1. Measurement Of Resistance
In electrical circuit, basically three are three elements viz, resistance, conductance and capacitance.
In fact each electrical equipment is consisting of these elements or combination of these elements.
This chapter mainly concentrates on the method of measurements of different resistances (i.e. low, medium or high value resistance) and inductance and capacitance.

2. Classification of resistance
By definition : Resistance is the property of a material by virtue of which it opposes the flow of current.
According to nature of supply resistance can be denoted as DC resistance and AC resistance.
DC resistance :
Ohms Law : The law states that the direct current flowing in a conductor is directly proportional to the potential difference between its ends provided that all the physical states of conductor remain same (dimensions and temperature).
It is usually formulated as I α V, where V is the potential difference, or voltage and I is the current.
The constant of proportionality is called the “resistance”, R measured in Ω (ohms).
Then the ohm’s law is given by:
I = V/R ....Amps (3.2.1)
The resistance R of a conductor of uniform cross section can be computed as,
R = ρl/a (3.2.2)
Where, l is the length of the conductor, it is measured in meters [m] a is the cross-sectional area of the current flow, measured in square m2 ρ (Greek : rho) is the electrical resistivity of the material, it is measured in ohm-meters (Ω m). Resistivity is a measure of the material’s ability to oppose electric current. For practical reasons, any connections to a real conductor will almost certainly mean the current density is not totally uniform. However, this formula still provides a good approximation for long thin conductor such as wires.

3. AC resistance :
If a wire conducts high-frequency alternating current then the effective cross sectional area of the wire is reduced because of the skin effect.
If several conductors are together, then due to proximity effect, the effective resistance of each is higher than that if conductors were alone.
According to the value of resistance : According to the value, resistances are mainly categorized into three parts viz low, medium and high resistance.

4. Low resistance :
When the value of resistance is below one ohm. (R<1 Ω) then it is called as low resistance e.g. resistance of armature winding of generator, resistance of series field winding of DC series generator, resistance of transformer winding, bus bar resistance, earth wire resistance etc.
Medium resistance:
When the value of resistance lies between 1Ω to 0.1 mega ohm (i.e. 1 Ω<R<0.1 M Ω) then it is called as medium resistance. e.g. resistance of field winding of DC shunt generator, resistance of long transmission line etc.
High resistance:
When the value of resistance ia greater than 0.1 mega ohm (i.e. R>0.1 MΩ).Then it is called as high resistance. e.g. resistance of cable insulation, resistance of insulatodisc of transmission line etc. Generally the resistance of conductor is under the category of low range and that of insulator is treated as high range.

5. Measurement of Medium Resistance

6. Hence V=Va + Vr .let the resistance of ammeter be Ra Ω s. V=IRa+IR
This method(a and b) is very common as voltmeter and ammeter is available in all labs.
There are two methods of connecting voltmeter and ammeter for measurement of resistances as shown in Fig (a) and(b). In both the cases the measured value of the unknown resistances is equal to the reading of voltmeter divided by reading of ammeter.
Let the reading of voltmeter is 'V and ammeter I. hence measured value of the resistances = Rm =V/I.
For connection in Fig (a) the reading of ammeter is equal to I = the current flowing through resistances.
Reading of voltmeter = the voltage across resistance + voltage across ammeter
Hence V=Va + Vr .let the resistance of ammeter be Ra Ω s.
V=IRa+IR
=I(Ra+R)
Hence Rm=V/I=I(Ra+R)/I
Rm=Ra+R
True value of resistance ’R’ is equal to Rm-Ra.
R=Rm-Ra
R=Rm(1-Ra/Rm) ……….(3.3.1)
From this expression R=Rm only when Ra=0.
This is the ideal case but practically ammeter has low resistance.
Hence to reduce the error in measurement this method is used for measurement medium and high resistances. As in this case ‘Rm’ will be very much greater than ‘Ra’ Hence the ratio ‘Ra/Rm’ is approximately = 0
From Fig ( b),
Rm =V/I.
The reading of ammeter is equal to I= Iv+ Ir let the resistance of voltmeter by ‘Rv’.
I=V/Rv+V/R
Reading of voltmeter V=Voltage across resistance VR.
Rm=V/(V/R+V/Rv )
=1/(1/R+1/Rv )=1/R+1/Rv
1/R=1/Rm-1/Rv
1/R=Rv-Rm/RmRv
R=RmRv/Rv-Rm
Divided by Rv;
R=Rm/(1-Rm/Rv)…………….(3.3.2)
From this expression the true value of resistance R=Rm only, when resistance of voltmeter is ‘∞’(This is the ideal case).
But practically Rv is in medium or high resistance class.
Hence this method is used for measurement of low resistance as in this case Rm /Rv will be approximately equal to zero.
In both cases % error in measurement of resistance is equal to true value R-measured value of Rm divided by product of R and Rm.

7. Whetstone's Bridge Method
Resistance ‘R’ is the unknown resistances under measurement.
It is connected in the whetstone's bridge circuit, formed by the ratio arm resistances ‘P’ and ‘Q’ standard arm resistances 'S'.
This bridge operates on D.C supply and the detector used to detect the balanced
condition of bridge is a sensitive galvanometer ‘G’.

9. Working :
The bridge is said to be under balanced condition when galvanometer shows zero deflection.
By using the equations of voltage drops at balanced condition, we can find the magnitude of unknown resistance, ‘R’.
connect the source and galvanometer as shown in Fig and unknown resistance between points ‘B’ and ‘C’ of the bridge.
Put ‘ON’ key ‘K1’ then put 'ON' key ‘K2'. Keep resistance ‘N’ at minimum position.
Observe the reading of galvanometer. If it is vary the resistance ‘N’ to the maximum resistance position. If there is a small deflection on G adjust ratio P/Q and variable resistance ‘S’ to get zero reading on the galvanometer.
Now the bridge is completely balanced hence points A and C are at same potential.
Hence voltage across AB is equal voltage across BC.

10. VAB = VBC
I1P = I2R ………………(3.3.5)
VAD = VCD
I3Q = I4S
I1 = I3 and I2 = I4 (bridge is at balance condition i.e. Ia=0)
I1Q = I2S ………………..(3.3.6)
Dividing Equation (3.3.5) by Equation (3.3.6),
I1P/I1Q=I2R/I2S
P/Q = R/S
R=PS/Q
Thus we can find magnitude of unknown resistance by ratio P/Q and the standard resistance ‘S’.
This method is accurate than voltmeter ammeter method because we are calculating the value of ‘R’ by method of comparison.
Hence this value is independent of error in the galvanometer.
This method is used for measurement of medium resistances very accurately.

11. Kelvin double Bridge Method
The Kelvin double bridge is the modification of the bridge and provides
greatly increased accuracy measurement of low resistance.
An understanding of the Kelvin bridge arrangement may be obtained by the study of the difficulties that arise in a Wheatstone bridge on account of the resistance of the leads and the contact resistances while measuring low valued resistance

12. Figure shows the schematic diagram of the Kelvin bridge
Figure shows the schematic diagram of the Kelvin bridge. The first of ratio arms is P and Q. the second set of ratio arms, p and q is used to connect the galvanometer to a point d at the appropriate potential between points m and n to eliminate effect of connecting lead of resistance r between the unknown resistance, R, and the standard resistance, S.
The ratio p/q is made equal to P/Q. under balance conditions there is no current through the galvanometer, which means that the voltage drop between a and b, Eab is equal to the voltage drop Eamd.
When the bridge is under balanced condition Ig = 0. Hence points H and F are at the same potential.
Hence voltage across ‘P’ = voltage across resistance R + voltage across’P’.
Hence,
VP = VR+ VP
I1 .P =I.R +I2 .p ……….. (3.4.3)
Similarly voltage across Q = voltage across ‘S’ + voltage across ’q’.
VQ=VS+Vq
I1.Q = I.S +I2.q ………(3.4.4)
As ’r’ is in parallel with p + q, voltage across ‘r’ = voltage across (p+q)
Vr = V(p+q)
(I-I2)r =I2 (p +q)
Ir = I2 (p+q+r) I2
Subatituting This value of I in Equation (3.4.3) and (3.4.4)
I1P = IR +()p …..…..(3.4.5)
I1Q =IS +()q …….(3.4.6)
Divide Equation (3.4.5) by Equation (3.4.6)
(S + ( ) q )= R + ( ) p
R = (S + ( ) q ) - ( ) p
= ……….(3.4.7)

13. Usually while balancing the bridge ratio P/Q is adjusted equal to p/q
Usually while balancing the bridge ratio P/Q is adjusted equal to p/q . Hence expression for resistance ‘R’ is simplified as  
Above equation is the usual working equation for the Kelvin Bridge. It indicates that the resistance of connecting lead, r, has no effect on the measurement, provided that the two sets of ratio arms have equal ratio. The former equation is useful, however, as it shows the error that is introduced in case the ratios are not exactly equal. It indicates that it is desirable to keep r as small as possible in order to minimize the errors in case there is a difference between ratios P/Q and p/Q.

15. Figure shows the schematic diagram of the Kelvin bridge
Figure shows the schematic diagram of the Kelvin bridge. The first of ratio arms is P and Q. the second set of ratio arms, p and q is used to connect the galvanometer to a point d at the appropriate potential between points m and n to eliminate effect of connecting lead of resistance r between the unknown resistance, R, and the standard resistance, S.
The ratio p/q is made equal to P/Q. under balance conditions there is no current through the galvanometer, which means that the voltage drop between a and b, Eab is equal to the voltage drop Eamd.
When the bridge is under balanced condition Ig = 0. Hence points H and F are at the same potential.
Hence voltage across ‘P’ = voltage across resistance R + voltage across’P’.
Hence,
VP = VR+ VP
I1 .P =I.R +I2 .p ……….. (3.4.3)
Similarly voltage across Q = voltage across ‘S’ + voltage across ’q’.
VQ=VS+Vq
I1.Q = I.S +I2.q ………(3.4.4)
As ’r’ is in parallel with p + q, voltage across ‘r’ = voltage across (p+q)
Vr = V(p+q)
(I-I2)r =I2 (p +q)
Ir = I2 (p+q+r) I2
Subatituting This value of I in Equation (3.4.3) and (3.4.4)
I1P = IR +()p …..…..(3.4.5)
I1Q =IS +()q …….(3.4.6)
Divide Equation (3.4.5) by Equation (3.4.6)
(S + ( ) q )= R + ( ) p
R = (S + ( ) q ) - ( ) p
= ……….(3.4.7)
Usually while balancing the bridge ratio P/Q is adjusted equal to p/q . Hence expression for resistance ‘R’ is simplified as
Above equation is the usual working equation for the Kelvin Bridge. It indicates that the resistance of connecting lead, r, has no effect on the measurement, provided that the two sets of ratio arms have equal ratio. The former equation is useful, however, as it shows the error that is introduced in case the ratios are not exactly equal. It indicates that it is desirable to keep r as small as possible in order to minimize the errors in case there is a difference between ratios P/Q and p/Q.