1. 10. Transmission Line 서강대학교 전자공학과 윤상원 교수
* “RF Circuit Design: Theory and Applications”, R. Ludwig & P. Bretchko

2. 차 례
10-1. Introduction
10-2. Transmission Lines
10-3. Equivalent Circuit
10-4. General Transmission Line Equation
10-5. Lossless transmission line
10-6. Microstrip Transmission Lines
10-7. Terminated Lossless line
10-8. Standing Waves
10-9. Special Termination Conditions
Sourced and loaded line
Power considerations for a line 3 4 6 8 11 14 20 23 28 35 38

3. 1-1. Introduction At RF and microwave frequencies
Physical size of circuit approaches to the wave- length - the phase of ac signal must be considered
At higher frequency range
For larger size of the circuits
Voltage and Current must be treated as waves
Phasor notation is very convenient
On the circuit board one dimensional analysis is possible
Distributed circuit approach must be used
Lumped element equivalent circuit approach enable us to use Basic Circuit Theory
Impedance is very important as in the Circuit Theory

4. 1-2. Transmission Lines
Two wire lines
Coaxial line

5. Transmission lines(2) Microstrip lines and Striplines
Parallel-plate transmission line

6. 1-3. Equivalent Circuit(1)

7. Equivalent circuit (2)

8. 1-4. General Transmission Line Equation
For a small segment of a transmission line
Lumped element equivalent circuit
Apply KVL and KCL

9. General Transmission Line Equation (2)
leads to the differential form as or
where propagation constant k given as

10. General Transmission Line Equation (3)
Voltage and current waves
where the characteristic impedance given as

11. 1-5. Lossless transmission line
and
Propagation constant becomes
Characteristic impedance becomes
Voltage and current waves become

12. 1-6. Microstrip Transmission Lines
Microstrip Line geometry
Assume that
‘t’ is negligible compared to ‘h’ ; t/h < 0.005
→ depend only on ‘w’, ‘h’ and er.

13. Microstrip Transmission Lines (2)
For a narrow lines ; w/h < 1
: characteristic line impedance
: wave impedance in free space
: effective dielectric constant

14. Microstrip Transmission Lines (3)
For a wide lines ; w/h > 1
Wavelength
: characteristic
impedance
: effective dielectric constant

15. Microstrip Transmission Lines (4)
Z0 and εeff are plotted as w/h and εr

16. Microstrip Transmission Lines (5)
Assuming an infinitely thin line conductor,
w/h ≤ 2 ;
w/h ≥ 2 ;

17. Microstrip Transmission Lines (6)
Corrections for nonzero strip thickness t ;

18. 1-7. Terminated Lossless line
Voltage Reflection Coefficient ;
Use standing wave concept

19. Terminated Transmission line (2)
Input impedance ;
Input impedance at ;
: Reflection coefficient
at load

20. Terminated Transmission line (3)
Reflection coeff. for various terminations ;
Open line :
Short circuit :
Impedance matched :
For a infinite transmission line ;
Phase constant :
Dispersion-free transmission line
Since

21. 1-8. Standing Waves
Shorted transmission line ;
in the time domain ;

22. Standing Waves(2)

23. Standing Waves(3) Standing wave expressions ;
Standing wave ratio(SWR) ;

24. Standing Waves(4)

25. Standing Waves(5) Graphical interpretation
Voltage standing wave ratio(VSWR)
or return loss used:
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26. 1-9. Special Termination Conditions
Input impedance of terminated line ; or

27. Special Termination Conditions(2)
Short Circuit Transmission Line

28. Special Termination Conditions(3)

29. Special Termination Conditions(4)
Open-circuit transmission line

30. Special Termination Conditions(5)

31. Special Termination Conditions(6)
Quarter-wave transmission line
in case
In case

32. Special Termination Conditions(7)
Quarter-wave transformer
impedance matching condition ;

33. 1-10. Sourced and loaded line
Phasor representation of source
Input voltage at plane AA’ ;

34. Sourced and loaded line(2)
The input reflection coeff. at plane AA’ ;
The source reflection coeff. at plane AA’ ;
The source reflection coeff. at plane BB’ ;

35. Sourced and loaded line(3)
Transmission coefficient at plane AA’ ;
At the load end (at plane BB’) ;

36. 1-11. Power considerations for a line
Time averaged power
The total power at plane AA’ ;
the complex input voltage :
input current :

37. Power considerations for a line(2)
In terms of generator voltage ;
The input and the generator impedances ;
The generator voltage in terms of

38. Power considerations for a line(3)
When the impedances are matched ;
: available power
When the source is not matched ;
: available power at AA’