Resonance in RLC Circuit

Communication via energy transfer

Introduction

There are 2 types of resonance circuit:

  • Serial RLC circuit
  • Parallel RLC circuit

Resonance is a condition in an RLC circuit in which the capacitive and inductive reactances are equal in magnitude, thereby resulting in a purely resistive impedance.

The impedance of a RLC circuit is:

$$ Z = R + j(X_L - X_C) $$
$X_L = 2\pi fr \times L$
$X_C = \frac{1}{2\pi fr \times C}$

When the RLC circuit is at resonance:
$$ X_L = X_C $$

Resonance in Serial RLC circuit

A serial resonance circuit is used as voltage amplifier.

The Impedance of serial RLC circuit:
$$Z_m = R_m + j\omega L_m + {1\over j\omega C_m}$$

$$= R_m + j(\omega L_m - {1\over \omega C_m})$$

The resonance frequency ($f_r$):
$$ f_r = {1\over 2\pi \sqrt {L_mC_m}} (Hz) $$

Quality factor (Q):
$$ Q = {\omega_r L_m\over R_m} = {1\over {\omega_r C_mR_m}} $$
$$ = {1\over R_m}\sqrt{L_m\over C_m} = \frac{{\omega}_r}{BW} $$
$BW$: bandwidth


Resonance in Parallel RLC circuit

A parallel resonance circuit is used as current amplifier.

The Impedance of parallel RLC circuit:
$$Y_p = G_p+{1\over j\omega L_p}+j\omega C_p$$
$$Z_p = {1\over Y_p} = {{\frac{1}{R_p} - j(\omega C_p - {1\over \omega L_p})}\over {{\frac{1}{R_p}}^2 + (\omega C_p - {1\over \omega L_p})^2}}$$

The resonance frequency ($f_r$):
$$ f_r = {1\over 2\pi \sqrt {L_pC_p}} (Hz) $$

Quality factor (Q):
$$ Q = {R_p\over {\omega_r L_p}} = {\omega_r C_pR_p} $$
$$ = R_p\sqrt{C_p\over L_p} = \frac{{\omega}_r}{BW} $$
$BW$: bandwidth


Applications:

Power Electronics

  • LLC resonant converters
  • ZVS/ZCS soft switching
  • Wireless charging

Ultrasonic Systems

  • PZT transducer matching
  • Ultrasonic cleaners
  • Ultrasonic scalpels

RF Systems

  • Antenna tuning
  • Radio receivers
  • Band-pass filters

Sensors

  • Metal detection
  • Resonant sensing


Wave Multiplication
Modulation