es-1 natural and forced response

8/8/2019 ES-1 Natural and Forced Response

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ELECTRICAL SCIENCES - I

01.09.08 Lecture – 12


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NATURAL & COMPLETE RESPONSE OF

FIRST-ORDER CIRCUITS

OBECTI!ES


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First-Order Circuits :-

t < 0 s

+V

−

R gS t


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* Switch S c#osed ≡ Short circuit

* For t


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Figure (

+

V

−

R g

t ≥ 0 s

+

V

−

S t ≥ 0 s

Cic ↓

+

v

−

R ↓ iR

R g

* Switch S opened ≡ Open circuit at t.0 s

* Switch S opened at t.0 s $instantaneous#y% and re'ains

open for t≥0 s


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Since vo#tage across capacitor can not change

instantaneous#y the vo#tage re'ains the sa'e at t.0 s that

is

v$0% . RV/$R+R g%

"#$t % '(t) *+r t,0

To find answer to this uestion

y 1C2

iC + iR . 0

&n ter's of vo#tage we write

C$dv/dt% + $v/R% . 0


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&n this euation varia,#e is v$t% and therefore dv$t%/dt

ie first derivative

* For this reason this euation is *%rt-+rer %**ere/t%$

eu$t%+/.

* Further the descri,ing euation is #++3e/e+u4 %/e$r

%**ere/t%$ eu$t%+/4 ,ecause every non-3ero ter' is of

*%rt e3ree in the dependent varia,#e ' and its derivative

* "ote that coefficients of v and its derivative are

constants


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To find v$t% that satisfies the descri,ing euation and the

initia# condition #et us rearrange descri,ing euation as

$dv/dt% . −$!/RC%v

Separating varia,#es ie dividing ,oth sides with ,y 4v5

$!/v%$dv/dt% . −$!/RC%

&ntegrating ,oth sides with respect to ti'e

∫ $!/v%$dv/dt%dt . ∫ −$!/RC%dt

y Chain ru#e of ca#cu#us we change integrating varia,#e

on 2eft 6and Side as

∫ $!/v%dvdt/dt . ∫ $!/v%dv . ∫ −$!/RC%dt


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∴ #n v$t% . −$!/RC%t + 1

where 1 is a constant of integration

Ta7ing e)ponent on ,oth sides

e#n v$t% . e $−t/RC + 1%

∴ v$t% . e$−t/RC%e1

8se initia# condition to deter'ine 1

9t t.0 s v$0% . e0e1 . e1

Su,stituting e1 . v$0% in e)pression for v$t% we get

v$t% . v$0%e$−t/RC% for t≥0 s


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9s shown ear#ier

v$0% . R/$R+R g%; V vo#ts

S7etch of v$t% for t≥0 s is shown ,e#ow

0 !RC (RC RC =RCt

v$t% . v$0%e-t/RC

v$0%e-!.0>?v$0%

v$0%e-(.0!@v$0%

v$0%e-.00@0v$0%

v$0%


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Ae see that initia##y $at t.0 s% the capacitor is charged to

v$0% vo#ts and for tB0 s the capacitor discharges through

the resistor e)ponentia##y

* Resistor current for t≥0 s is

iR $t% . v$t%/R . v$0%e$−t/RC%

/ R

* Current through capacitor for t≥0 s is

iC$t% . −iR $t% . −$v$0%/R%e

$−t/RC%

Or

iC$t% . C$dv$t%/dt% . Cd$v$0%e$−t/RC%%/dt

⇒ iC$t% . C$−!/RC%v$0%e$−t/RC%

⇒ iC$t% . −$v$0%/R%e$− t/RC%


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* nergy stored in capacitor is at t.0 s is

wC$t% . Cv($0%/( D

* nergy initia##y stored in the capacitor is eventua##y

dissipated as heat ,y resistor

The power a,sor,ed ,y the resistor is

pR $t% . RiR ($t% . R$v$0%e$−t/RC%/R%(

⇒ pR . $v($0%e$−(t/RC%/R%


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∴Tota# energy a,sor,ed ,y R is ∞ ∞

wR . ∫ pR $t%dt . ∫ $v($0%e$−(t/RC%/R%dt . Cv($0%/(

0 0⇒ wR . wC$0%

"ow RC . Ti'e constant of circuit . τ $seconds%

∴ v$t% . v$0%e$−t/ τ%

N$tur$ Re5+/e 6-

* For t≥

0 s the ,attery has no effect on the circuit* Therefore the circuit ,ehaves on its own ie ,ehaves

/$tur$7.

* The e)pressions for v$t% and i$t% for t≥0 s are said to

give/descri,e NATURAL RESPONSE of the circuit


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)a'p#e :- For the circuit shown the switch opens at ti'e

t.0 s Find v!$t% v($t% i!$t% i($t% and v$t% for a## ti'eE

+

!0 V −

$!/!(% F

(

t.0 s

$!/=% F

+ v! − →i!

+

v(

−

i(↓+

v

−


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For t


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y vo#tage division

v!$t% . !0/$+(% . > V

v($t% . (!0/$+(% . = V

∴v!$0% . > V v($0% . = V and

∴v$0% . v!$0% + v($0% . >+= . !0 V

For t≥0 s the circuit is shown ,e#ow :

+

!0 V

−

$!/!(% F

( $!/=% F

+ v! − →

i!+

v(−

i(↓+

v

−

0 9

→

0 9

←

0 9

→


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y 1C2

$!/!(%$dv!/dt% + $v!/% . 0 H $!/=%$dv(/dt% + $v(/(% . 0

$dv!/dt% + =v! . 0 H $dv(/dt% + (v( . 0

∴ v!$t% . v!$0%e

$−=t%

. >e

$−=t%

Vv($t% . v($0%e

$−(t% . =e$−(t% V

Thus

i!$t% . $!/!(%d$>e$−=t%%/dt; . −(e$−=t% 9

and i($t% . $!/=%d$=e$−(t%%/dt; . −(e$−(t% 9

y 1V2 v$t% . v!$t% + v($t% . >e$−=t% + =e$−(t% V


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6ence

v!$t% . > V for te$−=t% V for t≥0 s

v($t% . = V for t


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Su''ary

* First Order Circuit R-C c7t e)hi,its natura#

response on re'oving e)citation

* The e)act response is deter'ined ,y ti'e constantof the circuit

* "atura# response refers to e)ponentia# decay of

energy stored in the capacitor through resistor

* The response of 2-R c7t is deter'ined in a si'i#ar

fashion


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COMPLETE RESPONSE OF

FIRST-ORDER CIRCUITS


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Re'%e 6-

Co'p#ete response of first order circuits which inc#ude

a% "on-ho'ogeneous first-order differentia#

euation ,% 2inearity and Ti'e invariance


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F%rt-Orer C%rcu%t T#e C+5ete Re5+/e 6-

Consider RC circuit shown ,e#ow :

+

V

−

C

+ vR −

+

vC−

i ↓

R

,

aS +

vS

−


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* The switch S 'oves in 3ero ti'e fro' a to , at ti'e

t.0 s

vs $t%.0 t


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This function is depicted in figure shown ,e#ow :

vs$t% . Vu$t%

V

0t

* Ahen the vo#tage vs$t% . Vu$t% is app#ied to a series

RC co',ination the resu#ting vo#tage v$t% and currenti$t% are ca##ed te5 re5+/e.* Iescri,ing euation using noda# ana#ysis ,y 1C2 is

Vu$t% − vC;/R . C$dvC/dt%


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⇒ $dvC/dt% + $!/RC%vC . $V/RC%u$t%

So#ution to this differentia# euation for t


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The so#ution to this differentia# euation is

vC$t% . $V/RC%/$!/RC% + 9e$−t/RC% . V + 9e$−t/RC%

where 9 is constant of integration to ,e deter'ined fro'

%/%t%$ c+/%t%+/.

Ae 7now that

vC$t% . 0 for t


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⇒ 9 . −V

Thus at t.0 s fro' a,ove so#ution ,eco'es

vC

$0% . V + 9$e−0% . 0

∴The vo#tage across the capacitor is

vC$t% . V − Ve$−t/RC% . V$! − e$−t/RC%% for t≥0 s

Co',ining this resu#t with the fact that

vC$t% . 0 V for t


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The s7etch of this wavefor' is shown on ,e#ow

0

vC$t%.V$!−e−t/RC%u$t%

V

RC (RC RC

t


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C$5$c%t+r curre/t 6-* 8sing e)pression for vC$t% we can now find the

capacitor current* &n this case the capacitor current is eua# to theresistor current

* For t


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0

iC$t%.$V/R%$!−e−t/RC%u$t%

V/R

RC (RC RC

t

* S7etch of this function is shown on ,oard


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L%/e$r%t7 $/ T%e %/'$r%$/ce 6-

Consider the circuit shown in figure

* Suppose that switch S returns to 4a5 at ti'e t.! s

* 8nder this circu'stance the vo#tage vs$t% is the vo#tage

pu#se shown ,e#ow:

0 !s t

vs$t%

V


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* 6ere vC$t% . 0 vo#ts for t


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Vo#tage response to app#ied vo#tage pu#se is shown

,e#ow

0

t

vC$t% V$!−e−t/RC%

V$!−e−t/RC%

V$!−e−!/RC%e−$t−!%/RC%

!s


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va.Vu$t%

0t

V

0

−V

!s

v ,$t%.−

Vu$t−

!%

t

y #inearity the response to

vs$t% . va$t% + v ,$t%

is the su' of the responses to va$t% and v ,$t%

In particular, we can express the pulse vs(t) as the sum of two voltage steps.

va(t) = V.u(t) volts

vb(t) = −V.u(t−1) volts


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* Response to va$t% is

vCa$t% . V$!−e−t/RC%u$t%

* Ae have noted that the coefficients of non-3ero ter's in

descri,ing euation are c+/t$/t.

* 9s a conseuence this euation ,eco'es

t%e-%/'$r%$/t $corresponding circuits are ca##ed t%e-

%/'$r%$/t c%rcu%t%


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The property of t%e-%/'$r%$/ce $7 that

* Ie#aying an ec%t$t%+/ ,y a certain a'ount of ti'ede#ays the re5+/e ,y the sa'e a'ount of ti'e

* &n particu#ar the response to v ,$t% . −Vu$t−!% is

* "ow ,y #inearity we have that the response tovs$t% . va$t% + v ,$t% is

vC$t% . vCa$t% + vC,$t%

⇒ vC$t% . V$! − e−t/RC%u$t% − V$! − e−$t−!% /RC%u$t − !%

vC,$t% . −V! − e−$t−!%/RC;u$t −!%


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Fro' this so#ution we can o,tain so#ution/e)pression for

vC$t% for different ti'e do'ains as

! when t


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Ahen tJ! s then

u$t% . !

u$t − !% . !

∴ vC$t% . V$! − e−t/RC% − V$! − e−$t−!% /RC% for tJ! s

which agrees with s7etch of vs$t% over different ti'e

do'ains


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* The e)pression for current in any ti'e do'ain

can ,e o,tained as $fro' #ast e)pression%

i$t% . $V/R% e−t/RC u$t% +$−V/R%e−$t−!%/RCu$t − !%

⇒ i$t% . ia$t% + i ,$t%

es-1 natural and forced response

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