1. Chapter 2. Transmission Line Theory
Sept. 29th, 2008

2. 2.1 Transmission Lines
A transmission line is a distributed-parameter network, where voltages and currents can vary in magnitude and phase over the length of the line.
Lumped Element Model for a Transmission Line
Transmission lines usually consist of 2 parallel conductors.
A short segment Δz of transmission line can be modeled as a lumped-element circuit.

3. Figure Voltage and current definitions and equivalent circuit for an incremental length of transmission line. (a) Voltage and current definitions. (b) Lumped-element equivalent circuit.

4. R = series resistance per unit length for both conductors
L = series inductance per unit length for both conductors
G = shunt conductance per unit length
C = shunt capacitance per unit length
Applying KVL and KCL,

5. Dividing (2.1) by Δz and Δz  0,
 Time-domain form of the transmission line, or telegrapher, equation.
For the sinusoidal steady-state condition with cosine-based phasors,

6. Wave Propagation on a Transmission Line
By eliminating either I(z) or V(z):
where the complex propagation constant. (α = attenuation constant, β = phase constant)
Traveling wave solutions to (2.4):
Wave propagation in +z directon
Wave propagation in -z directon

7. Applying (2.3a) to the voltage of (2.6),
If a characteristic impedance, Z0, is defined as
(2.6) can be rewritten

8. Converting the phasor voltage of (2.6) to the time domain:
The wavelength of the traveling waves:
The phase velocity of the wave is defined as the speed at which a constant phase point travels down the line,

9. Lossless Transmission Lines
R = G = 0 gives or
The general solutions for voltage and current on a lossless transmission line:

10. The wavelength on the line:
The phase velocity on the line:

11. 2.2 Field Analysis of Transmission Lines
Transmission Line Parameters
Figure 2.2 (p. 53) Field lines on an arbitrary TEM transmission line.

12. The time-average stored magnetic energy for 1 m section of line:
The circuit theory gives 
Similarly,

13. Power loss per unit length due to the finite conductivity (from (1
Circuit theory 
(H || S)
Time-average power dissipated per unit length in a lossy dielectric (from (1.92))

14. Circuit theory 
Ex 2.1 Transmission line parameters of a coaxial line
Table 2.1

15. Eq. (2.3) can also be obtained from ME.
The Telegrapher Equations Derived form Field Analysis of a Coaxial Line
Eq. (2.3) can also be obtained from ME.
A TEM wave on the coaxial line: Ez = Hz = 0.
Due to the azimuthal symmetry, no φ-variation
 ə/əφ = 0
The fields inside the coaxial line will satisfy ME.
where

16. Since the z-components must vanish,
From the B.C., Eφ = 0 at ρ = a, b  Eφ = 0 everywhere

17. The voltage between 2 conductors
The total current on the inner conductor at ρ = a

18. The characteristic impedance of the coaxial line
Propagation Constant, Impedance, and Power Flow for the Lossless Coaxial Line
From Eq. (2.24)
For lossless media,
The wave impedance
The characteristic impedance of the coaxial line
Ex 2.1

19. Power flow ( in the z direction) on the coaxial line may be computed from the Poynting vector as
The flow of power in a transmission line takes place entirely via the E & H fields between the 2 conductors; power is not transmitted through the conductors themselves.

20. 2.3 The Terminated Lossless Transmission Lines
The total voltage and current on the line

21. The total voltage and current at the load are related by the load impedance, so at z = 0
The voltage reflection coefficient:
The total voltage and current on the line:

22. It is seen that the voltage and current on the line consist of a superposition of an incident and reflected wave.  standing waves
When Γ= 0  matched.
For the time-average power flow along the line at the point z:

23. When the load is mismatched, not all of the available power from the generator is delivered to the load. This “loss” is return loss (RL):
RL = -20 log|Γ| dB
If the load is matched to the line, Γ= 0 and |V(z)| = |V0+| (constant)  “flat”.
When the load is mismatched,

24. A measure of the mismatch of a line, called the voltage standing wave ratio (VSWR)
From (2.39), the distance between 2 successive voltage maxima (or minima) is l = 2π/2β = λ/2 (2βl = 2π), while the distance between a maximum and a minimum is l = π/2β = λ/4.
From (2.34) with z = -l,

25. At a distance l = -z,
 Transmission line impedance equation

26. Special Cases of Terminated Transmission Lines
Short-circuited line
ZL = 0  Γ= -1

27. Figure 2.6 (a) Voltage, (b) current, and (c) impedance (Rin = 0 or ) variation along a short-circuited transmission line.

28. Open-circuited line
ZL = ∞  Γ= 1

29. Figure 2.8 (a) Voltage, (b) current, and (c) impedance (Rin = 0 or ) variation along an open-circuited transmission line.

30. Terminated transmission lines with special lengths.
If l = λ/2, Zin = ZL.
If the line is a quarter-wavelength long, or, l = λ/4+ nλ/2 (n = 1,2,3…), Zin = Z02/ZL.  quarter-wave transformer

31. Figure 2.9 (p. 63) Reflection and transmission at the junction of two transmission lines with different characteristic impedances.

32. 2.4 The Smith Chart
A graphical aid that is very useful for solving transmission line problems.
Derivation of the Smith Chart
Essentially a polar plot of the Γ(= |Γ|ejθ).
This can be used to convert from Γto normalized impedances (or admittances), and vice versa, using the impedance (or admittance) circles printed on the chart.

33. Figure 2.10 (p. 65) The Smith chart.

34. If a lossless line of Z0 is terminated with ZL, zL = ZL/Z0 (normalized load impedance),
Let Γ= Γr +jΓi, and zL = rL + jxL.

35. The Smith chart can also be used to graphically solve the transmission line impedance equation of (2.44).
If we have plotted |Γ|ejθ at the load, Zin seen looking into a length l of transmission line terminates with zL can be found by rotating the point clockwise an amount of 2βl around the center of the chart.

36. Smith chart has scales around its periphery calibrated in electrical lengths, toward and away from the “generator”.
The scales over a range of 0 to 0.5 λ.

37. Ex 2.2 ZL = 40+j70, l = 0.3λ, find Γl, Γin and Zin
Figure (p. 67) Smith chart for Example 2.2.

38. The Combined Impedance-Admittance Smith Chart
Since a complete revolution around the Smith chart corresponds to a line length of λ/2, a λ/4 transformation is equivalent to rotating the chart by 180°.
Imaging a give impedance (or admittance) point across the center of the chart to obtain the corresponding admittance (or impedance) point.

39. Ex 2.3 ZL = 100+j50, YL, Yin ? when l = 0.15λ
Figure (p. 69) ZY Smith chart with solution for Example 2.3.

40. The Slotted Line
A transmission line allowing the sampling of E field amplitude of a standing wave on a terminated line.
With this device the SWR and the distance of the first voltage minimum from the load can be measured, from this data ZL can be determined.
ZL is complex  2 distinct quantities must be measured.
Replaced by vector network analyzer.

41. Figure 2.13 (p. 70) An X-band waveguide slotted line.

42. Assume for a certain terminated line, we have measured the SWR on the line and lmin , the distance from the load to the first voltage minimum on the line.
Minimum occurs when
The phase of Γ =
Load impedance

43. Ex 2.4
With a short circuit load, voltage minima at z = 0.2, 2.2, 4.2 cm
With unknown load, voltage minima at z = 0.72, 2.72, 4.72 cm
λ = 4 cm,
If the load is at 4.2 cm, lmin = 4.2 – 2.72 = 1.48 cm = 0.37 λ

44. Figure 2. 14 (p. 71) Voltage standing wave patterns for Example 2. 4
Figure (p. 71) Voltage standing wave patterns for Example 2.4. (a) Standing wave for short-circuit load. (b) Standing wave for unknown load.

45. Figure 2.15 (p. 72) Smith chart for Example 2.4.

46. 2.5 The Quarterwave Transformer
Impedance Viewpoint
For βl = (2π/λ)(λ/4) = π/2
In order for Γ = 0, Zin = Z0

47. Figure 2.16 (p. 73) The quarter-wave matching transformer.

48. Ex 2.5 Frequency Response of a Quarter-Wave Transformer
RL = 100, Z0 = 50

49. Figure (p. 74) Reflection coefficient versus normalized frequency for the quarter-wave transformer of Example 2.5.

50. The Multiple Reflection Viewpoint
Figure (p. 75) Multiple reflection analysis of the quarter-wave transformer.

52. Numerator

53. 2.6 Generator and Load Mismatches
Because both the generator and load are mismatched, multiple reflections can occur on the line.
In the steady state, the net result is a single wave traveling toward the load, and a single reflected wave traveling toward the generator.
In Fig. 2.19, where z = -l,

54. Figure 2.19 (p. 77) Transmission line circuit for mismatched load and generator.

55. The voltage on the line:
Power delivered to the load:
By (2.67) &

56. Case 1: the load is matched to the line, Zl = Z0, Γl = 0, SWR = 1, Zin = Z0,
Case 2: the generator is matched to the input impedance of a mismatched line, Zin = Zg
If Zg is fixed, to maximize Pl,

57. or
Therefore, Rin = Rg and Xin = -Xg, or Zin = Zg*
Under these conditions
Finally, note that neither matching for zero reflection (Zl = Z0), nor conjugate matching (Zin = Zg*), necessary yields a system with the best efficiency.

58. 2.7 Lossy Transmission Lines