# EE - Chapter 7 First-Order Circuit Items: 1. RC and RL...

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Chapter 7 First-Order Circuit

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1. RC and RL Circuits 2. First-order Circuit Complete Response 3. Initial and Final Conditions 4. First-order Circuit Sinusoidal Response Items:
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

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FORMULATING RC AND RL CIRCUIT EQUATIONS
Eq.(7-1) Eq.(7-2) Eq.(7-3) Eq.(7-4) Eq.(7- 5) Eq.(7-6) RC RL

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makes V T =0 in Eq.(7-3) we find the zero-input response Eq.(7-7) Eq.(7-7) is a homogeneous equation because the right side is zero. Eq.(7-8) where K and s are constants to be determined A solution in the form of an exponential ZERO-INPUT RESPONSE OF FIRST-ORDER CIRCUITS
Substituting the trial solution into Eq.(7-7) yields OR Eq.(7-9) characteristic equation a single root of the characteristic equation zero -input response of the RC circuit:

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Eq.(7-10) Fig. 7-3: First-order RC circuit zero-input response time constant TC=R T C
Graphical determination of the time constant T from the response curve

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Eq.(7-11) Eq.(7-12) The root of this equation The final form of the zero-input response of the RL circuit is Eq.(7-13)
EXAMPLE 7-1 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. 7-4

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SOLUTION:
EXAMPLE 7-2 Find the response of the state variable of the RL circuit in Figure 7-5 using L 1 =10mH, L 2 =30mH, R 1 =2k ohm, R 2 =6k ohm, and i L (0)=100mA Fig. 7-5

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SOLUTION:
2. First-order Circuit Complete Response When the input to the RC circuit is a step function** Eq.(7-15) 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 Zero-State Response . Since u(t)=1 for t 0 we can write Eq.(7-15) as Eq.(7-16)

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divide solution v(t) into two components: natural response forced response The natural response is the general solution of Eq.(7-16) when the input is set to zero.
The forced response is a particular solution of Eq. (7-16) when the input is step function. seek a particular solution of the equation Eq.(7- 19) The equation requires that a linear combination of V F (t) and its derivative equal a constant V A for t 0. Setting V F (t)=V A meets this condition since . Substituting V F =V A into Eq.(7-19) reduces it to the identity V A =V A . Now combining the forced and natural responses, we obtain

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