Frequency response

Coert Vonk

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

rlc resonator different real poles
\(s\)-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

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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 \(|p_2|\lt \omega\lt |p_1|\).

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Frequency response

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

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\(s\)-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

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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.

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Frequency response

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

rlc resonator complex poles
\(s\)-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

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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 \(|p|\).

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Output magnitude for different values of R

Continue reading about Bandwidth and Q-factor.