1. ELEC 401 MICROWAVE ELECTRONICS Lecture 3
Instructor: M. İrşadi Aksun
Acknowledgements:
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. Review of EM Wave Theory So, we need BOUNDRY CONDITIONS
Wave equations are the governing differential equations of EM waves
Wave equation
Helmholtz equation
Facts: Wave equations
have solutions representing wave nature of the fields;
are second order partial differential equations;
have unique solutions if two boundary conditions are introduced.
So, we need BOUNDRY CONDITIONS

4. Review of EM Wave Theory
Boundary Conditions on E and H:
as Dh0
as Dh0

5. Review of EM Wave Theory
Comments on the surface current density Js :
Volume current density must go to
as Dh goes zero.
Current Sheet: Current is
distributed over a very narrow
sheet of conductor.
Facts: 1. Current flow in a good conductor is practically confined to the layer next to the surface, whose thickness depends inversely on both the conductivity of the material and the frequency of the field.
2. Conductor is modeled as a sheet of current with a finite surface current density.
3. This layer of finite thickness is called the skin depth of that specific conducting material. z

6. Review of EM Wave Theory
Boundary Conditions on D and B:
as Dh0
Similarly, from

7. Review of EM Wave Theory
Let us review the boundary conditions with words:
1. Tangential E fields are continuous
2. Tangential H fields are discontinuous
by the amount of the surface current
3. Normal components of D are discontinuous
by the amount of the surface charge
4. Normal components of B are continuous

8. Review of EM Wave Theory
Special case: Dielectric-PEC interface
1. Tangential E field is zero
2. Tangential H field is equal to the surface current density
3. Normal component of D is equal to the surface charge density z
4. Normal component of B is zero

9. Review of EM Wave Theory
So far, we have derived the governing equation of the fields involved in waves, and the necessary boundary conditions;
Now, it is time to see them in action to determine the fields in a wave generated by a time-varying current source !!!!

10. Review of EM Wave Theory
Example: Assume an infinite sheet of electric surface current density A/m is placed on a z = 0 plane between free space for z < 0 and a dielectric medium with for z > 0. Find the resulting electric and magnetic fields in both regions.
About the problem:
The surface current density is uniform on the z = 0 plane;
Magnitude of the current density is J0 with time dependence;
We assume that layers are lossless, isotropic, homogenous, and semi-infinite in extent.

11. Review of EM Wave Theory
About the solution:
The best way to solve the wave equation in a piecewise homogeneous geometry is
to find the source-free solutions with unknown coefficients in each homogeneous sub-region;
to apply the necessary boundary conditions at the interfaces to account for the boundaries between different media; and then
to apply the boundary conditions at the sources to incorporate the influence of the sources into the solution.

12. Review of EM Wave Theory
So, let us implement these steps one-by-one:
Write the frequency-domain wave equation in a source-free medium as
From the source distribution and geometry, the functional form of the solutions can be predicted.
Source and geometry are independent of x and y, and so are the solutions!!!
WHY ?

13. Review of EM Wave Theory
As we have noted earlier, the field components must satisfy Maxwell’s equations, so using
Maxwell’s equations can be written as follows:
As a result, two decoupled set of solutions are obtained:

14. Review of EM Wave Theory
Note that these solutions are possible solutions; either only one of them or both with different weights can exist depending upon the boundary conditions.
- Since the surface current density is in x-direction, it gives rise to a discontinuity only on Hy .
- Therefore, it can be interpreted that the source excites only one of the two possible solutions, with the following governing equation:

15. Review of EM Wave Theory
Then, the solutions can be written as
where A, B, C, D are unknown coefficients, and k0 and k are the wavenumbers in free-space and materials, respectively.
The corresponding magnetic field Hy can be obtained from Faraday-Maxwell equations as follows:

16. Review of EM Wave Theory
There are four unknowns (A, B, C and D) to be determined by using the necessary boundary conditions:
1. Source is at z=0, and no wave can travel from infinity towards source, implying that A and D must be zero.
2. The remaining boundary conditions are at the interface:

17. Review of EM Wave Theory
Energy Flow and Poynting’s Vector
The general law of the conservation of energy states that if an object radiates electromagnetic waves (for example, light), it loses energy.
From Maxwell’s equations, one can get the following relation:
Energy flux density (W/m2)
Instantaneous Poynting’s vector

18. Review of EM Wave Theory
For most practical applications, the time-averaged energy and power quantities are required because most systems usually respond to the average power, rather than its instantaneous values.
: time-averaged power density
For time-harmonic fields, the instantaneous fields are written as