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

2. 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. 3 Wave reflection at discontinuities To satisfy boundary conditions between two dissimilar lines If the line is lossy, Z 0 will be complex.

4. 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. 5 Reflection coefficient at the load (2) Reflection coefficient A reflected wave will experience a reduction in amplitude and a phase shift Transmission coefficient

6. 6 Power transmission in terms of reflection coefficient W W W

7. 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. 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. 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. 10 Forms of voltage (2) let a load be position at z = 0 and the input wave amplitude is V 0, where

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

12. 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. 13 The locations where minimum and maximum voltage amplitudes occur (2) If  = 0,  is real and positive and Each z min are separated by multiples of one-half wavelength, the same applies to z max. The distance between z min and z max is a quarter wavelength. We can show that

14. 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 Z 0 = 50 .

15. 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. 16 Input impedance (1) Using and gives Using, we have

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

18. 18 Quarter wavelength lines It is used for joining two TL lines with different characteristic impedances If then we can match the junction Z 01, Z 02, and Z 03 by choosing Quarter-wave matching

19. 19 Complex loads Input complex impedance or loads may be modeled using simple resistor, inductor, and capacitor lump elements For example, Z L = 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 Z 0 = R 0 and Z L = jX L, then we have that a unity magnitude, so the wave is completely reflected.

20. 20 Ex2 From the circuit below, find  Power delivered to load v = 2.5 x 10 8 m/s Z in, P in

21. 21 b) If another receiver of 300  is connected in parallel with the load, what is b.1)  b.2) VSWR b.3) Z in b.4) input power

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

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

24. 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. 25 Smith chart derivation (1)  plane

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

27. 27 Smith chart derivation (3)

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

29. 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. 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)

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32. 32 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 Z in at Z = -l along the TL corresponds to Moving away from the load corresponds to moving in a clockwise direction on the Smith chart.

33. 33 Using the smith chart (2) Since is sinusoidal, it repeats for  every one turn (360  ) corresponds to Note: Follow Wavelength Toward Generator (WTG) V min and V max are locations where the load Z L is a pure resistance. V max occurs when r > 1 (R L > Z 0 ) at wtg = 0.25. V min occurs when r < 1 (R L < Z 0 ) at wtg = 0.

34. 34 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.

35. 35 Example using the smith chart 1. Normalize the value of the load impedance Z L to Z L /Zo

36. 36 2. Plot the normalized impedance z L on the Smith chart

37. 37 3. Draw a straight line from the center to the coordinate of z L, read the value of the reflection coefficient angle (  ) on the scale located on the perimeter “angle of reflection coefficient in degrees”. 4. The magnitude of the reflection coefficient |  | is found by taking the distance from the center of the chart to the point divided by the distance from the center of the chart to the periphery (|  | = 1).

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39. 39 5. Another way to obtain |  | is to read it directly from the scale provided at the bottom of the chart. For this method, you will need to measure the length of the highlighted line from page 37, then measure that length on the scale “RFL. Coeff. E or I”. The value you read will be |  |. But the angle (  ), you still need to read it off from the scale on the perimeter.

40. 40 6. Moving away from the load (toward the generator) corresponds to moving in a clockwise direction on the Smith chart. X.XXX

41. 41 At point c, z in = 0.175 - j0.08 Hence, denormalizing this value, we’ll get Z in = z in x Zo = 8.75 - j4 Ω

42. 42 7. When moving a point around 1 loop (360  ), it corresponds to moving along the transmission-line of distance /2

43. 43 8. The voltage standing wave ratio (VSWR) can be determined by reading the value of r at the   = 0  crossing the constant-|  L | circle.

44. 44 9. In addition, the voltage standing wave ratio (VSWR) can be determined by reading the scale (SWR) at the bottom of the Smith chart.

45. Vmax 0.xxx Vmin 10. V min and V max are locations where the load Z L is a pure resistance. 0.XXX Vmax