The ac bridge shown in Fig. 9.84 is known as a Maxwell bridge and is used for accurate measurement of inductance and resistance of a coil in terms of a standard capacitance Cs Show that when the bridge is balanced, Lx=R2R3Cs and Rx= R2/R1 R3 Find Lx and Rx for R1=40 kΩ, R2=1.6 kΩ, R3=4 kΩ, and

Question

The ac bridge shown in Fig. 9.84 is known as a Maxwell bridge and is used for accurate measurement of inductance and resistance of a coil in terms of a standard capacitance C_{s} Show that when the bridge is balanced,

\begin{aligned}&L_{x}=R_{2} R_{3} C_{s} \quad ext { and } \quad R_{x}=\frac{R_{2}}{R_{1}} R_{3}\\& ext { Find } L_{x} ext { and } R_{x} ext { for } R_{1}=40 \mathrm{k} \Omega, R_{2}=1.6 \mathrm{k} \Omega, R_{3}=4 \mathrm{k} \Omega, ext { and } C_{s}=0.45 \mu \mathrm{F}\end{aligned}

Accepted Answer

Let [latex]\quad {Z}_{1}=\mathrm{R}_{1} \| \frac{1}{\mathrm{j} \omega \mathrm{C}_{\mathrm{s}}}, \quad \mathrm{Z}_{2}=\mathrm{R}_{2}, \quad \mathrm{Z}_{3}=\mathrm{R}_{3},[/latex] and [latex]\mathrm{Z}_{\mathrm{x}}=\mathrm{R}_{\mathrm{x}}+\mathrm{j} \omega \mathrm{L}_{\mathrm{x}}\\[/latex]

[latex]{Z}_{1}=\frac{\frac{\mathrm{R}_{1}}{\mathrm{j} \omega \mathrm{C}_{\mathrm{s}}}}{\mathrm{R}_{1}+\frac{1}{\mathrm{j} \omega \mathrm{C}_{\mathrm{s}}}}=\frac{\mathrm{R}_{1}}{\mathrm{j} \omega \mathrm{R}_{1} \mathrm{C}_{\mathrm{s}}+1}\\[/latex]

since [latex]{Z}_{\mathrm{x}}=\frac{{Z}_{3}}{{Z}_{1}} {Z}_{2}\\[/latex]

[latex]\mathrm{R}_{\mathrm{x}}+\mathrm{j} \omega \mathrm{L}_{\mathrm{x}}=\mathrm{R}_{2} \mathrm{R}_{3} \frac{\mathrm{j} \omega \mathrm{R}_{1} \mathrm{C}_{\mathrm{s}}+1}{\mathrm{R}_{1}}=\frac{\mathrm{R}_{2} \mathrm{R}_{3}}{\mathrm{R}_{1}}\left(1+\mathrm{j} \omega \mathrm{R}_{1} \mathrm{C}_{\mathrm{s}}\right)\\[/latex]

Equating the real and imaginary components,

[latex]{R}_{\mathrm{x}}=\frac{{R}_{2} {R}_{3}}{{R}_{1}}\\[/latex]

[latex]\omega \mathrm{L}_{\mathrm{x}}=\frac{\mathrm{R}_{2} \mathrm{R}_{3}}{\mathrm{R}_{1}}\left(\omega \mathrm{R}_{1} \mathrm{C}_{\mathrm{s}}\right)\\[/latex] implies that

[latex]{{L}_{{x}}}={R}_{2} {R}_{3} {C}_{{s}}\\[/latex]

Given that [latex]\mathrm{R}_{1}=40 \mathrm{k} \Omega, \mathrm{R}_{2}=1.6 \mathrm{k} \Omega, \mathrm{R}_{3}=4 \mathrm{k} \Omega,[/latex] and [latex]\mathrm{C}_{\mathrm{s}}=0.45 \mu \mathrm{F}\\[/latex]

[latex]\begin{array}{l}\mathrm{R}_{\mathrm{x}}=\frac{\mathrm{R}_{2} \mathrm{R}_{3}}{\mathrm{R}_{1}}=\frac{(1.6)(4)}{40} \mathrm{k} \Omega=0.16 \mathrm{k} \Omega={160} \Omega \\\mathrm{L}_{\mathrm{x}}=\mathrm{R}_{2} \mathrm{R}_{3} \mathrm{C}_{\mathrm{s}}=(1.6)(4)(0.45)={2.88} \mathrm{H}\\end{array}[/latex]

Question 9.84: The ac bridge shown in Fig. 9.84 is known as a Maxwell bridg...

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