http://openelectrical.org/index.php?title=RL_Circuit_Switching&feed=atom&action=history
RL Circuit Switching - Revision history
2024-03-28T17:49:37Z
Revision history for this page on the wiki
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http://openelectrical.org/index.php?title=RL_Circuit_Switching&diff=154&oldid=prev
Jules at 08:23, 22 November 2020
2020-11-22T08:23:24Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:23, 22 November 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Introduction ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Introduction ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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 [[<del class="diffchange diffchange-inline">Short_Circuit_Calculation</del>|short circuit currents]]. </div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 [[<ins class="diffchange diffchange-inline">Short Circuit</ins>|short circuit currents]]. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Derivation ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Derivation ==</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Related Topics ==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:* [[Short Circuit]]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:* [[Transformer Inrush]]</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Modelling / Analysis]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Modelling / Analysis]]</div></td></tr>
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Jules
http://openelectrical.org/index.php?title=RL_Circuit_Switching&diff=153&oldid=prev
Jules: Created page with "== 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 computa..."
2020-11-22T08:21:52Z
<p>Created page with "== 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 computa..."</p>
<p><b>New page</b></p><div>== Introduction ==<br />
<br />
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_Calculation|short circuit currents]]. <br />
<br />
== Derivation ==<br />
<br />
[[Image:RL_Switching.PNG|right|thumb|400px|Figure 1. Basic RL switching circuit]]<br />
<br />
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 <math>t=0 \,</math>, the switch S will close and complete the circuit. Suppose the voltage source can be characterised as a sinusoid (as a function of time):<br />
<br />
: <math> V(t) = V_{m} \sin(\omega t + \theta) \,</math><br />
<br />
Where <math>\theta \,</math> is an arbitrary phase angle to capture the time of switching.<br />
<br />
At the point of switching, the voltage is given by [http://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws Kirchhoff's voltage law]:<br />
<br />
: <math> V_{m} \sin(\omega t + \theta) = R I(t) + L \frac{dI(t)}{dt} \,</math><br />
<br />
The current <math>I(t) \,</math> must reach a steady state current of:<br />
<br />
: <math>I_{s} = \frac{V}{Z} = \frac{V}{R + j \omega L} \,</math> ... Equ. (1)<br />
<br />
And have a steady state power factor:<br />
<br />
: <math> \cos{\phi_{s}} = \frac{R}{|Z|} = \frac{R}{\sqrt{R^{2} + \omega^{2} L^{2}}} \,</math><br />
<br />
However, at <math>t=0 \,</math>, <math>I(0) = 0 \,</math> and the inductance <math>L \,</math> 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 <math>I(0) = 0 \,</math> to the steady-state current <math>I(t_{s}) = I_{s} \,</math>.<br />
<br />
Equation (1) can be re-written as follows:<br />
<br />
: <math> V_{m} \left[ \sin(\omega t)\cos \theta + \cos(\omega t)\sin \theta \right] = R I(t) + L \frac{dI(t)}{dt} \,</math><br />
<br />
Taking the Laplace transform of both sides, we get:<br />
<br />
: <math> V_{m} \left( \frac{\omega \cos \theta}{s^{2} + \omega^{2}} + \frac{s \sin \theta}{s^{2} + \omega^{2}} \right) = R i(s) + s L i(s) - L I(0) \,</math><br />
<br />
<br />
We assume that the initial current <math>I(0) = 0 \,</math>, so therefore re-arranging the equation above we get:<br />
<br />
: <math>i(s) = \frac{V_{m}}{L} \left( \frac{1}{\frac{R}{L} + s} \right) \left( \frac{\omega \cos \theta}{s^{2} + \omega^{2}} + \frac{s \sin \theta}{s^{2} + \omega^{2}} \right) \,</math><br />
<br />
:: <math> = \frac{V_{m}}{L} \left[ \frac{\omega \cos \theta}{(s^{2} + \omega^{2})(\frac{R}{L} + s)} + \frac{s \sin \theta}{(s^{2} + \omega^{2})(\frac{R}{L} + s)} \right] \,</math> ... Equ. (2)<br />
<br />
<br />
It can be shown that the expression <math> \frac{1}{(s + \alpha)(s^{2} + \omega^{2})} </math> can be simplified as follows:<br />
<br />
: <math> \frac{1}{(s + \alpha)(s^{2} + \omega^{2})} = \frac{1}{\alpha^{2} + \omega^{2}} \left( \frac{1}{s + \alpha} - \frac{s}{s^{2} + \omega^{2}} + \frac{\alpha}{s^{2} + \omega^{2}} \right) \, </math><br />
<br />
<br />
Therefore, the [http://en.wikipedia.org/wiki/Laplace_transform#Table_of_selected_Laplace_transforms inverse Laplace transforms] of the terms in Equation (2) can be evaluated in a fairly straightforward manner:<br />
<br />
'''1st term:''' <math> \mathcal{L}^{-1} \left[ \frac{\omega \cos \theta}{(s^{2} + \omega^{2})(\frac{R}{L} + s)} \right] = \frac{\omega \cos \theta}{\left( \frac{R}{L} \right)^{2} + \omega^{2}} \left[ e^{-\frac{R}{L}t} - \cos (\omega t) + \frac{R}{\omega L} \sin (\omega t) \right] \,</math><br />
<br />
'''2nd term:''' <math> \mathcal{L}^{-1} \left[ \frac{s \sin \theta}{(s^{2} + \omega^{2})(\frac{R}{L} + s)} \right] = \frac{\sin \theta}{\left( \frac{R}{L} \right)^{2} + \omega^{2}} \left[ -\frac{R}{L} e^{-\frac{R}{L}t} + \omega \sin (\omega t) + \frac{R}{L} \cos (\omega t) \right] \,</math><br />
<br />
<br />
Combining the two terms together (and including the constants), we get the transient current:<br />
<br />
: <math> I(t) = \frac{V_{m}}{L \left( \frac{R}{L} \right)^{2} + \omega^{2}} \left[ (\omega \cos \theta -\frac{R}{L} \sin \theta) e^{-\frac{R}{L}t} + (\frac{R}{L} \cos \theta + \omega \sin \theta) \sin (\omega t) - (\omega \cos \theta - \frac{R}{L} \sin \theta) \cos (\omega t) \right] \,</math> ... Equ. (3)<br />
<br />
Earlier, we found that the steady state power factor is:<br />
<br />
: <math> \cos{\phi_{s}} = \frac{R}{\sqrt{R^{2} + \omega^{2} L^{2}}} \,</math><br />
<br />
This can be re-arranged as follows:<br />
<br />
: <math> \cos{\phi_{s}} = \frac{R}{L} \frac{1}{\sqrt{(\frac{R}{L})^{2} + \omega^{2}}} \,</math><br />
<br />
Likewise, the sine of the power angle is:<br />
<br />
: <math> \sin{\phi_{s}} = \frac{\omega}{\sqrt{(\frac{R}{L})^{2} + \omega^{2}}} \,</math><br />
<br />
Using these two equations above, we can simplify Equation (3) even further:<br />
<br />
: <math> I(t) = \frac{V_{m} \sqrt{(\frac{R}{L})^{2} + \omega^{2}}}{L \left[ \left( \frac{R}{L} \right)^{2} + \omega^{2} \right]} \left[ ( \cos \theta \sin \phi_{s} - \sin \theta \cos \phi_{s}) e^{-\frac{R}{L}t} + (\cos \theta \cos \phi_{s} + \sin \theta \sin \phi_{s}) \sin (\omega t) - (\cos \theta \sin \phi_{s} - \sin \theta \cos \phi_{s}) \cos (\omega t) \right] \,</math><br />
<br />
Using some [http://en.wikipedia.org/wiki/Trigonometric_identities#Angle_sum_and_difference_identities angle sum and difference trigonometric identities], we get:<br />
<br />
: <math> I(t) = \frac{V_{m}} {\sqrt{(R^{2} + \omega^{2} L^{2}}} \left[ ( - \sin (\theta - \phi_{s}) e^{-\frac{R}{L}t} + (\cos \theta \cos \phi_{s} + \sin (\theta - \phi_{s}) \sin (\omega t) + (\cos (\theta - \phi_{s}) \cos (\omega t) \right] \,</math><br />
<br />
Simplifying again with the same trig identities, we get the final equation:<br />
<br />
: <math> I(t) = \frac{V_{m}} {\sqrt{(R^{2} + \omega^{2} L^{2}}} \left[ ( \sin (\omega t + \theta - \phi_{s}) - \sin (\theta - \phi_{s}) e^{-\frac{R}{L}t} \right] \,</math> ... Equ. (4)<br />
<br />
== Interpretation ==<br />
<br />
[[Image:RL_Transient.png|right|thumb|400px|Figure 2. RL switching transient current]]<br />
<br />
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.<br />
<br />
[[Category:Modelling / Analysis]]</div>
Jules