RLC Resonator

RLC circuits are resonant circuits, as the energy in the system “resonates” between the inductor and capacitor. There is an exact analogy between the charge in an RLC circuit and the distance in a mechanical harmonic oscillator (mass attached to spring). We will examine the properties of a resonator consisting of series circuit of an inductor (L), capacitor (C) and resistor (R), where the output is taken across the resistor. The components are the same as in the RLC filter discussed earlier, except that the output is taken over the resistor.

Prerequisite reading includes Impedance and Transfer Functions.\(\)

own work
LCR schematic

Transfer Function

In the RLC circuit, the current is the input voltage divided by the sum of the impedance of the inductor \(Z_l=j\omega L\), capacitor \(Z_c=\frac{1}{j\omega C}\) and the resistor \(Z_r=R\). The output is the voltage over the capacitor and equals the current through the system multiplied with the capacitor impedance.
$$
\begin{align}
H(s) &= \frac{Z_r}{Z_l+Z_c+Z_r}\nonumber \\
&=\frac{R}{sL+\frac{1}{sC}+R}\nonumber \\
&= \frac{R}{L}\frac{s}{s^2+s\frac{R}{L}+\frac{1}{LC}}\label{eq:voltagedivider}
\end{align}
$$

The zeros of \(H(s)\) are the values of \(s\) such that H(s)=0. There is one zero at \(s=0\). The poles of \(H(s)\) are those values of \(s\) such that \(H(s)=\infty\) . By the quadratic formula, we find the system’s poles:
$$
H(s) = \frac{R}{L}\frac{(s-z)}{(s-p_1)(s-p-2)},\ z=0,\ p_{1,2} = -\frac{R}{2L}\pm\sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}}
\label{eq:transferpolynomial}
$$

The poles in \(\eqref{eq:transferpolynomial}\) may be real or complex conjugates. Highlight this by parameterizing the transfer function in terms of the damping ratio \(\zeta\), and natural frequency \(\omega_n\). The parameter choices will become evident as we examine complex conjugate poles. [MIT-me]
$$
H(s)=\frac{R}{L}\frac{s}{s^2+2\zeta\omega_ns+{\omega_n}^2},\ w_n=\frac{1}{\sqrt{LC}},\ \zeta=\frac{R}{2}\sqrt{\frac{C}{L}}
\label{eq:transferpoles}
$$
For resonators (narrow band-pass filters), we commonly use a \(Q\) factor instead of \(\zeta\). The Q factor expresses how under-damped a resonator is, and is defined as the frequency \(\omega\) multiplied with the quotient of the maximum energy stored and the power loss. The Q factor depends on frequency but it is most often quoted for the resonant frequency \(\omega_n\). The maximum energy stored can be calculated from the maximum energy in the inductor. Power is only dissipated in the resistor. For this series RLC circuit, the Q-factor is
$$
Q(\omega_n)=\omega_n\frac{L\, i_{rms}^2}{R\,i_{rms}^2}=\omega_n\frac{L}{R}=\frac{1}{\sqrt{LC}}\frac{L}{R}=\frac{1}{R}\sqrt{\frac{L}{C}}
$$

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, \(\zeta >1\))

In the overdamped case the two poles \(\eqref{eq:transferpolynomial}\) 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 \(\eqref{eq:transferpolynomial}\) gives the frequency response \(\dot{U}_o(s)\)
$$
\begin{align}
U_o(s)&=U_i(s)\,H(s)=H(s) \\
H(s)&= \frac{R}{L}\frac{s}{(s-p_1)(s-p_2)}\label{eq:case1b_multiplication}
\end{align}
$$

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 \(\eqref{eq:transferpolynomial}\)
$$
H(\omega)=\frac{R}{L}\frac{j\omega}{(j\omega-p_1)(j\omega-p_2)},\ p_{1,2} = -\frac{R}{2L} \pm \sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}},\ {R>2\sqrt\frac{L}{C}}
\label{eq:case1b_splane}
$$

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

own work
Evaluated at \(j=s\omega\)

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

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, \(\zeta =1\))

In the critically-dampled case the two poles \(\eqref{eq:transferpolynomial}\) 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

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

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

own work
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.

own work
Frequency response

Complex Poles (underdamped case, \(\zeta \lt 1\))

In the underdamped case the conjugate poles \(\eqref{eq:transferpolynomial}\) lay in the left half of the s-plane.
$$
p,\,p^* = -\frac{R}{2L} \pm j\sqrt{\left(\frac{1}{LC}-\frac{R}{2L}\right)^2},\ {R\lt 2\sqrt\frac{L}{C}}
$$ or in terms of \(\zeta\) and \(\omega_n\)
$$
p,\,p^*=\omega_n\left(-\zeta \pm j\sqrt{1-\zeta^2}\right),\ \zeta=\frac{R}{2}\sqrt{\frac{C}{L}}\lt 1,\ \omega_n=\sqrt{\frac{1}{LC}}
$$

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 \(\eqref{eq:transferpolynomial}\)
$$
H(s)=\frac{R}{L}\frac{s}{s^2+2\zeta\omega_ns+{\omega_n}^2},\
\zeta=\frac{1}{2}R\sqrt{\frac{C}{L}},\ \omega_n=\sqrt{\frac{1}{LC}}
$$
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
$$
H(s)=\frac{R}{L}\frac{j\omega}{(j\omega-p)(j\omega-p^*)},\
p,\,p^* = -\frac{R}{2L} \pm j\sqrt{\left(\frac{1}{LC}-\frac{R}{2L}\right)^2}
\label{eq:case3b_splane}
$$

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

own work
Evaluates at \(s=j\omega\)

The transfer function can be expressed in polar form as
$$
\left\{\begin{align}
|H(\omega)| &=\frac{R}{L}\frac{\omega}{\sqrt{({\omega_n}^2-\omega^2)^2+(2\zeta\omega_n\omega)^2}}\nonumber\\
\Rightarrow\ |H_{dB}(\omega)| &= 20\log\frac{R}{L}+20\log\omega -20\log\sqrt{({\omega_n}^2-\omega^2)^2+(2\zeta\omega_n\omega)^2}\nonumber\\
\angle{H(\omega)}&=\frac{\pi}{2}-\mathrm{atan2}({2j\zeta\omega_n\omega,{\omega_n}^2-\omega^2})\nonumber\\
\end{align}\right.
\label{eq:case3b_polar}
$$

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 \(|p|\).

own work
Output magnitude for different values of R

Bandwidth and Q Factor

Oscillators with a high quality factor oscillate with a smaller range of frequencies and are therefore more stable. The quality factor is defined as the natural frequency \(\omega_n\) multiplied with the ratio of the maximum energy stored and the power loss. The maximum energy stored can be calculated from the maximum energy in the inductor or capacitor. The equation below uses the maximum energy in the inductor \(LI_{rms}^2\). At the natural frequency \(\omega_n\), the impedance of the capacitor and inductor cancel each other and power is only dissipated in the resistor \(RI_{rms}^2\).
$$
Q=\omega_n\frac{L\, I_{rms}^2}{R\,I_{rms}^2}=\omega_n\frac{L}{R}=\frac{1}{R}\sqrt{\frac{L}{C}}=\frac{1}{2\zeta}\label{eq:qfactor1}
$$

The Q factor also relates the frequencies \(\omega_1\) and \(\omega_2\) where the dissipated power equals half the power stored. Consequently, the transfer function \(H(s)\) equals \(\frac{1}{\sqrt{2}}\) (-3dB) as shown in the illustration below

own work
Power

The half power bandwidth BW follows from solving the equation \(H(s)=\frac{1}{\sqrt{2}}\)
$$
\begin{align}
|H(\omega)|&=\frac{R}{\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}}\equiv\frac{1}{\sqrt{2}}\nonumber\\
\Rightarrow\ R^2&+\left(\omega L-\frac{1}{\omega C}\right)^2=2R^2\nonumber\\
\Rightarrow\ \omega^2 &\pm \frac{R}{L}\omega – \frac{1}{LC}=0\nonumber\\
\Rightarrow\ \omega_{1,2}\ &= \pm\frac{R}{2L}\pm\sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}},\ \omega_{1,2}>0\nonumber\\
&= \pm\frac{R}{2L}+\sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}}\nonumber\\
\Rightarrow\ BW&=\omega_1-\omega_2=\frac{R}{L}
\end{align}
$$

The Q factor equals the ratio of resonant frequency \(\omega_n\) to half power bandwidth \(\omega_2-\omega_1\).
$$
Q=\frac{\omega_n}{\omega_2-\omega_1}=\frac{1}{R}\sqrt{\frac{L}{C}}=\frac{1}{2\zeta}
\label{eq:qfactor2}
$$

High quality factor \(Q>1\) results in a sharp resonance peak.

own work
Bode magnitude for different \(\zeta\)

Note that the frequency-dependent definition can be uses to describe circuits with a single capacitor or inductor, opposed to the frequency-to-bandwidth ratio definition.

Appendix

Bode magnitude, in GNU/Octave

clc; close all; clear all; format short eng
L=47e-3; # 47mH
C=47e-9; # 47nF
#Rvector = [3.9e3]; # seperate real poles
#Rvector = [2e3]; # coinciting real poles
Rvector = [18 220 820]; # conjugate complex poles
f=logspace(1,6,200);
w=2*pi*f;
for R = Rvector
 wn=1/sqrt(L*C);
 zeta=(R/2)*sqrt(C/L)
 if (zeta < 1 ) # complex conjugate poles on left side of s-plane
 u=20*log10(R/L) + 20*log10(w) - 20*log10(sqrt((wn.^2-w.^2).^2+(2*zeta*wn*w).^2));
 hold on; h=semilogx(f,u); hold off;
 hold on; 
 hold off
 poles = [-wn*zeta+sqrt(1-zeta^2)*j, -wn*zeta-sqrt(1-zeta^2)*j ];
 endif
 if (zeta == 1) # coninciting real poles
 p=-R/(2*L); #1/sqrt(L*C);
 u=20*log10(R/L) + 20*log10(w) - 40*log10(sqrt(w.^2+p.^2)); 
 hold on; h=semilogx(f,u); 
 plot([wn/(2*pi) wn/(2*pi)], get(gca,'YLim'),'k--');
 text(wn/(2*pi),5,'|p|/2\pi');
 f1=-p/(2*pi);
 fmin=min(f);
 fmax=max(f);
 #asymp1=0 - 20*log10((f1-fmin)/fmin);
 #asymp2=0 - 20*log10((fmax-f1)/f1);
 #plot([fmin f1 fmax],[asymp1 0 asymp2],'k--');
 hold off
 poles=[-wn*zeta -wn*zeta];
 endif
 if (zeta > 1) # separate real poles
 p1=wn*(-zeta+sqrt(zeta^2-1));
 p2=wn*(-zeta-sqrt(zeta^2-1));
 u=20*log10(R/L) + 20*log10(w) - 20*log10(sqrt(w.^2+p1.^2)) - 20*log10(sqrt(w.^2+p2.^2)); 
 figure(1);
 hold on; h=semilogx(f,u); hold off;
 hold on; 
 f1=-p1/(2*pi); f2=-p2/(2*pi);
 plot([f1 f1], get(gca,'YLim'),'k--');
 plot([f2 f2], get(gca,'YLim'),'k--');
 fmin=min(f);
 fmax=max(f);
 asymp1=0 - 20*log10((f1-fmin)/fmin);
 asymp2=0 - 20*log10((fmax-f2)/f2);
 plot([min(f) f1 f2 fmax],[asymp1 0 0 asymp2],'k--');
 text(f1,5,'|p1|/2\pi');
 text(f2,5,'|p2|/2\pi');
 hold off
 poles=[p1+0j p2+0j];
 endif
 #figure(2);
 #grid off;
 ##axis([-1e4 1e4 -1e4 1e4]);
 ##hold on;
 #zplane([],poles');
 ##hold off;
endfor
figure(1);
grid off;
axis([min(f) max(f) -100 20]);
xlabel('frequency [Hz]'); ylabel('20log| H(t)|');
leg=strread(num2str(Rvector,4),'%s');
if (zeta>=1)
 leg=[leg;'asymptote'];
endif
t=['Bode Magnitude in dB(f'];
t2=['), C=' num2str(C*1e9) '\muF, L=', num2str(L*1e3),'mH'];
if(length(Rvector)==1)
 t=[t t2 ', R=' num2str(R/1e3) 'k\Omega']
else
 t=[t ',R' t2];
 legend(leg);
endif
if (zeta<1)
 hold on;
 plot([wn/(2*pi) wn/(2*pi)], get(gca,'YLim'),'k--');
 text(wn/(2*pi),5,'\omega_n/2\pi');
 hold off;
endif
t = [t ];
title(t, "fontsize", 15);
Embedded software developer
Passionately curious and stubbornly persistent. Enjoys to inspire and consult with others to exchange the poetry of logical ideas.

2 Replies to “RLC Resonator”

  1. Hi Coert,

    Excellent walk-through!

    Can I ask why your reason for using abs(H(t)) in your bode plots rather than abs(H(w))?

    Another question: I am putting together teaching material for internal education use at my company, is it okay if I use your pictures?

    Best regards,
    Daniel

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