1. ELEC 401 MICROWAVE ELECTRONICS Lecture 6
Instructor: M. İrşadi Aksun
Acknowledgements:
Materials, illustrations, and some examples have been taken from the following book:
Modern Microwave Circuits, by N. Kinayman and M. I. Aksun
Artech House , Boston, 2005

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. Transmission Lines - General
In general, circuits and networks are made up of
some components that perform operations on the signals, and
some others that only carry signals from one place to another with minimum distortion on the signals.
These components that are responsible for the transmission of signals are called Transmission Lines (TL), and they come in various shapes and forms:

4. Transmission Lines - General
Coaxial line
Two-wire line
Strip line
Microstrip line

5. Transmission Lines – General
For microwave circuits and antennas, TLs play important role, as apposed to their low-frequency applications.
TLs are extremely short for low-frequency circuits in terms of wavelength, and can be modeled by considering only the bulk resistance, which is close to zero in most cases.
For microwave and antenna applications, TLs are longer, and hence their influence on the signals passing through them needs to be accounted for.

6. Transmission Lines – General
For the characterization of transmission lines in high-frequency applications, a special method that utilizes circuit theory as well as the propagation aspect of the waves is developed and called the transmission line theory.
Although the theory can be developed strictly from the circuit analysis, the major assumptions of the theory, as well as its limitations, will be missing unless its derivation via Maxwell’s equations is introduced.
Therefore, a general field-analysis approach is first provided starting from Maxwell’s equations, and then the transmission line theory will be developed.

7. Field Analysis of General Cylindrical Waveguides
General Cylindrical Waveguides (WG)
guide signals in a preferred direction, called the longitudinal direction (z-direction);
confine the fields in transverse direction (transverse to the direction of propagation, the x-y plane).
Assumptions on the structure:
it is assumed to be uniform in the direction of propagation; and
it has an arbitrary cross-section in the transverse direction.

8. Field Analysis of General Cylindrical Waveguides
WGs vs. TLs:
TLs are made up of two conducting materials, while WGs may consists of only one conductor or many conductors;
TLs support a special type of waves called transverse electromagnetic (TEM) waves, while WGs can support any type of wave configuration.

9. Field Analysis of General Cylindrical Waveguides
Observations from the geometry:
The fields are supposed to be guided along the z-direction inside the waveguide between the two conductors;
the waveguide is uniform and infinite in the z-direction;
the walls are assumed to be a perfect conductor, and
the medium between the walls is filled with a dielectric material or air.
Goal is
to find governing equations and field expressions, as it may be possible to find a simpler version of the wave equation for such a specific class of geometries.

10. Field Analysis of General Cylindrical Waveguides
To achieve the goal, some common features of the geometry may be utilized in Maxwell’s equations or in the field expressions.
For example, because
the geometry is uniform and infinite in the longitudinal direction (z-direction), and
it is used for the propagation of waves in the z- direction,
the electric and magnetic fields should be in the following form:

11. Field Analysis of General Cylindrical Waveguides
Why ?
Because
the geometry supports propagation in z- direction, variations in z-direction are predicted as exponentials with an unknown factor (propagation constant),
there is no geometrical variation along the z-direction, the magnitudes of the fields are expected to be independent of z.

12. Field Analysis of General Cylindrical Waveguides
With this form of the fields, vector quantities in Maxwell’s curl equations can be written as follows:
Substituting these into Maxwell’s equations results in
transverse
longitudinal

13. Field Analysis of General Cylindrical Waveguides
Then, following the manipulations shown below:
making use of
We obtain

14. Field Analysis of General Cylindrical Waveguides
Let us review what we have achieved:
Once the longitudinal components of the fields, (Ez ,Hz), are obtained, the rest of the field components can be calculated from
Observations: Since Maxwell’s equations are linear in linear media, it is observed that the fields can be split into three independent modes:
Transverse Electric (TE) modes:
Transverse Magnetic (TM) modes:
Transverse ElectroMagnetic (TEM) mode:
Any field configuration in a waveguide can be written in terms of TE, TM, and TEM mode contributions, and they are calculated independently.

15. Field Analysis of General Cylindrical Waveguides
TE Waves -
Governing equation for TE waves

16. Field Analysis of General Cylindrical Waveguides
TM Waves -
Governing equation for TM waves

17. Field Analysis of General Cylindrical Waveguides
TEM Waves -
is there a trivial solution only?
Not if
Governing equation for TEM waves
B.C. on PEC

18. Field Analysis of General Cylindrical Waveguides
TEM Waves -
B.C. on PEC
transverse
the coupling of the electric and magnetic fields is established

19. Field Analysis of General Cylindrical Waveguides
Salient features of TEM Waves:
fields of TEM waves are like static fields in the transverse domain, evidenced by Laplace’s equation in the transverse domain;
there exists a unique scalar potential, in cases of more than one uniform conductor in the longitudinal direction;
B.C. on PEC
the scalar potential is equivalent to voltage in electrostatic fields;

20. Field Analysis of General Cylindrical Waveguides
Salient features of TEM Waves (Cont’d):
although there seems to be no coupling between the electric and magnetic fields in TEM waves, which is not possible for any wave representation, the dynamic nature of the fields is taken into consideration by the propagation of the fields in the z-direction;
TEM waves don’t suffer from DISPERSION
Independent of frequency

21. Dispersion
Dispersion causes the shape of a wave pulse to change as it travels
Acoustics and Vibration Animations Daniel A. Russell, Ph.D., Physics Department, Kettering University

22. Dispersion
A wave packet built up from a sum of 100 cosine functions with different frequencies. In a non-dispersive medium all of the different frequency components travel at the same speed so the wave function doesn't change at all as it travels.
When the medium is dispersive, the wave changes shape. For this example, the dispersion is such that lower frequencies travel faster than higher frequencies. As a result, the wave packet spreads out with the longer wavelengths moving faster and the shorter wavelengths lagging behind.
Acoustics and Vibration Animations Daniel A. Russell, Ph.D., Physics Department, Kettering University

23. Transmission Line Analysis via Circuit Theory
Can we model TEM supporting TLs per unit length by a circuit of lumped components?

24. Transmission Line Analysis via Circuit Theory
To devise a circuit model for a transmission line, we need to understand the physical mechanisms involved. Here are the mechanisms and their circuit models:
if the conductor of TL has finite conductivity, then there is heat loss, equivalent of a series resistor;
current flowing on both conductors in opposite directions induces magnetic flux density between the conductors, which can be modeled as an inductance along the line;
two conductors with finite surface areas and finite separation distance can naturally be modeled as a capacitor between the conductors; and finally,
finite conductivity of the dielectric medium between the two conductors (i.e., the loss) would cause the current to leak through the media, and modeled as a conductance between the conductors.

25. Transmission Line Analysis via Circuit Theory
What is the governing equations for the voltage and current on the transmission line based on the circuit model of the line?
Apply Kirchoffs’ voltage and current laws for the circuit:

26. Transmission Line Analysis via Circuit Theory
Rearranging the last two equations and taking the limit of result in the following expressions:
Governing Equations
where Z and Y are the series impedance and parallel admittance per unit length on the circuit equivalent of the transmission line, respectively:

27. Transmission Line Analysis
If R=G=0 (lossless line)
where

28. Transmission Line Analysis
Define
Define
as the characteristic impedance of the line

29. Transmission Line Analysis
Let us consider a special case of lossless line (R=G=0):
If we introduce a load ZL

30. Transmission Line Analysis
Lossless line (Cont’d):
If we introduce a load ZL