AC CIRCUITS

Ho Kyung Kim, Ph.D.

hokyung@pusan.ac.kr

School of Mechanical Engineering

Pusan National University

Basic Experiment and Design of Electronics

Outline

• Capacitor & inductor

• RC circuits

• RLC circuits

• Storing energy in a electric field (capacitive energy storage)

– in the form of a charge separation when appropriately polarized by an electric field

– i.e., voltage

• Acting as an open circuit in DC

• Capacitance, C/V = F

Capacitor

dielectricsCVQ

)()( tCvtq

Taking a differentiation;

t

CC dttiC

tv ')'(1

)(

We can have;

Or

00

')'(1

)( VdttiC

tvt

tCC

where

0

')'(1

)( 00

t

CC dttiC

ttvV

initial voltage due to some charge stored at time t0

dt

tdvCti

dt

tdq CC

)()(

)(

• Energy storage in capacitors

t

C

t

C

t

C

t

C

dttiCCC

dttiC

dttiC

dttiC

tvtvtvtv

')'(111

')'(1

')'(1

')'(1

)()()()(

321

321

321

''

)'()'(')'()'(')'()( dt

dt

tdvCtvdttitvdttPtW C

CCCCC)(

2

1)( 2 tCvtW CC

Series circuit Parallel circuit

• Storing energy in a magnetic field

– current flowing

• Acting as a short circuit in DC

• Inductance, Vs/A = Wb/A = H

Inductor

dt

tdiLtv L

L

)()(

t

LL dttvL

ti ')'(1

)(

00

')'(1

)( IdttvL

tit

tLL

0

')'(1

)( 00

t

LL dttvL

ttiI

We can have;

Or

where

• Energy storage in inductors

Series circuit Parallel circuit

dt

tdiLLL

dt

tdiL

dt

tdiL

dt

tdiL

tvtvtvtv

)(

)()()(

)()()()(

321

321

321

)(

2

1)()()()()( 2 tLi

dt

d

dt

tdiLtitvtitP L

LLLLL

')'(

2

1

'')'()( 2 dttLi

dt

ddttPtW LLL )(

2

1)( 2 tLitW LL

• To describe the behavior of a voltage or a current during the transition between two distinct steady-state conditions

Transient analysis

v or i

t

steady-state

region

steady-state

region

transient

region

RC circuit

v(t) C

R

0')'(1

)()(

t

CCS dttiC

tRitv)()( titi CR

Applying KVL;

Differentiating;

0)(1)()(

tiCdt

tdiR

dt

tdvC

CS

dt

tdv

Rti

RCdt

tdi SC

C )(1)(

1)(

Another way;

dt

tdvCti

R

tvtvti C

CCS

R

)()(

)()()(

)(1

)(1)(

tvRC

tvRCdt

tdvSC

C

In general, any circuit containing a single energy storage element;

)()()(

001 tfbtxadt

tdxa

)()()(

0

0

0

1 tfa

btx

dt

tdx

a

a

)()()(

tfKtxdt

tdxS

first-order ordinary differential equation

first-order system equation

time constant = RC = V/I Q/V [time/charge] [charge] = [time]

or = L/R = Vs/I/(V/I) [time]

DC gain or static sensitivity

0')'(1)(

)()(

t

S dttiCdt

tdiLtRitv 0')'(

1)()()(

t

S dttiCdt

tdiLtRitv

RLC circuit

v(t) C

R LApplying KVL;

Differentiating;

0)(1)()()(

2

2

tiCdt

tidL

dt

tdiR

dt

tdvS

dt

tdvti

Cdt

tidL

dt

tdiR S )(

)(1)()(

2

2

Another way;

)()()()(

2

2

tvtvdt

tvdLC

dt

tdvRC SC

CC

dt

tdvCti C )(

)(

In general, any circuit containing two energy storage elements;

)()()()(

0012

2

2 tfbtxadt

tdxa

dt

txda

)()()()(

0

0

0

1

2

2

0

2 tfa

btx

dt

tdx

a

a

dt

txd

a

a

)()()(2)(1

2

2

2tfKtx

dt

tdx

wdt

txd

wS

nn

2

0

a

awn

20

1 1

2 aa

a

second-order ordinary differential equation

second-order system equation

where natural frequency

damping ratio

• Capacitor voltages and inductor currents cannot change instantaneously

DC steady-state condition

dt

tdvCti C

C

)()(

dt

tdiLtv L

L

)()(

– instantaneous changing iC(t) or vL(t)

– PC or PL (impossible!!!) since power is energy per time or v i

v or i

tt = 0

abrupt change

infinite slope

• DC steady state refers to circuits that have been connected to a DC (voltage or current) source for a very long time ( t )

– assuming that all voltages and current in the circuits have become constant

– all derivatives in the governing equations go to zero

)()()(

tfKtxdt

tdxS )()(

)(2)(12

2

2tfKtx

dt

tdx

wdt

txd

wS

nn

dt

tdvCti C

C

)()(

dt

tdiLtv L

L

)()(

iC(t) 0 as t vL(t) 0 as t

Steady-state capacitor current; Steady-state inductor voltage;

Open circuit at DC Short circuit at DC

Transient response of first-order circuit

)()()(

tfKtxdt

tdxS

VB

t = 0R

C vC(t)

Considering that f(t) is a forcing function, switched on at time t = 0;

FKtxdt

tdxS )(

)(

0)()(

txdt

tdxN

N

)()( tx

dt

tdx NN

/)( t

N etx

for t 0 I.C., x(t =0) = x(0)

solution = natural response + forced response= homogeneous solution + particular solution

natural response when t = 0

or

x(t)

t

e-1

= 0.368

FKtxdt

tdxSF

F )()(

)()( xFKtx SF

)()()()( // xeFKetxtxtx t

S

t

FN

)()0()0( xxtx )()0( xx

)1)(()0(

)()]()0([)(

/

/

t

t

exx

xexxtx

forced response for t 0

or

Then, we have the complete response

applying I.C.;

Therefore,

t

x(t)

x()

x(t)

t

x(0) + x()

x(0)

Transient response of second-order circuit

0)()()( tvtiRtv CSTT

)()( tvtv LC dt

tdiLtv L

C

)()(

0)()()()(

tidt

tdvC

R

tvtvL

C

T

CT0)()()( tititi LCS

vT

R

C vC(t) L vL(t)

iC(t) iL(t)iS(t)

Applying KVL;

Applying KCL;

)()()(

)(1

2

2

tidt

tidLC

dt

tdi

R

Ltv

RL

LL

T

T

T

)(1

)()()(

2

2

tvR

tidt

tdi

R

L

dt

tidLC T

T

LL

T

L

Then, we have;

or

dt

tdv

R

Ltv

dt

tdv

R

L

dt

tvdLC T

T

CC

T

C )()(

)()(2

2

Why not vC(t)?

0)()()( tvtiRtv CSTT 0)()()( tititi LCSfrom and

0)()()()(

tidt

tdvC

R

tvtvL

C

T

CT

we can have

There is no unique method to arrive at the final equation!

)()]()([)( tvtitiRtv CLCTT

differentiating the above equation and using

① asymptotically tending to final value of 1

– due to KS = 1 or x(t) = f(t) when steady state (all derivatives are zero)

② oscillating with a approximate period of 6 s

– due to wn = 1 or T = 2p/wn 6.28 s

③ decaying as time goes

– due to

– when = 1, no longer overshoot like that of first-order system

)()()(2)(1

2

2

2tfKtx

dt

tdx

wdt

txd

wS

nn

t

No

rma

lize

d A

mp

litu

de

1

T = 2p/wn

Let's consider the response of second-order systemto switched unit input with KS = 1, wn = 1, = 0.2

)()()(2)(1

2

2

2tfKtx

dt

tdx

wdt

txd

wS

nn

solution = natural response + forced response= homogeneous solution + particular solution

(1) Natural response

0)()(2)(1

2

2

2

tx

dt

tdx

wdt

txd

wN

N

n

N

n

st

N etx )(

012

2

2

sww

s

nn

tsts

N eetx 21

21)(

14)2(2

1 222 nnnnn wwwwws

021 2

2

stst

n

st

n

eesw

esw

or

natural response

characteristic polynomial or equation

characteristic roots s1 and s2

① overdamped solution ( > 1; real and distinct roots)

1

1

1 2

nn ww

1

1

2 2

nn ww

where

21

2221

/

2

/

1

)1(

2

)1(

121)(

tt

twwtwwtsts

N

ee

eeeetx nnnn

1 = 2 = 1, wn = 1, = 1.5

② critically damped solution ( = 1; real and repeated roots)

where

1 = 2 = 1, wn = 1, = 1

/

2

/

1

212121)(

tt

twtwtsts

N

tee

teeteetx nn

nw

1

③ underdamped solution ( < 1; complex conjugate roots)

where1 = 2 = 1, wn = 1, = 1.5

tjwwtjwwtsts

Nnnnn eeeetx

)1(

2

)1(

121

2221)(

assuming that 1 = 2 = ;

twetwe

eeetx

d

tw

n

tw

tjwtjwtw

N

nn

nnn

cos2)1cos(2

)(

2

)1()1( 22

21 nd ww damped natural frequency

– wd wn as 0

– time constant for exponential decay, = 1/ wn

– (oscillation decays quickly) as (more damping)

(2) Forced response

)()()(2)(1

2

2

2tfKtx

dt

tdx

wdt

txd

wS

nn

FKtx SF )(

FKxtx SF )()(

switched DC sources (or excitations) constant excitation all derivatives are zero

forced response (DC steady-state solution)

(3) Complete response

① overdamped case

② critically damped case

③ underdamped case

)(

)()()(

)1(

2

)1(

1

22

xee

txtxtx

tjwwtjww

FN

nnnn

)(

)()()(

)1(

2

)1(

1

22

xee

txtxtx

twwtww

FN

nnnn

)(

)()()(

21

xee

txtxtx

twtw

FN

nn

• unknown constants 1 and 2

– from two I.C.'s

)0()0( xtx

)0(')(

0

xdt

tdx

t

Laplace-transformed models of circuit elements

R

L

C

Transfer function

• Called “Network function”

• Providing the system engineer with a great deal of knowledge about the systems operation, since dynamic properties of transfer functions are governed by the system poles

• In deriving a TF, all IC’s are set equal to zero

From the Laplace transform with all the IC’s = 0,

outputor response

network or transfer function

input orforcing function

- Zeros = roots of numerator- Poles = roots of denominator

Pole-zero plot

Recall the natural responses:

Second-order network:

XX

X

X

X

Example

vi(t) C

R L

vo(t)