Simple AC Bridge Circuit Formula and Theory of Operation

An ac bridge circuit is one of the simple application of RLC ac circuits along with phase shifter.

This bridge circuit is used in measuring the inductance L of an inductor or the capacitance C of a capacitor.

Make sure to read what is ac circuit first.

Make sure to read:

  1. What is phasor
  2. Impedance and admittance
  3. Kirchhoff’s laws for ac circuit
  4. Power calculation in ac circuit
  5. Three phase circuit

And its applications:

  1. Phase shifter circuit and formula
  2. AC op-amp
  3. Capacitance multiplier circuit
  4. Wien bridge oscillator

AC Bridge Circuit

It is similar in form to the Wheatstone bridge for measuring an unknown resistance. To measure L and C, however, an ac source is required as well as an ac meter instead of the galvanometer.

The ac meter may be a sensitive ac ammeter or voltmeter.

Consider the general ac bridge circuit displayed in Figure.(1).

ac bridge circuit
Figure 1. A general ac bridge

The bridge is balanced when no current flows through the meter. This means that V1V2. Applying the voltage division principle,

ac bridge


ac bridge


ac bridge

This is the balanced equation for the ac bridge.

Specific ac bridges for measuring L and C are drawn in Figure.(2), where Lx and Cx are the unknown inductance and capacitance to be measured while Ls and Cs are a standard inductance and capacitance (the values of which are known to great precision).

In each case, two resistors, R1 and R2, are varied until the ac meter reads zero. Then the bridge is balanced. From Equation.(3), we get

ac bridge


ac bridge

Notice that the balancing of the ac bridges in Figure.(2) does not depend on the frequency f of the ac source since f does not appear in the relationships in Equations.(4) and (5).

ac bridge
Figure 2. Spesific ac bridges: (a) for measuring L, (b) for measuring C


Read also : laplace transform circuit element

AC Bridge Circuit Example

For a better understanding let us review the example below :

1. The ac bridge circuit of Figure.(1) balances when Z1 is a 1 kΩ resistor, Z2 is a 4.2 kΩ resistor, Z3 is a parallel combination of a 1.5 MΩ resistor and a 12 pF capacitor, and f = 2kHz. Find: (a) the series components that makeup Zx, and (b) the parallel components that makeup Zx.

Solution :

From Equation.(3),

ac bridge

where Zx = Rx + jXx,

ac bridge


ac bridge


ac bridge

(a) Assuming that Zx is made up of series components, we substitute Equations.(1.2) and (1.3) in (1.1) and get

ac bridge

Equating the real and imaginary parts yields Rx = 5.993 MΩ and a capacitive reactance

ac bridge


ac bridge

(b) Zx remains the same as in Equation.(1.4) but Rx and Xx are in parallel. Assuming an RC parallel combination,

ac bridge

By equating the real and imaginary parts, we get

ac bridge

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