Question

We have a 10 microfarad capacitor and a 10 millihenry inductor and a 1k ohm resistance connected in series with an AC power supply.

a) What is the resonant frequency of the circuit?

b) What is the total impedance of the circuit at 100 Hz?

c) What is the phase shift between the applied voltage and the current at 100 Hz?

d) What is the quality factor, Q, of the resonance?

e) If I wanted a quality factor of 10^6, what would I have to change the resistance to?

Please give me a step by step solution. Give formulas and detials please.

Answer 1

given C = 10 micro farad = 10* =

L = 10 milli henry =

R = 1 Kilo Ohm = 1000 Ohm

a) Resonant Requency

Resonant frequency $\omega =\frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{10^{-2}*10^{-6}*10}}$

$\frac{1}{\sqrt{10^{-8}*10}}$        = $\frac{1}{10^{-4}*3.16}$         = 3164.6 Hertz

Resonant requency is 3164.6 Hertz

b) Impedance at frequenct 100 Hertz     $\omega =100$

Impedance Z = $\sqrt{R^{2}+\frac{1}{C^{2}R^{2}}$      = $\sqrt{10^{6}+\frac{1}{10^{-10}*10^{4}}}$

$\sqrt{10^{6}+10^{6}}$    = $\sqrt{2*10^{6}}$      =

Therfore impedance is 141.42 Ohm

C) Phase Shift   $\phi = tan ^{-1}\left ( \frac{1}{C\omega R} \right )$     here $\omega =100 hertz$

$\phi =tan^{-1}\left ( \frac{1}{10^{-5}*100*1000} \right )$

= $tan^{-1}\left ( \frac{1}{10^{-5}*10^{5}} \right )$       = $tan^{-1}\left ( 1 \right )$

therfore $\phi =45 degrees$

therefore Phase shift is 45 degrees

d) quality factor   Q = $\frac{\omega *L}{R}$      = $\frac{3164.6*10^{-2}}{1000}$

e) if Q is then what is R

$10^{6} = \frac{3164.6*10^{-2}}{R}$

Answer 2

given C= 10 micro farad, L=10 milli henry, R= 1 K ohm

now the a) resonant frequency f= $\frac{1}{2\pi \sqrt{LC}}$

                                             = 50.35 Hz

b) total impedance Z= $\sqrt{R^{2}+(Xl-Xc)^{2}}$ =1010.16

                              Xl=2*pi*f*L= 6.28 , Xc= 1/2*pi*f*C= 159.23

c) phase shifr $\varphi = Cos^{-l}(R/Z))$ = 1000/1010.16=8.133

d) quality factor at resonance frequency Q= 2*pi*f*l/R= 3161

e)quality factor to be 10^6 the resistance should be 0.03161 ohms