Find the voltage across the resistor R, and V. for the following values of C: 56.4 nF, 100 nF, 500 nF Save both voltage waveforms and compute the phase difference between the waveforms for each value of C. Explain your results. Did you get the results that you expected? XSC1 Ext Trid R1 67ΚΩ + C1 Vo 56.4nF V1 156Vpk 60Hz 0° L1 125H R2 47ΚΩ

Question

23-11-2022
Physics

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Find the voltage across the resistor R, and V. for the following values of C: 56.4 nF, 100 nF, 500 nF Save both voltage waveforms and compute the phase difference between the waveforms for each value of C. Explain your results. Did you get the results that you expected? XSC1 Ext Trid R1 67ΚΩ + C1 Vo 56.4nF V1 156Vpk 60Hz 0° L1 125H R2 47ΚΩ

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qwbrown qwbrown · Answer 1 · 2022-11-29T08:24:26-05:00

1. phase difference = 45.1318°

2. phase difference = 54.52°

3. phase difference = 62.153°

Case 1: C = 56.4 ₙF

V₁ = [tex]\frac{156}{\sqrt[]{2} }[/tex]

[tex]x_{c}[/tex] = [tex]\frac{1}{2\pi fc}[/tex]

[tex]x_{c}[/tex] = [tex]\frac{1}{2\pi * 60*56.4*10^{-9} }[/tex]

[tex]x_{c}[/tex] = 47.031 KΩ

here, [tex]V_{0} = V_{A}[/tex]

[tex]X_{L}[/tex] = 2πfL

[tex]X_{L}[/tex] = 2π x 60 x 125

[tex]X_{L}[/tex] = 94.247 KΩ

Apply KCV of node "A"

[tex]\frac{V_{A}-0 }{-jX_{c} +jX_{L}+R_{2} }[/tex] + [tex]\frac{V_{A} -V_{1} }{R_{1} }[/tex] = 0

[tex]V_{A}[/tex] ( [tex]\frac{1}{R_{2} +j(X_{L} -X_{C} )} + \frac{1}{R}[/tex] ) = [tex]\frac{V_{1} }{R_{1} }[/tex]

[tex]V_{A}[/tex] ( [tex]\frac{1}{47 + j(94.247 - 47.031)} + \frac{1}{67}[/tex] = [tex]\frac{156/\sqrt[]{2} }{67}[/tex]

[tex]V_{0} = V_{A} = \frac{84.227}{\sqrt{2} }[/tex]

[tex]V_{A}[/tex] = 84.227 sin (2π x 60t + 22.663) Volts

[tex]V_{R} _{1}[/tex] = [tex]V_{1}[/tex] = [tex]V_{A}[/tex] = [tex]\frac{84.706}{\sqrt{2} }[/tex]

V[tex]R_{1}[/tex] = 84.706 sin [2π x 60t + (-22.498°)] Volts

phase difference = 22.6333 - (-22.498)

phase difference = 45.1318°

Case 2: C = 100ₙF

[tex]X_{C}[/tex] = [tex]\frac{1}{2\pi * 60 * 100 * 10^{-9} }[/tex]

[tex]X_{C}[/tex] = 26.525 Ω

[tex]V_{0}[/tex] = [tex]\frac{156}{\sqrt{2} }[/tex] x [tex]\frac{R_{2} + j (X_{L} - X_{C}) }{R_{1} + R_{2} +j(X_{L} - X_{C} )}[/tex]

[tex]V_{0}[/tex] = [tex]\frac{96.98}{\sqrt{2} }[/tex] [tex]\\ 24.52[/tex]° Volts

[tex]V_{R1}[/tex] = [tex]V_{1}[/tex] - [tex]V_{0}[/tex] = [tex]\frac{78.82}{\sqrt{2} }[/tex] -30° Volts

phase difference = 24.52 - (-30)

phase difference = 54.52°

Case 3: C = 500ₙF

[tex]X_{C[/tex] = 5.305 KΩ

[tex]V_{0}[/tex] = [tex]\frac{156}{\sqrt{2} }[/tex] ([tex]\frac{47 + j 94.247 - j 5.305}{67 + j 94.247 - j 5.305 + 47}[/tex])

[tex]V_{0}[/tex] = [tex]\frac{108.533}{\sqrt{2} }[/tex] (24.186°)

[tex]V_{R1} = V_{1} - V_{0}[/tex] = [tex]\frac{72.28}{\sqrt{2} }[/tex] (-37.967°) Volts

phase difference = 24.186 - (-37.967) Volts

phase difference = 62.153°

To learn more about phase difference, please refer:

https://brainly.com/question/14894245

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