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

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),

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:

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

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),

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

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

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)

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

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

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

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

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

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

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.

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