1. ENE 428 Microwave Engineering
Lecture 5 Discontinuities and the manipulation of transmission lines problems RS

2. Review
Transmission lines or T-lines are used to guide propagation of EM waves at high frequencies.
Distances between devices are separated by much larger order of wavelength than those in the normal electrical circuits causing time delay.
General transmission line’s equation
Voltage and current on the transmission line
characteristic of the wave propagating on the transmission line

3. Wave reflection at discontinuities
To satisfy boundary conditions between two dissimilar lines
If the line is lossy, Z0 will be complex.

4. Reflection coefficient at the load (1)
The phasor voltage along the line can be shown as
The phasor voltage and current at the load is the sum of incident and reflected values evaluated at z = 0.

5. Reflection coefficient at the load (2)
A reflected wave will experience a reduction in amplitude and a phase shift
Transmission coefficient

6. Power transmission in terms of reflection coefficient

7. Total power transmission (matched condition)
The main objective in transmitting power to a load is to configure line/load combination such that there is no reflection, that means

8. Voltage standing wave ratio
Incident and reflected waves create “Standing wave”.
Knowing standing waves or the voltage amplitude as a function of position helps determine load and input impedances
Voltage standing wave ratio

9. Forms of voltage (1)
If a load is matched then no reflected wave occurs, the voltage will be the same at every point.
If the load is terminated in short or open circuit, the total voltage form becomes a standing wave.
If the reflected voltage is neither 0 nor 100 percent of the incident voltage then the total voltage will compose of both traveling and standing waves.

10. Forms of voltage (2)
let a load be position at z = 0 and the input wave amplitude is V0,
where

11. Forms of voltage (3) The maximum amplitude occurs when
we can show that
traveling wave
standing wave
The maximum amplitude occurs when
The minimum amplitude occurs when standing waves
become null,

12. The locations where minimum and maximum voltage amplitudes occur (1)
The minimum voltage amplitude occurs when two phase terms have a phase difference of odd multiples of .
The maximum voltage amplitude occurs when two phase terms are the same or have a phase difference of even multiples of .

13. The locations where minimum and maximum voltage amplitudes occur (2)
If  = 0,  is real and positive
and
Each zmin are separated by multiples of one-half wavelength, the same applies to zmax. The distance between zmin and zmax is a quarter wavelength.
We can show that

14. Ex1 Slotted line measurements yield a VSWR of 5, a 15 cm between successive voltage maximum, and the first maximum is at a distance of 7.5 cm in front of the load. Determine load impedance, assuming Z0 = 50 .

15. Transmission lines of finite length (1)
Consider the propagation on finite length lines which have load that are not impedance-matched.
Determine net power flow.
Assume lossless line, at load
we can write

16. Input impedance (1)
Using and gives
Using , we have

17. Input impedance (2) At z = -l, we can express Zin as
I. Special case if
then
II. Special case if

18. Quarter wavelength lines
It is used for joining two TL lines with different characteristic
impedances If
then we can match the junction Z01, Z02, and Z03 by choosing
Quarter-wave matching

19. Complex loads
Input complex impedance or loads may e modeled using simple resistor, inductor, and capacitor lump elements
For example, ZL = 100+j200  this is a 100  resistor in series
with an inductor that has an inductance of j200 .
Let f = 1 GHz,
What if the lossless line is terminated in a purely reactive load?
Let Z0 = R0 and ZL+jXL, then we have
that a unity magnitude, so the wave is
completely reflected.

20. Ex2 From the circuit below, find
Power delivered to load

21. b) If another receiver of 300  is connected in parallel with the load, what is
b.2) VSWR
b.3) Zin
b.4) input power

22. c) Where are the voltage maximum and minimum and what are they?
d) Express the load voltage in magnitude and phase?

23. Ex3 Let’s place another purely capacitive impedance of –j300  in parallel with two previous loads, find Zin and the power delivered to each receiver.

24. Smith chart A graphical tool used along with Transmission
lines and microwave circuit components
Circumventing the complex number arithmetic
required in TL problems
Using in microwave design

25. Smith chart derivation (1)
 plane

26. Smith chart derivation (2)
From
define
then
Now we replace the load along with any arbitrary length of
TL by Zin, we can then write

27. Smith chart derivation (3)

28. Smith chart derivation (4)
We can rearrange them into circular equations,

29. Normal resistance circle
Consider a normalized resistance r = 1, then we have
If r = 0, we have
so the circle represents all possible
points for  with ||  1

30. Normal reactance circle
Consider a normalized resistance x = 1, then we have
The upper half represents positive
reactance (inductance)
The lower half represents negative
reactance (capacitance)

31. Using the smith chart (1)
A plot of the normalized impedance
The magnitude of  is found by taking the distance from
the center point of the chart, divided by the radius of the
chart (|| = 1). The argument of  is measured from the
axis.
Recall
we see that Zin at Z = -l along the TL corresponds to
Moving away from the load corresponds to moving in a
clockwise direction on the Smith chart.

32. Using the smith chart (2)
Since is sinusoidal, it repeats for
every one turn (360) corresponds to
Note: Follow Wavelength Toward Generator (WTG)
Vmin and Vmax are locations where the load ZL is a pure
resistance.
Vmax occurs when r > 1 (RL > Z0) at wtg = 0.25.
Vmin occurs when r < 1 (RL < Z0) at wtg = 0.

33. Using the smith chart (3)
The voltage standing wave ratio (VSWR) can be
determined by reading the value of r at the  = 0 crossing
the constant-|L| circle.