# Frequency response

Shows the frequency response of a RLC resonator in the overdamped, critically-damped and underdamped cases. Part of the article RLC resonator.

## Frequency response

The dampening coefficient $$\zeta$$ determines the behavior of the system. With the physical assumption that the value of $$\frac{1}{LC}\gt 0$$ and $$\frac{R}{L}\geq0$$, we can identify four classes of pole locations.

Effect of the dampening coefficient on system behavior
Condition Pole location ζ Referred to as
$$R>2\sqrt\frac{L}{C}$$ different locations on the negative real axis $$\zeta>1$$ overdamped
$$R=2\sqrt\frac{L}{C}$$ coincite on the negative real axis $$\zeta=1$$ critically damped
$$R\lt 2\sqrt\frac{L}{C}$$ complex conjugate poles in the left half of the s-plane $$\zeta\lt 1$$ underdamped

The remainder of this post will determine determine the frequency response for each of these classes.

### Two Different Real Poles (overdamped case)

In the overdamped case the two poles are on separate locations on the negative real axis.

$$p_{1,2} = -\frac{R}{2L} \pm \sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}},\ {R>2\sqrt\frac{L}{C}}$$

Note that $$p_1\lt p_2\lt 0$$ and $$|p_1|\lt |p_2|$$, as visualized in the s-plane $$s$$-plane

The frequency response is the magnitude (or gain) as a function of the frequency. It describes how well the filter can distinguish between different frequency signals.

A cosinusoidal input signal $$u_i(t)$$ with angular frequency $$\omega$$, amplitude $$1$$ and with the value 1 at $$t=0$$, can be expressed as

\begin{align} u_i(t) &= cos(\omega t)=\Re\left\{e^{j\omega t}\right\}\nonumber\\ \Rightarrow\ U_i(s) &= 1\label{eq:frequency} \end{align}

Combining the cosinusoid input function $$\eqref{eq:frequency}$$ with the transfer function gives the frequency response $$\dot{U}_o(s)$$

\begin{align} U_o(s) &= U_i(s)\,H(s) = H(s) \nonumber \\ H(s) &= \frac{R}{L}\frac{s}{(s-p_1)(s-p_2)} \label{eq:case1b_multiplication} \end{align}

Therefore, the frequency response may be written in terms of the system poles and zeros by substituting $$s=j\omega$$ for $$s$$ directly into the factored form of the transfer function

\begin{align} H(\omega) &= \frac{R}{L}\frac{j\omega}{(j\omega-p_1)(j\omega-p_2)} \label{eq:case1b_splane} \\ \rm{where}\quad p_{1,2} &= -\frac{R}{2L} \pm \sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}},\ {R>2\sqrt\frac{L}{C}} \nonumber \end{align}

The poles and zero may be interpreted as vectors in the s-plane, originating from the zero or poles $$p_i$$ and directed to the point $$s=j\omega$$ at which the function is to be evaluated Evaluated at $$j=s\omega$$

The transfer function can be expressed in polar form

\begin{gather} \left\{ \begin{aligned} |H(\omega)| &= \frac{R}{L}\frac{\omega}{\sqrt{\omega^2+{p_1}^2}\sqrt{\omega^2+{p_2}^2}}\, p_{1,2}\in\mathbb{R} \nonumber \\ \Rightarrow\ |H_{dB}(\omega)| &= 20\log\frac{R}{L}+20\log\omega -20\log\sqrt{\omega^2+{p_1}^2} \nonumber \\ & \quad-20\log\sqrt{\omega^2+{p_2}^2},\ p_{1,2}\in\mathbb{R} \nonumber \\ \angle{H(\omega)}&=\frac{\pi}{2}-\mathrm{atan2}({\omega,-p_1 })-\mathrm{atan2}({\omega,-p_2 }),\ p_{1,2}\in\mathbb{R}\nonumber\\ &=\frac{\pi}{2}-arctan\left(\frac{\omega}{-p_1}\right) \\ &\quad -arctan\left(\frac{\omega}{-p_2}\right),\ p_{1,2}\lt 0\land p_{1,2}\in\mathbb{R}\nonumber \end{aligned} \right. \end{gather} \label{eq:case1b_polar}

The frequency response has -20 dB/decade drop-offs, and a relatively wide band-pass for $$|p_2|\lt \omega\lt |p_1|$$.

### Coinciting Real Poles (critically-damped case)

In the critically-dampened case the two poles coincite on the negative real axis.

$$p = -\frac{R}{2L},\ R=2\sqrt\frac{L}{C}$$

The poles and zero are on the left real axis $$p\lt 0$$, as visualized in the s-plane $$s$$-plane

The frequency response may be written in terms of the system poles and zeros by substituting $$j\omega$$ for $$s$$ directly into the factored form of the transfer function

$$H(s) = \frac{R}{L}\frac{j\omega}{(j\omega-p)^2},\ p=p_{1,2}=-\frac{R}{2L} \label{eq:case2b_splane}$$

The poles and zero may be interpreted as vectors in the s-plane, originating from the poles $$p$$ or zero $$z=0$$ and directed to the point $$s=j\omega$$ at which the function is to be evaluated Evaluated at $$s=j\omega$$

The transfer function can be expressed in polar form as

\left\{\begin{align} |H(\omega)| &=\frac{R}{L}\frac{\omega}{\sqrt{p^2+\omega^2}\sqrt{p^2+\omega^2}}&p\in\mathbb{R} \nonumber\\ \Rightarrow\ |H_{dB}(\omega)| &= 20\log\frac{R}{L}+20\log\omega -40\log\sqrt{\omega^2+{p}^2}&p\in\mathbb{R}\nonumber\\ \angle{H(\omega)}&=\mathrm{atan2}(\omega,0)-2\mathrm{atan2}({\omega,-p })&p\in\mathbb{R}\nonumber\\ &=\frac{\pi}{2}-2\arctan\left(\frac{\omega}{-p}\right)& p\lt 0\land p\in\mathbb{R}\nonumber \end{align}\right. \label{eq:case2b_polar}

The magnitude of the frequency response has -20 dB/decade drop-offs, and an apparent resonance at $$|p|$$, but not sharp enough.

### Complex Poles (underdamped case)

In the underdamped case the conjugate poles lay in the left half of the s-plane.

\begin{align} p,\,p^* &= -\frac{R}{2L} \pm j\sqrt{\left(\frac{1}{LC}-\frac{R}{2L}\right)^2} \\ \rm{where}\quad R &\lt 2\sqrt\frac{L}{C} \nonumber \end{align}

or in terms of $$\zeta$$ and $$\omega_n$$

\begin{align} p,\,p^* &= \omega_n\left(-\zeta \pm j\sqrt{1-\zeta^2}\right) \\ \rm{where}\quad \zeta &= \frac{R}{2}\sqrt{\frac{C}{L}}\lt 1, \nonumber \\ \omega_n &= \sqrt{\frac{1}{LC}} \nonumber \end{align}

Note that the poles are each others conjugates $$p=p^*$$. If $$p=\sigma+j\omega$$ then $$p^*=\sigma-j\omega$$, as visualized in the $$s$$-plane $$s$$-plane

Apply the parameters $$\zeta$$ and $$\omega_n$$ to the transfer function

\begin{align} H(s) &= \frac{R}{L}\frac{s}{s^2+2\zeta\omega_ns+{\omega_n}^2} \\ \rm{where}\quad \zeta &= \frac{1}{2}R\sqrt{\frac{C}{L}},\ \omega_n=\sqrt{\frac{1}{LC}} \nonumber \end{align}

The frequency response may be written in terms of the system poles and zeros by substituting $$j\omega$$ for $$s$$ directly into the transfer function

\begin{align} H(s) &= \frac{R}{L}\frac{j\omega}{(j\omega-p)(j\omega-p^*)} \\ \rm{where}\quad p,\,p^* &= -\frac{R}{2L} \pm j\sqrt{\left(\frac{1}{LC}-\frac{R}{2L}\right)^2} \nonumber \end{align} \label{eq:case3b_splane}

The poles and zero may be interpreted as vectors in the s-plane, originating from the poles (pi) and zero and directed to the point s=jω at which the function is to be evaluated Evaluates at $$s=j\omega$$

The transfer function can be expressed in polar form as

\begin{gather} \left\{ \begin{aligned} |H(\omega)| &= \frac{R}{L}\frac{\omega}{\sqrt{({\omega_n}^2-\omega^2)^2+(2\zeta\omega_n\omega)^2}} \nonumber \\ \Rightarrow\ |H_{dB}(\omega)| &= 20\log\frac{R}{L}+20\log\omega – \\ &\quad 20\log\sqrt{({\omega_n}^2-\omega^2)^2+(2\zeta\omega_n\omega)^2} \nonumber \\ \angle{H(\omega)}&=\frac{\pi}{2}-\mathrm{atan2}({2j\zeta\omega_n\omega,{\omega_n}^2-\omega^2}) \nonumber \end{aligned} \right. \end{gather} \label{eq:case3b_polar}

The graph shows the magnitude of the output for different values of $$R$$. Note that the voltage amplification around the natural frequency $$\omega_n$$ . The magnitude of the frequency response has -20 dB/decade drop-offs, and a sharp resonance at $$|p|$$.