The ac bridge circuit of Fig. 9.85 is called a Wien bridge. It is used for measuring the frequency of a source. Show that when the bridge is balanced, f= 1/2π √R2 R4 C2 C4

Question

The ac bridge circuit of Fig. 9.85 is called a Wien bridge. It is used for measuring the frequency of a source. Show that when the bridge is balanced,

f=\frac { 1 }{ 2\pi \sqrt { { R }_{ 2 }{ R }_{ 4 }{ C }_{ 2 }{ C }_{ 4 } } }

Accepted Answer

Let [latex]\quad {Z}_{1}=\mathrm{R}_{1}, \quad {Z}_{2}=\mathrm{R}_{2}+\frac{1}{\mathrm{j} \omega \mathrm{C}_{2}}, \quad {Z}_{3}=\mathrm{R}_{3},[/latex] and [latex]{Z}_{4}=\mathrm{R}_{4} \| \frac{1}{\mathrm{j} \omega \mathrm{C}_{4}}\\[/latex]

[latex]{Z}_{4}=\frac{{R}_{4}}{j \omega {R}_{4} C_{4}+1}=\frac{-j {R}_{4}}{\omega R_{4} C_{4}-{j}}\\[/latex]

since [latex]{Z}_{4}=\frac{{Z}_{3}}{{Z}_{1}} {Z}_{2} \longrightarrow{Z}_{1} {Z}_{4}={Z}_{2} {Z}_{3}\\[/latex]

[latex]\begin{array}{l}\frac{-\mathrm{jR}_{4} \mathrm{R}_{1}}{\omega \mathrm{R}_{4} \mathrm{C}_{4}-\mathrm{j}}=\mathrm{R}_{3}\left(\mathrm{R}_{2}-\frac{\mathrm{j}}{\omega \mathrm{C}_{2}}\right) \\\frac{-\mathrm{jR}_{4} \mathrm{R}_{1}\left(\omega \mathrm{R}_{4} \mathrm{C}_{4}+\mathrm{j}\right)}{\omega^{2} \mathrm{R}_{4}^{2} \mathrm{C}_{4}^{2}+1}=\mathrm{R}_{3} \mathrm{R}_{2}-\frac{\mathrm{jR}_{3}}{\omega \mathrm{C}_{2}}\\end{array}[/latex]

Equating the real and imaginary components,

[latex]\frac{\mathrm{R}_{1} \mathrm{R}_{4}}{\omega^{2} \mathrm{R}_{4}^{2} \mathrm{C}_{4}^{2}+1}=\mathrm{R}_{2} \mathrm{R}_{3} (1)\\[/latex]

[latex]\frac{\omega \mathrm{R}_{1} \mathrm{R}_{4}^{2} \mathrm{C}_{4}}{\omega^{2} \mathrm{R}_{4}^{2} \mathrm{C}_{4}^{2}+1}=\frac{\mathrm{R}_{3}}{\omega \mathrm{C}_{2}} (2)\\\[/latex]

Dividing (1) by (2),

[latex]\begin{array}{l}\frac{1}{\omega \mathrm{R}_{4} \mathrm{C}_{4}}=\omega \mathrm{R}_{2} \mathrm{C}_{2} \\\omega^{2}=\frac{1}{\mathrm{R}_{2} \mathrm{C}_{2} \mathrm{R}_{4} \mathrm{C}_{4}} \\\omega=2 \pi \mathrm{f}=\frac{1}{\sqrt{\mathrm{R}_{2} \mathrm{C}_{2} \mathrm{R}_{4} \mathrm{C}_{4}}} \\{f}=\frac{1}{2 \pi \sqrt{{R}_{2} {R}_{4} \mathrm{C}_{2} \mathrm{C}_{4}}}\end{array}[/latex]

Question 9.85: The ac bridge circuit of Fig. 9.85 is called a Wien bridge. ...

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