1. Advanced Microwave Measurements
ECE 451
Advanced Microwave Measurements
Transmission Lines
Jose E. Schutt-Aine
Electrical & Computer Engineering
University of Illinois

2. Maxwell’s Equations Faraday’s Law of Induction Ampère’s Law
Gauss’ Law for electric field
Gauss’ Law for magnetic field
Constitutive Relations

3. Why Transmission Line? Wavelength : l propagation velocity l =z
Wavelength : l
propagation
velocity l =
frequency

4. Why Transmission Line? In Free Space At 10 KHz : l = 30 km At 10 GHz :
= 3 cm
Transmission line behavior is prevalent when the structural dimensions of the circuits are comparable to the wavelength.

5. Justification for Transmission Line
Let d be the largest dimension of a circuit
circuit z l
If d << l, a lumped model for the circuit can be used

6. Justification for Transmission Line
circuit z l
If d ≈ l, or d > l then use transmission line model

7. Telegraphers’ Equations
L: Inductance per unit length.
C: Capacitance per unit length.
Assume
time-harmonic
dependence

8. TL Solutions

9. TL Solutions
(Frequency Domain)
forward wave
backward wave

10. TL Solutions Propagation constant Propagation velocity Wavelength
Characteristic impedance

11. Reflection Coefficient
At z=0, we have V(0)=ZRI(0)
But from the TL equations:
Which gives
where
is the load reflection coefficient

12. Reflection Coefficient
- If ZR = Zo, GR=0, no reflection, the line is matched
- If ZR = 0, short circuit at the load, GR=-1
- If ZR inf, open circuit at the load, GR=+1
V and I can be written in terms of GR

13. Impedance transformation equation
Generalized Impedance
Impedance transformation equation

14. Generalized Impedance
- Short circuit ZR=0, line appears inductive for 0 < l < l/2
- Open circuit ZR  inf, line appears capacitive for 0 < l < l/2
- If l = l/4, the line is a quarter-wave transformer

15. Generalized Reflection Coefficient
transformation equation

16. Voltage Standing Wave Ratio (VSWR)
We follow the magnitude of the
voltage along the TL
Maximum and minimum
magnitudes given by

17. Voltage Standing Wave Ratio (VSWR)
Define Voltage Standing Wave Ratio as:
It is a measure of the interaction between
forward and backward waves

18. VSWR – Arbitrary Load
Shows variation of amplitude along line

19. VSWR – For Short Circuit Load
Voltage minimum is reached at load

20. VSWR – For Open Circuit Load
Voltage maximum is reached at load

21. VSWR – For Open Matched Load
No variation in amplitude along line

22. Application: Slotted-Line Measurement
Measure VSWR = Vmax/Vmin
Measure location of first minimum

23. Application: Slotted-Line Measurement
At minimum,
Therefore,
So,
Since
then
and

24. Summary of TL Equations
Voltage
Current
Impedance Transformation 
Reflection Coefficient Transformation 
Reflection Coefficient – to Impedance 
Impedance to Reflection Coefficient 

25. Example 1
Consider the transmission line system shown in the figure above.
(a) Find the input impedance Zin.

26. Example 1 – Cont’
(b) Find the current drawn from the generator

27. Example 1 – Cont’
c) Find the time-average power delivered to the load

28. Example 1 – Cont’
In the system shown above, a line of characteristic impedance 50 W is terminated by a series R, L, C circuit having the values R = 50 W, L = 1 mH, and C = 100 pF.
(d) Find the source frequency fo for which there are no standing waves on the line.

29. Example 1 – Cont’
No standing wave on the line if ZR = R = Zo which occurs at the resonant frequency of the RLC circuit. Thus,