1. Lecture 3

2. Transmission line Propagating waves: Wavevector:
ki, Phase constant, rad/m
kr, attenuation constant, nep/m
Characteristic impedance:
Wavelength:
Phase velocity:

3. 1.5 The terminated lossless transmission line
What is a voltage reflection coefficient?
Assume an incident wave ( ) generated from a source at z < 0. We have seen that the ratio of voltage to current for such a traveling wave is Z0, the characteristic impedance. But when the line is terminated in an arbitrary load ZL  Z0, the ratio of voltage to current at the load must be ZL. Thus, a reflected wave must be excited with the appropriate amplitude to satisfy this condition.

4. What is a voltage reflection coefficient?
Total voltage and current on the line (superposition of incident and reflected waves):
(V0+: incident; V0-: reflected)
The total voltage and current at the load are related by the load impedance, so at z = 0, we must have
Γ0 = 0 (Zl = Z0)
Voltage reflection coefficient Γ:
(Phase difference: π)

5. What is a voltage reflection coefficient?
The total voltage and current waves on the line :
Consider the time-average power flow along the line at the point z:
which can be simplified:
Constant average power flow at any point on the line;
Total power delivered to the load = incident power – reflected power

6. What is a voltage reflection coefficient?
(a). Standing wave ( = -1)
(b). Voltage standing wave ratio (  < 1)
(z = -l)
(1  SWR < ， where SWR=1 implied a match load.)
(return loss)

7. Input impedance, Zin The reflection coefficient at z = -l:
(l)
(0)
The reflection coefficient at z = -l:
At a distance l from the load, the input impedance seen looking toward the load is
A more usable form of input impedance:
Input impedance of a portion of transmission line with an arbitrary load impedance.
Transmission line impedance Equation.

8. 1.5The terminated lossless transmission line
Voltage and current in the line:

9. (1). Short circuit transmission line (ZL=0, Γ=-1)
d = l
Zin
voltage
current
impedance

10. (2). Open circuit transmission line (ZL=, Γ=1)
voltage
current
impedance
Zin

11. (3). Quarter-wave transmission line

12. (4). Interface of two transmission lines
Reflection coefficient at z = 0:
Transmission coefficient at z = 0:
Insertion loss:

13. 1.6 Sourced and loaded transmission lines
Total input voltage:
Zin

14. The incident input voltage:
(1)
Here the input impedance is
(2)
Substitute (2) and (3) into (1), eliminate Zin
and the reflection coefficient seen looking into the generator is
(3)

15. Power consideration:

16. Homework
3. A lossless transmission line of electrical length l = 0.3, is terminated with a complex load impedance as shown below. Find the reflection coefficient at the load, the SWR on the line, the reflection coefficient at the input of the line, and the input impedance to the line.
ZL = 30 –j20  ZL
l = 0.3 
Zin
Z0= 75 
4. A radio transmitter is connected to an antenna having an impedance 80 + j40  with a 50  coaxial cable. If the 50  transmitter can deliver 30 W when connected to a 50  load. How much power is delivered to the antenna?