Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
Question
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Chapter 11, Problem 58P
To determine

Calculate the length of the shorted shunt-stub tuner in terms of λ.

Expert Solution & Answer
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Answer to Problem 58P

The length of the shorted shunt-stub tuner is l=0.0867λ and l=0.0819λ.

Explanation of Solution

Calculation:

Consider the general formula for transformed load impedance at stub position z=d.

Z(z=d)=Zo(ZL+jZotZo+jZLt)        (1)

Where,

t={XL±(RLZo)[(ZoRL)2+XL2]RLZoRLZoXL2ZoRL=Zo        (2)

Consider the general formula for load admittance.

Y=1Z=G+jB        (3)

Consider the general formula for distance.

dλ={12πtan1(t)tZo12π[π+tan1(t)]t<Zo        (4)

Consider the general formula for length of the short-circuit stub.

lλ=12πtan1(YoB)        (5)

Given that, the load impedance is,

ZL=75+j100        (6)

Consider the general formula for load impedance.

ZL=RL+jXL        (7)

s

On comparing Equation (6) and (7),

RL=75

XL=100

Substitute 75 for RL, 50 for Zo, and 100 for XL in Equation (2).

t=100±(7550)[(5075)2+1002]7550RLZo=100±126.225

Therefore,

t=9.048s and t=1.048s

Find the length of the stub for t=9.048s and t=1.048s.

Consider t=9.048s.

Substitute 50 for Zo, 9.048 for t, and 75+j100 for ZL in Equation (1).

Z=(50)[(75+j100)+j(50)(9.048)(50)+j(75+j100)(9.048)]=(50)[75+j100+j452.450904.8+j678.6]=(50)[75+j552.4854.8+j678.6]=13.04j21.95Ω

Substitute 13.04j21.95 for Z in Equation (3).

Y=113.04j21.95

Y=0.02+j0.033        (8)

Rewrite Equation (3) as follows,

Y=Yo+jB{G=Yo}        (9)

On comparing Equation (8) and (9),

Yo=0.02

B=0.033

Substitute 0.02 for Yo and 0.033 for B in Equation (5).

lλ=12πtan1(0.020.033)lλ=0.0867l=0.0867λ

Consider t=1.048s.

Substitute 50 for Zo, 1.048 for t, and 75+j100 for ZL in Equation (1).

Z=(50)[(75+j100)+j(50)(1.048)(50)+j(75+j100)(1.048)]=(50)[75+j100j52.450+104.8j78.6]=(50)[75+j47.6154.8j78.6]=13.053j22Ω

Substitute 13.053j22 for Z in Equation (3).

Y=113.053j22

Y=0.019+j0.0336        (10)

On comparing Equation (9) and (10),

Yo=0.019

B=0.0336

Substitute 0.019 for Yo and 0.0336 for B in Equation (5).

lλ=12πtan1(0.0190.0336)lλ=0.0819l=0.0819λ

Conclusion:

Thus, the length of the shorted shunt-stub tuner is l=0.0867λ and l=0.0819λ.

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