Fundamentals of Electromagnetic Fields |
31 |
|||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
e2 |
|
|
2 |
|
|
|
|||
|
cos u |
1 − √ |
|
|
|
|
− sin |
|
u1 |
|
||||||||
r = |
e1 |
|
(2.91) |
|||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
e2 |
− sin2 u 1 + cos u 1 |
||||||||||||||
|
|
|
|
|
||||||||||||||
|
|
|
|
|
||||||||||||||
|
√e1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
t = |
|
|
|
|
|
2 cos u |
1 |
|
|
|
|
(2.92) |
||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
e2 |
− sin2 u 1 + cos u 1 |
|||||||||||||||
|
|
|
|
|||||||||||||||
|
|
|
|
|||||||||||||||
|
√e1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
Figure 2.8 shows the behavior of the reflection coefficient as a function |
||||||||||||||||||
of the angle of incidence for both polarizations, when n 1 |
< n 2 , that is, |
|||||||||||||||||
e1 < e2 . In case of the parallel polarization, the reflection coefficient is equal |
||||||||||||||||||
to zero at Brewster’s angle |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
uB = arcsin |
|
|
|
|
|
|
e2 |
|
|
|
(2.93) |
|||||||
√e2 |
+ e 1 |
|
|
|||||||||||||||
|
|
|
|
|
|
|
|
|
|
|||||||||
If e1 > e2 , a total reflection occurs at angles of incidence |
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
√ |
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
e 1 |
|
|
|
|
||||
|
u 1 ≥ arcsin |
|
|
|
|
e2 |
|
|
|
(2.94) |
||||||||
|
|
|
|
|
|
|
|
|
|
|||||||||
2.7 Energy and Power
Let us consider the principle of energy conservation in volume V, which is enclosed by a surface S. The medium filling the volume V is characterized
Figure 2.8 The reflection coefficient for the parallel ( r || ) and perpendicular ( r ) polarization as a function of the angle of incidence u 1 , when e1 < e2 .
32 Radio Engineering for Wireless Communication and Sensor Applications
by er , mr , and s. Let us assume that in the volume V there are the electromagnetic sources J and M (M is the magnetic current density; see Section 2.1), which cause fields E and H. The complex power that these sources produce is
2 |
E |
|
|
|
Ps = − |
1 |
|
(E ? J * + H* ? M) dV |
(2.95) |
|
|
|||
V
In a sinusoidal steady-state case, the time-averaged stored electric energy in the volume V is
|
|
4 |
E |
|
4 |
|
E |
|
|
|
|
W |
e |
= |
e 0 |
|
e ′ E ? E* dV = |
e0 |
|
|
e ′ | E |2 |
dV |
(2.96) |
|
|
|
|||||||||
|
|
|
|
r |
|
|
|
r |
|
|
|
|
|
|
|
V |
|
|
|
V |
|
|
|
Accordingly, the time-averaged stored magnetic energy in the volume V is
|
|
4 |
|
E |
|
4 |
|
E |
|
|
|
|
W |
m |
= |
m 0 |
|
|
m ′ H ? H* dV = |
m 0 |
|
|
m ′ | H |2 |
dV |
(2.97) |
|
|
|
|
|||||||||
|
|
|
|
|
r |
|
|
|
r |
|
|
|
|
|
|
|
|
V |
|
|
|
V |
|
|
|
Using Poynting’s vector we can calculate the power flow out of the closed surface S
2 |
|
R |
|
|
|
Po = |
1 |
Re |
|
E × H* ? d S |
(2.98) |
|
|
||||
S
Power dissipated in the volume V due to conduction, dielectric, and magnetic losses is
P l = 12 Es | E |2 dV + v2 EXe 0 er″ | E |2 + m0 mr″ | H |2 C dV (2.99)
V V
According to the energy conservation principle, the power delivered by the sources in the volume V is equal to the sum of the power transmitted through the surface S and power dissipated in the volume, plus 2v times the net reactive energy stored in the volume. This principle is called Poynting’s theorem, which can be written as
Fundamentals of Electromagnetic Fields |
33 |
Ps = Po + P l + 2 jv(Wm − We ) |
(2.100) |
Equation (2.100) combines the powers and energies presented in (2.95)– (2.99).
References
[1]Feynman, R. P., R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II, Reading, MA: Addison-Wesley, 1964.
[2]Collin, R. E., Foundations for Microwave Engineering, 2nd ed., New York: IEEE Press, 2001.
[3]Gardiol, F. E., Introduction to Microwaves, Dedham, MA: Artech House, 1984.
[4]Kong, J. A., Electromagnetic Wave Theory, New York: John Wiley & Sons, 1986.
[5]Kraus, J., and D. Fleisch, Electromagnetics with Applications, 5th ed., Boston, MA: McGraw-Hill, 1998.
[6]Pozar, D. M., Microwave Engineering, 2nd ed., New York: John Wiley & Sons, 1998.
[7]Ramo, S., J. Whinnery, and T. van Duzer, Fields and Waves in Communication Electronics, New York: John Wiley & Sons, 1965.
[8]Van Bladel, J., Electromagnetic Fields, Washington, D.C.: Hemisphere Publishing, 1985.