EE292_ch05 _Sine_Part1_em3

Electrical Engineering: Principles and Applications, 6e Allan R. Hambley

Copyright 2014 by Pearson Education, Inc. All rights reserved.

EE292

Chapter 5. Steady State Sinusoidal Analysis

(part 1)



V = V0 exp(jwt+f),

= wV

I = I0 exp(jwt+f),

= wI

.

Steady-State Sinusoidal Current & Voltage



EE292 Capacitor complex impedance

Cvq

dt

dvCi 0

0

1tvdtti

Ctv

t

t

Complex Impedance & Phasor form

901

)2

exp(1

C

jCj

Zw

wC

CCC IZV

00

tqdttitq

t

t



EE292

00

1tidttv

Lti

t

t

dtdi

Ltv

Inductor complex impedance

Complex Impedance & Phasor form

90)2

exp( LjLLjZ w

wwL

LLL IZV



5.1 SINUSOIDAL CURRENTS AND

VOLTAGES

Vm is the peak value

is the angular frequency in radians per second is the phase angle T is the period

T

w

2

fw 2

90cossin zz

Tf

1



Root-Mean-Square Values (3 operations)

dttvT

V

T

2

0

rms

1

R

VP

2

rmsavg

dttiT

I

T

2

0

rms

1

RIP 2rmsavg

2rms

mVV RMS Value of a Sinusoid



EE292

Figure 5.1 A sinusoidal voltage waveform given by v(t) = Vm cos (t + ). Note: Assuming that is in degrees, we have tmax= T. For the waveform shown, is 45.

360



EE292 Figure 5.2 Voltage and power versus time for Example 5.1.



5.2 Phasors

111 cos :function Time tVtv

111 :Phasor VV

1V



EE292

Phasor as Rotating Vector A sinusoid can be represented as the real part of a vector rotating counterclockwise in the complex plane.

Sinusoids can be

visualized as the real-

axis projection of

vectors rotating in the

complex plane. The

phasor for a sinusoid is

a snapshot of the

corresponding rotating

vector at t = 0.



Adding Sinusoids Using Phasors

Step 1: Determine the phasor for each term.

Step 2: Add the phasors using complex

arithmetic.

Step 3: Convert the sum to polar form.

Step 4: Write the result as a time function.



Examples 5.3: Using Phasors to Add

Sinusoids

45cos201 ttv w

60cos102 ttv w

45201 V

30102 V



7.3997.29

14.1906.23

5660.814.1414.14

30104520

21s

j

jj

VVV

7.39cos97.29 ttvs w



Phase Relationships

To determine phase relationships from a

phasor diagram, consider the phasors to

rotate counterclockwise. Then when standing

at a fixed point, if V1 arrives first followed by

V2 after a rotation of , we say that V1 leads V2 by . Alternatively, we could say that V2 lags V1 by . (Usually, we take as the smaller angle between the two phasors.)



To determine phase relationships between

sinusoids from their plots versus time, find

the shortest time interval tp between positive

peaks of the two waveforms. Then, the

phase angle is

= (tp/T ) 360. If the peak of v1(t) occurs first, we say that v1(t) leads v2(t) or

that v2(t) lags v1(t).

Determining Phase Relationships



EE292

Example: Because the vectors rotate counterclockwise, V1 leads V2 by 60 (or, equivalently, V2 lags V1 by 60).



EE292 The peaks of v1(t) occur 60 before the peaks of v2(t). In other words, v1(t) leads v2(t) by 60.



EE292 5.3 Complex Impedance

9011

CCj

Zww

C

90 LLjZ wwL

RRZ 0LR

C

L



EE292 For a pure resistance, current and voltage are in phase.

RRZ 0L



EE292 For a pure inductance: Current lags voltage by 90

90 LLjZ wwL



EE292 Pure capacitance: Current leads voltage by 90

9011

CCj

Zww

C



EE292

5.4 Circuit Analysis with Phasors and Complex Impedances



Kirchhoffs Laws in Phasor Form We can apply KVL directly to phasors.

The sum of the phasor voltages equals

zero for any closed path.

The sum of the phasor currents entering a

node must equal the sum of the phasor

currents leaving.

1. Replace the time descriptions of the

voltage and current sources with the

corresponding phasors. (All of the sources

must have the same frequency.)



2. Replace inductances by their complex

impedances ZL = jL. Replace capacitances by their complex impedances ZC = 1/(jC). Resistances have impedances equal to their

resistances.

3. Analyze the circuit using any of the

techniques studied earlier in Chapter 2,

performing the calculations with complex

arithmetic.



454.141

10010050150100

1040500

13.0500100

6

jjj

jj

ZZRZ CL

15707.0454.141

30100

Z

VI s

Simple Circuit Example 1



CRj

CRjLj

CjR

CjR

LjZR

RZZZ

w

ww

w

ww

11

1

C

CL

Simple Circuit Example 2



KVL, KCL

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EE292_ch05 _Sine_Part1_em3