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2020-11-22T08:20:36Z

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== Classical Derivations ==

=== Series Resonance ===

[[Image:Series_Resonance_Basic.png|right|thumb|264px|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 <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 ===

[[Image:Parallel_Resonance_Basic1.png|right|thumb|292px|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 <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 ===

[[Image:Practical_Series_Resonance.png|330px|Typical series resonance circuit]]

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 ===

[[Image:Practical_Parallel_Resonance.png|350px|Typical parallel resonance circuit]]

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.

[[Category:Fundamentals]]
[[Category:Modelling / Analysis]]

Resonance - Revision history

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