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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:

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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

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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 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.

Eq.(7-8)

where K and s are constants to be determined

A solution in the form of an exponential

RC Circuit:

ZERO-INPUT RESPONSE OF FIRST-ORDER CIRCUITS

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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=RTC

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Graphical determination of the time constant T from the response curve

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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)

  

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EXAMPLE 7-1 The switch in Figure 7- 4 is closed at t=0, connecting a capacitor with an initial voltage of 30V to th

e resistances shown. Find the responses vC(t), i(t), i1(t) and i2(t) for t  0.  

Fig. 7-4

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SOLUTION:

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EXAMPLE 7-2Find the response of the state variable of the RL circuit in Figure 7-5 using L 1=10mH, L2=30mH, R1=2k ohm, R2=6k ohm, and iL(0)=100mA

Fig. 7-5

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SOLUTION:

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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)

 

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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.

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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

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using the initial condition:

K=(VO-VA)

The complete response of the RC circuit:

Eq.(7-20)

Fig. 7-12: Step response of first-order RC circuit

The zero-state response of the RC circuit:

)1()( / CRtA

TeVtv t0

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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

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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:

)1()( / CGtA

NeIti t0

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The complete response of a first-order circuits depends on three quantities:

1. The amplitude of the step input (VA or IA)

2. The circuit time constant(RTC or GNL)

3. The value of the state variable at t=0 (VO or IO)

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EXAMPLE 7-4

Find the response of the RC circuit in Figure 7-13

SOLUTION:

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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

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EXAMPLE 7-6The 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?

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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                         

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The first parts of the above equations are Zero-input response and the second parts are Zero-state response.

COMPLETE RESPONSE

What is s step response?

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

Fig. 7-15

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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.

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3. Initial and Final Conditions

Eq.(7-23)

the general form :

                                                                                                             

 

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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.

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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 t0

Fig. 7-18: Solving a switched dynamic circuit using the initial and final conditions

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There is another way to find the nonstate variables.

1. Get f(0) from initial value of state variable

2. Get f()---use equivalent circuit

3. Get TC---calculate the equivalent resistance Re, TC=ReC or L/ Re

Then,

)())()0(()(

fefftf TC

t

Generally, method of “three quantities” can be applied in step response on any branch of First-order circuit.

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How to get initial value f(0)?

1. the capacitor voltage and inductor current are always continuous in some condition. Vc(0+)=Vc(0-); IL(0+)=IL(0-)

2. ---use 0+ equivalent circuit C: substituted by voltage source; L: substituted by current source

3. 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.

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)())()0(()(

fefftf TC

t

forced

response

natural

response

)1)(()0()( TC

t

TC

t

efeftf

Zero-input

response

Zero-state

response

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EXAMPLE 7-8The 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

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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

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another way to solve the problem:

0

)())()0(()(

0)(

)0(

2

21

2

0000

20

21

20

teRR

VR

VeVVtV

CRTCV

RR

VRV

CR

t

A

TC

t

A

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4. First-Order Circuit Sinusoidal Response

If the input to the RC circuit is a casual sinusoid

Eq.(7-24)

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where

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EXAMPLE 7-12The 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.

Fig. 7-26

SOLUTION:

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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.

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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