Question

Figure shows a balanced ac bridge excited by a voltage source of fixed frequency. The impedance of the unknown arm Z, is

• A. an R and L in series
• B. an R and C in series
• C. an R and C in parallel
• D. an R and L in parallel

Answer 1

The correct option is C an R and C in parallel

Given, Bridge balance is obtained.
Voltage source is of fixed frequency.

Now at balance,
$R_1{R}_2=(R_3+j\omega L_3)Z$

To obtain bridge balance z should be capacitive in nature.

Case.I. z = R and C in series

$\Rightarrow z=R+\frac{1}{j\omega C}$

$\Rightarrow R_1{R}_2=(R_3+j\omega L_3)(R+\frac{1}{j\omega C})$

$\Rightarrow R_1{R}_2=R_{3}R+\frac{L_3}{R}$ ...(i)

$and0=j[\omega L_{3}R-\frac{R_3}{\omega C}]$

$\Rightarrow \omega L_{3}R=\frac{R_3}{\omega C}$ ...(ii)

To obtain bridge balance we need to select $R_{3}and{L}_3$ as variable as source frequency is fixed. Since $R_{3}and{L}_3$ term is present in both equations i.e., (i) and (ii), so obtaining bridge balance is difficult.

Case.II. z = R and C in parallel

$\Rightarrow R_1{R}_2=(R_3+j\omega L_3)\left(\frac{R}{j\omega CR+1}\right)$

$\Rightarrow R_1{R}_2+j\omega CR{R}_1{R}_2=R_{3}R+j\omega L_{3}R$

$\Rightarrow R_1{R}_2=R_{3}R$

$\Rightarrow R=\frac{R_3}{R_1{R}_2}$ ...(iii)

$and\omega C{R}_1{R}_1{R}_2=\omega L_{3}R$

$\Rightarrow C=\frac{L_3}{R_1{R}_2}$ ...(iv)

To obtain bridge balance we will select $R_{3}and{L}_3$ as variable. So, we will vary $R_3$ only for R and vary $L_3$ only for C.

Out of both cases I and II, bridge balance can be obtained but this bridge is seldon used for case I as bridge balance will be difficult.