1) The document discusses Poynting's theorem and reciprocity relations for discontinuous electromagnetic fields across three cases: (1) a dielectric-PEC interface sustaining only electric surface current, (2) a dielectric-dielectric interface sustaining no surface current, and (3) an arbitrary media-media interface sustaining both electric and magnetic surface currents. 2) It introduces the concept of a characteristic function and sense of distributions to describe fields on both sides of an interface, and defines a surface impedance Z tensor to relate the electric and magnetic surface currents. 3) For Case 3 with Z=ZT, the interface satisfies reciprocity and can be treated as a reciprocal element,

1. Poynting and Reciprocity
on Discontinuous Field
( J. W. Hsu ) R98941103
( P. E. Chang ) R97943086
Panel - 15 -
Main Reference:
B. Polat, “On Poynting’s Theorem and Reciprocity Relations for Discontinuous Fields”

2. Poynting Reciprocity
Interface

3. Sense of Distributions
A(r) = {A(r)} + [A(r)]s
Regular Singular
Component Component

4. Why?
Boundary Conditions
ARE NOT postulation

5. Why?
Rigorous Treatment
in Mathematics

6. Why?
Being able to Describe Interface
( Boundary condition is weak to do this )

7. Review Boundary Trick
Choose W infinitely close to 0
to get four B.C.
Gaussian
Surface
H
/
Contour D E
W B H

8. Three Cases
which will be discussed later

9. Cases 1
Dielectric PEC
interface sustained only electric current.
no magnetic current

10. Cases 2
Dielectric 1 Dielectric 2
Interface sustained no current

11. Cases 3
arbitrary arbitrary
media media
infinite thin film
sustained electric and magnetic current

12. Review Poynting’s Thm
pin = pd + pc + pv
pi = E ⋅ Ji + H ⋅ Mi
i = d, c, v
dissipated convection
conduction

13. Cases Disscussion

14. Prerequisite 1
Characteristic function
U(f) = 1 as f > 0
U(f) = 0 as f < 0

15. Prerequisite 2
A = A1U(f) + A2U(-f) A = E, D, H, B
on Surface, It converges to Average
of Field of both side of interface.

16. Cases 1 Dielectric PEC
pin = - div P1U(f)
- div P2U(-f)
- n (P1 - P2) δ(S)
pd = (E1 Jd1 + H1 Md1)U(f)
+ (E2 Jd2 + H2 Md2)U(-f)
pc = Es Js δ(S)

17. Cases 1 Dielectric PEC
- div Pi = Ei Jdi + Hi Mdi i = 1, 2
-n (P1 - P2) = Es Js

18. Cases 2 Dielectric Dielectric
- div Pi = Ei Jdi + Hi Mdi i = 1, 2
-n (P1 - P2) = 0

19. Requisite on Surface
Media Media
Ms = - n × Z Js

20. Cases 3 Media Media
- div Pi = Ei Jdi + Hi Mdi, i = 1, 2
-n (P1 - P2) = Es Js + Hs Ms
= Z(n × Hs)2

21. Review Reciprocity
div( Ea × Hb - Eb × Ha )
=
( Eb Jaf - Hb Maf ) - ( Ea Jbf - Ha Mbf )
<a, b> <b, a>

22. Cases Discussion

23. Cases 1 Dielectric PEC
div ( Eai × Hbi - Ebi × Hai ) =
( Ebi Jai - Hbi Mai ) -
( Eai Jbi - Hai Mbi ) i = 1, 2
n × [(Ea1 × Hb1 - Eb1 × Ha1) -
(Ea2 ×Hb2 - Eb2 ×Ha2)]s
= Ebs Jas - Eas Jbs

24. Cases 2 Dielectric Dielectric
div ( Eai × Hbi - Ebi × Hai ) =
( Ebi Jai - Hbi Mai ) -
( Eai Jbi - Hai Mbi ) i = 1, 2
n × [(Ea1 × Hb1 - Eb1 × Ha1) -
(Ea2 ×Hb2 - Eb2 ×Ha2)]s
=0

25. Cases 3 Media Media
div ( Ea1 × Hb1 - Eb1 × Ha1 ) =
( Eb1 Ja1 - Hb1 Ma1 ) -
( Ea1 Jb1 - Ha1 Mb1 )
n × [(Ea1 × Hb1 - Eb1 × Ha1)]s
= (Ebs Jas - Ebs Mas)
- (Eas Jbs - Has Mbs)

26. Requisite on Surface
Media Media
Ms = - n × Z Js

27. Cases 3 application
Ebs Jas - Ebs Mas =
[(n × Has) (Z - ZT) (n × Hbs)]/2
Eas Jbs - Has Mbs =
[(n × Hbs) (Z - ZT) (n × Has)]/2

28. Cases 3 application
When Z = ZT
n × [(Ea1 × Hb1 - Eb1 × Ha1)]s = 0.
From the aspect of microwave engineering,
we could say this is a reciprocal element, so
that we say this is a Reciprocal Interface.

29. Conclusion
With sense of distribution, there is no
Boundary, only Fields Distribution on
Volume and Interface.
You can derive Z tensor from material
parameters of both side and replace
boundary conditions with Z tensor.

30. Conclusion
In the past, Fields react on interface
COULD NOT be described with
boundary conditions.
With sense of distribution, interface
could be described and treated as part
of space or as an element.

31. Simple Application 1
Regular Field Regular Field
Distribution Distribution
Z [ABCD]

32. Simple Application 2
In FDTD simulation
programs, it will cause
field broken if grid point
just locate on boundary.
Using Z tensor to
calculate effective
parameters help us to
Z εeμeσe avoid this status.

33. Thank for your attention