16.360 Lecture 3

• Phasor

VR(t)

Vs(t) VC(t)

i (t)

Vs(t) = V0Sin(t+0),

VR(t) = i(t)R,

VC(t) = i(t)dt/C,

Vs(t) = VR(t) +VC(t),

V0Sin(t+0) = i(t)dt/C + i(t)R, Integral equation,

Using phasor to solve integral and differential equations

16.360 Lecture 3

• Phasor

Z(t) = Re( Z ejt

)

Z is time independent function of Z(t), i.e. phasor

Vs(t) = V0Sin(t+0)

)j(0 - /2)= Re(V0 e

jte

jte= Re(V ),

V = V0 e j(0 - /2) ,

16.360 Lecture 3

• Phasor

i(t) = Re( I ejt

)

), = Re(I jte

i(t)dt = Re( I e jt )dt

j1

V0Sin(t+0) = i(t)dt/C + i(t)R,

time domain equation:

phasor domain equation:

)(tf f

)(tfdt

dfj

dttf )( fj

Time Phasor

VR(t)

Vs(t) VC(t)

i (t)

V + I R , = IjC

1

16.360 Lecture 3

• Phasor domain

Back to time domain:

V + I R , = IjC

1

I = V

R + 1/(jC)

= R + 1/(jC)

V0 e j(0 - /2)

,

i(t) = Re( I ejt

) = Re ( jt

) R + 1/(jC)

V0 e j(0 - /2)

e

VR(t)

Vs(t) VC(t)

i (t)

V0Sin(t+0) = i(t)dt/C + i(t)R,

16.360 Lecture 3

• An Example :

VL(t)

Vs(t) = V0Sin(t+0),

VR(t) = i(t)R,

VL(t) = Ldi(t)/dt,

Vs(t) = VR(t) +VL(t),

V0Sin(t+0) = Ldi(t)/dt + i(t)R, differential equation,

Using phasor to solve the differential equation.

VR(t)

Vs(t)

i (t)

16.360 Lecture 3

• Phasor

i(t) = Re( I ejt

)

), = Re(Ijt

e

di(t)/dt = Re(d I e jt )/dt

j

V0Sin(t+0) = Ldi(t)/dt + i(t)R,

time domain equation:

phasor domain equation:

jte Re(V ) Re( I e

jt), )L + = Re(I

jtej

16.360 Lecture 3

• Phasor domain

Back to time domain:

V + I R, = I jL

I = V

R + (jL)

= R + jL)

V0 e j(0 - /2)

,

i(t) = Re( I ejt

) = Re ( jt

) R + (jL)

V0 e j(0 - /2)

e

16.360 Lecture 3

• Steps of transferring integral or differential equations to linear equations using phasor.

1. Convert the given expressions to cosine function2. Express time-dependent variables as phsaor.3. Rewrite integral or differential equations in phasor domain.4. Solve phasor domain equations5. Change phasors variable to their time domain value

16.360 Lecture 3

• Waves in phasor domain

Recall waves, traveling wave in time domain

)22

cos(),( 0

tT

xAtxy

In phasor domain

02

)(

xjAexy + x direction

- x direction02

)(

xjAexy

16.360 Lecture 3

• A question

Answer: a traveling wave in phasor domain

What’s this?

xjAexy

2

)(

Complex amplitude

16.360 Lecture 3

• Electromagnetic spectrum.

Recall relation: f = v.

• Some important wavelength ranges:

1. Fiber optical communication: = 1.3 – 1.5m.2. Free space communication: ~ 700nm – 980nm.3. TV broadcasting and cellular phone: 300MHz – 3GHz. 4. Radar and remote sensing: 30GHz – 300GHz

Relations for Complex Numbers

Learn how to perform these with your calculator/computer

Summary