“Circuit Elements in the Phasor Domain”

Chapter I

Sinusoidal Steady State Analysis

Objectives

2

• To introduce Phasors and convert the time domain sinusoidal

waveform into Phasors.

• To develop the phasor relationships for the basic circuit

elements.

• To solve electric circuits in phasor domain.

Sinusoid-Phasor Transformation

3

Phasor Domain Sources

• Convert time domain elements and sources into phasors

Time Domain Phasor Domain

Circuit Elements in Frequency Domain

5

VV

i

I

0 t

RV

icos( )mi I t

cos( )

cos( )

m

m

v Ri RI t

V t

rms rmsV RI

rms m

rms m

V VR

I I

Then：

1. Resistance

IV

O0

6

V, i have the same

frequency，V leads i

by 90o

i

Vπ 2π

0 ωt

2. Inductance

7

Relationship between voltage and current

sinmi I t

0cos sin( 90 )m m

div L LI t V t

dt

rms rmsV LI m mV LI

Reactance XL

sinmi I t 0sin( 90 )mv V t

Sinusoidal response of Inductance

8

Effective Value

9

i

v C

sinmv V t

0

0

sin( 90 )

sin( 90 )

m

m

dvi C C U t

dt

I t

Current leads voltage by 90°

2m m mI CV fCV Vi

20

3. Capacitors

Relationship between voltage and current

11

Summary of voltage-current relationship

Element Time domain Frequency

domain

R

L

C

Riv RIV

dt

diLv LIjV

dt

dvCi

Cj

IV

Example

14

Answer: i(t) = 30 cos(100t + 60o) mA

15

The Impedance

Kirchhoff’s Laws in the Frequency Domain

16

• Both KVL and KCL are hold in the phasor domain or more commonly

called frequency domain.

• Moreover, the variables to be handled are phasors, which are complex

numbers.

• Series and parallel combinations are the same as in D.C. circuits

analysis.

• All the mathematical operations involved are now in complex domain.

Example

Refer to Figure below, determine v(t) and i(t).

Answer

Series and parallel Impedances

Series parallel

Example

Solution

Step 1: Construct the phasor domain

circuit

Step 2: Apply circuit analysis tehnique

Impedance Combinations

20

Determine the input impedance of the circuit in figure below at ω =10 rad/s.

Answer: Zin = 32.38 – j73.76