How do EM Fields Propagate?

Transmission Line LOAD How do EM Fields Propagate? • Transmission line equations • Characteristic impedance of a line • How much is reflected at a load? • How would one eliminate this reflection? • What about transient pulses?

+ + + + + + + + + How do static charges interact? (Electrostatics) F2 F1 r Coulomb’s Law Force  q1q2/r2 q2 q1 Gauss’s Law Flux  q1 + q2 + .. Interaction in materials (Polarization)

E = lim F/q q  0 Vector Fields(A “disturbance in the force”) q

Non-negligible q Vector Fields(A disturbance in the force)

I1 i21 I dl2 i12 R I2 dl1 r H How do magnetic fields interact?(Magnetostatics) Biot-Savart’s Law Current  magnetic field H ~ I/r Another current senses it Force ~ i2 x H

I r H Time variation couples E and H Ampere’s Law Varying E produces H Faraday’s Law Varying H produces E

E E E E H H H Varying E produces H produces E produces H .. Faraday’s Law Ampere’s Law Coulomb/Gauss’ Law Gauss’ law for magnets Electrodynamics Maxwell’s Eqns.

Faraday’s Law Ampere’s Law Coulomb/Gauss’ Law Gauss’ law for magnets Electrodynamics Don’t worry about memorizing these yet – We will come back to these later. But meanwhile, let’s try to understand them qualitatively....

I r H Deciphering Maxwell’s equations Q Magnetic fields curl but don’t diverge Electric fields diverge but don’t curl

I r H Deciphering Maxwell’s equations Q Magnetic fields curl but don’t diverge (they loop on themselves since there are no magnetic poles) Electric fields diverge but don’t curl (they start and end on charges or ‘poles’)

I r H We thus have Maxwell’s equations in theirsimplest form (for static sources, in vacuum) Q Curl(H)  I Div(H) = 0 Div(E)  Q Curl(E) = 0 We will define Div and Curl precisely later on. For now, think of them as the number of diverging and curling lines respectively

E • E • E • E • H • H • H Note how E and H equations are independent of each other !! This is true for static sources For dynamic sources (time-dependent currents), you also get dH/dt terms for the E equations and dE/dt terms for the H equations, which couple them. • Varying E produces H produces E produces H .. We thus have Maxwell’s equations in theirsimplest form (for static sources, in vacuum) Curl(H)  I Div(H) = 0 Div(E)  Q Curl(E) = 0

Consequences of Maxwell’s equations Waves Radiation

Traveling Waves Direction of propagation Wavefront f+(x-vt)

f+(x) t=0 x0=vt0 f+(x-x0) x1=vt1 t=t0 f+(x-x1) t=t1 xt=vt f+(x-xt) t x1 xt x0 Wave propagation f+(x-vt)

Sinusoid ASin(x-vt) For propagation direction, only relative sign between x and t matters

Sinusoid ASin(x+vt) For propagation direction, only relative sign between x and t matters

Decaying Sinusoid Ae-axSin(x-vt) Propagation in a lossy medium

Electromagnetic Spectrum Application determined by wavelength http://lectureonline.cl.msu.edu/~mmp/applist/Spectrum/s.htm

Wavelength determines application

Periodicity/Wavelength y = Asin[2p(t/T – x/l)] Frequency f = 1/T Angular Frequency w=2pf = 2p/T Wavenumber n = 1/l Wavevector/Prop const • = 2p/l y = Asin[2pt/T] y = Asin[2px/l] ~ y = Asin[wt-bx]

ejq = cos(q) + jsin(q) Euler’s Formula Profound!! Connects algebra with trig! Converts PDEs into algebraic equations!! d/dq[ej(aq)] = a[ej(aq)] ∫dqej(aq) = ej(aq)/a Trig identities become algebraic identities!! ejq1.ejq2 = ej(q1+q2) Complex #s, Phasors

ejq = cos(q) + jsin(q) • e-jq = cos(q) - jsin(q) • cos(q) = [ejq+e-jq]/2 • sin(q) = [ejq-e-jq]/2j sin(A+B) = [ej(A+B)-e-j(A+B)]/2j = [ejA.ejB-e-jA.e-jB]/2j = ([cosA+jsinA][cosB+jsinB]-[cosA-jsinA][cosB-jsinB])/2j = ([cosAcosB+jsinAcosB+jcosAsinB-sinAsinB] -[cosAcosB-jsinAcosB-jcosAsinB+sinAsinB])/2j = sinAcosB-cosAsinB

ejq = cos(q) + jsin(q) z = |z|ejq = |z|q z* = |z|e-jq = |z|-q z1z2= (|z1|ejq1)(|z2|ejq2) = |z1||z2|(q1 + q2) Complex #s, Phasors • zn=|z|nnq • z1/2=|z|1/2q/2 (q unknown upto2pm) • z-1=|z|-1-q • z1/z2= (|z1|ejq1)/(|z2|ejq2) = |z1|/|z2|(q1 - q2)

z = a + jb z* = a – jb  Complex Conjugate zz* = a2 + b2 |z| = zz* = (a2+b2)  Magnitude/Norm/Amplitude 1/z = z*/|z|2 = (a-jb)/(a2+b2)  Rationalizing z=|z|ejq= |z|(cosq + jsinq)  Phasor notation |z|cosq = a, |z|sinq = b  Components |z| = (a2+b2), q = tan-1(b/a) Complex #s

Phasor examples LC PhasorRC PhasorRL PhasorRLC Circuit

R L coswt+jsinwt V0cos(wt) ~ ~ i =Re(iejwt) ~ V = Re(Vejwt) ~ ~ [R + jLw] i = V ~ ~ ~ i = V/[R+jLw] = V[R-jLw]/{R2+L2w2} cos(wt) = Re[ejwt] sin(wt) = Im[ejwt] = Re[-jejwt] = Re[ej(wt-p/2)] Ri + Ldi/dt = V0cos(wt) i = Re(V0ejwt[R-jLw]/{R2+L2w2}) = V0(Rcoswt+Lwsinwt)/{R2+L2w2}

R L V0cos(wt) ~ ~ i =Re(iejwt) ~ V = Re(Vejwt) ~ ~ [R + jLw] i = V ZR ZL(w) Impedance Ri + Ldi/dt = V0cos(wt) ZC(w) = 1/jwC = -j/wC Try an LC circuit !!

How do EM Fields Propagate?