Chapter 7 First-Order Circuit SJTU

First-order Circuit Complete Response Initial and Final Conditions Items: RC and RL Circuits First-order Circuit Complete Response Initial and Final Conditions First-order Circuit Sinusoidal Response SJTU

1. RC and RL Circuits Two major steps in the analysis of a dynamic circuit use device and connection equations to formulate a differential equation. solve the differential equation to find the circuit response. SJTU

FORMULATING RC AND RL CIRCUIT EQUATIONS   SJTU

Eq.(7-1) RC Eq.(7-2) Eq.(7-3) Eq.(7-4) . Eq.(7-5) RL Eq.(7-6) SJTU

ZERO-INPUT RESPONSE OF FIRST-ORDER CIRCUITS RC Circuit: makes VT=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. A solution in the form of an exponential Eq.(7-8) where K and s are constants to be determined SJTU

characteristic equation 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: SJTU

Fig. 7-3: First-order RC circuit zero-input response Eq.(7-10) time constant TC=RTC Fig. 7-3: First-order RC circuit zero-input response SJTU

Graphical determination of the time constant T from the response curve SJTU

RL Circuit: 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)   SJTU

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 vC(t), i(t), i1(t) and i2(t) for t  0.   Fig. 7-4 SJTU

SOLUTION: SJTU

EXAMPLE 7-2 Find the response of the state variable of the RL circuit in Figure 7-5 using L1=10mH, L2=30mH, R1=2k ohm, R2=6k ohm, and iL(0)=100mA Fig. 7-5 SJTU

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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)   SJTU

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. SJTU

seek a particular solution of the equation 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 VF(t) and its derivative equal a constant VA for t  0. Setting VF(t)=VA meets this condition since . Substituting VF=VA into Eq.(7-19) reduces it to the identity VA=VA. Now combining the forced and natural responses, we obtain SJTU

Fig. 7-12: Step response of first-order RC circuit using the initial condition:  K=(VO-VA) The complete response of the RC circuit: Eq.(7-20) The zero-state response of the RC circuit: t0 Fig. 7-12: Step response of first-order RC circuit SJTU

the initial and final values of the response are The RL circuit in Figure 7-2 is the dual of the RC circuit Eq.(7-21) Setting iF=IA SJTU

The constant K is now evaluated from the initial condition:                                                          The initial condition requires that K=IO-IA, so the complete response of the RL circuit is Eq.(7-22)   The zero-state response of the RC circuit: t0 SJTU

The complete response of a first-order circuits depends on three quantities: The amplitude of the step input (VA or IA) The circuit time constant(RTC or GNL)  The value of the state variable at t=0 (VO or IO) SJTU

Find the response of the RC circuit in Figure 7-13 EXAMPLE 7-4 Find the response of the RC circuit in Figure 7-13 SOLUTION: SJTU

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EXAMPLE 7-5 Find the complete response of the RL circuit in Figure 7-14(a). The initial condition is i(0)=IO Fig. 7-14 SJTU

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(a) What is the circuit time constant? EXAMPLE 7-6 The state variable response of a first-order RC circuit for a step function input is                                                           (a) What is the circuit time constant? (b) What is the initial voltage across the capacitor? (c) What is the amplitude of the forced response? (d) At what time is VC(t)=0? SJTU

(b) The initial (t=0) voltage across the capacitor is SOLUTION: (a) The natural response of a first-order circuit is of the form        . Therefore, the time constant of the given responses is Tc=1/200=5ms (b) The initial (t=0) voltage across the capacitor is                                                                  (c) The natural response decays to zero, so the forced response is the final value vC(t).                                                                     (d) The capacitor voltage must pass through zero at some intermediate time, since the initial value is positive and the final value negative. This time is found by setting the step response equal to zero:                                  which yields the condition                          SJTU

COMPLETE RESPONSE The first parts of the above equations are Zero-input response and the second parts are Zero-state response. What is s step response? SJTU

EXAMPLE 7-7 Find the zero-state response of the RC circuit of Figure 7-15(a) for an input                                                 Fig. 7-15 SJTU

The first input causes a zero-state response of                                                The second input causes a zero-state response of                                                            The total response is the superposition of these two responses.                                        Figure 7-15(b) shows how the two responses combine to produce the overall pulse response of the circuit. The first step function causes a response v1(t) that begins at zero and would eventually reach an amplitude of +VA for t>5RC. However, at t=T<5TC the second step function initiates an equal and opposite response v2(t). For t> T+5RC the second response reaches its final state and cancels the first response, so that total pulse response returns to zero. SJTU

3. Initial and Final Conditions Eq.(7-23) the general form :                                                                                                                SJTU

The state variable response in switched dynamic circuits is found using the following steps: STEP 1: Find the initial value by applying dc analysis to the circuit configuration for t<0 STEP 2: Find the final value by applying dc analysis to the circuit configuration for t>0. STEP 3: Find the time constant TC of the circuit in the configuration for t>0 STEP 4: Write the step response directly using Eq.(7-23) without formulating and solving the circuit differential equation. SJTU

Example: The switch in Figure 7-18(a) has been closed for a long time and is opened at t=0. We want to find the capacitor voltage v(t) for t0 Fig. 7-18: Solving a switched dynamic circuit using the initial and final conditions SJTU

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There is another way to find the nonstate variables. Generally, method of “three quantities” can be applied in step response on any branch of First-order circuit. Get f(0) from initial value of state variable Get f()---use equivalent circuit Get TC---calculate the equivalent resistance Re, TC=ReC or L/ Re Then, SJTU

How to get initial value f(0)? the capacitor voltage and inductor current are always continuous in some condition. Vc(0+)=Vc(0-); IL(0+)=IL(0-) ---use 0+ equivalent circuit C: substituted by voltage source; L: substituted by current source Find f(0) in the above DC circuit. How to get final value f(∞)? Use ∞ equivalent circuit(stead state) to get f(∞). C: open circuit; L: short circuit How to get time constant TC? The key point is to get the equivalent resistance Re. SJTU

forced response natural response Zero-input response Zero-state response SJTU

EXAMPLE 7-8 The switch in Figure 7-20(a) has been open for a long time and is closed at t=0. Find the inductor current for t>0. SOLUTION: Fig. 7-20 SJTU

EXAMPLE 7-9 The switch in Figure 7-21(a) has been closed for a long time and is opened at t=0. Find the voltage vo(t) Fig. 7-21 SJTU

another way to solve the problem: SJTU

4. First-Order Circuit Sinusoidal Response If the input to the RC circuit is a casual sinusoid Eq.(7-24) SJTU

where SJTU

EXAMPLE 7-12 The switch in Figure 7-26 has been open for a long time and is closed at t=0. Find the voltage v(t) for t  0 when vs(t)=[20 sin 1000t]u(t)V. SOLUTION: Fig. 7-26 SJTU

Summary Circuits containing linear resistors and the equivalent of one capacitor or one inductor are described by first-order differential equations in which the unknown is the circuit state variable. The zero-input response in a first-order circuit is an exponential whose time constant depends on circuit parameters. The amplitude of the exponential is equal to the initial value of the state variable. The natural response is the general solution of the homogeneous differential equation obtained by setting the input to zero. The forced response is a particular solution of the differential equation for the given input. For linear circuits the total response is the sum of the forced and natural responses. SJTU

Summary For linear circuits the total response is the sum of the zero-input and zero-state responses. The zero-input response is caused by the initial energy stored in capacitors or inductors. The zero-state response results form the input driving forces. The initial and final values of the step response of a first and second-order circuit can be found by replacing capacitors by open circuits and inductors by short circuits and then using resistance circuit analysis methods. For a sinusoidal input the forced response is called the sinusoidal steady-state response, or the ac response. The ac response is a sinusoid with the same frequency as the input but with a different amplitude and phase angle. The ac response can be found from the circuit differential equation using the method of undetermined coefficients Homework: zero-input: 7.6 7.8 7.14 7.21 step response:7.35 7.36 7.55 op amp : 7.597.62 SJTU