Resonant Circuits

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A resonant circuit refers to an electrical circuit using circuit elements such as an inductor (L) and a capacitor (C) to cause resonance at a specific frequency.

There are two types of resonant circuits:

  • series resonant circuits
  • parallel resonant circuits

In a series resonant circuit, the impedance of the circuit reaches its minimum value at resonance, whereas in a parallel resonant circuit, the impedance reaches its maximum value

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antiresonance

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Resonant Frequency

\(\zeta \lt 1\): Complex-Conjugate Poles, but not resonant peak

\(\zeta \lt \sqrt{2}/2\): resonant peak

image-20251205220247644

[https://lpsa.swarthmore.edu/Bode/underdamped/underdampedApprox.html]

image-20251205233053573

Prof. M. Green / U.C. Irvine EECS 270C / Winter 2013 [pdf]

image-20251205232150381 \[ s^2 + \frac{R}{L}s + \frac{1}{LC_L} = s^2 + 2\zeta \omega_n s + \omega_n^2 \] where \(\omega_n = \frac{1}{\sqrt{LC_L}}\) and \(\zeta=\frac{R}{2}\sqrt{\frac{C_L}{L}}\)

Resonant frequency is \[ \omega_r = \omega_n \sqrt{1-2\zeta^2} = \frac{1}{\sqrt{LC_L}}\left(1-\frac{C_LR^2}{2L}\right) \] To have no resonant \(\zeta^2 >\frac{1}{2}\), i.e \[ \frac{L}{C_LR^2} < \frac{1}{2} \]

LC Resonator

image-20240826223955851

Complex Conjugate Zeros

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Complex Conjugate Poles

\(\zeta \to 0\) push \(|G(s)\approx \frac{1}{2\zeta} \to+\infty\)

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Frequency selectivity

EEE 211 ANALOG ELECTRONICS [https://www.ee.bilkent.edu.tr/~eee211/LectureNotes/Chapter%20-%2004.pdf]

Parallel resonance

image-20251213165220671

assuming \(i(t) = I_p\cos\omega_0 t\), where \(\omega_0 =1/\sqrt{LC}\) , suppose all current flow into \(R\) \[ V(t) = I_pR\cdot \cos\omega_0 t \] \(I_C\), the current flow through \(C\) \[ \color{red}I_C(t)=C\frac{\mathrm{d}V(t)}{\mathrm{d}t}=-C\omega_0\cdot I_pR\cdot \sin\omega_0 t \] Then, we have voltage between \(L\), given \(I_L = -I_C\) \[ V_L(t) = L\frac{\mathrm{d}I_L(t)}{\mathrm{d}t} = LC\omega_0^2\cdot I_pR\cdot \cos\omega_0 t = I_pR\cdot \cos\omega_0 t \]

Series resonance

image-20251213165337258

assuming \(V(t)=V_s\cos\omega_0t\), where \(\omega_0 =1/\sqrt{LC}\) , suppose all current flow into \(V_C+V_L=0\) \[ V_R(t) = V(t) = V_s\cos\omega_0t \] then \[ I_s(t) = \frac{V_s}{R}\cos\omega_0 t \] \(V_L(t)\) is obtained \[ V_L(t) = L\frac{\mathrm{d}I_s(t)}{\mathrm{d}t} = -L\omega_0\cdot \frac{V_s}{R}\sin\omega_0 t \] Then \[ V_C(t) = V(t) - (V_L(t) + V_R(t)) = -V_L(t) \] Therefore, \(I_C\) current flow through \(C\) \[ I_C(t) = C\frac{\mathrm{d}V_C(t)}{\mathrm{d}t}= LC\omega_0^2\cdot \frac{V_s}{R}\cos\omega_0 t= \frac{V_s}{R}\cos\omega_0 t \] voltage potential between \(L\) and \(C\) \[ \color{red}V_m(t) = V_R(t) + V_L(t) = V_s\cos\omega_0t -L\omega_0\cdot \frac{V_s}{R}\sin\omega_0 t = V_s\sqrt{1+L/R^2C}\cos(\omega_0t+\phi) \] image-20251213180419366

Non ideal capacitor & inductor

Tank Circuits/Impedances [https://stanford.edu/class/ee133/handouts/lecturenotes/lecture5_tank.pdf]

Resonant Circuits [https://web.ece.ucsb.edu/~long/ece145b/Resonators.pdf]

Series & Parallel Impedance Parameters and Equivalent Circuits [https://assets.testequity.com/te1/Documents/pdf/series-parallel-impedance-parameters-an.pdf]

ES Lecture 35: Non ideal capacitor, Capacitor Q and series RC to parallel RC conversion [https://youtu.be/CJ_2U5pEB4o]

Non ideal Capacitor

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\[ Q_s = \frac{X_s}{R_s} = X_p\frac{Q_p^2}{Q_p^2+1}\cdot \frac{Q_p^2+1}{R_p} =\frac{Q_p^2}{R_p/X_p}=Q_p \]

So long as \(Q_s\gg 1\) \[\begin{align} R_p &\approx Q_s^2R_s \\ C_p &\approx C_s \end{align}\]

image-20251011224853381

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Non ideal Inductor

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So long as \(Q_s\gg 1\) \[\begin{align} R_p &\approx Q_s^2R_s \\ L_p &\approx L_s \end{align}\]

SRF (Self-Resonant Frequency)

[Understanding RF Inductor Specifications, https://www.ece.uprm.edu/~rafaelr/inel5325/SupportDocuments/doc671_Selecting_RF_Inductors.pdf]

[RFIC-GPT Wiki, https://wiki.icprophet.net/]

image-20240802210109935

\[ f_\text{SRF} = \frac{1}{2\pi \sqrt{LC}} \] The SRF of an inductor is the frequency at which the parasitic capacitance of the inductor resonates with the ideal inductance of the inductor, resulting in an extremely high impedance. The inductance only acts like an inductor below its SRF

image-20241221092745311

  • For choking applications, chose an inductor whose SRF is at or near the frequency to be attenuated

  • For other applications, the SRF should be at least 10 times higher than the operating frequency

    it is more important to have a relatively flat inductance curve (constant inductance vs. frequency) near the required frequency

RLC inspection

image-20260512003152012

For analyzing RLC circuits, Log-Log is indeed the best choice.

image-20260512003526175

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% Parameters
R = 100; L = 0.1; C = 10e-6;
f = logspace(1, 4, 1000); % Frequency range: 10Hz to 10kHz
w = 2 * pi * f;

% Calculations
fr = 1 / (2 * pi * sqrt(L * C)); % Resonant frequency (approx 159.15 Hz)
Z_mag = abs(R + 1j*w*L + 1./(1j*w*C)); % Total Impedance Magnitude

% Plotting setup
plot_funcs = {@plot, @semilogx, @semilogy, @loglog};
titles = {'Linear Plot', 'Semilog-X', 'Semilog-Y', 'Log-Log'};

for i = 1:4
    subplot(2,2,i);
    plot_funcs{i}(f, Z_mag, 'LineWidth', 1.5); 
    hold on;
    
    % Vertical line for Resonant Frequency
    xline(fr, '--r', sprintf(' f_r = %.2f Hz', fr), ...
        'LabelVerticalAlignment', 'bottom', 'LineWidth', 1.2);
    
    % Horizontal line for Minimum Impedance (Z = R)
    yline(R, '--g', sprintf(' |Z| = R = %d \\Omega', R), ...
        'LabelHorizontalAlignment', 'left', 'LineWidth', 1.2);
    
    title(titles{i});
    xlabel('Frequency (Hz)'); ylabel('|Z| (Ohms)');
    grid on;
end

reference

Hossein Hashemi, RF Circuits, [https://youtu.be/0f3yZMvD2Jg]

Resonant Circuits: Resonant Frequency and Q Factor [https://techweb.rohm.com/product/circuit-design/electric-circuit-design/18332/]

J. Nako, G. Tsirimokou, C. Psychalinos and A. S. Elwakil, "Approximation of First–Order Complex Resonators in the Frequency–Domain," in IEEE Access, vol. 13, pp. 54494-54503, 2025 [pdf]

How to generate complex poles without inductor? [https://a2d2ic.wordpress.com/2020/02/19/basics-on-active-rc-low-pass-filters/]

Visvesh Sathe. Resonant Clock Design for a Power-efficient, High-volume x86 -64 Microprocessor [https://ewh.ieee.org/r5/denver/sscs/Presentations/2012_05_Sathe.pdf]

Pozar, David M. Microwave Engineering. 4th ed. Wiley, 2012. [pdf]