# Difference between revisions of "RL Circuit Switching"

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− | The RL switching / closing transient is one of the most common electrical transients that is encountered in practice, and is also the basis for the computation of [[ | + | The RL switching / closing transient is one of the most common electrical transients that is encountered in practice, and is also the basis for the computation of [[Short Circuit|short circuit currents]]. |

== Derivation == | == Derivation == | ||

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The figure right depicts a plot of the transient current in Equation (4) for the parameters R/L = 40 and switching angle <math>\theta</math> = 0<sup>o</sup>. Here we see the classic transient current waveform for an RL switching (closing) circuit with the time constant R/L. | The figure right depicts a plot of the transient current in Equation (4) for the parameters R/L = 40 and switching angle <math>\theta</math> = 0<sup>o</sup>. Here we see the classic transient current waveform for an RL switching (closing) circuit with the time constant R/L. | ||

+ | |||

+ | == Related Topics == | ||

+ | |||

+ | :* [[Short Circuit]] | ||

+ | :* [[Transformer Inrush]] | ||

[[Category:Modelling / Analysis]] | [[Category:Modelling / Analysis]] |

## Latest revision as of 08:23, 22 November 2020

## Introduction

The RL switching / closing transient is one of the most common electrical transients that is encountered in practice, and is also the basis for the computation of short circuit currents.

## Derivation

Consider the basic switching circuit in the figure to the right, consisting of an AC voltage source V, a switch S, a resistance R and an inductance L (all ideal circuit elements). At time , the switch S will close and complete the circuit. Suppose the voltage source can be characterised as a sinusoid (as a function of time):

Where is an arbitrary phase angle to capture the time of switching.

At the point of switching, the voltage is given by Kirchhoff's voltage law:

The current must reach a steady state current of:

- ... Equ. (1)

And have a steady state power factor:

However, at , and the inductance will prevent the circuit from reaching the steady-state current instantaneously. Therefore, there must be some transient that will provide a continuous transition path from to the steady-state current .

Equation (1) can be re-written as follows:

Taking the Laplace transform of both sides, we get:

We assume that the initial current , so therefore re-arranging the equation above we get:

- ... Equ. (2)

It can be shown that the expression can be simplified as follows:

Therefore, the inverse Laplace transforms of the terms in Equation (2) can be evaluated in a fairly straightforward manner:

**1st term:**

**2nd term:**

Combining the two terms together (and including the constants), we get the transient current:

- ... Equ. (3)

Earlier, we found that the steady state power factor is:

This can be re-arranged as follows:

Likewise, the sine of the power angle is:

Using these two equations above, we can simplify Equation (3) even further:

Using some angle sum and difference trigonometric identities, we get:

Simplifying again with the same trig identities, we get the final equation:

- ... Equ. (4)

## Interpretation

The figure right depicts a plot of the transient current in Equation (4) for the parameters R/L = 40 and switching angle = 0^{o}. Here we see the classic transient current waveform for an RL switching (closing) circuit with the time constant R/L.