RLC circuit: harmonic response

In this article, a resonant RLC circuit will be analyzed. This circuit contains a capacitor and an inductor that continuously exchange their energy with one another. As a result this kind of device can manifest an oscillator behaviour and its transfer function includes a second order factor in the denominator. Another interesting feature of this circuit is the ability to discriminate the frequencies given in input, and provide higher amplitudes in the output voltage, just for frequencies that very close to the circuit resonance frequency.

As usual, the calculations will be carried out in the Laplace domain, for the sake of simplicity. Using the voltage divider rule, we can write:

V_u=V_i\frac{Z_C}{Z_L+Z_R+Z_C}=

=V_i\frac{\frac{1}{sC}}{sL+R+\frac{1}{sC}}=

If we evaluate the sum of the elements in the denominator, we get:

=V_i\frac{\frac{1}{sC}}{\frac{1+RCs+LCs^2}{sC}}=

=V_i\frac{1}{1+RCs+LCs^2}

It’s now time to calculate the parameters in the transfer function: in this instance we have just a second order factor in the denominator. As usual, we will take the following formula as a reference.

K\cdot\frac{(1+T_1s)\cdot...(1+\frac{2\zeta_1s}{\rho_{n1}}+\frac{s^2}{\rho_{n1}^2})\cdot...}{s^n\cdot(1+\tau_1s)\cdot...(1+\frac{2\xi_1s}{\omega_{n1}}+\frac{s^2}{\omega_{n1}^2})\cdot...}

In order to calculate the parameters, we have to find the solutions of the following system of equations:

\begin{cases} \frac{2\xi_1s}{\omega_{n1}}=RCs \\ \frac{s^2}{\omega_{n1}^2} = LCs^2 \end{cases}

Where the variables are: $\omega_{n1}$ and $\xi_1$ . The easiest way is to start with the second equation.

\frac{s^2}{\omega_{n1}^2} = LCs^2

\frac{1}{\omega_{n1}^2} = LC

\frac{1}{LC} = \omega_{n1}^2

\omega_{n1}= \frac{1}{\sqrt{LC}}

We can now substitute the value of $\omega_{n1}$ in the second equation, so that we can find $\xi_1$ :

2\xi_1s\sqrt{LC} = RCs

2\xi_1\sqrt{LC} = RC

\xi_1=\frac{RC}{2\sqrt{LC}}

Parametro	Valore
K	1
$\omega_1$	$\frac{1}{\sqrt{LC}}$
$\xi_1$	$\frac{RC}{2\sqrt{LC}}$