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
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
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
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)
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
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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