1. 4th Week Seminar
Sunryul Kim
Antennas & RF Devices Lab.

2. MICROWAVE ENGINEERING
CONTENTS
Lumped-Element Circuit Model
Field Analysis of Transmission Lines
Terminated Lossless Transmission Line
Smith Chart
Slotted Line
Quarter-Wave Transformer
Generator and Load Mismatches
Lossy Transmission Lines
MICROWAVE ENGINEERING
David M . Pozar
Antennas & RF Devices Lab.

3. Lumped-Element Circuit Model
Transit Time Effect
At some instant let the voltage at 𝐀𝐀′ be 𝑽 𝑷 .
Then 𝑽 𝑷 will appear at 𝐁 𝐁 ′ only after 𝒕 𝒓 .
However, during this time the voltage at 𝐀𝐀′ changes to 𝑽 𝑸 .
Transit time
𝒕 𝒓 = 𝒍 𝒗
The transit time effect can be neglected if
𝑻≫ 𝒕 𝒓 → 𝛌≫𝒍
Antennas & RF Devices Lab.

4. 𝑅=Ω/m , 𝐿=H/m , 𝐺=S/m , 𝐶=F/m
Lumped-Element Circuit Model
Transmission line Equations
FIGURE 2.1 Voltage and current definitions and equivalent circuit for an incremental length of transmission line.
(a) Voltage and current definitions. (b) Lumped-element equivalent circuit.
KVL (Kirchhoff’s voltage law)
KCL (Kirchhoff’s current law)
Antennas & RF Devices Lab.

5. Lumped-Element Circuit Model
Transmission line Equations
Phasor
General solution
(2.3a) , (2.6a)
Characteristic impedance
Complex propagation constant
Antennas & RF Devices Lab.

6. Lumped-Element Circuit Model
Transmission line Equations (Lossless Line)
General
Lossless
‘Real’
Antennas & RF Devices Lab.

7. Field Analysis of Transmission Lines
Parameters
Chapter 1
Circuit theory
FIGURE 2.2 Field lines on an arbitrary TEM transmission line.
Magnetic energy
Conductivity power loss
Electric energy
Dielectric power loss
Antennas & RF Devices Lab.

8. Field Analysis of Transmission Lines
Parameters
TABLE 2.1 Transmission Line Parameters for Some Common Lines
Antennas & RF Devices Lab.

9. Field Analysis of Transmission Lines
Coaxial Line (TEM Wave)
𝜕/𝜕𝜙=0
𝐸 𝑧 = 𝐻 𝑧 =0
𝜕 𝜕𝜌 𝜌 𝐸 𝜙 =0
Independent of 𝜌
Antennas & RF Devices Lab.

10. Field Analysis of Transmission Lines
Coaxial Line (TEM Wave)
Voltage between the two conductors
Eliminate
𝒉(𝒛) and 𝒈(𝒛)
Current on the inner conductor (𝜌=𝑎)
TABLE 2.1
Antennas & RF Devices Lab.

11. Terminated Lossless Transmission Line
Total Voltage and Current
FIGURE 2.4 A transmission line terminated in a load impedance 𝑍 𝐿 .
Load impedance
Rearrange
Reflection coefficient
Antennas & RF Devices Lab.

12. Terminated Lossless Transmission Line
Return Loss
Complex conjugate
→ zero
Incident power
Reflected power
Return loss
𝑉 𝑍 0
𝑉 𝑍 0 𝛤 2
𝚪=𝟎 → RL =∞
𝚪=𝟏 → RL =0
Second order!
Antennas & RF Devices Lab.

13. Terminated Lossless Transmission Line
Standing Wave Ratio
𝑉 𝑧 = 𝑉 0 + 𝑒 −𝑗𝛽𝑧 +𝛤 𝑉 0 + 𝑒 𝑗𝛽𝑧
Euler formula
Standing Wave Ratio
Antennas & RF Devices Lab.

14. Terminated Lossless Transmission Line
Input Impedance
𝑒 𝑗𝜃 = cos 𝜃 +𝑗 sin 𝜃
Antennas & RF Devices Lab.

15. Terminated Lossless Transmission Line
Short Circuit
FIGURE 2.5 A transmission line terminated in a short circuit. =∞
Antennas & RF Devices Lab.

16. Terminated Lossless Transmission Line
Open Circuit
FIGURE 2.6 A transmission line terminated in a open circuit. =∞
Antennas & RF Devices Lab.
Antennas & RF Devices Lab.

17. Terminated Lossless Transmission Line
Half-Wavelength Line & Quarter-Wavelength Line
Half-Wavelength Line
𝛽= 2𝜋 𝜆
Quarter-Wavelength Line
“Quarter-wave transformer”
=−𝛤(0)
180° phase shift !
Antennas & RF Devices Lab.

18. Terminated Lossless Transmission Line
Junction of Two Transmission Lines
FIGURE 2.9 Reflection and transmission at the junction of two transmission lines
with different characteristic impedances.
Equal when
𝑧=0
Return loss
Transmission coefficient
Insertion loss
Antennas & RF Devices Lab.

19. Smith Chart
Normalize
Rearrange
Rearrange
Antennas & RF Devices Lab.

20. Smith Chart
Antennas & RF Devices Lab.

21. Smith Chart Antennas & RF Devices Lab.
Rotate the point clockwise an amount 2𝛽ℓ
FIGURE 2.11 Smith chart for Example 2.2.
Antennas & RF Devices Lab.

22. Slotted Line Antennas & RF Devices Lab.
SWR, distance of the first voltage minimum from the load can be measured.
Load impedance
Antennas & RF Devices Lab.

23. Quarter-Wave Transformer
The Impedance Viewpoint
𝑍 𝐿 , 𝑍 0 are both real
FIGURE 2.16 The quarter-wave matching transformer.
ℓ= 𝜆 4
Antennas & RF Devices Lab.

24. Quarter-Wave Transformer
Multiple Reflection Viewpoint
180° phase shift !
numerator
FIGURE 2.18 Multiple reflection analysis of
the quarter-wave transformer.
Antennas & RF Devices Lab.

25. Quarter-Wave Transformer
Multiple Reflection Viewpoint
numerator =0
numerator =0
This is the same result as (2.63)
Antennas & RF Devices Lab.

26. Generator and Load Mismatches
FIGURE 2.19 Transmission line circuit for mismatches load and generator.
Voltage distribution
Antennas & RF Devices Lab.

27. Generator and Load Mismatches
FIGURE 2.19 Transmission line circuit for mismatches load and generator.
Standing wave ratio
Power delivered to the load
Antennas & RF Devices Lab.

28. Generator and Load Mismatches
Load Matched to Line
Lossless line → 𝑍 0 is real
→ 𝑍 in is real
Antennas & RF Devices Lab.

29. Generator and Load Mismatches
Generator Matched to Loaded Line
The power delivered to the load may be less than ‘Load Matched to Line’
Load Matched to Line
Antennas & RF Devices Lab.

30. Generator and Load Mismatches
Conjugate Matching
*Find the maximum value through derivative.
Antennas & RF Devices Lab.

31. Lossy Transmission Lines
Low-Loss Line
Taylor series expansion
Low-loss line can be closely approximated by considering the line as lossless.
Antennas & RF Devices Lab.

32. Lossy Transmission Lines
Terminated Lossy Line
FIGURE 2.20 A lossy transmission line terminated in the impedance 𝑍 𝐿 .
Lossless
tanh 𝑧 = 𝑒 𝑧 − 𝑒 −𝑧 𝑒 𝑧 + 𝑒 −𝑧
Antennas & RF Devices Lab.

33. Lossy Transmission Lines
Terminated Lossy Line
FIGURE 2.20 A lossy transmission line terminated in the impedance 𝑍 𝐿 .
Both terms increase as 𝛼 increses.
Antennas & RF Devices Lab.

34. Lossy Transmission Lines
Perturbation Method for Calculating Attenuation
*No use Transmission line parameters ( 𝐿, 𝐶, 𝑅, 𝐺 ).
*Assume that the fields of the lossy line are not greatly different from the fields of the lossless line.
Antennas & RF Devices Lab.

35. THANK YOU