1. 16.360 Lecture 3 Phasor V R (t) Vs(t)V C (t) i (t) Vs(t) = V 0 Sin(  t+  0 ), V R (t) = i(t)R, V C (t) = i(t)dt/C, Vs(t) = V R (t) +V C (t), V 0 Sin(  t+  0 ) = i(t)dt/C + i(t)R,Integral equation, Using phasor to solve integral and differential equations

2. 16.360 Lecture 3 Phasor Z(t) = Re( Z e jtjt ) Z is time independent function of Z(t), i.e. phasor Vs(t) = V 0 Sin(  t+  0 ) ) j(  0 -  /2) = Re(V 0 e jtjt e jtjt e = Re(V), V = V 0 e j(  0 -  /2),

3. 16.360 Lecture 3 Phasor i(t) = Re( I e jtjt ) ), = Re(I jtjt e i(t)dt = Re( I e jtjt )dt jj 1 V 0 Sin(  t+  0 ) = i(t)dt/C + i(t)R, time domain equation: phasor domain equation: jj jj TimePhasor V R (t) Vs(t)V C (t) i (t) V + I R, = I jCjC 1

4. 16.360 Lecture 3 Phasor domain Back to time domain: V + I R, = I jCjC 1 I = V R + 1/(j  C) = V 0 e j(  0 -  /2), i(t) = Re( I e jtjt ) = Re ( jtjt ) R + 1/(j  C) V 0 e j(  0 -  /2) e V R (t) Vs(t)V C (t) i (t) V 0 Sin(  t+  0 ) = i(t)dt/C + i(t)R,

5. 16.360 Lecture 3 An Example : V L (t) Vs(t) = V 0 Sin(  t+  0 ), V R (t) = i(t)R, V L (t) = Ldi(t)/dt, Vs(t) = V R (t) +V L (t), V 0 Sin(  t+  0 ) = Ldi(t)/dt + i(t)R,differential equation, Using phasor to solve the differential equation. V R (t) Vs(t) i (t)

6. 16.360 Lecture 3 Phasor i(t) = Re( I e jtjt ) ),= Re(I jtjt e di(t)/dt = Re(d I e jtjt )/dt jj V 0 Sin(  t+  0 ) = Ldi(t)/dt + i(t)R, time domain equation: phasor domain equation: jtjt e Re(V) Re( I e jtjt ), )L + = Re(I jtjt e jj

7. 16.360 Lecture 3 Phasor domain Back to time domain: V + I R, = I jLjL I = V R + (j  L) = R + j  L) V 0 e j(  0 -  /2), i(t) = Re( I e jtjt ) = Re ( jtjt ) R + (j  L) V 0 e j(  0 -  /2) e

8. 16.360 Lecture 3 Steps of transferring integral or differential equations to linear equations using phasor. 1.Convert the given expressions to cosine function 2.Express time-dependent variables as phsaor. 3.Rewrite integral or differential equations in phasor domain. 4.Solve phasor domain equations 5.Change phasors variable to their time domain value

9. 16.360 Lecture 3 Waves in phasor domain Recall waves, traveling wave in time domain In phasor domain + x direction - x direction

10. 16.360 Lecture 3 A question Answer: a traveling wave in phasor domain What’s this? Complex amplitude

11. 16.360 Lecture 3 Electromagnetic spectrum. Recall relation: f = v. Some important wavelength ranges: 1.Fiber optical communication: = 1.3 – 1.5  m. 2.Free space communication: ~ 700nm – 980nm. 3.TV broadcasting and cellular phone: 300MHz – 3GHz. 4.Radar and remote sensing: 30GHz – 300GHz

12. Relations for Complex Numbers Learn how to perform these with your calculator/computer

14. Summary