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How do EM Fields Propagate?

This text provides an overview of how electromagnetic fields propagate and interact in transmission lines, electrostatics, and magnetostatics. It covers topics such as transmission line equations, characteristic impedance, reflection at loads, elimination of reflections, interaction of static charges, and Maxwell's equations.

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How do EM Fields Propagate?

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  1. 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?

  2. + + + + + + + + + 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)

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

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

  5. 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

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

  7. 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.

  8. 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....

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

  10. 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’)

  11. 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

  12. 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

  13. Consequences of Maxwell’s equations Waves Radiation

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

  15. 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)

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

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

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

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

  20. Wavelength determines application

  21. 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]

  22. 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

  23. 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

  24. 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)

  25. 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

  26. Phasor examples LC PhasorRC PhasorRL PhasorRLC Circuit

  27. 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}

  28. 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 !!

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