2._Three-Phase_Circuits_2011

# 2._Three-Phase_Circuits_2011 - Three Phase Circuits Revised...

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EEL 3211 © 2011, Henry Zmuda 2. Three-Phase Circuits 1 Three – Phase Circuits Revised Tuesday, May 17, 2011

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EEL 3211 © 2011, Henry Zmuda 2. Three-Phase Circuits 2 Preliminary Comments and a quick review of phasors . We live in the time domain. We also assume a causal (non- predictive) world. Real-world signals are always of the form: Where f ( t ) is real and, No one has ever measured an imaginary physical quantity. ( ) ft ( ) 0, o t t =<
EEL 3211 © 2011, Henry Zmuda 2. Three-Phase Circuits 3 Preliminary Comments and a quick review of phasors . In some applications, such as systems whose bandwidth is narrow, or because of the nature of the excitation, the signals are well-approximated by sinusoids. Furthermore, if we assume that we wait a sufficiently long time for any transient behavior to decay to zero, then all signals (voltage, current, etc.) will take the form: Here the amplitude A and the phase shift θ are real quantities. ( ) ( ) cos ft A t ωθ =

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EEL 3211 © 2011, Henry Zmuda 2. Three-Phase Circuits 4 Preliminary Comments and a quick review of phasors . To ease our computational burdens, classical circuit theory reveals that we may express sinusoidal signals as where is known as a phasor . A cos ! t "# ( ) = Re Ae j t " # ( ) \$ % & ' = Re Ae " j e j t \$ % & ' = Re ! Ae j t \$ % & ' ! A = Ae ! j "
EEL 3211 © 2011, Henry Zmuda 2. Three-Phase Circuits 5 Preliminary Comments and a quick review of phasors . In EEL 3111 you learned how to express the excitation(s) in phasor form, perform the circuit analysis required to obtain the phasor form of the quantities sought, and then return to the quantity as it would be measured by multiplying the phasor form of the answer by exp[ j ω t ] then taking the real part. We review this process with a simple example.

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EEL 3211 © 2011, Henry Zmuda 2. Three-Phase Circuits 6 Preliminary Comments and a quick review of phasors . For the circuit shown, find the voltage v ( t ) across the capacitor C. R ~ C L + v C ( t ) _ V o sin( ω t )
EEL 3211 © 2011, Henry Zmuda 2. Three-Phase Circuits 7 Preliminary Comments and a quick review of phasors . V o sin ! t = V o cos t " # 2 \$ % & ' ( ) = Re V o e " j 2 e j t * + , - . / 0 ! V o = V o e " j 2 = " jV o cos sin j ej θ θθ ± Remember:

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EEL 3211 © 2011, Henry Zmuda 2. Three-Phase Circuits 8 Preliminary Comments and a quick review of phasors . ~ o jV L Zj L ω = R ZR = 1 C Z jC = + ! V C ! ! V C = Z C Z R + Z L + Z C ! V o