b).Yes
Given circuit is wien's bridge , it is designed to
measure frequency of a given system.
i).
When detector detects zero deflection then this is
called balanced bridge
During balanced condition
![Z_1Z_4=Z_2Z_3](//img.homeworklib.com/questions/da028a00-d355-11ea-815b-8961f1f6ff16.png?x-oss-process=image/resize,w_560)
![Z_1=R_1-\frac {j}{\omega C_1}, \ Z_4=R_4, \ Z_3=\frac{R_3}{1+j\omega C_3 R_3}, \ Z_2= R_2](//img.homeworklib.com/questions/da4bed60-d355-11ea-b109-b3cff147e905.png?x-oss-process=image/resize,w_560)
![(R_1-\frac {j}{\omega C_1})R_4=\frac{R_3}{1+j\omega C_3 R_3}\times R_2](//img.homeworklib.com/questions/daa6aa70-d355-11ea-b6ff-577a043d8771.png?x-oss-process=image/resize,w_560)
![(1+j\omega C_1R_1)({1+j\omega C_3 R_3})R_4=j\omega C_1R_2R_3](//img.homeworklib.com/questions/dafdbc00-d355-11ea-9ddb-e3e2d1ade8b7.png?x-oss-process=image/resize,w_560)
Separate real and imaginary parts
Real parts
![(1-{\omega^2}C_1C_3R_3R_1)R_4=0](//img.homeworklib.com/questions/db50f6c0-d355-11ea-83bc-eba4b2ba66a3.png?x-oss-process=image/resize,w_560)
![\omega = \frac{1}{\sqrt{C_1C_3R_1R_3}}](//img.homeworklib.com/questions/db9cefb0-d355-11ea-a2b5-f343a6eebc6d.png?x-oss-process=image/resize,w_560)
![f = \frac{1}{2 \pi \sqrt{C_1C_3R_1R_3}}](//img.homeworklib.com/questions/dbe62140-d355-11ea-bcb0-ad022407379a.png?x-oss-process=image/resize,w_560)
Imaginary parts
![\frac{C_3}{C_1}+\frac{R_1}{R_3}=\frac{R_2}{R_4}](//img.homeworklib.com/questions/dc34fff0-d355-11ea-a941-f5ea61f0ebf3.png?x-oss-process=image/resize,w_560)
ii).
Given
![f=398Hz](//img.homeworklib.com/questions/dc7db0c0-d355-11ea-9908-87123c665162.png?x-oss-process=image/resize,w_560)
![R_1=400,\ R_2=1000, \ R_3=800](//img.homeworklib.com/questions/dcc6f280-d355-11ea-ad52-1b9ec6dc7818.png?x-oss-process=image/resize,w_560)
![C_1=2C_3](//img.homeworklib.com/questions/dd1d2c80-d355-11ea-9408-8b342657ffeb.png?x-oss-process=image/resize,w_560)
We know
![f = \frac{1}{2 \pi \sqrt{C_1C_3R_1R_3}}](//img.homeworklib.com/questions/dbe62140-d355-11ea-bcb0-ad022407379a.png?x-oss-process=image/resize,w_560)
![f = \frac{1}{2 \pi \sqrt{2C_3^2R_1R_3}}](//img.homeworklib.com/questions/ddada340-d355-11ea-8909-75ffe5638209.png?x-oss-process=image/resize,w_560)
![398 = \frac{1}{2 \pi \sqrt{2C_3^2\times 400\times 800}}](//img.homeworklib.com/questions/ddfbc6d0-d355-11ea-943a-39235b835ef8.png?x-oss-process=image/resize,w_560)
![C_3=4.99858\times 10^{-7} \approx 5 \times 10^{-7}F](//img.homeworklib.com/questions/de5904d0-d355-11ea-9c7c-01061509987f.png?x-oss-process=image/resize,w_560)
![C_1=2C_3=10^{-6}](//img.homeworklib.com/questions/deb11880-d355-11ea-9e8e-11d276826fff.png?x-oss-process=image/resize,w_560)
and from the relation
![\frac{C_3}{C_1}+\frac{R_1}{R_3}=\frac{R_2}{R_4}](//img.homeworklib.com/questions/dc34fff0-d355-11ea-a941-f5ea61f0ebf3.png?x-oss-process=image/resize,w_560)
![\frac{C_3}{2C_3}+\frac{400}{800}=\frac{1000}{R_4}](//img.homeworklib.com/questions/df3de450-d355-11ea-acce-4f2247c38017.png?x-oss-process=image/resize,w_560)
![\frac{1}{2}+\frac{1}{2}=\frac{1000}{R_4}](//img.homeworklib.com/questions/df8e85f0-d355-11ea-a2d4-bf4d1d298c2d.png?x-oss-process=image/resize,w_560)
![R_4=1000](//img.homeworklib.com/questions/dfdf7220-d355-11ea-bc0d-a1efb94c6ada.png?x-oss-process=image/resize,w_560)
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