1. EEE340Lecture 291 7-5: Electromagnetic Boundary Conditions E 1t =E 2t, always. B 1n =B 2n, always. For PEC, the conductor side H 2 =0, E 2 =0. (7.66b) (7.66c) (7.66a) (7.66d) (7.68)

2. EEE340Lecture 292 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. EEE340Lecture 293 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. EEE340Lecture 294 7-6.2: Source-Free Wave Equations In a region, where, Maxwell’s equations becomes (7.79a) (7.79b) (7.79c) (7.79d)

5. EEE340Lecture 295 Taking both sides on (7.79a), Hence, In the same fashion, (7.80) (7.82)

6. EEE340Lecture 296 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(  )cos(  t+  ) where I is the magnitude,  =2  f. Circuit where e(t) = E cos  t is the electromotive force (emf). In the phasor form (taking the Fourier transform), or (7.84) (7.91)

7. EEE340Lecture 297 Example 7-6: Convert a time domain expression into the phasor form (complex exponential) Solution: In comparison with (7.92a), Therefore, (7.92a)

8. EEE340Lecture 298 The complex form And the phasor form, after dropping the time conversion e j  t is

9. EEE340Lecture 299 Rules: From phasor to time-domain From time-domain to phasor See the example 3. Similar to the Laplace transform

10. EEE340Lecture 2910 7-7.2: Time-Harmonic Electromagnetics Using the rule We have the Maxwell’s equations in phasor form as The Lorentz gauge (7.98) (7.94)

11. EEE340Lecture 2911 The nonhomogeneous wave equations where The wavenumber The phasor solutions (7.100) (7.99) (7.97) (7.96) (7.95)

12. EEE340Lecture 2912 Note: Wavenumber (7.102)