7-5: Electromagnetic Boundary Conditions

1

• 7-5 Electromagnetic Boundary Conditions
• E1tE2t, always.
• B1nB2n, always.
• For PEC, the conductor side H20, E20.

(7.66a)
(7.66b)
(7.66c)
(7.66d)
(7.68)
2

• 7-6 Wave Equations and Time-Domain Solutions
• Retarded scalar and vector potentials are the
  solution
• to the non-homogeneous wave equation (7-65) and
  (7-63),
• respectively
• where is the propagation velocity,
  and

(7.77)
(7.78)
3

• In contrast to the static case of instantaneous
• response of
• The wave nature of (7.77) and (7.78) shows the
• retarded response to the source.

(3-61)
(6-23)
4

• 7-6.2 Source-Free Wave Equations
• In a region, where ,
  Maxwells equations
• becomes

(7.79a)
(7.79b)
(7.79c)
(7.79d)
5

• Taking both sides on (7.79a),
• Hence,
• In the same fashion,

(7.80)
(7.82)
6

• 7-7 Time-Harmonic Fields
• 7-7.1 Phasors
• Phasor (frequency domain) converts the ordinary
  differential equations (ODEs) into algebraic
  equations.
• Let I(t)I(w)cos(wtj) where I is the magnitude,
  w2pf.
• Circuit
• where e(t) E cos wt is the electromotive force
  (emf).
• In the phasor form (taking the Fourier
  transform),
• or

(7.84)
(7.91)
7

• Example 7-6 Convert a time domain expression
• into the phasor form (complex exponential)
• Solution
• In comparison with (7.92a),
• Therefore,

(7.92a)
8

• The complex form
• And the phasor form, after dropping the time
  conversion ejwt is

9

• Rules
• From phasor to time-domain
• From time-domain to phasor
• See the example
• 3. Similar to the Laplace transform

10

• 7-7.2 Time-Harmonic Electromagnetics
• Using the rule
• We have the Maxwells equations in phasor form
  as
• The Lorentz gauge

(7.94)
(7.98)
11

• The nonhomogeneous wave equations
• where
• The wavenumber
• The phasor solutions

(7.95)
(7.96)
(7.97)
(7.99)
(7.100)
12

• Note
• Wavenumber

(7.102)