Parallel Resonance: Understanding RLC Circuits

Demonstrative Video

PARALLEL RESONANCE

also called Tank circuit

\begin{matrix} Y & = H (ω) = \frac{I}{V} Y & = \frac{1}{R} + j (ω C - \frac{1}{ω L}) ω C - \frac{1}{ω L} = 0 ω_{0} & = \frac{1}{\sqrt{L C}} rad / s \end{matrix}

$\begin{aligned} \mathbf{Y}&=H(\omega) =\frac{\mathbf{I}}{\mathbf{V}}\\ \mathbf{Y} & =\frac{1}{R}+j\left(\omega C-\frac{1}{\omega L}\right) \\ & \omega C-\frac{1}{\omega L}=0 \\ \omega_{0} & =\frac{1}{\sqrt{L C}} \operatorname{rad} / \mathrm{s} \\ \end{aligned}$

$\begin{aligned} \omega_{1} &=\omega_{0} \sqrt{1+\left(\frac{1}{2 Q}\right)^{2}}-\frac{\omega_{0}}{2 Q}, \quad \omega_{2}=\omega_{0} \sqrt{1+\left(\frac{1}{2 Q}\right)^{2}}+\frac{\omega_{0}}{2 Q} \end{aligned}$
Half-power frequencies
$\begin{aligned} \omega_{1} & \simeq \omega_{0}-\frac{B}{2}, \quad \omega_{2} \simeq \omega_{0}+\frac{B}{2} \end{aligned}$
circuits For high

Resonant RLC circuits

Characteristic	Series circuit	Parallel circuit
Resonant freq, $\omega_{0}$	$\dfrac{1}{\sqrt{L C}}$	$\dfrac{1}{\sqrt{L C}}$
Quality factor, Q	$\dfrac{\omega_{0} L}{R} \text { or } \dfrac{1}{\omega_{0} R C}$	$\dfrac{R}{\omega_{0} L} \text { or } \omega_{0} R C$
Bandwidth, B	$\dfrac{\omega_{0}}{Q}$	$\dfrac{\omega_{0}}{Q}$
Half-power freq., $\omega_{1,2}$	$\omega_{0} \sqrt{1+\left(\dfrac{1}{2 Q}\right)^{2}} \pm \dfrac{\omega_{0}}{2 Q}$	$\omega_{0} \sqrt{1+\left(\dfrac{1}{2 Q}\right)^{2}} \pm \dfrac{\omega_{0}}{2 Q}$
For $Q \geq 10, \omega_{1}, \omega_{2}$	$\omega_{0} \pm \dfrac{B}{2}$	$\omega_{0} \pm \dfrac{B}{2}$

Problem

Let $R=8~k \Omega$ , $L=0.2~\mathrm{mH}$ , and $C=8~\mathrm{\mu F}$ . calculate

$\omega_{0,1,2}, ~Q, ~B$
power dissipated at $\omega_{0,1,2}$

\begin{matrix} ω_{0} & = \frac{1}{\sqrt{L C}} = 25 k r a d / s Q & = \frac{R}{ω_{0} L} = 1600 B & = \frac{ω_{0}}{Q} = 15.625 r a d / s ω_{1} & = ω_{0} - \frac{B}{2} = 24, 992 r a d / s ω_{2} & = ω_{0} + \frac{B}{2} = 25, 008 r a d / s \end{matrix}

$\begin{aligned} \omega_0 & = \dfrac{1}{\sqrt{LC}} = 25~\mathrm{krad/s} \\ Q & = \dfrac{R}{\omega_0 L} = 1600 \\ B & = \dfrac{\omega_0}{Q} = 15.625 ~\mathrm{rad/s} \\ \omega_1 & = \omega_0 - \dfrac{B}{2} = 24,992~\mathrm{rad/s} \\ \omega_2 & = \omega_0 + \dfrac{B}{2} = 25,008~\mathrm{rad/s} \\ \end{aligned}$