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Chapter 7

- First-Order Circuit

Items

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

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.

FORMULATING RC AND RL CIRCUIT EQUATIONS

Eq.(7-1)

Eq.(7-2)

RC

Eq.(7-3)

Eq.(7-4)

Eq.(7-5)

RL

Eq.(7-6)

ZERO-INPUT RESPONSE OF FIRST-ORDER CIRCUITS

RC Circuit

makes VT0 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

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Eq.(7-10)

time constant TCRTC

Fig. 7-3 First-order RC circuit zero-input

response

Graphical determination of the time constant T

from the response curve

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)

EXAMPLE 7-1 The switch in Figure 7- 4 is closed

at t0, 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

SOLUTION

EXAMPLE 7-2 Find the response of the state

variable of the RL circuit in Figure 7-5 using

L110mH, L230mH, R12k ohm, R26k ohm, and

iL(0)100mA

Fig. 7-5

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)

divide solution v(t) into two components

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 VF(t) and its derivative equal a constant VA

for t ? 0. Setting VF(t)VA meets this condition

since . Substituting VFVA into Eq.(7-19)

reduces it to the identity VAVA.

Now combining the forced and natural responses,

we obtain

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

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 iFIA

The constant K is now evaluated from the initial

condition

The initial condition requires that KIO-IA, so

the complete response of the RL circuit is

Eq.(7-22)

The zero-state response of the RC circuit

t?0

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 t0 (VO or IO)

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 7-6 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?

SOLUTION (a) The natural response of a

first-order circuit is of the form .

Therefore, the time constant of the given

responses is Tc1/2005ms (b) The initial (t0)

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

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?

EXAMPLE 7-7 Find the zero-state response of the

RC circuit of Figure 7-15(a) for an input

Fig. 7-15

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 tgt5RC.

However, at tTlt5TC the second step function

initiates an equal and opposite response v2(t).

For tgt T5RC the second response reaches its

final state and cancels the first response, so

that total pulse response returns to zero.

3. Initial and Final Conditions

Eq.(7-23)

the general form

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 tlt0

STEP 2 Find the final value by applying dc

analysis to the circuit configuration for

tgt0. STEP 3 Find the time constant TC of the

circuit in the configuration for tgt0 STEP 4

Write the step response directly using Eq.(7-23)

without formulating and solving the circuit

differential equation.

Example The switch in Figure 7-18(a) has been

closed for a long time and is opened at t0. 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

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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,

TCReC or L/ Re

Then,

- 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(8)? Use 8 equivalent

circuit(stead state) to get f(8). 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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EXAMPLE 7-8 The switch in Figure 7-20(a) has been

open for a long time and is closed at t0. Find

the inductor current for tgt0.

SOLUTION

Fig. 7-20

EXAMPLE 7-9 The switch in Figure 7-21(a) has

been closed for a long time and is opened at t0.

Find the voltage vo(t)

Fig. 7-21

another way to solve the problem

4. First-Order Circuit Sinusoidal Response

If the input to the RC circuit is a casual

sinusoid

Eq.(7-24)

where

EXAMPLE 7-12 The switch in Figure 7-26 has been

open for a long time and is closed at t0. Find

the voltage v(t) for t 0 when vs(t)20 sin

1000tu(t)V.

SOLUTION

Fig. 7-26

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.

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