1. 16.360 Lecture 3 Last lecture: Magnetic field by constant current r I B = 2r2r II    =  r  0,  r: relative magnetic permeability  r =1 for most materials = 2r2r I   H =  B

2. 16.360 Lecture 3 Last lecture: Traveling wave y(x,t) = Acos(2  t/T-2  x/ ),  (x,t) = 2  t/T-2  x/, y(x,t) = Acos  (x,t),

3. 16.360 Lecture 3 Last lecture: Traveling wave y(x,t) = Acos(2  t/T+2  x/ ), Velocity = 0.6 /0.6T = /T Vp = dx/dt = - /T Phase velocity:

4. 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

5. 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),

6. 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tjt e Re(V) Re( I e jtjt ), )/C + = Re(I jtjt e jj 1

7. 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

8. 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)

9. 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

10. 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

11. 16.360 Lecture 3 Steps of transferring integral or differential equations to linear equations using phasor. 1.Express time-dependent variables as phsaor. 2.Rewrite integral or differential equations in phasor domain. 3.Solve phasor domain equations 4.Change phasors variable to their time domain value

12. 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

13. 16.360 Lecture 3 Transmission lines 1.Transmission line parameters, equations 2.Wave propagations 3.Lossless line, standing wave and reflection coefficient 4.Input impedence 5.Special cases of lossless line 6.Power flow 7.Smith chart 8.Impedence matching 9.Transients on transmission lines

14. 16.360 Lecture 3 Today 1.Transmission line parameters, equations Vg(t) V BB’ (t) V AA’ (t) A A’ B’ B L V AA’ (t) = Vg(t) = V0cos(  t), V BB’ (t) = V AA’ (t-t d ) = V AA’ (t-L/c) = V0cos(  (t-L/c)), V BB’ (t) = V AA’ (t) Low frequency circuits: Approximate result

15. 16.360 Lecture 3 1.Transmission line parameters, equations Vg(t) V BB’ (t) V AA’ (t) A A’ B’ B L V BB’ (t) = V AA’ (t-t d ) = V AA’ (t-L/c) = V0cos(  (t-L/c)) = V0cos(  t- 2  L/ ), Recall:  =c, and  = 2  If >>L, V BB’ (t)  V0cos(  t) = V AA’ (t), If <= L, V BB’ (t)  V AA’ (t), the circuit theory has to be replaced.

16. 16.360 Lecture 3 Next lecture 1.Types of transmission lines 2.Lumped-element model 3.Transmission line equations 4.Wave propagation