1. Notes 8 ECE 5317-6351 Microwave Engineering Waveguides Part 5:
Fall 2012
Prof. David R. Jackson
Dept. of ECE
Notes 8
Waveguides Part 5:
Transverse Equivalent Network (TEN)

2. Waveguide Transmission Line Model
Our goal is to come up with a transmission line model for a waveguide mode. z a b x y
The waveguide mode is not a TEM mode, but it can be modeled as a wave on a transmission line. + - z

3. Waveguide Transmission Line Model (cont.)
For a waveguide mode, voltage and current are not uniquely defined. z a b x y x y a b A B
TE10 Mode
The voltage depends on x!

4. Waveguide Transmission Line Model (cont.)
For a waveguide mode, voltage and current are not uniquely defined.
TE10 Mode x y a b x1 x2 z a b x y
Current on top wall:
Note: If we integrate around the entire boundary, we get zero current.
The current depends on the length of the interval!

5. Waveguide Transmission Line Model (cont.)
Examine the transverse (x, y) fields:
Modal amplitudes
The minus sign arises from:
Wave impedance

6. Waveguide Transmission Line Model (cont.)
Introduce a defined voltage into field equations:
We may use whatever definition of voltage we wish here.
where or

7. Waveguide Transmission Line Model (cont.)
Introduce a defined current and then from this define a characteristic impedance:
We may use whatever definition of current we wish here.
where

8. Waveguide Transmission Line Model (cont.)
Summary:

9. Waveguide Transmission Line Model (cont.)
Note on Z0:
We can define voltage and current, and this will determine the value of Z0.
Or, we can define voltage and Z0, and this will determine current.

10. Waveguide Transmission Line Model (cont.)
The transmission-line model is called the transverse equivalent network (TEN) model of the waveguide + - z

11. Waveguide Transmission Line Model (cont.)
Power flow down the waveguide:

12. Waveguide Transmission Line Model (cont.)
Set
Then

13. Waveguide Transmission Line Model (cont.)
Summary of Constants (for equal power)
Once we pick Z0, the constants are determined.
The most common choice:

14. Waveguide Transmission Line Model (cont.)
We have two constants (C1 and C2)
Here are possible constraints we can choose to determine the constants:
We can define the voltage
We can define the current
We can define the characteristic impedance
We can impose the power equality condition
Any two of these are sufficient to determine the constants.

15. Rectangular Waveguide
Example: TE10 Mode of
Rectangular Waveguide z a b x y
Method 1:
Define voltage
Define current
(This determines Z0)
Method 2:
Choose Z0 = ZTE
Assume power equality

16. Example: TE10 Mode (cont.)z a b x y
Method 1
Define:

17. Example: TE10 Mode (cont.)z a b x y
Hence
Since we have defined both voltage and current, the characteristic impedance is not arbitrary, but is determined.

18. Example: TE10 Mode (cont.)z a b x y

19. Example: TE10 Mode (cont.)z a b x y

20. Example: TE10 Mode (cont.)z a b x y
Method 2

21. Example: TE10 Mode (cont.)z a b x y
Take the conjugate of the second one and multiply the two together.
Solution:
The solution is unique to within a common phase term.

22. Example: TE10 Mode (cont.)z a b x y
(as expected)

23. Example: Waveguide Discontinuity
For a 1 [V/m] (field at the center of the guide) incident TE10 mode E-field in guide A, find the TE10 mode fields in both guides, and the reflected and transmitted powers. b a z
z = 0 x y A B
a = cm
b = cm
r = 2.54
f = 10 GHz

24. Example (cont.)
TEN
Convention:
Choose Z0 = ZTE
Assume power equality

25. Example (cont.) TEN Equivalent reflection problem:
Note: The above TL results come from enforcing the continuity of voltage and current at the junction, and hence the tangential electric and magnetic fields are continuous in the WG problem.

26. Example (cont.)
TEN
Recall that for the TE10 mode:

27. Example (cont.)
TEN
Hence, we have that

28. Example (cont.)
Substituting in, we have

29. Example (cont.)
Substituting in, we have
(wave impedance)
(our choice)

30. Example (cont.)
Substituting in, we have

31. Example (cont.)
Substituting in, we have
(our choice)

32. Example (cont.)
Summary of Fields

33. Example (cont.) Power Calculations: Alternative:
Note: In this problem, Z0 and  are real.
Alternative:

34. Example (cont.)
For a 1 [V/m] incident TE10 mode E-field in guide A (field at the center of the guide) , find the TE10 mode fields in both guide, and the reflected and transmitted powers.
Final Results: b a z
z = 0 x y A B
a = cm
b = cm
r = 2.54
f = 10 GHz

35. Discontinuities in Waveguide
Note: Planar discontinuities are modeled as purely shunt elements.
Rectangular Waveguide
(end view)
The equivalent circuit gives us the correct reflection and transmission of the TE10 mode.

36. Discontinuities in Waveguide (cont.)
Inductive iris in air-filled waveguide
Top view: z x
TE10
Higher-order mode region 1 T 
Because the element is a shunt discontinuity, we have
TEN Model
Z0TE Lp

37. Discontinuities in Waveguide (cont.)
Much more information can be found in the following reference:
N. Marcuvitz, Waveguide Handbook, Peter Perigrinus, Ltd. (on behalf of the Institute of Electrical Engineers), 1986.
Equivalent circuits for many types of discontinuities
Accurate CAD formula for many of the discontinuities
Graphical results for many of the cases
Sometimes, measured results