1. EKT241 – ELECTROMAGNETICS THEORY
Chapter 5 Transmission Lines

2. Chapter Objectives Introduction to transmission lines
Lump-element model that represent TEM lines
Lossless line
Smith Chart to analyze transmission line problem

3. Chapter Outline 5-1) General Considerations Lumped-Element Model
Transmission-Line Equations
Wave Propagation on a Transmission Line
The Lossless Transmission Line
Input Impedance of the Lossless Line
Special Cases of the Lossless Line
Power Flow on a Lossless Transmission Line
The Smith Chart
Impedance Matching
Transients on Transmission Lines
5-2)
5-3)
5-4)
5-5)
5-6)
5-7)
5-8)
5-9)
5-10)
5-11)

4. 5-1 General Considerations
Transmission lines connect a generator circuit to a load circuit at the receiving end.
Transverse electromagnetic (TEM) lines have waves that propagate transversely.

5. 5-2 Lumped-Element Model
Transmission lines can be represented by a lumped-element circuit model.

6. 5-2 Lumped-Element Model
Lumped-element circuit model consists 4 transmission line parameters:
R’  (Ω/m)
L’  (H/m)
G’  (S/m)
C’  (F/m)

7. 5-2 Lumped-Element Model
In summary,
All TEM transmission lines share the relations:
where µ, σ, ε = properties of conductor

8. 5-3 Transmission-Line Equations
Transmission line equations in phasor form is given as

9. 5-4 Wave Propagation on a Transmission Line
The wave equation is derived as
γ has real part α (attenuation constant) and imaginary part β (phase constant).
Complex propagation constant

10. 5-4 Wave Propagation on a Transmission Line
Characteristic impedance Z0 of the line is
Phase velocity for propagating wave is
where f = frequency (Hz)
λ = wavelength (m)

11. Example 5.1 Air Line
An air line is a transmission line for which air is the
dielectric material present between the two conductors, which renders G’ = 0. In addition, the conductors are made of a material with high conductivity so that R’ ≈0. For an air line with characteristic impedance of 50 and phase constant of 20 rad/m at 700 MHz, find the inductance per meter and the capacitance per meter of the line.

12. Solution 5.1 Air Line The following quantities are given:
With R’ = G’ = 0,
The ratio is given by
We get L’ from Z0

13. 5-5 The Lossless Transmission Line
Low R’ and G’ for transmission line is called lossless transmission line.
Using relation properties,

14. 5-5 The Lossless Transmission Line
Wavelength is given by
where εr = relative permittivity
For the lossless line, there are 2 unknowns in the equations for the total voltage and current on the line.

15. 5-5.1 Voltage Reflection Coefficient
The relations for lossless are
A load that is matched to the line when ZL = Z0, Γ = 0 and V0−= 0.

16. Example 5.2 Reflection Coefficient of a Series RC Load
A 100-Ω transmission line is connected to a load
consisting of a 50-Ω resistor in series with a 10-pF
capacitor. Find the reflection coefficient at the load for
a 100-MHz signal.

17. Solution 5.2 Reflection Coefficient of a Series RC Load
The following quantities are given
The load impedance is
Voltage reflection coefficient is

18. 5-5.2 Standing Waves 3 types of voltage standing-wave patterns:
(a) Matched load
(b) Short-circuited line
(c) Open-circuited line

19. Standing Waves
To find maximum and minimum values of voltage magnitude, we have

20. 5-5.2 Standing Waves First voltage maximum occurs at
First voltage minimum occurs at
Voltage standing-wave ratio S is defined as

21. Example 5.4 Standing-wave Ratio
A 50- transmission line is terminated in a load with ZL = (100 + j50)Ω . Find the voltage reflection coefficient and the voltage standing-wave ratio (SWR).
We have,
S is given by
Solution

22. 5-6 Input Impedance of the Lossless Line
Voltage to current ratio is called input impedance Zin.
The input impedance at z = −l is given as
and

23. Example 5.6 Complete Solution for v(z, t) and i(z, t)
A 1.05-GHz generator circuit with series impedance
Zg = 10Ω and voltage source given by
is connected to a load ZL = (100 + j50) through a
50-Ω, 67-cm-long lossless transmission line. The phase
velocity of the line is 0.7c, where c is the velocity of light
in a vacuum. Find v(z, t) and i(z, t) on the line.

24. Solution 5.6 Complete Solution for v(z, t) and i(z, t)
We find the wavelength from
and
The voltage reflection coefficient at the load is
The input impedance of the line

25. Solution 5.6 Complete Solution for v(z, t) and i(z, t)
Rewriting the expression for the generator voltage,
Thus the phasor voltage is
The voltage on the line is
and phasor voltage on the line is

26. Solution 5.6 Complete Solution for v(z, t) and i(z, t)
The instantaneous voltage and current is

27. 5-7 Special Cases of the Lossless Line
Special cases has useful properties.
For short-circuited line at z = −l,
Short-Circuited Line

28. Example 5.7 Equivalent Reactive Elements
Choose the length of a shorted 50- lossless transmission line (Fig. 5-16) such that its input impedance at 2.25 GHz is equivalent to the reactance of a capacitor with capacitance Ceq = 4 pF. The wave velocity on the line is 0.75c.

29. Solution 5.7 Equivalent Reactive Elements
We are given
The phase constant is
2nd quadrant is
4th quadrant is
Any length l = 4.46 cm + nλ/2, where n is a positive integer, is also a solution.

30. Open-Circuited Line
With ZL = ∞, it forms an open-circuited line.

31. 5-7.3 Application of Short-Circuit and Open-Circuit Measurements
Product and ratio of SC and OC equations give the following results:
Radio-frequency (RF) instruments measure the impedance of any load.

32. Example 5.8 Measuring Z0 and β
Find Z0 and β of a 57-cm-long lossless transmission
line whose input impedance was measured as Zscin = j40.42Ω when terminated in a short circuit and as Zocin = −j121.24Ω when terminated in an open circuit. From other measurements, we know that the line is between 3 and 3.25 wavelengths long.

33. Solution 5.8 Measuring Z0 and β
We have,
True value of βl is
and

34. Example 5.9 Quarter-Wave Transformer
A 50-Ω lossless transmission line is to be matched to a resistive load impedance with ZL = 100Ω via a quarter-wave section as shown, thereby eliminating reflections along the feedline. Find the characteristic impedance of the quarter-wave transformer.

35. Solution 5.9 Quarter-Wave Transformer
To eliminate reflections at terminal AA’, the input impedance Zin looking into the quarter-wave line should be equal to Z01, the characteristic impedance of the feedline. Thus, Zin = 50 .
Since the lines are lossless, all the incident power will end up getting transferred into the load ZL.

36. 5-8 Power Flow on a Lossless Transmission Line
We shall examine the flow of power carried by incident and reflected waves.
Instantaneous power is the product of instantaneous voltage and current.
More interested in time-averaged power flow.
Instantaneous Power
Time-Average Power

37. 5-8.2 Time-Average Power There are 2 types of approach:
1) Time-Domain Approach
Incident power and reflected wave power are
For net average power delivered to the load,

38. 5-8.2 Time-Average Power 5-9 Smith Chart 2) Phasor-Domain Approach
Time-average power for any propagating wave is
The Smith Chart is used for analyzing and designing transmission-line circuits.
5-9 Smith Chart

39. 5-9 Smith Chart Impedances represented by normalized values, Z0.
Reflection coefficient is
Normalized load admittance is

40. Example 5.11 Determining ZL using the Smith Chart
Given that the voltage standing-wave ratio is S = 3 on
a 50-Ω line, that the first voltage minimum occurs at 5 cm from the load, and that the distance between successive minima is 20 cm, find the load impedance.
Solution
The first voltage minimum is at

41. Solution 5.11 Determining ZL using the Smith Chart
From Smith Chart,
The normalized load
impedance at point C is
Multiplying by Z0 = 50Ω ,
we obtain

42. 5-10 Impedance Matching
Transmission line is matched to the load when Z0 = ZL.
Alternatively, place an impedance-matching network between load and transmission line.

43. Example 5.12 Single-Stub Matching
50-Ω transmission line is connected to an antenna
with load impedance ZL = (25 − j50). Find the position and length of the short-circuited stub required
to match the line.
The normalized load impedance is
Located at point A.
Solution

44. Solution 5.12 Single-Stub Matching
Value of yL at B is which locates at position 0.115λ on the WTG scale.
At C, located at 0.178λ on the WTG scale.
Distant B and C is
Normalized input admittance at
the juncture is

45. Solution 5.12 Single-Stub Matching
Normalized admittance of −j 1.58 at F and position 0.34λ on the WTG scale gives
At point D,
Distant B and C is
Normalized input admittance at G.
Rotating from point E to point G, we get

46. Solution 5.12 Single-Stub Matching

47. 5-11 Transients on Transmission Lines
Transient response is a time record of voltage pulse.
An example of step function is shown below.

48. Transient Response
Steady-state voltage V∞ for d-c analysis of the circuit is
where Vg = DC voltage source
Steady-state current is