Wheatstone bridge ac circuit explanation and equation solution

     Resistance can be measured by direct-current Wheatstone bridge having resistance in each arm. Inductance and capacitance can also be measured by a similar four-arm bridge, but instead or using a source of direct current alternating current is employed and galvanometer is replaced by a vibrating galvanometer (for commercial frequencies or by telephone detector if frequencies are higher 500 to 2000 Hz.). The general form of an ac bridge arms Z₁, Z₂, Z₃ and Z₄ are indicated as unspecified impedances and the detector is represented by headphones. During the balance condition in this ac bridge is reached when the detector response is zero or indicated a null. The condition for bridge balance requires the potential difference from A to C to zero.

At balance con condition,

E_BA = E_BC or I₁Z₁ = I₂Z₂ ———————–(i)

I₁ = E/(Z₁ + Z₄), I₂ = E/(Z₂ + Z₃)—————–(ii)

From equation (i) and (ii)

(Z₁E)/(Z₁+Z₄) = (Z₂E)/(Z₂+Z₃)

Or, Z₁Z₃ = Z₂Z₄——————————-(iii)

      Equation (iii) states that the product of impedances of one pair of opposite arms must equal the product of impedance of the other pair of opposite arms.
If the impedance are expressed in complex notation, i.e

.                                                              Z = Z<θ

Or, (Z₁< θ₁)(Z₃< θ₁) = (Z₂< θ₂)(Z₄< θ₄)

Z₁Z₃< θ₁+ θ₃ = Z₂Z₄< θ₂ + θ₄ ——————————-(iv)

From equation (iv), we can also write,

Z1Z3 = Z2Z4

The products of the magnitudes of the opposite arms must be equal.

<θ₁ + <θ₃ = <θ₂ + <θ₄

The sum of the phase angles of the opposite arms must be equal.