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Unformatted text preview: SJTU 1 Chapter 7 FirstOrder Circuit SJTU 2 1. RC and RL Circuits 2. Firstorder Circuit Complete Response 3. Initial and Final Conditions 4. Firstorder Circuit Sinusoidal Response Items: SJTU 3 1. RC and RL Circuits 1. use device and connection equations to formulate a differential equation. 2. solve the differential equation to find the circuit response. Two major steps in the analysis of a dynamic circuit SJTU 4 FORMULATING RC AND RL CIRCUIT EQUATIONS SJTU 5 Eq.(71) Eq.(72) Eq.(73) Eq.(74) Eq.(7 5) Eq.(76) RC RL SJTU 6 makes V T =0 in Eq.(73) we find the zeroinput response Eq.(77) Eq.(77) is a homogeneous equation because the right side is zero. Eq.(78) where K and s are constants to be determined A solution in the form of an exponential ZEROINPUT RESPONSE OF FIRSTORDER CIRCUITS SJTU 7 Substituting the trial solution into Eq.(77) yields OR Eq.(79) characteristic equation a single root of the characteristic equation zero input response of the RC circuit: SJTU 8 Eq.(710) Fig. 73: Firstorder RC circuit zeroinput response time constant TC=R T C SJTU 9 Graphical determination of the time constant T from the response curve SJTU 10 Eq.(711) Eq.(712) The root of this equation The final form of the zeroinput response of the RL circuit is Eq.(713) SJTU 11 EXAMPLE 71 The switch in Figure 7 4 is closed at t=0, connecting a capacitor with an initial voltage of 30V to the resistances shown. Find the responses v C (t), i(t), i 1 (t) and i 2 (t) for t 0. Fig. 74 SJTU 12 SOLUTION: SJTU 13 EXAMPLE 72 Find the response of the state variable of the RL circuit in Figure 75 using L 1 =10mH, L 2 =30mH, R 1 =2k ohm, R 2 =6k ohm, and i L (0)=100mA Fig. 75 SJTU 14 SOLUTION: SJTU 15 2. Firstorder Circuit Complete Response When the input to the RC circuit is a step function** Eq.(715) The response is a function v(t) that satisfies this differential equation for t ≥ 0 and meets the initial condition v(0). If v(0)=0, it is ZeroState Response . Since u(t)=1 for t ≥ 0 we can write Eq.(715) as Eq.(716) SJTU 16 divide solution v(t) into two components: natural response forced response The natural response is the general solution of Eq.(716) when the input is set to zero. SJTU 17 The forced response is a particular solution of Eq....
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 Fall '08
 PingLi
 Inductor, RC circuit, RL circuit

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