1. MICROWAVE ENGINEERING by David Pozar Chapter 1.2 – 1.7
Harim KIM

2. Contents 1. Electromagnetic Theory 1.2 Maxwell’s Equations
1.3 Fields in Media and Boundary Conditions
1.4 The Wave Equation and Basic Plane Wave Solutions
1.5 General Plane Wave Solutions
1.6 Energy and Power
1.7 Plane Wave Reflection from a Media Interface
Antennas & RF Devices Lab.

3. Differential Form of Maxwell Equations
In free-space the following simple relations between the electric and magnetic field intensities and flux densities.

4. 1.2 Maxwell Equations

5. 1.3 Fields in Media and Boundary Condition
Loss tangent is defined as

6. 1.3 Fields in Media and Boundary Condition
Fields at a General Material Interface
Case of plane interface between two media. Maxwell’s Equations can be used to deduce normal and tangential fields at surface.
Applying Maxwell’s equations in integral form can be used to deduce conditions involving the normal and tangential fields at this interface.

7. 1.3 Fields in Media and Boundary Condition
Fields at a General Material Interface
Applying Maxwell’s equations in integral form of (1.6).
The Dirac delta function can be used to write (1.34) or

8. 1.3 Fields in Media and Boundary Condition
Fields at a Dielectric Interface
At an interface between two lossless dielectric materials, no charge or surface current densities will ordinarily exist
Fields at the Interface with a Perfect Conductor (Electric Wall)
In this case of a perfect conductor, all field components must be zero inside the conducting region. This result can be seen by considering a conductor with finite conductivity(σ < ∞) and noting that the skin depth goes to zero as σ → ∞.

9. 1.3 Fields in Media and Boundary Condition
The Magnetic Wall Boundary Condition
This boundary does not really exist in practice but may be approximated by a corrugated surface or in certain planar transmission line problem.
The Radiation Condition
This boundary condition is known as the radiation condition and is essentially a statement of energy conservation. It states that, at an infinite distance from a source, the fields must either be vanishingly small (i.e., zero) or propagating in an outward direction.

10. 1.4 The Wave Equation and basic Plane Wave Solutions
The Helmholtz Equation
We can derive
Plane Waves in a Lossless Medium
From Helmholtz Equation

11. 1.4 The Wave Equation and basic Plane Wave Solutions
Plane Waves in a General Lossy Medium
Plane Waves in a Good Conductor
Skin depth

12. 1.5 General Plane Wave Solutions
This equation can be solved by the method of separation of variables, a standard technique for treating such partial differential equations.
The magnetic field can be found from Maxell’s equation,
to give

13. 1.5 General Plane Wave Solutions
Circularly Polarized Plane Waves
The time domain form of this field is

14. 1.6 Energy and Power
In the sinusoidal steady-state case, the time-average stored electric energy in a volume V is given by
Similarly, the time-average magnetic energy stored in Volume V is
And now we can derive Poynting’s theorem, which leads to energy conservation for electromagnetic fields and sources.
by using two results in vector identity gives
by using two results in vector identity gives

15. 1.6 Energy and Power
First integral on the right-hand side of (1.88) represents complex power flow out of the closed surface S. This power can be expressed as

16. Power Absorbed by a Good Conductor
1.6 Energy and Power
Power Absorbed by a Good Conductor
Interface between a lossless medium and a good conductor. The real average power is
where

17. 1.7 Plane Wave Reflection from a Media Interface
A number of problems in electromagnetic fields at the interface of a plane wave normally incident from free-space onto a half-space of an arbitrary material.
General Medium
And the reflection fields can be written
And the transmitted fields can be written as

18. 1.7 Plane Wave Reflection from a Media Interface
Lossless Medium
The wavelength and phase velocity in the dielectric is
Power conservation for the incident, reflection and transmitted waves. And computing Poynting vectors. For z < 0 the complex Poynting vector is

19. 1.7 Plane Wave Reflection from a Media Interface
Good Conductor
from section (1.4)
Perfect Conductor
It has zero real part and this indicates that no real power is delivered to the perfect conductor.

20. 1.7 Plane Wave Reflection from a Media Interface
The Surface Impedance Concept
The surface impedance concept is an approximate but very convenient. a plane wave normally incident on this conductor is mostly reflected, and the power that is transmitted into the conductor is dissipated as heat within a very short distance from the surface. There are some ways to compute this power.
Another way is to compute the power flow into the conductor using the Poynting vector since all power entering the conductor at z = 0 is dissipated.

21. Thank you for your attention
Antennas & RF Devices Lab.