fundamentals of electric circuit analysis

8/12/2019 fundamentals of electric circuit analysis

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ELECTRIC CIRCUIT

(BEL10103) LECTURE #06Oleh:

Dr. Soon Chin Fhong

Electronic Engineering Department

Faculty of Electrical and Electronic Engineering

Universiti Tun Hussein Onn Malaysia


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Chapter 6:

FIRST-ORDERCIRCUITS


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Lecture Contents

Introduction

The source-free RL circuit

The source-free RC circuit

Singularity Functions


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6.1 First order circuits

First order circuitsare electrical circuits thatcontain a single energy storage element (either acapacitor or an inductor).

The number of capacitors and inductors in acircuit determine the differential orderof theequation that represents the network.

Hence afirst order differential equationisrequired to solve a circuit containing a singleenergy storage element.


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The total response of a first order circuit

is made up of two parts; the transientresponseand the steady state response

Total Response

Transient Response+ Steady State Response


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Transient response

- is the initial response of the circuit to a

change at its input. This response decays

exponentially, and will eventually disappear.

Steady state response

- is the response of the circuit after all

conditions have stabilized.


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6.2 The source-free RC circuit

A source-free RC circuit

occurs when its dc

source is suddenly

disconnected.

The energy already

stored in the capacitor is

released to the resistors.


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Since the capacitor is initially charged, we canassume that at time t = 0, the initial voltage is:

(0) = V0 (6.1)

with the corresponding value of the energy stored as

w(0) = (6.2)2

0

2

1CV


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dt

dvC

R

v

dt

dvC

R

v

dt

dv

RC

v

Applying KCL at the top

node of the circuit

iC+ iR= 0 (6.3)

By definition, iC= and iR= .Thus

+ = 0 (6.4a)

or + = 0 (6.4b)


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This is a first-order differential equation, since onlythe first derivative of is involved. To solve it, we

rearrange the terms as

Integrating both sides, we get

where lnAis the integration constant

(6.6)

dtRCv

dv 1

ARC

tv lnln

RC

t

A

vln


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Taking powers of eproduces

But from the initial conditions,

(0) = A = V0. Hence,

(6.7)

RCteAtv )(

RCt

eVtv

0

)(

This shows that the voltage response of the RC

circuit is an exponential decay of the initial voltage.


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Since the response is due to the initial energy

stored and the physical characteristics of the

circuitand not due to some external voltage orcurrent source, it is called the natural response

of the circuit.

The voltage respon se

of the RC circui t

Natural respo nse: the

behavior (in terms of

voltages and currents) of

the circuit itself,

with no external sources of

excitation.


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e

1

RCeV

0

10

eV

The time constant,, of a circuit is the time

required for the response to decay by a factor of

or 36.8 % of its initial value.

This implies that at t = , Eq. (6.7) becomes

= = 0.368 V0

or = RC (6.8)


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In terms of the time constant, Eq. (6.7) can be

written as

The capacitor is fully discharged (or charged)after five time constants.

(*refer to Table 7.1 page 252,text book)

In other words, it takes 5 for the circuit toreach its final state or steady statewhen nochanges take place with time.

)(tv teV 0

(6.9)


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Graphical determination of the time constant

from the response curve.


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With the voltage in Eq. (6.9), we can find the

current iR(t),

(6.10)

The power dissipated in the resistor is

(6.11)

t

e

R

V

R

tvti

R

0)()(

teR

VivtpR

22)( 0


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The energy absorbed by the resistor up to time tis

Notice that as t , , which is thesame as , the energy initially stored in thecapacitor. The energy that was initially stored inthe capacitor is eventually dissipated in the

resistor.

t t

R dtte

RVdtptw

0 0

0 22

)(

RC

000 ),2

1(2212

2

2

teCV

teR

Vt

)(Rw

2

0

2

1CV

)0(C

w


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Example 1:

In figure below, let C(0) = 15 V. Find C,

X, and iXfor t > 0.


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Ae

Veev

VevvV

t

tt

tC

5.2xx

5.25.2

x

x

5.24.0

t

-

t

-

75.0

12

vi

Finally,

9)15(6.0812

12

v

sodivision;voltageusecanwe,get vTo

15,ev(0)ev

Thus,

0.4s4(0.1)ReqCis,constantTime

4520

520Req

0.1 F

8

125

iX

X_

+

_

+

VC

_

+

VC0.1 FReq

#

#

#


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6.3 The source-free RL circuit

To determine the circuit response, we will

assume the current i(t) through the inductor.


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At t= 0, we assume that the inductor has

an initial current I0, ori (0) = I0 (6.13)

with the corresponding energy stored inthe inductor as

w (0) = (6.14)202

1 IL


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Applying KVL around the loop in the figureabove,

L+ R= 0 (6.15)

But L= and R= iR. Thus

+ Ri = 0

or (6.16)

dtdiL

dt

diL

0 iL

R

dt

di


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Rearranging terms and integrating gives

or (6.17)

dtL

R

i

di ttiI 0

)(

0

ttiI L

tRi 0)(

0ln 0Ilni(t)ln 0 L

tR

L

tR

I

ti

0

)(

ln


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Taking the powers of e, we have

(6.18)L

Rt

eIti

0)(

Current response

of the RL circuit


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Therefore, from Eq. (6.18), the time

constant for the RL circuit is = (6.19)

Thus Eq. (6.18) may be written as

(6.20)

R

L

teIti 0

)(


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With the current in Eq. (6.20), we can find

the voltage across the resistor as

(6.21)RitvR

)( t

eRI

0

The power dissipated in the resistor is

(6.22)

t

IivpR

2Re2

0


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The energy absorbed by the resistor is

(6.23))1(

2

1)(

2

0

2

teLItw

R

Note that as t , wR() ,

which is the same as wL(0).

2

0

2

1LI


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Example 2:

The switch in the circuit has been closedfor a long time. At t = 0, the switch is opened.

Calculate i(t) for t > 0.

t = 01 2

42

12 1640 V 2 H


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6.4 Singularity Functions

Singu lar i ty fun ct ions(also called swi tch ingfunct ions) are very useful in circuit analysis.

They serve as good approximations to the

switching signals that arise in circuits with

switching operations.

The most widely used singularity functions incircuit analysis are the un i t stepand impulse

(or del ta) fun ct ion s.


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The un i t step funct ion u(t )is 0 for negative values

of t and 1 for positive values of t.In mathematical terms,

u(t) = (6.23)

0,0

0,1

t

t

Fig.1:The unit step functiont

u(t)

1


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If the abrupt change occurs at t = t0(where t0>0)

instead of t = 0, the unit step functions becomes;

u(t - t0) = (6.24)

0tt,0

0

tt,1

Fig.2 :The unit step functiont

u(t - t0)

1

t0

u(t) is delayed by t0 seconds


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If the change occurs at t = - t0(where t0


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We use the step function to represent an abrup t

change in vo l tage or cu rrent,

eg. changes occurs in the circuits of control

systems and digital computers.

v(t) = (6.26)

v(t) = V0 u(tt0)

If t0=0 v(t) = V0 u(t)

0

tt,0

0

tt,

0

V


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+

-

a

b

+

-

a

b

t = 0

VoV0u(t)

In Fig. 4 (b), terminals a-b are short-circuited (v=0)

for t0.

(a) Voltage source of V0u(t) (b) Its equivalent circuit

Figure 4


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a

b

a

b

t = 0

IoI0u(t)

Similarly, in Fig. 5 (b) terminals a-b are open-

circuited (i=0) for t0.

(a) Current source of I0u(t) (b) Its equivalent circuit

i

Figure 5


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The derivat iveof the uni t step func t ion u(t)is the uni t

impu lse funct ion, ,which we write as)(t

0t

0t

0t

0

Undefined

0

)t(udt

d)t(

t0

)(t 1

Fig. 6:The unit impulse function

(6.27)


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0

01)(t

Unit impulse may be expressed mathematically as

0 1 2 3-1-2

)3(4 t

)(10 t)2(5 t

t

Fig. 7: Three impulse function

Strength of impulse function=10

(6.28)


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0

00)(

)()()(

t

t

ttr

or

ttudttutrt

Integratingthe uni t s tep funct ionu(t)results in the

uni t ramp func t ion, r(t ), we write

1

1

Fig.8: The unit ramp function

r(t)

0 t

The unit ramp functionis zero for negative values of t

and has a unit slope for positive values of t

(6.29)


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For the delayed uni t ramp fu nct ion,

0

0

0

0

0)(

tt

tt

tt

ttr

t0+1

1

Fig. 9: The unit ramp

function delayed by t0

r(t-t0)

0 tt0

(6.30)


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For the advanced un i t ramp funct ion,

0

0

0

0

0)(

tt

tt

tt

ttr

-t0

1

Fig. 10: The unit ramp

function advanced by t0

r(t+t0)

0 t-t0+1

(6.31)


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Summary

Three singularity functions (impulse, step and

ramp) are related by differentiation as:

Or by integration as

dttutrdtttu

dt

tdrtu

dt

tdut

tt

)()(,)()(

)()(,

)()(

(6.32)

(6.33)


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6.5 Step Response RC Circuit

When the dc source of an RC circuit issuddenly applied, the voltage or current

source can be modeled as a step funct ion

and the response is known as a step

response, as in Figure 1.

The step responseof a circuit is its behavior

when the excitation is the step function,

which may be a voltage or a current source.

v(0 -) = voltage across


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Fig. 1: Step response of RC circuit

Initial condition:

v(0-) = v(0+) = V0

Applying KCL,

or

Where u(t) is the unit-step function

0)(

R

tuVv

dt

dvc

s

)(tuRC

Vv

dt

dv s

v(0) = voltage across

capacitor just before

switching

v(0+) = voltage immediately

after switching

Since thecapacitor

voltage cannot

change

instantaneously


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The expression of step response of RC circuit is

given as:

*refer text book pg 274

0)(

0)(

/0

0

teVVV

tVtv

tss

This is known as the complete

responseor(total respo nse)of the RC

circuit to a sudden

application of a dc voltage

source, assuming

the capacitor is initially charged. Fig. 2: Step response of an RC circui t w ith

in i t ial ly c harged capaci tor


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If the capacitor is assumed notcharged initially, therefore V0= 0,

Which can be written alternatively as:

This is the complete step response of

the RC circuit when the capacitor isinitially uncharged.

0t)e1(V

0t0)t(v

/t

s

)t(u)e1(V)t(v /ts

Fig. 3: Step response of an RC circui t wi th

in i t ial ly un charged capaci tor


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The current through the capacitor is obtained

from i(t) = C dv/dt, we get;

for t > 0)t(u)e(R

V

)t(i/ts

Fig. 3: Step response of an RC circu i t wi th

in i t ial ly un charged capaci tor current

response


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Final valueat t ->

Initial valueat t = 0

Source-freeResponse

Complete Response = Natural response + Forced Response(stored energy) (independent source)

= V0et/ + Vs(1e

t/)

0)(

0)(

/

0

0

teVVV

tVtv

t

ss


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Three stepsto find out the step response of an

RC circuit:

1. The initial capacitor voltage,v(0).

2. The final capacitor voltage,v() DC voltage

across C.

3. The time constant,.

/)]()0([)()( tevvvtv

Therefore, step response form in general is:

Note: This equation applies only to step response, that is when the

input excitation is constant.


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The end.

fundamentals of electric circuit analysis

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