AC Circuits

Sinusoidal Steady-State Analysis

Phasor

DC & AC

DC : Direct Current

AC : Alternating Current

Sinusoidal function

Sinusoidal function

Sine and Cosine Functions

Sine and Cosine Functions

Advancing in time

Delaying in time

Oscilloscope

Differential Equation with AC Source

:

cos ,

cos , (0) 1

KVL

dvRi v e t i C

dt

dvRC v t v

dt

Forced response(particular solution), steady state:

2 2

cos sin

cos sincos sin cos

sin cos cos sin cos

cos sin cos

1, 0

, 1

1,

1 1

v A t B t

d A t B tRC A t B t t

dt

RCA t RCB t A t B t t

RCB A t B RCA t

RCB A B RCA

B RCA RC RCA A

RCA B

RC RC

Forced response(particular solution), steady state:

2 2

2

1

2

1

2

cos sin

1cos sin

1 1

1cos( tan )

1

1cos( tan )

1

v A t B t

RCt t

RC RC

RCt RC

RC

t RCRC

Complete response

2 2

1

2

2

2 2

2

2 2 2

1( ) cos sin

1 1

1cos( tan )

1

1(0) 1,

1 1

1( ) cos sin

1 1 1

t

RC

t

RC

t

RC

RCv t Ke t t

RC RC

Ke t RCRC

RCv K K

RC RC

RC RCv t e t t

RC RC RC

Check Initial Condition

Check

2

2 2 2

2 2

2 2 2

2 2

2 2

0

1( ) cos sin

1 1 1

( ) 1sin cos

1 1 1

( ) 10

1 1

t

RC

t

RC

t

RC RCv t e t t

RC RC RC

RCdv t RCe t t

dt RC RC RC RC

RCdv t RC

dt RC RC RC

Phasors and Sinusoids

Sinusoidal source(a known fixed single frequency)

Steady state, forced response

Euler’s formula

Time domain, Frequency domain

cos sinje j

Euler Formula

2 3 4 5

2 4 3 5

12! 3! 4! 5!

12! 4! 3! 5!

cos sin

jj j j j

e j

j

j

Motivating Example: RC circuit

:

cos ,

cos , (0) 1

KVL

dvRi v e t i C

dt

dvRC v t v

dt

Motivating Example: RC circuit

Euler’s formula

Assume the solution:

cos sin

cos Re

cos

j t

j t

j t

e t j t

t e

dv dvRC v t RC v e

dt dt

( )j t j j tv Ve Ve e

Motivating Example: RC circuit

Assume the solution:

( )

( )( )

( ) ( )

tan

2

1 1 1

1 1

1 1

j t j j t

j tj t j t

j t j t j t j j t j j t j t

j j j

j

v Ve Ve e

dVeRC Ve e

dt

j RCVe Ve e j RCVe e Ve e e

j RCVe Ve j RC Ve

Ve ej RC RC

1

( ) 1

2

1Re cos( tan )

1

RC

j tVe t RCRC

Arithmetic using Phasors

KVL in Phasor

Impedances

Ohm’s Law for AC circuits

Impedance and admittance

( ) ( ) ( )

1 ( )( )

( ) ( )

V Z I

IY

Z V

Capacitor

( ) cos( ) V

( ) ( ) sin( ) cos( 90 ) A

C

C C

v t A t

di t C v t C A t C A t

dt

Inductor

( ) cos( ) A

( ) ( ) sin( ) cos( 90 ) V

L

L L

i t A t

dv t L i t L A t L A t

dt

Resistor

( ) cos( )

( )( ) cos( )

R

RR

v t A t

v t Ai t t

R R

Node analysis

KCL : ( ) ( ) ( )

48 75 ( ) ( ) ( )

50 80 80

( ) ( ) 50 5048 75 50 ( ) 1 ( )

80 80 80 80

C RI I I

V V V

j j

V V j jj V V

j j

Node analysis(MATLAB)

50 5048 75 1 ( )

80 80

j jV

j

>> V=48*exp(j*75*pi/180)/(j*50/(-j*80)+(j*50)/80+1)

V =

63.3158 +18.1122i

>> abs(V)

ans =

65.8555

>> angle(V)*180/pi

ans =

15.9638

Series Impedance

Parallel Impedance

1 21 2

1 2

1 2

1||

1 1eq

Z ZZ Z Z

Z Z

Z Z

Voltage and Current Division

In the frequency domain

( 80)80|| 40 40

80 80

|| 40 40 12 20

|| 50 (40 40) 17

(1.372 59.04 )(48 75 ) 65.9 15.96

C

CS S S

L C

jZ R j

j

Z R j jV V V V

Z Z R j j