A C Bridges Practice Questions

The condition for balanced bridge

Maxwell's bridge is used for measurement of inductance. Wein's bridge is used for measurement of frequency Kelvin's bridge is used for measurement of low value of resistance. Schering bridge is used for measurement of capacitance, dilectric loss and permittivity etc.

Redrawing the given bridge circuit, we have:

At balance, current through galavanometer: $I_g=0$ and $|Z_1||Z_4|=|Z_2||Z_3|$ $\therefore \left ( \frac{1}{\omega C_1} \right ) \times (105K)=(35K)\left ( \frac{10^6}{0.1\omega } \right )$ $C_1=\frac{105 \times 0.1}{35 }\mu F$ $\;\;=0.3\mu F$

Given bridge is shown in figure From the given bridge circuit, we see that product of opposite arm impedance are equal

Hence, the bridge is balanced, i.e. no current flows through the voltage source V. Now,

$\therefore$ Net current supplied by the source is $I=\frac{100\sqrt{2}}{(0.5+Z_{eq})}A$ $Z_{eq}=0$ (i.e. net impedance of whole bridge=0) $\therefore \;\; I=\frac{100\sqrt{2}}{0.5}=141.42A$ $\therefore \;\;\frac{I_2}{2}=70.71A$ Current flow through each parallel path of the bridge circuit, therefore $V=V_a-V_b$ $\;\;=\left ( \frac{I}{2} \times -j \right )-\left ( \frac{I}{2} \times j \right )$ $\;\;=-jI=-j141.42$ volt (rms value) $\therefore$ Reading of the voltmeter in volt (rms) $=V=V_{ab}=141.42$

Heaciside Campbell bridge method is commonly used for finding mutual inductance.

The bridge is Maxwell bridge. Element is an inductor. Inductance $=L_x$ effective resistance of the inductor $=R_x$ $Q=\frac{\omega L_x}{R_x}=\omega C_1R_1 \;\;...(i)$ The bridge is limited to measurement of low Q inductor ( $1 \lt Q \lt 10$ ). It is clear from equation (i) that the measurement of high Q coils demands a large value of resistance $R_1$ , perhaps $10^5$ or $10^6 \Omega$ . The resistance boxes of such high value are very expensive. Thus for value of $Q \gt 10$ , the bridge is unsuitable. The bridge is also unsuited for coils with a very low value of Q(i.e. $Q \lt 1$ ).

$Z_1=R+j\omega L$ $Z_2=R_2$ $Z_3=R_3$ $Z_4=R_4||\left ( \frac{-j}{\omega C_4} \right )$ $\;\;=R_4||\left ( \frac{1}{\omega C_4} \right )$ $\;\;=\frac{R_4}{1+j\omega C_4R_4}$ At balance, $Z_1Z_4=Z_2Z_3$ $(R+ j\omega L)\left ( \frac{R_4}{1+j\omega C_4R_4} \right )=R_2R_3$ $R_4(R+ j\omega L)=R_2R_3(1+j\omega C_4R_4)$ $RR_4+j\omega L R_4=R_2R_3+j\omega R_2R_3R_4C_4$ Equating real and imaginary terms $RR_4=R_2R_3$ $R=\frac{R_2R_3}{R_4}$ $\omega LR_4=\omega R_2R_3R_4C_4$ $L=R_2R_3C_4$

$Z_{AB}=500\Omega$ $Z_{CD}=Z$ $Z_{BC}=R_{BC}+\frac{1}{j\omega C}$ $\;\;=300-\frac{j}{2 \pi \times 2 \times 10^3 \times 0.398 \times 10^{-6}}$ $Z_{BC}=300-j200\Omega$ $Z_{AD}=R_{AD}+j\omega L$ $Z_{AD}=300+j2\pi\times 2 \times 10^3 \times 15.91 \times 10^{-3}$ $\;\;=300+j200\Omega$ At balance, $Z_{AB} \times Z_{CD}=Z_{BC} \times Z_{AD}$ $Z_{CD}=Z$ $\;\;=\frac{Z_{BC} \times Z_{AD}}{Z_{AB}}$ $\Rightarrow \;Z=\frac{(300-j200)\times (300+j200)}{500}$ $Z=(260+j0)\Omega$

$x_1=\omega L_1$ and $x_4=\frac{1}{\omega C_4}$ $z_1=R_1+jx_1=R_1+j\omega L_1$ $z_2=R_2$ $z_3=R_3$ $z_4=R_4-jx_4=R_4-\frac{j}{\omega C_4}$ Under balanced condition $z_1z_4=z_2z_3$ $(R_1+j\omega L_1)(R_4-jx_4)=R_2R_3$ $\left ( R_1R_4+\frac{L_1}{C_4} \right )+j\left ( \omega L_1R_4-\frac{R_1}{\omega C_4} \right )=R_2R_3$ Equating real and imaginary terms, we obtain $R_1R_4+\frac{L_1}{C_4} =R_2R_3$ $\omega L_1R_4-\frac{R_1}{\omega C_4}=0$ Solving above equations, we get $L_1=\frac{R_2R_3C_4}{1+\omega ^2C_4^2R_4^2}$ and $R_2=\frac{\omega ^2R_2R_3R_4C_4^2}{1+\omega ^2C_4^2R_4^2}$ Q-factor of the coil $Q=\frac{\omega L_1}{R_1}=\frac{1}{\omega C_4R_4}$ Therefore, $L_1=\frac{R_2R_3C_4}{1+\left ( \frac{1}{Q} \right )^2}\;\;...(i)$ $R_1=\frac{\omega ^2R_2R_3R_4C_4^2}{1+\left ( \frac{1}{Q} \right )^2}\;\;...(ii)$ For a value of a greater than 10, the term $(1/Q)^2$ will be smaller than 1/1000 and can be neglected. Therefore equation (i) and (ii) reduces to $L_1=R_2R_3C_4 \;\;...(iii)$ $R_1=\omega ^2R_2R_3R_4C_4^2\;\;...(iv)$ $R_4$ appears only in equation (iv) and $R_2$ appears in both equation (iii) and (iv). So first $R_2$ is adjusted and then $R_4$ is adjusted.

Wheat stone bridge is used for measurement of medium resistance. Kelvin double bridge is used for meserement of low resistance. Schering bridge is used for meserement of low value of capacitances. Wein's bridge is used for meserement of the frequency Hay's bridge is used for meserement of inductance of a coil with a large time-constant. Carey-foster bridge is used for comparision of resistances which are nearly equal.