1. ELEC 401 MICROWAVE ELECTRONICS Lecture 3
Instructor: M. İrşadi Aksun
Acknowledgements:
An art work on the illustration of uniform plane wave was taken from
Animation on the visualization of EM waves was taken from the following web page:

2. Outline
Chapter 1: Motivation & Introduction
Chapter 2: Review of EM Wave Theory
Chapter 3: Plane Electromagnetic Waves
Chapter 4: Transmission Lines (TL)
Chapter 5: Microwave Network Characterization
Chapter 6: Smith Chart & Impedance Matching
Chapter 7: Passive Microwave Components

3. Plane Electromagnetic Waves
So far, we have only considered waves in the form of planes, that is, the field components of the wave are traveling in the z direction and have no x and y dependence.
To visualize such a wave, it would look like a uniform plane of electric and magnetic fields on the x-y plane moving in the z direction, and hence such waves are called uniform plane waves.

4. Plane Electromagnetic Waves
The source of a plane wave is supposed to be uniform over an infinite plane in order to generate uniform fields over a plane parallel to the source plane.
There is no actual uniform plane wave in nature.
However, if one observes an incoming wave far away from a finite extent source, the constant phase surface of the fields (wavefront) becomes almost spherical. Hence, the wave looks like a uniform plane wave over a small area of a gigantic sphere of wavefront, where the observer is actually located.

5. Plane Electromagnetic Waves
- It is rather easy to visualize the plane waves when the fields and propagation direction coincide with the planes of the Cartesian coordinate system:
where E0 and H0 are, in general, complex constant vectors of the electric and magnetic fields, respectively, r is the position vector (or radius vector), and k is a real propagation vector for the lossless medium
- Note that the expressions of the uniform plane waves are written based on the physical interpretation of the visualized waves.

6. Plane Electromagnetic Waves
Since these fields have to satisfy Maxwell’s equations, some relations connecting the amplitudes and propagation vectors of such fields may exit.
Substitute the mathematical descriptions of the uniform plane waves into Maxwell’s equations.
Let us start with Gauss’s law in a source-free, homogeneous and anisotropic medium:
Using the vector identity

7. Plane Electromagnetic Waves
Additional relation can be obtained by implementing the Faraday-Maxwell law as follows:
Wave number:
Intrinsic Impedance:
Propagation vector k and the electric and magnetic field vectors E0 and H0 are all orthogonal to each other.

8. Plane Electromagnetic Waves
Propagation vector k and the electric and magnetic field vectors E and H are all orthogonal to each other. E H k

9. Plane Electromagnetic Waves
Example (from “Fundamentals of Applied Electromagnetics” by F. T. Ulaby): An x-polarized electric field of a 1 GHz plane wave travels in the +z-direction in free-space, and has its maximum at (t=0, z=5 cm) with the value of 1.2p (mV/cm). Using these information, find the expressions of Electric and Magnetic field in time-domain.
For maximum

10. Plane Electromagnetic Waves
Remember that electric and magnetic fields in a uniform plane wave are related as
Substituting the electric field expression
results in the following magnetic field expression:
Direction of propagation
Intrinsic impedance

11. Plane Electromagnetic Waves
Example: Arbitrarily directed uniform plane wave. With E0 and k being the appropriate constants, which of the following expressions represents the complex electric field vector of a uniform plane time-harmonic electromagnetic wave propagating in free space along the main diagonal of the first octant of the Cartesian coordinate system, so that its direction of propagation makes equal angles with all three coordinate axes?

12. Plane Electromagnetic Waves
Example: Magnetic field from electric field in time domain. The electric field of an electromagnetic wave propagating through free space is given by E = cos(3 × 108t + x) V/m (t in s ; x in m). The magnetic field intensity vector of the wave is

13. Plane Electromagnetic Waves in Lossy Media
So far, it was assumed that the free charge density and the free current density are zero.
Such a restriction is perfectly reasonable when wave propagation is taking place in vacuum or insulating materials.
In cases of conducting materials, the flow of charge, and in general , is certainly not zero.
Ohm’s law
Any initial free charge density dissipates in a characteristic time t=e/s. That is, free charges on a conductor will flow out to the edges.
Not interested in transient, and wait for free charge to disappear.

14. Plane Electromagnetic Waves in Lossy Media
Hence, and we have
Then, a plane wave solution will look like

15. Plane Electromagnetic Waves in Lossy Media
In general,
Attenuation constant in Neper/m
Propagation constant in rad/m

16. Plane Electromagnetic Waves in Lossy Media
As an example, let us consider a uniform plane wave as
In lossy media, the magnetic field lags the electric field by a phase lag f.

17. Plane Electromagnetic Waves in Lossy Media
Explicit expressions for the basic parameters:

18. Plane Electromagnetic Waves in Lossy Media
In many applications, we deal with either good insulators or good conductors. Therefore, it is usually not necessary to use the exact expressions of a. b. and hc.
For good dielectrics: s << we
In practice, the losses in dielectrics are defined by the loss tangent of the material rather than conductivity:

19. Plane Electromagnetic Waves in Lossy Media
For good conductors: s >> we

20. Plane Electromagnetic Waves in Lossy Media
Example: Fields and Poynting vector of a wave in a lossy case The instantaneous magnetic field intensity vector of a wave propagating in a lossy non-magnetic medium of relative permittivity er=10 is given by
Determine (a) the attenuation coefficient, (b) the instantaneous electric field intensity vector, and (c) the time-average Poynting vector of the wave. a)

21. Plane Electromagnetic Waves in Lossy Mediab) c)

22. Plane Electromagnetic Waves in Lossy Media
Example: Shielding effectiveness of an aluminum foil vs. frequency. In order to prevent the electric and magnetic fields from entering or leaving a room, the walls of the room are shielded with a 1-mm thick aluminum foil. The best protection is achieved at a frequency of
(A) 1 kHz.
(B) 10 kHz.
(C) 100 kHz.
(D) 1 MHz.
(E) No difference.

23. Plane Electromagnetic Waves - Polarization
The polarization of a plane wave is defined as
the locus of the tip of the electric field
as a function of time
at a given space point
in the plane perpendicular to the direction of propagation Ex Ey Ex Ey Ex Ey
Circular
Elliptical
Linear

24. Plane Electromagnetic Waves - Polarization
Let us assume a uniform plane wave propagating in z-direction:
no z-polarized E or H field ( for uniform plane waves)
electric field may have both x- and y-components;
- Assume a sinusoidal time variation
- Since wave polarization depends on the relative position of Ey with respect to Ex at a constant position in the direction of propagation, it must be independent of the absolute phases of Ex and Ey

25. Plane Electromagnetic Waves - Polarization
The same field is represented in frequency domain (phasor form) as
Question: How can we get the locus of the tip of the electric field as a function of time at a fixed propagation distance (z=0 is usually chosen for convenience)?
Answer:
- write the electric field in time domain
- calculate the magnitude of the field
- calculate the direction of the field

26. Plane Electromagnetic Waves - Polarization
Linear Polarization: A wave is linearly polarized if both components are in phase, that is q=0.
A and B are real constants
Step 1: Write the time-domain representation
Step 2: Calculate the magnitude of the electric field vector
Step 3: Find the direction of the electric field vector

27. Plane Electromagnetic Waves - Polarization
Linear Polarization:
A=0; B=1
A=ax; B= -ay

28. Plane Electromagnetic Waves - Polarization
Circular Polarization: A wave is circularly polarized if both components of the electric field have the same magnitude (A=B) with 90-degree phase difference ( ).
Step 1: Write the time-domain representation
Step 2: Calculate the magnitude of the electric field vector
Step 3: Find the direction of the electric field vector

29. Plane Electromagnetic Waves - Polarization
Circular Polarization:
A=a; B=a;q= -900
A=a; B=a; q= 900
Left-Hand Circularly Polarized (LHCP)
Right-Hand Circularly Polarized (RHCP)

30. Plane Electromagnetic Waves - Polarization
Elliptic Polarization: A wave is elliptically polarized if both components of the electric field have different magnitudes (A B 0) with arbitrary phase difference Ex Ey
Elliptical

31. Plane Electromagnetic Waves - Polarization
Example: Determination of polarization state of a plane wave. Determine the type (linear, circular, or elliptical) of the polarization of an electromagnetic wave whose instantaneous electric field intensity vector is given by E(x, t) = [2 cos(wt + bx) − sin(wt + bx) ] V/m, where w and b are the angular frequency and phase coefficient of the wave. The polarization of the wave is
A) Linear
B) Circular
C) Elliptical
D) Need more information

32. Plane Electromagnetic Waves - Polarization
Example: Determination of polarization state of a plane wave. Determine the type (linear, circular, or elliptical) of the polarization of an electromagnetic wave whose instantaneous electric field vector is given by E(x, t) = [2 cos(wt + bx) − cos(wt + bx) ] V/m. The polarization of the wave is
A) Linear
B) Circular
C) Elliptical
D) Need more information

33. Plane Electromagnetic Waves - Polarization
Example: Determination of polarization state of a plane wave. Determine the type (linear, circular, or elliptical) of the polarization of an electromagnetic wave whose instantaneous electric field vector is given by E(x, t) = [ cos(wt + bx) − sin(wt + bx) ] V/m. The polarization of the wave is
A) Linear
B) Circular
C) Elliptical
D) Need more information

34. Plane Electromagnetic Waves - Polarization
Example: Polarization handedness for the magnetic field vector. Polarization handedness (right- or left-handed) of an elliptically polarized wave determined by considering the magnetic field vector of the wave is opposite to that obtained by viewing the electric field vector.
A) True
B) False