## Heaviside-Campbell Equal Ratio Bridge

## Heaviside-Campbell Equal Ratio Bridge

It is a mutual induction bridge and is used for measuring self-inductance over a wide range in terms of mutual inductometer readings. The connections for Heaviside’s bridge employing a standard variable mutual inductance are shown in figure 1. The primary of the mutual inductometer is inserted in the supply circuit and the secondary having self-inductance L2 and resistance R2 is put in arm 2 of the bridge. The unknown inductive impedance having self inductance of L1 and resistance R1 is placed in arm 1. The other two arms have pure resistances of R3 and R4.

Balance is obtained by varying mutual inductance M and resistance R3 and R4.

For balance, **I3R3 = I2R4……………………..(i)**

**I1(R1 + jωL1) = I2(R2 + jωL2) + jωMI………..(ii)**

Since **I = I1 + I2**, hence putting the value of I in equation (ii), we get

**I1[R1 + jω(L1-M)] = I2[R2 + jω(L2 + M)]……….(iii)**

Dividing equation (iii) by (i), we have

**R3[R2 + jω(L2 + M)] = R4[R1 + jω(L1-M)]**

Equation the reals and imaginaries, we have **R2R3 = R1R4…………(iv)**

Also, **R3(L2 + M) = R4(L1-M).**

If **R3 = R4, then L2 + M = (L1 - M)**

Therefore, **L1 - L2 = 2M……………..(v)**

This bridge, as modified by Campbell, is shown in figure 2. Here **R3 = R4.** A balancing coil or a test coil of self-inductance equal to the self-inductance L2 of the secondary of the inductometer and of resistance slightly greater then R2 is connected in series with the unknown inductive impedance (R1 and L1) in arm 1. A non-inductive resistance box along with a constant-inductance rheostat is also introduced in arm 2 as shown.

Balance is obtained by varying M and r. Two reading are taken; one when Z1 is in circuit and second when Z1 is removed or short-circuited across its terminals.

With unknown impedance Z1 still in circuit, suppose for balance the values of mutual inductance and r are M1 and r1. With Z1 short-circuited, let these values be M2 and r2. Then

**L1 = 2(M1 – M2) and R1 = r1 – r2.**

By this method, the self – inductance and resistance of the leads are eliminated.