resistance
6
Therefore, the deflection of the galvanometer,
θ = S ve = S v
ES ∆R
(R + S )2
(1.20)
The bridge sensitivity S B is defined as the deflection of the galvanometer per unit
fractional change in unknown resistance.
The bridge sensitivity,
θ
S v ES R
=
∆R/R (R + S )2
S vE
S vE
= R
= P
S
+2+ R
+2+
S
Q
SB =
(1.21)
(1.22)
Q
P
The bridge sensitivity will be maximum if and only if the bridge has all equal arms
i,e P = Q = R = S . The maximum bridge sensitivity is
S Bmax =
1.1.1.4
S vE
4
(1.23)
Galvanometer Current:
The current through the galvanometer can be found by the Thevenin equivalent
circuit. The Thevenin voltage E0 between terminal b and d is given by
E(R + ∆R)
EP
E0 = Ead − Eab = I2 (R + ∆R) − I1 P =
−
R + ∆R + S P + Q
#
"
(R + ∆R)
P
−
= E
R + ∆R + S P + Q
R+ R
a
Ro
P
d
b
c
Q
S
Figure 1.3: (a) Finding Thevenin resistance
(1.24)
Eo
G
(b) Thevenin equivalent
circuit
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.1. Measurement of Medium Resistance
For a bridge arm with equal arms R = S = P = Q. Therefore,
"
#
"
#
(R + ∆R)
R
(R + ∆R) 1
E0 = E
−
=E
−
R + ∆R + S R + S
2R + ∆R 2
∆R
≈ E( )
4R
7
(1.25)
(1.26)
Since, ∆R << R the Thevenin equivalent resistance of the bridge,
R0 =
RS
PQ
+
R+S P+Q
(1.27)
For equal arms P = Q = R = S , R0 = R and the current in through the galvanometer
is
E0
R0 + G
(1.28)
E(∆R/4R)
R+G
(1.29)
Ig =
For a bridge with equal arms,
Ig =
We know that the deflection of the galvanometer,
θ = S ve = S v
ES ∆R
(R + S )2
(1.30)
But, if S i be the current sensitivity of the galvanometer, then
Sv =
Therefore,
θ = S ve =
Si
R0 + G
(1.31)
S i ES ∆R
(R0 + G)(R + S )2
(1.32)
S i E∆R
4R(R + G)
(1.33)
For a bridge with equal arms,
θ = S ve =
And the bridge sensitivity,
SB =
θ
S i ES R
=
∆R/R (R0 + G)(R + S )2
(1.34)
For a bridge with equal arms the bridge sensitivity,
SB =
S iE
4(R + G)
(1.35)
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.1. Measurement of Medium Resistance
1.1.1.5
8
Precision Measurement of Medium Resistance with Wheatstone Bridge
In case of medium resistance measurements with Wheatstone bridge, the following
factors should be taken into consideration:
i. Resistance of connecting leads
ii. Thermoelectric effects
iii. Temperature effects
iv. Contact resistances
In precision measurements, the accurate comparisons are made on an equal ratio
bridge with a fixed standard nominally equal to the resistance under test. The
problem is further reduced by determining the exact ratio of R to S or the difference
between them. A slide wire bridge is used to determine that difference between the
standard and unknown resistance.
1.1.1.6
Carey Foster Slide Wire Bridge
This bridge is suitable for comparing two nearly equal resistances. Resistance P
and Q are adjusted so that the ratio P/Q is approximately equal to the ratio R/S .
Let, balanced point d is obtained at a distance l1 as shown in figure (1.4) . Therefore
at balance condition,
P
R + l1 r
=
Q
S + (L − l1 )r
P
R + l1 r + S + (L − l1 )r
R + S + Lr
+1 =
=
Q
S + (L − l1 )r
S + (L − l1 )r
(1.36)
(1.37)
where, r is the resistance per unit length of the slide wire. Then R and S are
interchanged and balanced obtained again at a distance l2 . Similarly for second
balance point,
P
S + l2 r
=
Q
R + (L − l2 )r
P
S + l2 r + R + (L − l2 )r
R + S + Lr
+1 =
=
Q
R + (L − l2 )r
R + (L − l2 )r
(1.38)
(1.39)
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.1. Measurement of Medium Resistance
9
b
Q
P
G
a
c
S
R
d
l2
l1
L
E
Figure 1.4: Carey Foster bridge to measure medium resistance
From equation (1.37) and (1.39)
S + (L − l1 )r = R + (L − l2 )r
S − R = (l1 − l2 )r
(1.40)
(1.41)
Thus the difference between S and R is obtained form the resistance per unit length
of the slide wire together with the difference (l1 − l2 ) between the two slide wire
lengths at balance.
The slide wire is calibrated, .i.e. r is obtained by shunting either S or R by a known
resistance and again determining the difference in length (l10 − l20 ).
Suppose, S is known and S 0 is its value when shunted by known resistance. After
shunting S equation (1.41) becomes
S 0 − R = (l10 − l20 )r
(1.42)
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.2. Measurement of Low Resistance
10
Therefore, equation (1.41) and (1.42) yield
S −R
S0 − R
= 0
l1 − l2
l1 − l20
S (l10 − l20 ) − S 0 (l1 − l2 )
R =
l10 − l20 − l1 + l2
(1.43)
(1.44)
The equation (1.44) shows that this method gives the direct comparison between
R and S in terms of length only and the resistances of P and Q contact resistance,
and the resistances of connecting leads are eliminated.
1.1.1.7
Limitations of Wheatstone bridge:
• few Ω to several MΩ
• upper limit is set by reducing the sensitivity to unbalance caused by resistance values
• upper limit can be extended by increasing emf that causes heat
• inaccuracy due to leakage out of insulation
• contact resistance presents a source of uncertainty that is difficult to overcome
1.2
1.2.1
Measurement of Low Resistance
Method of Measurement of Low Resistance
i. Ammeter-Voltmeter method
ii. Kelvin Double bridge method
iii. Potentiometer method
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.2. Measurement of Low Resistance
11
b
Q
P
G
a
c
S
R
m
n
d
r1
r2
r
E
Figure 1.5: Kelvin bridge method for low resistance measurement
1.2.1.1
Kelvin Double Bridge Method of Measurement of Low Resistance
Kelvin bridge, a modification of Wheatstone bridge, method increases the accuracy
in measurement of low resistance and remove the effect of connecting leads and
contact resistance. As shown in figure (1.5), r represents the resistance of lead that
connects the unknown resistance R and standard resistance S .
The galvanometer is connected at point d that divides the resistance r into r1 and r2
such that,
r1
P
=
r2 Q
(1.45)
Using equation (1.45) we have,
r1
P
P
=
⇒ r1 =
r
r1 + r2 P + Q
P+Q
r1 + r2 P + Q
Q
⇒
=
⇒ r2 =
r
r2
Q
P+Q
⇒
(1.46)
(1.47)
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.2. Measurement of Low Resistance
12
From the figure at balance condition
P
R + r1
=
S + r2
Q
P
(S + r2 )
R + r1 =
Q
P
P
Q
R+
r =
(S +
r)
P+Q
Q
P+Q
P
R = .S
Q
(1.48)
(1.49)
(1.50)
(1.51)
So, connecting the galvanometer at point d, the resistance of leads does not affect
the result. But, the problems with the above method are
• the method is not practical
• difficult to find correct galvanometer null point
To solve the above problems, two actual resistance unit of correct ratio is connected
between points m and n as shown in figure (1.6) which is the original Kelvin Double
bridge. The ratio arm of p and q is connected at d to eliminate the effect of conb
Q
P
G
d
p
a
R
q
n
m
r
I
S
c
I
Rb
E
Figure 1.6: Kelvin Double bridge method for low resistance measurement
necting leads between R and S . The value of P, Q, p and q are like that p/q = P/Q.
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.2. Measurement of Low Resistance
13
Under balance condition there is no current through galvanometer, which means
Eab = Eamd . Where,
Eab =
and
P
Eac
P+Q
(p + q)r
Eac = I R + S +
p+q+r
"
#
and
"
Eamd
#
r
= I R+
p
p+q+r
At balance condition,
⇒
⇒
⇒
⇒
Eab = Eamd
"
#
P
r
Eac = I R +
p
P+Q
p+q+r
"
#
"
#
r
P
(p + q)r
I R+S +
p
= I R+
P+Q
p+q+r
p+q+r
"
#
qr
P p
P
−
R= S +
Q
p+q+r Q q
p P
P
since, =
R= S
Q
q Q
(1.52)
(1.53)
(1.54)
(1.55)
(1.56)
The equation (1.56) shows that, the resistance of connecting leads has no effect but
error may be introduced in the ratio arms, i.e.
p
q
=
P
Q
may not equal. Thermoelectric
effect can be removed by reversing the battery connection, and true value of R will
be the mean of two readings.
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.3. Measurement of High Resistance
1.3
14
Measurement of High Resistance
Examples of high resistance
i. insulation resistance of components, machine, cables
ii. high resistance circuit elements, vacuum tubes
iii. leakage resistance of capacitor
iv. surface resistance
v. volume resistance
1.3.1
Use of Guard Circuit
The main problem in measurement of high resistance is the leakage resistance.
To eliminate the errors due to leakage resistance some form of guard circuits are
generally used as shown in the figure (1.8). In figure (1.7) the high resistance
mounted on a piece of insulating material is measured by the ammeter-voltmeter
method. The micro-ammeter measures the sum of the current through the resistor
IR+IL
μA
IR
E
V
Resistance
Terminal
IL
R
Figure 1.7: High resistance measurement without Guard circuit
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.3. Measurement of High Resistance
15
(IR ) and the current through the leakage path around the resistor (IL ). Therefore,
the measured value of the resistor calculated form the reading of the voltmeter and
micro-ammeter will not be the true value of the resistor but will be in error. In
figure (1.8) a guard terminal is added to the resistance terminal block and microammeter is bypassed. Now, micro-ammeter will measure only the current through
the resistor R that allows to determine the correct measurement of the resistor.
IL
μA
IR
IR
E
V
Guard
Terminal
IL
R
Figure 1.8: High resistance measurement with Guard circuit
1.3.2
Methods of Measurement of High Resistance
i. Direct deflection method
ii. Loss of Charge method
iii. Megohm bridge
iv. Meggar
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.3. Measurement of High Resistance
1.3.3
16
Loss of Charge Method
In this method the insulation resistance R to be measured is connected in parallel with a capacitor C and a electrostatic voltmeter. The capacitor is charged to
some suitable voltage by means of supply voltage V and then allowed to discharge
through the resistance.
V
V
R
C
Figure 1.9: Circuit for Loss of Charge Method
V
v=Ve-t/RC
Figure 1.10: Circuit for Loss of Charge Method
The voltage across the capacitor v at any instant t after application of supply voltage
V as shown in figure (1.9) is given by
v = Ve−t/RC
(1.57)
Therefore, the insulation resistance,
R=
t
0.4343t
=
Cloge (V/v) Clog10 (V/v)
(1.58)
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.3. Measurement of High Resistance
17
More accurate result may be obtained by using change in voltage V − v directly and
calling this change e, i.e.
R=
0.4343t
V
Clog10 V−e
(1.59)
This method is suitable for high resistance measurement but it requires a capacitor
having very high leakage resistance as high as the resistance being measured. This
method is also time consuming.
Actually, in this method the effect of the resistance of electrostatic voltmeter is
ignored and the leakage resistance of the capacitor is assumed infinite. In practical, correction must be applied. In figure (1.11), R1 represents the resistance of
voltmeter and capacitor.
V
C V
R1
R
Figure 1.11: Circuit for Loss of Charge Method
The measured resistances,
R0 =R1 ||R = 0.4343t/Clog10 (V/v)
R1 =0.4343t/Clog10 (V/v)
(1.60)
R0 =RR1 /R + R1
where, R0 represents the resistance when two resistances are in operation, test is
repeated disconnecting the resistance R. Then the true value R is obtained by using
the equation (1.60).
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.3. Measurement of High Resistance
1.3.4
18
Methods of Measurement of Earth Resistance
The earth resistance should be as minimum as possible used to: protect various
parts of insulations, high voltage discharge and for stabilizing 3 − φ circuit. The
methods of measuring earth resistance: i. Fall of Potential Method and ii. Earth Tester
1.3.4.1
Fall of Potential Method
A
V
V
Auxiliary Electrode
Auxiliary Electrode
Earth Electrode
I
(a)
A
V
V
E
D
C
B
(b)
V
VBC
VED
E
D
(c)
C
B
Figure 1.12: Fall of Potential Method
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.3. Measurement of High Resistance
19
As shown in figure (1.12) a current is passed through earth electrode E to another electrode B. The lines of first electrode current diverge and those of second
electrode current converge. As a result the current density is much greater in the
vicinity of the electrodes than at a distance from them. The potential distribution
between the electrodes is shown in figure 1.12(c). It is obvious from the curve that
the potential rises in the proximity of electrodes E and B and is constant along the
middle section. The resistance of earth therefore,
Earth Resistance
RE = V/I = VEA /I
RE
Distance between electrodes E and A
Figure 1.13: Earth resistance
1.3.4.2
Localization of Cable Faults
Fault occurring in cables which are in use on lower distribution voltage. The
common faults are: Ground fault(core of the cable to ground) and short circuit
fault(core of one cable to that of another cable). The methods for localizing of
these type of cable faults are
i. Murray Loop Test
ii. Varley Loop Test
Murray Loop Test: The connection diagram of this method is shown in figure
(1.14). The resistances P, Q, R and X forms essentially a Wheatstone bridge.
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.3. Measurement of High Resistance
P
20
Sound Cable
R
G
l
E
Q
X
l1
Faulty Cable
Figure 1.14: Murray Loop Test
Under balance condition,
P Q
X Q
X
Q
=
⇒ =
⇒
=
R X
R
P
X+R P+Q
Q
X=
(R + X)
P+Q
(1.61)
(1.62)
If l1 represents the length of the fault from the test end and l is the length of each
cable then
l1 =
Q
.2l
P+Q
(1.63)
Therefore, the position of the fault may be located if the length of the cable is
known. In this test it seen that the fault resistance does not alter the balance condition because it enters the battery circuit. But, if the fault resistance is high, this may
reduce the sensitivity of the bridge and accurate measurement will be impossible.
This high resistance effect may be reduced by applying high dc or ac voltage that
may carbonize the insulation of the cable at the point of fault.
Varley Loop Test: The necessary connection diagram for this test is shown in figure
(1.15). An SPDT switch K is set at position 1 and balance is obtained by varying
S.
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
1.3. Measurement of High Resistance
21
Sound Cable
P
R
G
l
Q
S
X
l1
Faulty Cable
1
K
E
2
Figure 1.15: Varley Loop Test
Let, for first case the value of S is S 1 . Under balance condition,
Q
R+X
P
P
=
⇒
=
R+X
S1
S1
Q
P
R+X =
S1
Q
(1.64)
(1.65)
The switch K is thrown to position 2 and bridge is rebalanced. Then, the balance
condition gives for the value of S is S 2
P
Q
=
R
X + S2
R
P
=
X + S2
Q
P+Q
R + X + S2
=
X + S2
Q
(R + X)Q − S 2 P
X =
P+Q
(1.66)
(1.67)
(1.68)
(1.69)
The value of X is obtained from equation (1.69) with the help of equation(1.65).
For the cables of same cross section and resistivity, the resistances are proportional
to the length. Now, if X = lr and R + X = 2lr then
X
l1
=
R+X
2l
l1 =
X
2l
R+X
(1.70)
(1.71)
Md. Alamgir Hossain, Assistant. Professor, Dept. of EEE, KUET, Bangladesh
Bibliography
[1] A. K. Sawhney,"A Course in Electrical and Electronic Measurements and Instrumentation”.
(Not cited.)