circuit B

circuit B - ECSE 210 Electric Circuits 2 Chapter 14 Circuit...

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ECSE 210: Electric Circuits 2 Chapter 14 Circuit Analysis in the s Circuit Analysis in the s-Domain Domain
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Element and Kirchhoff’s Laws the s omain in the s-Domain We have analyzed circuits in the steady steady-state state using the frequency domain in which s= j Next we will analyze dynamic circuits in the generalized s-domain in which s= + j
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Element and Kirchhoff’s Laws the s omain in the s-Domain irchhoff’s current and voltage laws Kirchhoff s current and voltage laws are unchanged in the s-domain. Element laws remain unchanged when all initial conditions are equal to zero. ll other properties derived from All other properties derived from these are also unchanged. s=j is replaced by s= +j
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A Simple Example What is V o as a function of V I ? R V o 1 Element Laws Initial onditions V I + - C s    j R Z sC C Z conditions are zero Voltage divider (based on Kirchhoff’s Laws): R V o 1 sC 1 V I 1 1 sRC V I R sC
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s-Domain Circuit Laplace s-domain models of circuit elements DC voltage and current sources remain unchanged DC source is a constant, which is transformed to a 1/s function in the Laplace domain
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Circuit Element Models Apart from the transformations R R, L sL, C 1 sC Zero initial conditions we must model the s-domain equivalents of the circuit elements when there are initial conditions (i.c.) Unlike resistors, both inductors and capacitors are able to store energy
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Resistor egin with the time domain relation Begin with the time domain relation for the element: = R i( v(t) R i(t) Laplace transform for the above xpression: expression V(s) = R I(s) ence a resistor R in the time Hence a resistor, R, in the time domain is simply that same resistor, , in the - omain R, in the s domain
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Capacitor Time domain relation for the element: dv(t) Laplace transform of the above i(t)=C dt expression: I(s) = s C V(s) – C v(0) Interpretation: a charged capacitor (a capacitor with non-zero initial conditions at t=0) is equivalent to an uncharged capacitor at t=0 in parallel with an impulsive current source current source with strength C·v(0)
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Capacitor Rearranging the above expression for the capacitor: V(s)= I(s) + v(0) Interpretation: a charged capacitor can be sC s replaced by an uncharged capacitor in series with a step-function voltage source whose height is v(0) wh h ght ( ) Circuit representations of the Laplace transformation of the capacitor appear on the next slide
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Capacitor + Time i C ( t ) C v C ( t ) Domain I C ( s ) I ( s ) 1/ s C + V C ( s ) + ) + V ( s ) 1/ s C C v (0) C v (0) s C Laplace (Frequency) Domain Equivalents
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Inductor im d m in l ti n f th l m nt: Time domain relation for the element: di(t) aplace transform the above expression: v(t) L dt Laplace transform the above expression V(s) = s L I(s) – L i(0) nterpretation: an energized inductor Interpretation: an energized inductor (an inductor with non-zero initial onditions) is equivalent to an conditions) is equivalent to an
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circuit B - ECSE 210 Electric Circuits 2 Chapter 14 Circuit...

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