A series R-L-C circuit is designed to resonate at $\omega_s = 10^5 rad/s$ , have a bandwidth of $0.15\omega_s$ , and draw 16 W from a 120 V source at resonance.
a. Determine the value of R.
b. Find the bandwidth in hertz.
c. Find the nameplate values of L and C.
d. Determine the $Q_s$ of the circuit.
e. Determine the fractional bandwidth.

Question

A series R-L-C circuit is designed to resonate at [latex]\omega_s = 10^5 rad/s[/latex], have a bandwidth of [latex]0.15\omega_s[/latex], and draw 16 W from a 120 V source at resonance.
a. Determine the value of R.
b. Find the bandwidth in hertz.
c. Find the nameplate values of L and C.
d. Determine the [latex]Q_s[/latex] of the circuit.
e. Determine the fractional bandwidth.

Accepted Answer

a. [latex]P=\frac{E^2}{R}\hspace{30 pt} and \hspace{30 pt} R=\frac{E^2}{P}=\frac{(120V)^2}{16W}=900\Omega [/latex]

b. [latex]f_s=\frac{\omega_s}{2\pi}=\frac{10^5rad/s}{2\pi}= 15,915.49 Hz\[0.5cm] BW=0.15f_s=0.15(15,915.49 Hz) = 2387.32 Hz [/latex]

c. Eq. (21.19):

[latex]BW=f_2-f_1=\frac{R}{2\pi L} [/latex] (21.19)

[latex]BW=\frac{R}{2\pi L} \hspace{30 pt} and\hspace{30 pt} L=\frac{R}{2\pi BW}=\frac{900 Ω}{2\pi(2387.32 Hz)} = 60 mH \[0.5cm] f_s=\frac{1}{2\pi\sqrt{LC} }\hspace{30 pt} and\hspace{30 pt} C=\frac{1}{4\pi^2f^{2}_{s}L}\[0.5cm] =\frac{1}{4\pi^2(15,915.49 Hz)^2(60\times10^{-3})}= 1.67 nF[/latex]

d. [latex]Q_s=\frac{X_L}{R}=\frac{2\pi f_sL}{R}=\frac{2\pi(15,915.49 Hz)(60 mH)}{900 Ω}=6.67 [/latex]

e. [latex]\frac{f_2-f_1}{f_s}=\frac{BW}{f_s}=\frac{1}{Q_s}=\frac{1}{6.67}=0.15 [/latex]

Question 21.5: A series R-L-C circuit is designed to resonate at ωs = 10^5 ...

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