Parallel Resonant Circuit

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A parallel resonant RLC circuit is shown in Figure 1.

Parallel RLC Circuit

Figure 1. Parallel RLC circuit excited with a current source

The input admittance of a parallel R-L-C circuit is given by,

(1)   \begin{eqnarray*} Y_{in}(j\omega) &=& {1\over R} + {1 \over j\omega L} + j\omega C \\ Z_{in}(j\omega) &=& {R \over 1+R \left( \frac{1}{j\omega L}+j\omega C \right)} \end{eqnarray*}

By definition, at resonant frequnecy (\omega_o) inductive and capacitive susceptances are equal in magnitude and opposite in phase. Therefore they cancel each other.

(2)   \begin{equation*} \omega_o C = \frac{1}{\omega_o L} \Rightarrow \omega_o = \frac{1}{\sqrt{LC}} \end{equation*}

Quality factor(Q) of a parallel R-L-C circuit at resonance is

(3)   \begin{eqnarray*} Q_{p} &=& 2\pi \quad {\mbox{Energy stored} \over \mbox{Energy dissipated per cycle}}\\ &=& 2\pi \frac{{1 \over 2} C V_{pk}^2 }{{1 \over 2} {V_{pk}^2 \over R}/f} = {\omega R C}\\ &=& 2\pi \frac{{1 \over 2} L \left(\frac{V_{pk}}{\omega L}\right)^{\tiny 2}}{\frac{1}{2} \left(\frac{V_{pk}^{\tiny 2}}{R}\right)/f} = \frac{R}{\omega L} \end{eqnarray*}

So the Q-factor at resonance is

(4)   \begin{equation*} \boxed{Q_{po} = \omega_o C R =\frac{R}{\omega_o L}= R{\sqrt{C\over L}}} \end{equation*}

From Eq-(1), Eq-(2) and Eq-(4),

(5)   \begin{eqnarray*}Y_{in}(j\omega) &=& j\omega C + \frac{1}{j\omega L } + \frac{1}{R} = \frac{1}{R}\left( 1 + j\omega_o C R \left( \frac{\omega}{\omega_o} - \frac{\omega_o}{\omega}\right)\right) \\ Y_{in}(j\omega) &=& \frac{1}{R}\left( 1 + jQ_{po} \left( \frac{\omega}{\omega_o} - \frac{\omega_o}{\omega}\right) \right)\\ Z_{in}(j\omega) &=& \frac{R}{1 + jQ_{po} \left( \frac{\omega}{\omega_o} - \frac{\omega_o}{\omega}\right)} \end{eqnarray*}

The transsfer function is

(6)   \begin{eqnarray*} H(j\omega) = {I_o(j\omega) \over I_s(j\omega)} = {{1\over R}\over {\frac{1}{R}\left( 1 + jQ_{po} \left( \frac{\omega}{\omega_o} - \frac{\omega_o}{\omega}\right) \right)}} = {1\over \left( 1 + jQ_{po} \left( \frac{\omega}{\omega_o} - \frac{\omega_o}{\omega}\right) \right)} \end{eqnarray*}

At resonant frequency |H(j\omega_o)|=1, which is real and minimum.

The input impedance of parallel RLC circuit value deceases by \frac{1}{\sqrt{2}} at half power frequencies (\omega_{lc} \mbox{ and } \omega_{uc}).

From Eq-(6) the condition for \omega_{uc} is \left( \frac{\omega_{uc}}{\omega_o} - \frac{\omega_o}{\omega_{uc}}\right) = \frac{1}{Q_{po}} and for \omega_{lc} is \left(\frac{\omega_{lc}}{\omega_o} - \frac{\omega_o}{\omega_{lc}}\right) =\frac{-1}{Q_{po}}

(7)   \begin{equation*} \Rightarrow \omega_{uc}=\underbrace{{\omega_o \over 2Q_{po}} + \omega_o \sqrt{1+ {1 \over 4Q_{po}^2}}}_{\mbox{positive frequency}} \quad \underbrace{{\omega_o \over 2Q_{po}} - \omega_o \sqrt{1+ {1 \over 4Q_{po}^2}}}_{\mbox{negative frequency}} \end{equation*}

(8)   \begin{equation*} \Rightarrow \omega_{lc}=\underbrace{-{\omega_o \over 2Q_{po}} + \omega_o \sqrt{1+ {1 \over 4Q_{po}^2}}}_{\mbox{positive frequency}} \quad \underbrace{-{\omega_o \over 2Q_{po}} - \omega_o \sqrt{1+ {1 \over 4Q_{po}^2}}}_{\mbox{negative frequency}} \end{equation*}

The bandwidth of the circuit is BW=\omega_{uc}-\omega_{lc}. Using Eq-(7) and Eq-(8), bandwidth is given as BW = {\omega_o \over Q_{po}}. The resonant frequency is geometric mean of upper and lower 3dB frequencies, \omega_o=\sqrt{\omega_{lc}\omega_{uc}}.

Quality factor is \boxed{Q = {\omega_o \over BW}}

For small deviations in frequency (\Delta \omega) from resonant frequency (\omega_o), \omega = \omega_o + \Delta \omega, input impedance is given by,

(9)   \begin{equation*} Z_{in}(j\omega) = \frac{R}{\left( 1 + jQ \left( \frac{2 \Delta\omega \omega_o + (\Delta\omega)^{\tiny 2}} {\omega_o^{\tiny 2}}\right) \right)}\end{equation*}

or when the deviations are very small compared to \omega_o, input impedance is approximated as

(10)   \begin{equation*}\boxed{Z_{in}(j\omega) \approx \frac{R} {1 + jQ\frac{2\Delta\omega}{\omega_o}}}\end{equation*}

From the above equation, at resonant frequency, the input impedance is R and is maximum at that frequency. Below \omega_o it is inductive and above \omega_o it is capacitive.

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