1. 1st Week Seminar
Sunryul Kim
Antennas & RF Devices Lab.

2. MICROWAVE ENGINEERING
Contents
1. Maxwell’s Equations
2. Boundary Conditions
3. Constitutive Parameters and Relations
4. Wave Equation
5. Energy & Power
MICROWAVE ENGINEERING
David M . Pozar
ADVANCED ENGINEERING
ELECTROMAGNETICS
Constantine A . Balanis
Antennas & RF Devices Lab.

3. Maxwell’s Equations Antennas & RF Devices Lab.
TABLE 1-1 Maxwell’s equations and the continuity equation in differential and integral forms for time-varying fields
appleii.tistory.com
Divergence : Diverging Component
Curl : Rotating Component
Antennas & RF Devices Lab.

4. Maxwell’s Equations Differential Form Antennas & RF Devices Lab.
TABLE 1-1
en.wikipedia.org
It is used to describe and relate the field vectors, current
densities, and charge densities at any point in space at any time.
Antennas & RF Devices Lab.

5. Maxwell’s Equations Integral Form Antennas & RF Devices Lab.
TABLE 1-1
The integral form of Maxwell’s equations describes the relations of the field vectors, charge
densities, and current densities over an extended region of space.
Antennas & RF Devices Lab.

6. Boundary Condition General Medium Antennas & RF Devices Lab.
FIGURE 1.6 Closed surface S for equation
(1.29)
(1.30)
Generalize
(1.31)
Boundary conditions at an arbitrary interface of materials and/or surface currents
(1.32)
(1.32) No free magnetic charge
Antennas & RF Devices Lab.

7. Boundary Condition General Medium Antennas & RF Devices Lab.
FIGURE 1.7 Closed contour C for equation
(1.33)
(1.35)
Generalize
(1.36)
Boundary conditions at an arbitrary interface of materials and/or surface currents
(1.37)
Antennas & RF Devices Lab.

8. Boundary Condition General Medium
*Intuitive understanding by conservation law
The difference in H field between medium 1 and medium 2 is caused by to the rotational component of the H field due to the electric current density.
The difference in D field between medium 1 and medium 2 is caused by to the divergent component of the D field due to the electric charge density.
Antennas & RF Devices Lab.

9. Boundary Condition General Medium (Source Free Condition)
(1.31)
(1.38a)
(1.32)
(1.38b)
(1.36)
(1.38c)
(1.37)
(1.38d)
Normal components of and are continuous across the interface.
Tangential components of and are continuous across the interface.
Antennas & RF Devices Lab.

10. Medium 1 perfect conductor
Boundary Condition
General Medium (Perfect Conductor)
(1.31)
(1.39a)
(1.32)
(1.39b)
(1.36)
Medium 1 perfect conductor
(1.39c)
(1.37)
(1.39d)
Tangential components of are “shorted out” (electric wall)
Antennas & RF Devices Lab.

11. Constitutive Parameters and Relations
Constitutive relations (Isotropic)
𝐷 = 𝜖 0 𝐸 + 𝑃 𝑒
(1.17)
𝛻× 𝐻 =𝑗𝜔 𝐷 + 𝐽
(1.20)
= 𝜖 0 𝐸 + 𝜖 0 𝜒 𝑒 𝐸
=𝑗𝜔𝜖 𝐸 +𝜎 𝐸
Total effective conductivity
= 𝜖 0 (1+ 𝜒 𝑒 ) 𝐸
=𝑗𝜔𝜖′ 𝐸 +(𝜔𝜖′′+𝜎) 𝐸
𝜖= 𝜖 ′ −𝑗𝜖′′
=( 𝜖 ′ −𝑗𝜖′′) 𝐸
Loss tangent is the ratio of the real(lossy reaction) to the imaginary(lossless reaction) part
=𝜖 𝐸
𝜒 𝑒 : Electric susceptibility
𝜖 0 : Vacuum permittivity
𝜖 𝑟 : Relative permittivity
tan 𝛿 = 𝜔𝜖′′+𝜎 𝜔𝜖′
(1.21)
(Loss tangent)
Dielectric : tan 𝛿≫1 →Polarization occurs in the dielectric and does not respond quickly to external fields
Conductor : The ratio of the conducting current density to the displacement current density
Antennas & RF Devices Lab.

12. Constitutive Parameters and Relations
Constitutive relations (Anisotropic)
Isotropic : The medium with the same direction of 𝑃 𝑒 and 𝐸 .
Anisotropic : The medium which the direction of 𝑃 𝑒 and 𝐸 are different.
(1.22)
Tensor of rank two (Dyad)
(1.26)
Antennas & RF Devices Lab.

13. Wave Equation Helmholtz Equation Antennas & RF Devices Lab.
Source-free
Linear
Isotropic
homogeneous
(1.41a)
(1.41b)
(B.14)
(Source-free)
(1.42)
Wavenumber or Propagation constant
Antennas & RF Devices Lab.

14. Wave Equation Plane Waves (Lossless Medium) Antennas & RF Devices Lab.
(1.42)
(1.44)
*Laplacian
Phasor
(1.45)
Time domain
(1.46)
*Laplacian
constant
𝑧= 𝜔𝑡− 𝑘
Antennas & RF Devices Lab.

15. Wave Equation Plane Waves (Lossless Medium) Antennas & RF Devices Lab.
Propagation Velocity
(1.47)
Wave length
(1.48)
H field
(1.45)
(1.49)
(1.41a)
Intrinsic impedance
Wave impedance; for planes waves
Antennas & RF Devices Lab.

16. Wave Equation Plane Waves (Lossless Medium) * 𝜔𝑡−𝑘𝑧=constant Why?
cos (𝜔𝑡−𝑘𝑧)
𝑡= 𝑡 1
𝑡= 𝑡 2
Position is the ‘constant’ ∆𝑧 ∆𝑡
Time and z Relation in specific phasor
Antennas & RF Devices Lab.

17. Wave Equation Plane Waves (Lossy Medium) Antennas & RF Devices Lab.
(1.51)
𝛾= Complex Propagation Constant
𝛼= Attenuation Constant
𝛽= Phase Constant
(1.52)
(1.53)
Antennas & RF Devices Lab.

18. Wave Equation Plane Waves (Lossy Medium) Antennas & RF Devices Lab.
(1.53)
(1.54)
Propagation Velocity
Phasor
Wavelength
Time domain
𝛼 : Exponential
damping factor
H field
(1.50a)
(1.56)
(1.54)
(1.58)
Antennas & RF Devices Lab.

19. Wave Equation Plane Waves (Good Conductor) Antennas & RF Devices Lab.
Conductive Current
Displacement Current
Skin depth
(1.60)
(1.59)
Amplitude
The skin depth depends on the frequency and conductivities.
As the frequency or conductivity increases, the skin effect increases and the skin depth decreases.
𝛾= Complex Propagation Constant
𝛼= Attenuation Constant
𝛽= Phase Constant
Antennas & RF Devices Lab.

20. Wave Equation General Plane Waves ( 𝑖=𝑥, 𝑦, 𝑧 )
(1.62)
(1.63)
( 𝑖=𝑥, 𝑦, 𝑧 )
(1.64)
Separation of variables
𝑓 ′′ 𝑔ℎ+𝑓 𝑔 ′′ ℎ+𝑓𝑔 ℎ ′′ + 𝑘 0 2 𝑓𝑔ℎ=0
must be a constant because it is only a function of 𝑥
(1.65)

21. Wave Equation General Plane Waves
*the terms + signs result in waves traveling in the negative 𝑥, 𝑦 or 𝑧 direction
*the terms − signs result in waves traveling in the negative 𝑥, 𝑦 or 𝑧 direction
(1.65)
𝑓 ′′ 𝑓 =− 𝑘 𝑥 2 𝑔 ′′ 𝑔 =− 𝑘 𝑦 2 ℎ ′′ ℎ =− 𝑘 𝑧 2
𝑑 2 𝑓 𝑑 𝑥 2 + 𝑘 𝑥 2 𝑓=0 𝑑 2 𝑔 𝑑 𝑦 2 + 𝑘 𝑦 2 𝑔=0 𝑑 2 ℎ 𝑑 𝑦 2 + 𝑘 𝑧 2 ℎ=0
𝑓=𝑒 ±𝑗 𝑘 𝑥 𝑥 𝑔= 𝑒 ±𝑗 𝑘 𝑦 𝑦 ℎ= 𝑒 ±𝑗 𝑘 𝑧 𝑧
𝐸 𝑥 𝑥, 𝑦, 𝑧 =𝑓𝑔ℎ
=𝐴 𝑒 −𝑗( 𝑘 𝑥 𝑥+ 𝑘 𝑦 𝑦+ 𝑘 𝑧 𝑧)
(1.68)
(1.66)
(1.69)
(1.70)
(1.67)
(1.71)

22. Wave Equation General Plane Waves H field
(1.71)
(1.72)
(1.75)
(1.73)
(1.76)
Directions of propagation and E-field are orthogonal
(1.74)

23. Energy & Power Poynting Vector Antennas & RF Devices Lab.
(1.56)
(1.67a)
Power density ( 𝑊 𝑚 2 )
(1.67b)
(*) indicates complex conjugate
(1.68)
(1.69)
Sinusoidal → Average=0
Time-average Poynting vector
(1.70)
Antennas & RF Devices Lab.

24. Energy & Power Poynting Theorem
(1.71a)
(1.72a)
(1.71b)
(1.72b)
(1.73)
(1.74)
(1.75)
‘Conservation of energy equation’ in differential form

25. Energy & Power Poynting Theorem Divergence theorem
(1.75)
Divergence theorem
(1.76)
(1.76b)
(1.76a)
(1.76c)
(1.76d)
‘Conservation of energy equation’ in integral form
(1.76e)
(1.76f)

26. Thank you