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 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 2

 Consider the propagation on finite length lines which have load that are not impedance-matched.  Determine net power flow. 3 Assume lossless line, at load we can write

4 Using and gives Using,we have

5 At z = -l, we can express Z in as I. Special case if then II. Special case if then

6 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

 Input complex impedance or loads may e modeled using simple resistor, inductor, and capacitor lump elements 7 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.

a) Power delivered to load 8

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 9

c) Where are the voltage maximum and minimum and what are they? d) Express the load voltage in magnitude and phase? 10

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12 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

13  plane

14 From define then Now we replace the load along with any arbitrary length of TL by Z in, we can then write

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16 We can rearrange them into circular equations,

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

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

19 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.

20 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 = V min occurs when r < 1 (R L < Z 0 ) at wtg = 0.

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

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23 Ex5 Z L = 80-j100  is located at z = 0 on a lossless 50  line, given the signal wavelength = 2 m, find a) If the line is 0.8 m in length, find Z in. = 1.5+2j b) VSWR = 4.6 c) What is the distance from load to the nearest voltage maximum =0.451 d) what is the distance from the input to the nearest point at which the remainder of the line could be replaced by a pure resistance? =0.199