Elements of Electromagnetics
Elements of Electromagnetics
7th Edition
ISBN: 9780190698669
Author: Sadiku
Publisher: Oxford University Press
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Chapter 11, Problem 34P
To determine

Show that the matrix ABCD is [coshγlZosinhγl1Zosinhγlcoshγl].

Expert Solution & Answer
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Explanation of Solution

Calculation:

Refer to the mentioned Figure given in the textbook.

The relation between the input and output variables matrix is,

[V1I1]=[ABCD][V2I2]

Consider the general expression for terminal conditions.

V1=Vs(z=0)=Vo++Vo        (1)

V2=Vs(z=l)=Vo+eγl+Voeγl        (2)

I1=Is(z=0)=Vo+ZoVoZo        (3)

I2=Is(z=l)=Vo+Zoeγl+VoZoeγl        (4)

Adding Equation (1) and (3). Therefore,

Vo+=12(V1+ZoI1)        (5)

Subtracting Equation (1) and (3). Therefore,

Vo=12(V1ZoI1)        (6)

Substitute Equation (5) and (6) in Equation (2).

V2=[12(V1+ZoI1)]eγl+[12(V1ZoI1)]eγl=12(V1eγl+ZoI1eγl)+12(V1eγlZoI1eγl)=12(eγl+eγl)V1+12Zo(eγleγl)I1

V2=coshγlV1ZosinhγlI1{cosh(ix)=12(eix+eix),sinh(ix)=12(eixeix)}        (7)

Substitute Equation (5) and (6) in Equation (4).

I2=[12(V1+ZoI1)]Zoeγl+[12(V1ZoI1)]Zoeγl=12Zo(V1+ZoI1)eγl+12Zo(V1ZoI1)eγl=12Zo(V1eγl+ZoI1eγl)+12Zo(V1eγlZoI1eγl)=12ZoV1eγl12ZoZoI1eγl+12ZoV1eγl12ZoZoI1eγl

The above equation becomes,

I2=12Zo(eγleγl)V112(eγl+eγl)I1

I2=1ZosinhγlV1coshγlI1        (8)

From Equation (7) and (8),

[V2I2]=[coshγlZosinhγl1Zosinhγlcoshγl][V1I1]

But,

[coshγlZosinhγl1Zosinhγlcoshγl]1=[coshγlZosinhγl1Zosinhγlcoshγl]

Therefore,

[V1I1]=[coshγlZosinhγl1Zosinhγlcoshγl][V2I2]

Compare the above equation with the given input and output variables matrix. Therefore, the matrix ABCD is,

[ABCD]=[coshγlZosinhγl1Zosinhγlcoshγl]

Conclusion:

Thus, the matrix ABCD is [coshγlZosinhγl1Zosinhγlcoshγl] and it is proved.

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Chapter 11 Solutions

Elements of Electromagnetics

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