1. ELEC 401 MICROWAVE ELECTRONICS Lecture 2
Instructor: M. İrşadi Aksun
Acknowledgements:
Artworks describing the fields in dielectric and magnetic materials were taken from the Charged Particle Acceleration download site:

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
Frequency Domain Representations of Maxwell’s equations
Integral form
Differential form
Differential forms are easier to work with

4. Review of EM Wave Theory
ElectroMagnetic waves (EM waves) are characterized by the vector fields in Maxwell’s equations.
What forms of EM waves do we commonly encounter in practice?
Electrical signals in high-frequency circuits (voltage and current may not be enough to analyze such circuits),
signals transmitted by antennas to carry information in wireless communication systems,
data carried by any transmission medium for internet operation or for wired communication systems,
optical data carried by fiber-optic cables or in free-space, and
lots of other forms

5. Review of EM Wave Theory
How many unknowns do we need to find to characterize a system with EM waves?
E, B, H, D are all vector unknowns to be determined;
There are 12 unknowns in the form of scalar functions;
J and r are related and known source functions.
However,
Differential form
The divergence equations (the last two) are dependent on the curl equations (the first two);
No information on the medium has been introduced yet.

6. Review of EM Wave Theory
Therefore,
we have two linearly independent vector equations and four vector unknowns !!!
Independent Maxwell’s
equations
no material properties are introduced !!!
Material properties are introduced as relations between D and E, and B and H;
The simplest relations are obtained for isotropic and homogeneous media, which are the most common type of media used in practice; D = eE, B=mH, where e and m are constants.

7. Review of EM Wave Theory Constitutive Relations
Relations between D and E, and B and H in a material are known as constitutive relations.
Fields are modified by the existence of material bodies.
Behavior of Dielectric Medium

8. Review of EM Wave Theory Constitutive Relations
Behavior of Magnetic Medium

9. Review of EM Wave Theory Constitutive Relations
Conducting Materials:
- the current density J is proportional to the force per unit charge
Ohm’s law
EM, chemical
or gravitational
Conductivity
of the material
Density r of mobile charges in a piece of copper wire of 1mm in diameter (assuming each atom contributes one free-electron):
Average electron velocity:

10. Review of EM Wave Theory Constitutive Relations
Dielectric Materials:
For anisotropic and inhomogeneous
For anisotropic and homogeneous
For isotropic and homogeneous
constants
functions of space coordinates
constants

11. Review of EM Wave Theory Constitutive Relations
Dielectric Materials:
For isotropic and homogeneous
The permittivity of a medium is usually a
complex number, whose imaginary part
accounts for the loss in the material
mr=1 for non-magnetic materials

12. Review of EM Wave Theory
Let us go back to the solutions of Maxwell’s equations:
Remember that
there are two linearly independent
vector equations and four vector unknowns
medium parameters are introduced
via constitutive parameters
Now there are two vector equations and two vector unknowns

13. Review of EM Wave Theory
Wave equation in a source free, homogeneous and isotropic region
Wave equation
Helmholtz equation

14. Review of EM Wave Theory
Why do we call these equations Wave Equations?
Because the solutions represent waves, and let us see on a simple example!!!
Let us find the solution of a one-dimensional1 time-domain wave equation.
- Assume
and the medium is free-space with no source:
If the time dependence of the solution is assumed to be f(t), then the
functional form of the solutions will be
Can you see this?
1 One-dimensional refers to the fields varying in one space coordinate only.

15. Review of EM Wave Theory
Let us consider one of the solutions,
represents waves propagating in –z direction

16. Review of EM Wave Theory
Let us do the same calculation in the frequency domain
- Assume
is the time-harmonic representation of the
solution, and the medium is free-space with no source.
Helmholts Equation:
General solution in frequency domain:
General solution in time domain: