Introduction to Electric Circuits

Chapter 10 Sinusoidal Steady-State AnalysisPart 1

Sinusoidal function

Sinusoidal function

Sine and Cosine Functions

Sine and Cosine Functions

Advancing in time

Delaying in time

Oscilloscope

Phasors and Sinusoids

Sinusoidal source(a known fixed single frequency) Steady state, forced response

Euler’s formula

Time domain, Frequency domain

cos sinje j

Motivating Example: RC circuit

:

cos ,

cos , (0) 1

KVLdvRi v e t i Cdt

dvRC v t vdt

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 edt dt

( )j t j j tv Ve Ve e

Motivating Example: RC circuit Assume the solution:

( )

( )( )

( ) ( )

tan

2

1 1 11 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 edVeRC Ve e

dtj RCVe Ve e j RCVe e Ve e ej 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 IIY

Z V

Capacitor( ) cos( ) V

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

C

C C

v t A tdi t C v t C A t C A tdt

Inductor( ) cos( ) A

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

L

L L

i t A tdv t L i t L A t L A tdt

Resistor( ) cos( )

( )( ) cos( )

R

RR

v t A tv t Ai t t

R R

Series Impedance

Parallel Impedance

1 21 2

1 2

1 2

1|| 1 1eqZ ZZ Z Z

Z ZZ Z

Voltage and Current Division

In the frequency domain

( 80)80|| 40 4080 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 jj

Z R j jV V V VZ Z R j j