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AC bridge balance and Schering bridge

Short answer

General AC bridge balance requires Z1Z3=Z2Z4 with equal magnitudes and angles. The Schering bridge measures unknown capacitance Cx and series loss rx using standard capacitor C2. At balance, Cx = C2R4/R3, rx = R3C4/C2, D=ωC4R4. Used for insulator testing.

Obtain the balance equation for an ac bridge and explain with diagram how Schering bridge can be used to measure unknown capacitance.

Balance equation of an AC bridge

Four impedance arms Z1, Z2, Z3, Z4 form the bridge; the AC source is across A–C and the detector across B–D.

         A
        / \
      Z1   Z4
      /     \
     B--[D]--D
      \     /
      Z2   Z3
        \ /
         C

At balance, the potentials of B and D are equal, so the detector current is zero:

  • Voltage drop across Z1 = drop across Z4, and across Z2 = drop across Z3.
  • I1 Z1 = I4 Z4 and I1 Z2 = I4 Z3 (same current in each series pair).

Dividing: Z1 / Z2 = Z4 / Z3, or Z1 Z3 = Z2 Z4.

In polar form: magnitudes |Z1||Z3| = |Z2||Z4| and angles θ1 + θ3 = θ2 + θ4. Both conditions must be met, so two adjustments are needed.

Schering bridge for unknown capacitance

          A
        /   \
 Cx,rx(Z1)   R3 (Z4)
      /         \
     B---[D]----D
      \         /
   C2 (Z2)   R4 || C4 (Z3)
        \     /
          C
  • Arm AB (Z1): unknown capacitor Cx with its series loss resistance rx.
  • Arm BC (Z2): standard loss-free capacitor C2.
  • Arm CD (Z3): resistor R4 in parallel with variable capacitor C4.
  • Arm DA (Z4): variable resistor R3.

Using Z1 Z3 = Z2 Z4:

(rx + 1/jωCx) · R4/(1 + jωC4R4) = R3/(jωC2)

(rx + 1/jωCx) R4 = R3/(jωC2) + R3R4C4/C2

Equating real parts: rx = R3 C4 / C2

Equating imaginary parts: Cx = C2 R4 / R3

Dissipation factor: D = ωCx·rx = ωC4R4.

Procedure

Adjust C4 and R3 alternately until the detector reads null; read R3, R4, C4, C2 and compute Cx. The scale of C4 can be calibrated directly in D. The bridge is widely used for capacitor testing, insulator and cable testing, where very small losses must be found.

✓ Answer checked by Claude Opus 5.5 (high effort)

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