# Schering Bridge with Formula

The capacitance can be measured using Schering bridge. It is a widely used bridge for the measurement of capacitance. This bridge though is used for measuring capacitance is very useful in measuring insulation properties i.e. for phase angles very near to 90°. The schematic representation to this bridge is shown in Figure. One of the ratio arms contains a parallel combination of capacitor and resistor C1 and R1. The other ratio arm consists of a variable resistor R2. The standard arm consists of a capacitance of fixed value C3. Mica capacitor or well designed air dielectric type capacitor are used. The reason for the choice is the phase angle of the two having a value nearing 90°.

To obtain balance the sum of the phase angles of arms 1 and 4 must be equal to the sum of the phase angles of arms 2 and 3. As we have a standard capacitor in arm 3 the sum of the phase angles of arm 2 and 3 will be 0° + 90°. In order to obtain the 90° phase angle needed for balance the sum of angles of arm 1 and arm 4 must be equal to 90°. As the unknown capacitor will have a phase angle smaller than 90° arm must be provided with a small capacitive angle. This is done by providing a capacitor C1 parallel with R1. Schering bridge formula is derived as follows
The balance equation can be derived as follows :

Substituting the impedance and admittance values in the general formula, Zx = Z2 Z3 Y1
or

Rx –j/ωCx = R2(-j/ωC3)(1/R1 + jωC1)

Expanding Rx –j/ωCx = R2C1/C3 – jR2/ωC3R1 --------------------------- 1

Equating real terms:

Rx = R2 C1/C3 ----------------------------------- 2

Equating imaginary terms,

Cx = C3 R1/R2 ----------------------------- 3

The two variables chosen for the balance adjustment are C1 and resistor R2. The dial of the capacitor C2 can be calibrated in terms of dissipation factor D, by keeping the value of R1 fixed. This can be explained as follows :

The power factor of a series RC combination is defined as the cosine of the phase angle of the circuit. Therefore the power factor of the unknown equals PF = Rx/Zx. For phase angles very close to 90°, the reactance is very nearly equal to the impedance and with a close approximation the power factor can be taken

PF = Rx/ Xx = ωCxRx ---------------------------- 4

The dissipation factor of a series RC circuit is explained as cotangent of the phase angle and hence by explanation the dissipation factor.

D = Rx/ Xx = ωCxRx     ---------------------------------------- 5

The quality factor of a coil is defined by Q = XL/RL, the dissipation factor, D is the reciprocal of the quality factor, Q and therefore D = I/Q. If we substitute the value of Cx in equation (3), and of Rx in equation (2), into the expression for the dissipation factor we obtain.

D = ωR1C1 -------------------------------- 6

Therefore keeping the value of R1 in the bridge fixed, the dial of capacitor C1 can be calibrated directly in dissipation factor D. One point to be noted here is that the term co appears in the equation for the dissipation factor D. Hence the calibration of the capacitor C1 is valid for one frequency at which it is calibrated. When a different frequency is used a correction can be made by multiplying the C1 dial reading by the ratio of the two frequencies. This is the advantage of using Schering bridge.