# Resonance

## Contents

## Classical Derivations

### Series Resonance

The classical circuit to demonstrate series resonance is the RLC circuit shown in the figure right, which shows a voltage source connected to R, L and C impedances in series. Given a fixed ac voltage source U operating at angular frequency [math] \omega [/math], the current in the circuit is given by the following:

- [math]I = \frac{U}{Z} = \frac{U}{R + j \left(\omega L - \frac{1}{\omega C} \right)} [/math]

- [math]= \frac{U}{R + j \left(\frac{\omega^{2} LC - 1}{\omega C} \right)} [/math]

The current is at a maximum when the impedance is at a minimum. So given constant R, L and C, the minimum impedance occurs when:

- [math]\omega^{2} LC - 1 = 0 \, [/math]

or

- [math]\omega = \frac{1}{\sqrt{LC}} [/math]

This angular frequency is called the **resonant frequency** of the circuit. At this frequency, the current in the series circuit is at a maximum and this is referred to as a point of series resonance. The significance of this in practice is when harmonic voltages at the resonant frequency cause high levels of current distortion.

### Parallel Resonance

The classical circuit to demonstrate series resonance is the RLC circuit shown in the figure right, which shows a current source connected to R, L and C impedances in parallel. Given a fixed ac current source I operating at angular frequency [math] \omega [/math], the voltage across the impedances is given by the following:

- [math]V = IZ = \frac{I}{\frac{1}{R} + j \left(\omega C - \frac{1}{\omega L} \right)} [/math]

- [math]= \frac{I}{\frac{1}{R} + j \left(\frac{\omega^{2} LC - 1}{\omega L} \right)} [/math]

The voltage is at a maximum when the impedance is also at a maximum. So given constant R, L and C, the maximum impedance occurs when:

- [math]\omega^{2} LC - 1 = 0 \, [/math]

or

- [math]\omega = \frac{1}{\sqrt{LC}} [/math]

Notice that the resonant frequency is the same as that in the series resonance case. At this resonant frequency, the voltage in the parallel circuit is at a maximum and this is referred to as a point of parallel resonance. The significance of this in practice is when harmonic currents at the resonant frequency cause high levels of voltage distortion.

## Resonance in Practical Circuits

### Series Resonance

Here a distorted voltage at the input of the transformer can cause high harmonic current distortion ([math]I_{h}[/math]) at the resonant frequency of the RLC circuit.

### Parallel Resonance

In this more common scenario, a harmonic current source ([math]I_{h}[/math]) can cause high harmonic voltage distortion on the busbar at the resonant frequency of the RLC circuit. The harmonic current source could be any non-linear load, e.g. power electronics interfaces such as converters, switch-mode power supplies, etc.