# Resonance

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## Classical Derivations

### Series Resonance

Figure 1. Classical series resonance circuit

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 ${\displaystyle \omega }$, the current in the circuit is given by the following:

${\displaystyle I={\frac {U}{Z}}={\frac {U}{R+j\left(\omega L-{\frac {1}{\omega C}}\right)}}}$
${\displaystyle ={\frac {U}{R+j\left({\frac {\omega ^{2}LC-1}{\omega C}}\right)}}}$

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:

${\displaystyle \omega ^{2}LC-1=0\,}$

or

${\displaystyle \omega ={\frac {1}{\sqrt {LC}}}}$

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

Figure 2. Classical parallel resonance circuit

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 ${\displaystyle \omega }$, the voltage across the impedances is given by the following:

${\displaystyle V=IZ={\frac {I}{{\frac {1}{R}}+j\left(\omega C-{\frac {1}{\omega L}}\right)}}}$
${\displaystyle ={\frac {I}{{\frac {1}{R}}+j\left({\frac {\omega ^{2}LC-1}{\omega L}}\right)}}}$

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:

${\displaystyle \omega ^{2}LC-1=0\,}$

or

${\displaystyle \omega ={\frac {1}{\sqrt {LC}}}}$

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 (${\displaystyle I_{h}}$) at the resonant frequency of the RLC circuit.

### Parallel Resonance

In this more common scenario, a harmonic current source (${\displaystyle I_{h}}$) 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.