上海交大基本电路理论第九章

Chapter 9 Sinusoids and Phasors

Sinusoids

A sinusoid is a signal that has the form of the sine or cosine function.

angle

phase um ent t frequency

angular am plitude

Vm where t V v m ==+==+=φφ??φ?arg )

cos(

)

cos(φ?+=t V v m φ

ω

t

f

T

ππω22==radians/second (rad/s)

f is in hertz(Hz)

)

cos()()cos()(222111φωφω+=+=t V t v t V t v m m Phase difference:

φ

θφθθθφφφωφωθby v lags v by v leads v phase in are v and v phase

of out are v and v if t t 210210210210)()(2

121<>=≠-=+-+=

Complex Number

form

l exponentia form sinusoidal form

polar form r rectangula φ

φφφj re

z jrsin rcos z r z jy x z =+=∠=+

=φ

Phasor

a phasor is a complex number representing the amplitude and phase angle of a sinusoidal voltage or current.

Eq.(8-1)

and Eq. (8-2) Eq.(8-3)

When Eq.(8-2) is applied to the general sinusoid we obtain

E q.(8-4)

The phasor V is written as

Eq.(8-5)

Fig. 8-1 shows a graphical representation commonly called

a phasor diagram.

Fig. 8-1: Phasor diagram Two features of the phasor concept need emphasis:

1.Phasors are written in boldface

type like V or I1 to distinguish

them from signal waveforms

such as v(t)and i1(t).

2. A phasor is determined by

amplitude and phase angle and

does not contain any

information about the frequency

of the sinusoid.

In summary, given a sinusoidal signal , the corresponding phasor representation is . Conversely, given the phasor , the corresponding sinusoid is found by multiplying the phasor by and reversing the steps in Eq.(8-4) as follows:

E q.(8-6)

)

cos()(φ?+=t V t v m φ

∠=Vm V Time domain

representation

Phase-domain representation

Properties of Phasors

?additive property

Eq.(8-7)

Eq.(8-8)

Eq.(8-9)

?derivative property

Eq.(8-10)

V

j dt

dv ω?∴Time domain representation

Phase-domain representation

?Integral property

Time domain representation

Phase-domain

representation

??ω

j V

vdt The differences between v(t) and V:

V(t) is the instantaneous or time-domain representation, while V is the frequency or phasor-domain representation.

2.V(t) is a real signal which is time dependent, while V is just a supposed value to simplify the analysis

The complex exponential is sometimes called a rotating phasor, and the phasor V is viewed as a snapshot of the situation at t=0.

Fig. 8-2: Complex exponential

+ j + real

-real

-j ωt

V m

θ= 0θ= 90 or π/2

θ= -90 or -π/2

θ= 180 or π1510

5

05

10

1515

13.5

12

10.597.564.5

31.5010V r ms ac signal at 0.5 Hz

voltage in volts

a n g u l a r f r e q u e n c y t i m e s t i m e i n r a d i a n s

12.566

-ω-t n

?14.142

14.142

-v rea l t ()n

5

10

15

1512

96303

69

1215

10V rms ac signal at 0.5 Hz

angular frequency times time in radians

V o l t a g e i n v o l t s

14.14214.142

-v im a g t n ()

12.566

0ωt ?()n

)

sin(is axis )(imaginary j on the phasor rotating the of projection The t V v m ima g ω?=)

cos(is axis real on the phasor rotating the of projection The t V v m rea l ω?=()()case

particular In this 5.02cos 102cos t t f V v m ??=??=ππ

EXAMPLE 8-1

(a)Construct the phasors for the following signals:

(b) Use the additive property of phasors and the phasors

found in (a) to find v(t)=v1(t)+v2(t).

SOLUTION

(a) The phasor representations of v(t)=v1(t)+ v2(t) are

(b) The two sinusoids have the same frequent so the additive property of phasors can be used to obtain their sum:

The waveform corresponding to this phasor sum is

V1

V2

1

j

V

EXAMPLE 8-2

(a)Construct the phasors representing the following signals:

(b) Use the additive property of phasors and the phasors found

in (a) to find the sum of these waveforms.

SOLUTION:

(a) The phasor representation of the three sinusoidal currents are

(b) The currents have the same frequency, so the additive property of phasors applies. The phasor representing the sum of these current is

Fig. 8-4

EXAMPLE 8-3

Use the derivative property of phasors to find the time derivative of v(t)=15cos(200t-30°).

The phasor for the sinusoid is V=15∠-30 °.According to

the derivative property, the phasor representing the dv/dt is

found by multiplying V by j

ω.

SOLUTION:

The sinusoid corresponding to the phasor jω

V is

Device Constraints in Phasor Form

V oltage-current relations for a resistor in the: (a) time domain, (b) frequency domain.

Resistor:

Re

jIm

I V

I

V m m R

I V

φφ==