Problem 2.53 A lossless 50-Ω transmission line is terminated in a load withZL = (50+ j25) Ω. Use the Smith chart to find the following:

(a) The reflection coefficientΓ.

(b) The standing-wave ratio.

(c) The input impedance at 0.35λ from the load.

(d) The input admittance at 0.35λ from the load.

(e) The shortest line length for which the input impedance is purely resistive.

(f) The position of the first voltage maximum from the load.

0.1

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.8

0.9

0.9

0.9

1.0

1.0

1.0

1.2

1.2

1.2

1.4

1.4

1.4

1.6

1.6

1.6

1.8

1.8

1.8

2.0

2.0

2.0

3.0

3.0

3.0

4.0

4.0

4.0

5.0

5.0

5.0

10

10

10

20

20

20

50

50

50

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

1.0

1.0

1.0

1.0

20-20

30-30

40-40

50

-50

60

-60

70

-70

80

-80

90

-90

100

-100

110

-110

120

-120

130

-130

140

-140

150

-150

160

-160

170

-170

180

±

0.04

0.04

0.05

0.05

0.06

0.06

0.07

0.07

0.08

0.08

0.09

0.09

0.1

0.1

0.11

0.11

0.12

0.12

0.13

0.13

0.14

0.14

0.15

0.15

0.16

0.16

0.17

0.17

0.18

0.18

0.190.19

0.2

0.2

0.21

0.210.22

0.220.23

0.230.24

0.24

0.25

0.25

0.26

0.26

0.27

0.27

0.28

0.28

0.29

0.29

0.3

0.3

0.31

0.31

0.32

0.32

0.33

0.33

0.34

0.34

0.35

0.35

0.36

0.36

0.37

0.37

0.38

0.38

0.39

0.39

0.4

0.4

0.41

0.41

0.42

0.42

0.43

0.43

0.44

0.44

0.45

0.45

0.46

0.46

0.47

0.47

0.48

0.48

0.49

0.49

0.0

0.0

AN

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F R

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CO

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WA

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TH

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D G

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<—

WA

VE

LE

NG

TH

S T

OW

AR

D L

OA

D <

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E R

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CO

MPO

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T (+jX

/Zo), O

R CAPACIT

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CAPACITIVE REACTANCE COM

PONEN

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/Zo), O

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SU

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(-j

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Yo

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RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)

Z-LOAD

SWRZ-IN

θr

0.350 λ

0.106 λ

Figure P2.53: Solution of Problem 2.53.

Solution: Refer to Fig. P2.53. The normalized impedance

zL =(50+ j25) Ω

50 Ω= 1+ j0.5

is at pointZ-LOAD.(a) Γ = 0.24ej76.0 The angle of the reflection coefficient is read of that scale at

the pointθr.(b) At the pointSWR: S= 1.64.(c) Zin is 0.350λ from the load, which is at 0.144λ on the wavelengths to generator

scale. So pointZ-IN is at 0.144λ + 0.350λ = 0.494λ on the WTG scale. At pointZ-IN:

Zin = zinZ0 = (0.61− j0.022)×50 Ω = (30.5− j1.09) Ω.

(d) At the point on the SWR circle oppositeZ-IN,

Yin =yin

Z0=

(1.64+ j0.06)50 Ω

= (32.7+ j1.17) mS.

(e) Traveling from the pointZ-LOAD in the direction of the generator (clockwise),the SWR circle crosses thexL = 0 line first at the pointSWR. To travel fromZ-LOAD to SWRone must travel 0.250λ −0.144λ = 0.106λ . (Readings are on thewavelengths to generator scale.) So the shortest line length would be 0.106λ .

(f) The voltage max occurs at pointSWR. From the previous part, this occurs atz= −0.106λ .