Question 5.7: The parameters of an RLC parallel circuit excited by a curre......

Given that, R = 40 Ω, L = 2 mH and C = 3 µF

Angular resonant frequency, $\omega_{ r }=\frac{1}{\sqrt{ LC }}$

$=\frac{1}{\sqrt{2 \times 10^{-3} \times 3 \times 10^{-6}}}=12909.9445 \approx 12910~ rad / s$

Resonant frequency ,$f_{ r }=\frac{\omega_{ r }}{2 \pi}=\frac{12910}{2 \pi}=2054.7 ~Hz =\frac{2054.7}{1000} kHz =2.0547~ kHz$

Quality factor at resonance , $Q _{ r }=\frac{ R }{\omega_{ r } L }=\frac{40}{12910 \times 2 \times 10^{-3}}=1.5492$

Bandwidth,  $\beta=\frac{1}{ RC }=\frac{1}{40 \times 3 \times 10^{-6}}=8333.3333 ~rad / s \approx 8333~ rad / s$

Bandwidth in Hz $=\frac{\beta}{2 \pi}=\frac{8333.3333}{2 \pi}=1326.3 ~Hz =\frac{1326.3}{1000}~ kHz =1.3263~ kHz$

Higher cut – off frequecy, $f _{ h }= f _{ r }\left[\frac{1}{2 Q _{ r }}+\sqrt{1+\frac{1}{4 Q _{ r }^2}}\right]$

$\begin{aligned} & =2.0547 \times\left[\frac{1}{2 \times 1.5492}+\sqrt{1+\frac{1}{4 \times 1.5492^2}}\right] \\ & =2.8222~ kHz \end{aligned}$