Chapter 3, Dipoles, Arrays, and Long-Wire Antennas Video Solutions, Antennas and Radiowave Propagation (Mcgraw Hill Series in Electrical and Computer Engineering)

Problem 1

Let a transmission line have inductance $L$ and capacitance $C$ per unit length. Then the following relations bold:
$$
\begin{aligned}
& k_0=\omega\left(\mu_0 \epsilon_0\right)^{1 / 2}=\omega(L C)^{1 / 2} \\
& Z_*=(L / C)^{1 / 2}
\end{aligned}
$$
Use these relations, along with Eq. (3.10), to show that for a biconical antenna
$$
\begin{aligned}
& L=\left(\mu_0 \epsilon_s\right)^{1 / 2} Z_e=\frac{\mu_0}{\pi} \operatorname{nn} \cot \frac{\theta_0}{2} \\
& C=\left(\mu_0 \epsilon_\theta\right)^{3 / 2} Y_s=\frac{\pi \epsilon_0}{\ln \cot \left(\theta_0 / 2\right)}
\end{aligned}
$$

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Problem 2

A biconical antenna has an input impedance of $130+j 40 \Omega$ and is connected to a transmission line with characteristic impedance of $158 \Omega$. Find the power reflection coefficient and the VSWR on the input transmission line.

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Problem 3

Find the radiation pattern of a dipoic antenna of total length $3 \lambda_s / 2$. The antenna is located along the $z$ axis between $=3 \lambda_d / 4$ and $3 \lambda_0 / 4$. The current on the antenna is given by $I_0 \cos k_n z$. Sketch the radiation pattern and compare the field strength in the $\theta=\pi / 2$ plane with that produced by a dipole $\lambda_0 / 2$ long. Give a physical reason why the longer antenna does not produce a larger field strength.

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03:32

Problem 4

Find the maximum field strength along the $x$ axis for the V -shaped dipole antenna shown in Fig. P3.4. Consider the two cases when $I_0=\lambda_0 / 4$ and $I_0=3 \lambda_0 / 4$. Compare the field strength for the two cases with that obtained in Prob. 3.3 when $a=30$ and $45^{\circ}$.
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Keshav Singh

Numerade Educator

Problem 5

A two-wire transmission line is required to have a characteristic impedance of 730. The conductors have a diameter of 0.5 in . Find the required spacing D. See Fig. II-2 in Appendix II. Is this a practical transmission line?

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Problem 6

For the transmission line circuit shown in Fig. 3.10 let $I(z)=I_0 \sin \left(a-k_g z\right) / \sin m$. Use $\Delta I / \partial z=-j \omega C_0 V$ to find $V(z)$. At $z=U 2$ the terminal condition $I=j \omega C V$ must hold. Use these results to derive Eq. (3.33) where $X_s=1 / \omega C$.

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Problem 7

The transmission-line model of the dual-band dipole antenna shown in Fig. 3.11 is like that shown in Fig. 3.8 but with $L_0 / 2$ replaced by the parallel resonant $L_1 C_3$ circuit. It is desired to operate this antenna at a frequency $f_1$ and also at $f_2=1.5 f_1$. Find the required lengths $I_1, I$ and the parameters $L_1, C_1$ when $f_1=50 \mathrm{MHz}$ and the antenna characteristic impedance $Z_e$ equals $600 \Omega$. A practical value to use for $\sqrt{L_1 / C_1}$ is 600 f .

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Problem 8

A uniform line array of five elements has a spacing of $d=0.4 \lambda_n$. Find the phasing in order to produce a beam at $45^{\circ}$ to the array axis, that is, at $\psi=\pi / 4$. Plot the array lactor $|F|$ and show the visible region. Sketch the main lobe radiation pattern.

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Problem 9

A lime array is required that will produce beams at $\psi=0, \pi / 2$, and $v$. Find the required spacing $d$ and the phase of excitation of each element.

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Problem 10

An antenna produces a field $E_\theta=C \cos ^3 \theta\left(e^{-i \theta} / 4 \pi r\right)$ where $C$ is a constant. The pattern consists of beams along the $\pm z$ axis.
(a) Find the total radiated power.
(b) Find the directivity $D$ at $\theta=0$.
(c) Find the hall-power beam width.
(d) Find the solid angle occupied by the beam up to the half-power angle. This is given by the area intercepted by the beam on a sphere of unit radius.
(e) Estimate the directivity by dividing $4 \pi$ by the solid angle of the two beams up to the hatt-power angle and compare with the exact result obtained in (b).

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Problem 11

Consider the two-element array, shown in Fig. P3.11, consisting of hall-wave dipoles at $x=0, x=d$ on the $x$ axis. The current in the two dipoles is $I_0$ and $J_0 e^{m o t}$. Find $a$ and the spacing $d$ so that zero radiation oceurs in the $-x$ direction and maximum radiation occurs in the $+x$ direction. Sketch the radiation pattern in the $x y$ plane.
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Problem 12

Use the Fourier series method to find a seven-element array thal will produce a least-mean-square error approximation to the array factor $F_d$ slsown in Fig. P3.12. Sketch the approximate pattern. Assume $d=\lambda d / 2$.
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Problem 13

Use the array polynomial method to design a six-element broadside array with a radiation pattern having nulls at $\theta=0, \pi / 6, \pi / 3,2 \pi / 3,5 \pi / 6$ and $\pi$, Choose $d=\lambda_\pi / 2$ and find the relative values of the required current in each element. Sketch the array factor. Note that the zeros at $\theta=0, \pi$ both correspond to $Z=-1$.

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Problem 14

Design a seven-element broadside array having double zeros in its radiation pattern at $\theta=0, \pi / 4,3 \pi / 4$, and $\pi$. Assume that $d=\lambda \sqrt{2}$. Find the relative value of the required current in each element. Sketch the array factor. Note that the zeros at $\theta=0$, $\pi$ both correspond to $Z=-1$. Repeat the design when $d=A d 3$ and nine elements are used.

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Problem 15

Design a seven-element Chebyshev array with $d=\lambda \sqrt{2}$ having side lobes 26 dB below the main lobe. Find the excitation coefficients and the beam width. Sketch the array factor $|F(u)|$.

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Problem 16

Find the excitation coefficients and side-lobe level for a five-element broadside Chebyshev array with element spacing $d=\lambda_0 / 2$ and having a beam width of $55^{\circ}$.

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Problem 17

Find the currents for a five-element superdirective array of length $L=A_p / 4$ and with side lobes 20 dB down. Find the beam width. Sketch the array factor. Find the eflective radiating current in the direction of the main lobe. You will need six-figure accuracy in the computations.

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Problem 18

Repeat Prob. 3.17 but use an array length $L=\lambda_0$. What is the peak value of $F(u)$ in the invisible region? This occurs when $x-a+b \cos u=a+b \cos \pi=a-b$ and may be found from $T_2(x)$ for this value of $x$.

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Problem 19

Design a five-element optimum Chebyshev array having the specifications given in Example 3.3. Use the design formulas given by Eq. (3.87). Note that the same results as in Example 3.3 are oblained.

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Problem 20

Design an optimum end-fire array with five elements using the design formulas given by Eq. (3.90). Carry out the design for the three cases- $k_0 d=0.6 \pi, 0.4 \pi$, and $0.1 \pi$-and compare the beam widths in the three cases. The smaller spacings result in a supergain design with extreme values for the currents. The side lobe ratio $R=10$.

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Problem 21

For the four-element array driven through the Butler beam-forming matrix shown in Fig. 3.37 verify that the relative aperture phase distributions are as given in the text. If the element spacing $d$ equals $\lambda_0 / 2$, find the directions in space of the four beams.

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Problem 22

Consider the two-element array shown in Fig. P3.22. This array is fed through a $90^{\circ}$ phase-lag hybrid junction, and a $-90^{\circ}$ phase shifter is incorporated in one feed line. The element spacing $d=3 \lambda_0 / 4$. Find the directions of the beams that are formed by exciting ports 1 and 2.
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Lainey Roebuck

Numerade Educator

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Problem 23

Find the optimum angle $\psi_0$ for a $V$ antenna, having arms $2 \lambda_0$ long, which will produce maximum radiation at an elevation angle of $10^{\circ}$,

Eli Heath

Numerade Educator

Problem 24

For the inverted V antenna shown in Fig. P3.24 find the optimum angle $\psi y$ in order to ohtain maximum radiation at an elevation angle of $0^{\circ}$. Each leg of the antenna is $3 \lambda_0 / 2$ long. Can the lobes be aligned for maximum radiation at a finite elevation angle $y$ ?
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Problem 25

Carry out the basic design of a traveling-wave thombic antenna with atms that are $4 \lambda_0$ long. Specily the angles that will align the lobes in the plane of the rhombus. Will this angle cause the feld radiated from each arm of the rhombus to add in phase in the desired direction? Find the angles that will result in the correct phase.

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