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# Capacitance Bridge Working Principle

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AC bridges are used for measurement of inductances and capacitances. All AC bridge circuits are based on Wheatstone bridge.

Figure 1(a) shows the circuit of a simple capacitance bridge. Cs is a precise standard capacitor, Cx is an unknown capacitance, and Q and P are standard resistors, one or both of which is adjustable. An AC supply is used, and the null detector (D) must be an AC instrument. A low-current rectifier ammeter is frequently employed as a null detector. Q is adjusted until the null detector indicates zero, and when this is obtained, the bridge is said to be balanced. Fig.1: (a) Simple Capacitance Bridge

## Working Principle of Capacitance Bridge

When the detector indicates null, the voltage drop across Cs must equal that across Cx, and similarly, the voltage across Q must be equal to the voltage across P. therefore,

\begin{align} & {{V}_{cs}}={{V}_{cx}} \\ & or \\ & \begin{matrix} {{i}_{1}}{{X}_{cs}}={{i}_{2}}{{X}_{cs}} & \cdots & (1) \\\end{matrix} \\\end{align}

And

\begin{align} & {{V}_{Q}}={{V}_{P}} \\ & or \\ & \begin{matrix} {{i}_{1}}Q={{i}_{2}}P & \cdots & (2) \\\end{matrix} \\\end{align}

Dividing equation (1) by equation (2):

$\begin{matrix} \frac{{{X}_{cs}}}{Q}=\frac{X{}_{cx}}{P} & \cdots & (3) \\\end{matrix}$

Referring to equation (3) and figure 1(b), the general balance equation for all AC bridges can be written as: Fig.1(b): General circuit diagram for an AC bridge

$\begin{matrix} \frac{{{Z}_{1}}}{{{Z}_{2}}}=\frac{{{Z}_{3}}}{{{Z}_{4}}} & \cdots & (4) \\\end{matrix}$

Substituting 1/ωCs for Xcs , and 1/ωCx for Xcx in equation (3),

\begin{align} & \frac{1}{\omega {{C}_{s}}Q}=\frac{1}{\omega {{C}_{x}}P} \\ & or \\ & {{C}_{x}}=\frac{Q\omega {{C}_{s}}}{P\omega } \\\end{align}

Giving

Formula for Unknown Capacitance $\begin{matrix} {{C}_{x}}=\frac{Q{{C}_{s}}}{P} & \cdots & (5) \\\end{matrix}$

It is seen that the unknown capacitance Cx can now be calculated from the known values of Q, Cs, and P.

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