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Thursday, 14 February 2019

Measurement of Inductance

Before examining the bridge methods for measuring inductance it is important to know the general form of the AC bridge and know the general balance equation for the bridge.

(a) General form and General Equation for Balance of AC Bridge:

The general form of an AC bridge is shown in Figure. It consists of four bridge arms consisting of impedances Z1, Z2, Z3, and Z4 that are not specified. An alternating voltage is applied to the bridge at points B and D. The detector shown is a head phone set, connected between points A and C.
When the potential difference from A to C is zero the bridge will be balanced. For this condition the voltage drop across BA must be equal to the voltage drop across BC in both magnitude and phase.

Using complex notation we can write :

EAB = EBC or I1Z1 = I2Z2   -----------------1

To satisfy balance condition the currents are:

I1 = E/(Z1 + Z3)   ------------- 2
I2 = E/(Z2 + Z4)   --------------3

Substituting equations (2) and (3), in equation (1), gives
Z1 Z4 = Z2Z3  ---------------- 4

Using admittances in place of impedances :
Y1 Y4= Y2Y3 ----------------- 5

Equation (4) is the general equation for balance of the AC bridge. It can be stated from this equation that the product of impedances of one pair of opposite arms must be equal to the product of the impedances of the other pair of opposite arms, the impedances expressed in the complex notation. Representing in this manner means that both the magnitude and the phase angles of the impedances and phase angle must be taken into consideration. Equation (4) is convenient to be used when dealing with series elements of a bridge circuit. Equation (5) is used in dealing with parallel elements in the bridge.

Representing in polar form the impedances can be written as Z = Z θ; where Z represents the magnitude and θ represents the phase angle of the complex impedance. Equation (4) can be represented in the form:

(Z1 θ1)(Z4 θ4) = (Z2 θ2)(Z3 θ3)      ----------------------------6

Hence the balance condition can be represented as:

Z1Z4 θ1 + θ4 = Z2Z3 θ2 + θ3  --------------------  7

From equation (7), it can be observed that two conditions are to be satisfied simultaneously to balance an AC bridge.

The first condition is :

Z1 Z4 = Z2Z3  ------------------ 8

That is the magnitude of the impedances must be equal as shown in equation (8). The second condition is that the phase angles of the impedances should satisfy the relation:

θ1 + θ4 = θ2 + θ3 ------------------ 9

The phase angles are positive for inductive impedance and negative for capacitive impedance. Working in terms of rectangular coordinates we can express

Z1 = R1+ jX1
Z2 = R2 + jX2
Z3 = R3 + jX3

And Z4 = R4 + jX4

From equation (8), for balance

Z1 Z4 = Z2 Z3

or (R1 + jX1) (R4 + jX4) = (R2 + jX2) (R3 + jX3)
or R1 R4 – X1 X4 + j (X1 R4 + X4 R1) = R2 R3 - X2 X3 + j (X2 R3 + X3 R2) ------------------ 10

Equation (10) is a complex equation. A complex equation will be satisfied only if real and imaginary parts of each side are separately equal in the equation. Therefore for balance:

R1 R4 – X1 X4 = R2 R3 - X2 X3 --------------------- 11
also X1R4 + X4 R1 = X2 R3 + X3 R2 ----------------------12

The above two independent conditions for balance must be satisfied. To satisfy both conditions for balance and for convenience of adjustment the bridge must have two variable elements in its configuration. For better results each of the balance equations must contain one variable element alone. Then the equations are called independent equations. If the variables chosen to balance the bridge do not yield independent equations the bridge has poor convergence of balance and gives an effect called 'sliding balance '. Sliding balance is a condition of interaction between the two controls of a bridge.

In an AC bridge circuit the balance equations are independent of frequency. This is an advantage, as the frequency of the source need not be known. One point is to be noted here pertaining to the balance condition of an AC bridge. If a bridge is balanced for a fundamental frequency it should also be balanced for the harmonic frequencies also. The reason is that inductance or capacitance is frequency conscious and hence the balance made at fundamental frequency is not valid at harmonic frequencies. In order to avoid such difficulties the wave form of the source should be pure and free from harmonics. The detector can be tuned to the fundamental frequency in all such cases. Some bridges use the frequency of the source to an advantage and in such cases the purity of waveform is of prime importance.

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