L16 - ECE 524: Transients in Power Systems Session 16; Page...

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ECE 524: Transients in Power Systems Session 16; Page 1/2 Spring 2012 ECE 524: Lecture 16 Derivation for Undamped TRV V m cos ω t () v c 0 L t it d d 1 C t d = Substitute the following for i(t) in the equation above C t v c t d d = Resulting equation: V m cos ω t v c 0 LC 2 t v c t d d 2 v c t v' c 0 = Where v' c 0 () is the derivative of v c t () at t = 0 (due to substitution into the integra)l We know that: 1 ω 0 2 = Divide by sides by LC, but and substitute with ω 0 2 and rearrange terms 2 t v c t d d 2 ω 0 2 v c t ω 0 2 V m cos ω t ω 0 2 v c 0 ω 0 2 v' c 0 = Now take LaPlace Transform s 2 V c s ω 0 2 V c s ω 0 2 V m s s 2 ω 2 ω 0 2 s V c 0 ω 0 2 V' c 0 = From initial conditions: v c 0 () 0 = bolted fault, same at t = 0- and t = 0+ since capacitor voltage i L 0 () i c 0 = 0 = inductor current zero when breaker clears, and current will flow though capacitor. As a result
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ECE 524: Transients in Power Systems Session 16; Page 2/2 Spring 2012 i c 0 () C t v c t () d d = at t= 0 is 0, so v' c 0 ()=0 These also map to 0 in the LaPlace domain. So the equation simplifies to:
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L16 - ECE 524: Transients in Power Systems Session 16; Page...

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