Series RLC Circuit

The input impedance of a series RLC circuit, shown in Figure 1 is given by,

(1)   \begin{equation*} Z_{in}(j\omega) = R + j\omega L + \frac{1}{j\omega C} \end{equation*}

Figure 1. Series R-L-C circuit

Figure 1. Series R-L-C circuit

At resonant frequency, (\omega = \omega_o), \omega_o L - \frac{1}{\omega_o C}=0. Therefore, resonant frequency is given by,

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

Quality factor(Q) of series R-L-C circuit is,

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

where I_{pk} is the peak current in the circuit
The characteristic impedance of the circuit is Z_o = \sqrt{\frac{L}{C}}
From Eq-(3), the Q-factor of series R-L-C circuit at resonance is

(4)   \begin{equation*} \boxed{Q_{so} = {\omega_o L \over R}={1 \over \omega_oCR}={1 \over R}\sqrt{L \over C}} \end{equation*}

Substituting the relations Eq.(2) and Eq.(3) in Eq.(1), the input impedance of the series resonant circuit as a function of Q_{so} and R is given by,

(5)   \begin{equation*}Z_{in} = R\left( 1 + jQ_{so} \left( \frac{\omega}{\omega_o}-\frac{\omega_o}{\omega}\right)\right) \end{equation*}

The input impedance of Series RLC circuit is shown in Fig.2 as a function of (1/Q_{so}).

At ‘\omega_o‘, the impedance seen by the source is equal to R which is the minimum and real

For small deviation in frequencies from center frequency,(\omega = \omega_o \pm \Delta \omega), the input impedance is

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

For \Delta\omega << \omega_o, Z_{in}(j\omega) is approximated as,

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

The magnitude transfer function of series rlc circuit is,

(8)   \begin{eqnarray*} | H(j\omega)| = \left|{V_o(j\omega) \over V_s(j\omega)}\right| =\left|{R \over Z_{in} }\right| = \left| { 1 \over 1 + jQ_{so} \left( \frac{\omega}{\omega_o}-\frac{\omega_o}{\omega}\right)} \right| \end{eqnarray*}

Figure 3 illustrates the H(j\omega) for different Q-factors of the circuit. At resonant frequency |H(j\omega_o)=1| irrespective of resistance (or Q-factor).

At resonance, voltage across resistance is maximum and therefore power dissipation(P_d = V_o^2 / 2R) is also maximum in resistor. When voltage drop across resistor drops by 1 \over \sqrt{2}, the power dissipation drops by 1 \over 2. The frequencies at which power drop by half from resonant frequency are called half-power frequencies.
Since the power at these frequencies is 3dB less or half that of resonant frequency, these frequencies are called 3dB frequencies or half-power frequencies. At 3dB frequencies,

(9)   \begin{equation*} {1 \over \sqrt{2}} = {1 \over \sqrt{1 + Q_{so}^2 \left({\omega^2 - \omega_o^2 \over \omega\omega_o}\right) }} \end{equation*}

The two roots of the equation are

(10)   \begin{eqnarray*} \omega_{uc} &=& {\omega_o\over 2Q_{so}}+\omega_o\sqrt{1+\left({1\over 4Q_{so}^2}\right)}\\ \omega_{lc} &=& {-\omega_o\over 2Q_{so}}+\omega_o\sqrt{1+\left({1\over 4Q_{so}^2}\right)} \end{eqnarray*}

Bandwidth is the given by

(11)   \begin{equation*} BW = \omega_{uc} - \omega_{lc} = {\omega_o\over Q_{so}} \end{equation*}

and resonant frequency is the geometric mean of 3dB frequencies

(12)   \begin{equation*} \omega_o = \sqrt{\omega_{uc} ~ \omega_{lc}} \end{equation*}

With decreasing R, the Q-factor increases whereas the bandwidth decreases

. High Q means higher selectivity or sensitivity. High Q circuits has low BW at a given center frequency, a desired feature in many filtering applications.

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