Maxwell, time harmonic, transversetoz

1
Maxwell, time harmonic, transverse-to-z

• collecting all the terms,
• assuming time harmonic solutions
• using Ohms law
• assuming there is no component of either E or H
  in the z direction
• Maxwells equations reduce to
• things to notice
• Ey is connected to Hx via d/dz and w
• Ex is connected to Hy via d/dz and w
• Hy is connected to Ex via d/dz and w
• Hx is connected to Ey via d/dz and w
• Ey and Ex are connected via d/dx and d/dy
• Hy and Hx are connected via d/dx and d/dy

2
Statics transverse to z solutions

• in STATICS
• now consider a two-dimensional statics problem
  that is transverse to the z axis, i.e., the
  symmetry is such that there is no field variation
  with respect to the z coordinate, and
  furthermore, there is NO field in the z direction
• Ex ? 0, Ey ? 0, Ez 0
• in this case, the curl will simplify
• and so the STATICS curl E 0 in this geometry
  gives

3
Statics and plane waves

• now compare the statics transverse to z
  requirement on E and the Maxwell transverse to z
  plane wave requirement on E
• the two requirements are identical!!!
• the requirements on the TEM wave guarantee that
  the field also satisfies the statics equation
• there is a uniqueness theorem, so if you have one
  solution satisfying the boundary conditions, you
  have the ONLY solution!
• the STATIC solution in a 2-d problem will also be
  the shape of the field for the TEM-to-z wave!!!!

4
Transmission line plane wave summary

• the propagation constant is
• the wavelength is
• inversely dependent on frequency
• inversely dependent on square root of m and e
• the phase velocity is
• independent of frequency
• inversely dependent on square root of m and e
• the ratio of E to H is
• and the fields satisfies the 2-D statics equation

5
L-C transmission line summary

• lossless L-C transmission line

6
Transmission line applets

• various
• http//www.eecg.toronto.edu/bradel/projects/trans
  missionline/
• http//fermi.la.asu.edu/w9cf/tran/
• matching (single stub) http//home3.netcarrier.com
  /chan/EM/PROGRAMS/STUBMATCH/
• Smith chart
• pointer on chart gives gamma and impedance
• http//www.ntu.edu.sg/eee/eee5/el/smithchart/
• http//www.ee.ust.hk/eee/course/elec241/java/Smith
  Chart.html
• local link
• how tos
• http//www.web-ee.com/primers/files/SmithCharts/sm
  ith_charts.htm
• http//cnx.rice.edu/content/m1060/latest/