(a)
Prove that the generated far-field expressions for a thin dipole of length
(a)
Explanation of Solution
Calculation:
Consider the mentioned equation given in the textbook,
Refer to the mentioned Figure given in the textbook, the integration limits must be
Refer to the integral tables of Appendix A.8,
Apply equation (3) in (2),
Applying the limits to the above equation as follows,
Consider the general relationship between magnetic flux density and magnetic field intensity.
Rearrange the equation as follows,
Where,
Therefore, equation (4) can be rewritten as follows,
Consider for a far-field expression, only
and the relationship between electric field intensity and magnetic field intensity is,
Conclusion:
Thus, the generated far-field expressions for a thin dipole of length
(b)
Sketch the group pattern
(b)
Answer to Problem 9P
The group pattern
Explanation of Solution
Calculation:
Refer to Part (a),
From the above expression, the group pattern is,
Consider the expression for the phase constant.
Here,
For
Substitute
MATLAB code to plot the group pattern
theta=-360:pi./180:360;
c_n1=(cos(pi.*cos(theta)))+1;
s_n1=sin(theta);
f1=abs(c_n1./s_n1);
polar(theta,f1)
MATLAB output:
Figure 1 shows the polar plot for the group pattern
For
Substitute
MATLAB code to plot the group pattern
theta=-360:pi./180:360;
c_n2=cos((3.*pi./2).*cos(theta));
s_n2=sin(theta);
f2=abs(c_n2./s_n2);
polar(theta,f2)
MATLAB output:
Figure 2 shows the polar plot for the group pattern
For
Substitute
MATLAB code to plot the group pattern
theta=-360:pi./180:360;
c_n3=cos(2.*pi.*cos(theta))-1;
s_n3=sin(theta);
f3=abs(c_n3./s_n3);
polar(theta,f3)
MATLAB output:
Figure 3 shows the polar plot for the group pattern
Conclusion:
Thus, the group pattern
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Chapter 13 Solutions
Elements Of Electromagnetics
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