This is an AC bridge. This Bridge uses mutual inductance to measure self-inductance over a wide range. Heaviside Campbell Equal Ratio Bridge employees a standard variable mutual inductance for the purpose. We connect the primary of a mutual inductometer in series with the supply. Now let us consider the secondary of the mutual inductometer has a self-inductance L_{2} and the series resistance R_{2}. In the figure below, the secondary portion of the inductometer forms one arm of the bridge circuit. Say it is AB. We connect the coil of unknown inductance and resistance across the arm BC of the bridge. In the other two arms, we will use only resistance. This resistances are R_{4} and R_{3} respectively.

Now, we connect a detector between node A and C. We obtain the balance condition of the Heaviside Campbell Equal Ratio Bridge by adjusting the mutual inductance of the inductometer.

## Theory of Heaviside Campbell Equal Ratio Bridge

Let us consider at balanced condition there will be a supply current i ampere. The bridge circuit divides the supply current into two components i_{1} and i_{2}. The current i_{1} flows through arms BC and CD of the bridge. The current i_{2} flows through the arms BA and AD. Hence, the voltage drop across arm AD is i_{2}R_{4} and the same for arm CD is i_{1}R_{3}. At balanced condition the potential of node A and C equal. For satisfying this condition we can write.

For the same reason, we can write also

Now, i = i_{1} + i_{2}, so we can write

By dividing (ii) by (i) we get,

Now, by considering real and imaginary parts of the equation (iii) separately, we get

If we make,

Then, we can write

### Modified Version of Heaviside Campbell Equal Ratio Bridge

Campbell further modified this bridge. He introduces one balancing coil in series with the coil under measurement in the arm BC. Also, He inserted one resistance box in series with the secondary of inductometer in arm AB. Then he obtained the balance of the bridge by varying the mutual inductance M and resistance r of the resistance box.

Here we have to find out the inductance and resistance of the coil under measurement i.e. L_{1} and R_{1}.

In this modified version of the bridge, we have to take the reading twice. That means we have to obtain the balanced condition of the bridge twice.

Once we do this by inserting the coil under measurement in the circuit.

Then we do this by replacing the coil under measurement by a short circuit.

Suppose M_{1} and r_{1} are the value of mutual inductance and resistance of the resistance box for balancing the bridge circuit with the coil under measurement. On the other hand, M_{2} and r_{2} are the mutual inductance and resistance of the resistance box for balancing the bridge circuit with short-circuiting the coil under measurement. Then we can write,

By modifying the Heaviside Campbell Equal Ratio Bridge, Campbell was able to eliminate the lead resistance and inductance the coil under measurement.

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