Wire and Loop Antennas

Introduction

Wire antennas are among the most fundamental radiators in electromagnetic engineering. Dipoles, monopoles, loops, helices, long wires and terminated travelling-wave structures are used from the low-frequency and HF bands through microwave frequencies. Their apparent simplicity conceals important interactions among current distribution, conductor geometry, feed structure, ground properties, loss and nearby objects.

This chapter develops the radiation fields and impedances of representative wire and loop antennas. Unless stated otherwise, phasors use the time convention \(e^{j\omega t}\), currents are peak phasors, the surrounding medium is homogeneous free space, and

\[

k=\frac{2\pi}{\lambda}=\omega\sqrt{\mu\epsilon},\qquad

\eta=\sqrt{\frac{\mu}{\epsilon}}\approx120\pi\ \Omega.

\]

Learning Objectives

After completing this chapter, the reader should be able to:

1. Relate wire current distributions to radiated fields.

2. Derive the fields and radiation resistance of electrically short dipoles.

3. Analyse resonant dipoles and monopoles using sinusoidal current models.

4. Derive the far fields and radiation resistance of electrically small loops.

5. Explain travelling-wave radiation and termination requirements.

6. Evaluate input impedance, efficiency, grounding, balancing and mutual coupling.

7. Select practical wire-antenna implementations from HF through microwave frequencies.

1. Thin-Wire Radiation Fundamentals

A thin wire is one whose radius is much smaller than its length and wavelength. If a wire lies along the \(z\)-axis and carries current \(I(z')\), its magnetic vector potential is

\[

A_z(\mathbf r)=\frac{\mu}{4\pi}\int I(z')\frac{e^{-jkR}}{R}\,dz',

\qquad R=|\mathbf r-\mathbf r'|.

\]

In the far zone, \(R\approx r-z'\cos\theta\) in phase and \(R\approx r\) in amplitude. Therefore,

\[

A_z\approx \frac{\mu e^{-jkr}}{4\pi r}

\int I(z')e^{jkz'\cos\theta}\,dz'.

\]

The resulting far electric field is

\[

E_\theta\approx

\frac{j\eta k e^{-jkr}\sin\theta}{4\pi r}

\int I(z')e^{jkz'\cos\theta}\,dz',

\qquad H_\phi=\frac{E_\theta}{\eta}.

\]

This radiation integral shows that an antenna pattern is the spatial Fourier transform of its current distribution, multiplied by the element factor \(\sin\theta\).

For a perfectly conducting wire, the total tangential electric field on its surface must vanish. The unknown current therefore satisfies a thin-wire electric-field integral equation. Analytical current distributions are available only for simplified cases; practical antennas are commonly solved using the method of moments, finite-element methods or finite-difference time-domain methods.

2. Electrically Short Dipoles

2.1 General effective-length result

Consider a centre-fed dipole of total length \(L\ll\lambda\). Because \(kL\ll1\),

\[

e^{jkz'\cos\theta}\approx1.

\]

Define the transmitting effective length

\[

l_{\mathrm{eff}}=\frac{1}{I_0}\int_{-L/2}^{L/2}I(z')\,dz',

\]

where \(I_0\) is the feed current. The far fields become

\[

E_\theta=\frac{j\eta kI_0l_{\mathrm{eff}}}{4\pi r}

e^{-jkr}\sin\theta,

\qquad

H_\phi=\frac{E_\theta}{\eta}.

\]

The time-average radiated power density is \(|E_\theta|^2/(2\eta)\). Integrating over a sphere and using

\[

\int_0^{2\pi}\int_0^\pi\sin^3\theta\,d\theta d\phi=\frac{8\pi}{3}

\]

gives

\[

P_{\mathrm{rad}}=\frac{\eta k^2|I_0|^2l_{\mathrm{eff}}^2}{12\pi}.

\]

Since \(P_{\mathrm{rad}}=|I_0|^2R_r/2\),

\[

\boxed{R_r=80\pi^2\left(\frac{l_{\mathrm{eff}}}{\lambda}\right)^2}.

\]

2.2 Triangular-current short dipole

A practical short, centre-fed dipole approximately has a triangular current:

\[

I(z)=I_0\left(1-\frac{2|z|}{L}\right),\qquad |z|\leq L/2.

\]

Its current area is

\[

\int_{-L/2}^{L/2}I(z)\,dz=\frac{I_0L}{2},

\]

so \(l_{\mathrm{eff}}=L/2\). Consequently,

\[

\boxed{E_\theta=rac{j\eta kI_0L}{8\pi r}e^{-jkr}\sin\theta}

\]

and

\[

\boxed{R_r=20\pi^2\left(\frac{L}{\lambda}\right)^2}.

\]

This differs from a Hertzian element of length \(L\), whose assumed uniform current gives \(R_r=80\pi^2(L/\lambda)^2\). The distinction is caused entirely by the different current distributions.

Worked Example 1: Short-dipole resistance

A short dipole has \(L=0.05\lambda\). Its triangular-current radiation resistance is

\[

R_r=20\pi^2(0.05)^2=0.493\ \Omega.

\]

If conductor and loading losses total \(2.0\ \Omega\), its radiation efficiency is

\[

\eta_r=\frac{R_r}{R_r+R_{\mathrm{loss}}}

=\frac{0.493}{2.493}=0.198,

\]

or approximately \(19.8\%\). Electrically short antennas can therefore be inefficient even when constructed from good conductors.

3. Resonant Dipoles

For a thin, straight dipole with total length \(2l\), a useful sinusoidal approximation is

\[

I(z)=I_m\sin[k(l-|z|)],\qquad |z|\leq l,

\]

where \(I_m\) is the sinusoidal-current amplitude. The feed current is \(I_f=I_m\sin(kl)\). Substitution into the radiation integral gives

\[

E_\theta=rac{j\eta I_m e^{-jkr}}{2\pi r}

\frac{\cos(kl\cos\theta)-\cos(kl)}{\sin\theta}.

\]

For a half-wave dipole, \(2l=\lambda/2\), so \(kl=\pi/2\) and \(I_m=I_f\). Thus,

\[

E_\theta=rac{j\eta I_f e^{-jkr}}{2\pi r}

\frac{\cos[(\pi/2)\cos\theta]}{\sin\theta}.

\]

The thin half-wave dipole has approximately:

• radiation resistance: \(73.1\ \Omega\);
• directivity: \(1.64\), or \(2.15\ \mathrm{dBi}\);
• maximum radiation broadside to the wire;
• nulls along the wire axis.

An exactly half-wave thin dipole is slightly inductive. Practical resonant length is usually about \(0.47\lambda\) to \(0.49\lambda\), depending on conductor diameter, insulation, end geometry and surroundings.

3.1 Input impedance

The feed-point impedance may be represented as

\[

Z_{\mathrm{in}}=R_r+R_{\mathrm{loss}}+jX_{\mathrm{in}}.

\]

Radiation resistance is not a physical resistor; it represents power carried away by electromagnetic waves. Loss resistance accounts for conductor, dielectric and ground dissipation. Resonance occurs when \(X_{\mathrm{in}}=0\), possibly after adding a matching or tuning network.

The radiation efficiency is

\[

\eta_r=\frac{R_r}{R_r+R_{\mathrm{loss}}}.

\]

A matching network can eliminate feed-line mismatch but cannot recover power already dissipated in antenna losses.

3.2 Folded and loaded dipoles

A two-wire folded dipole with equal conductor diameters has an input resistance approximately four times that of an ordinary dipole, or about \(292\ \Omega\), under ideal symmetric conditions. Unequal diameters change this transformation ratio.

Inductive loading permits physical shortening, but increases stored energy, loss sensitivity and quality factor. Capacitive hats improve current uniformity and can provide greater efficiency than a concentrated loading coil for the same overall height.

4. Monopoles and Ground Systems

A vertical monopole of height \(h\) above an infinite perfectly conducting plane can be analysed by image theory. The monopole and its image form an equivalent dipole of total length \(2h\).

In the upper hemisphere, the monopole field equals the field of the corresponding dipole for equal feed current. No field exists below the ideal conducting plane. Hence the monopole radiates half the total power of the corresponding isolated dipole, not twice its power. Its radiation resistance is therefore one-half of the corresponding dipole value.

For a quarter-wave monopole:

\[

R_r\approx\frac{73.1}{2}=36.5\ \Omega.

\]

Because the radiation is concentrated into one hemisphere, its directivity is twice that of a half-wave dipole:

\[

D\approx3.28=5.15\ \mathrm{dBi}.

\]

4.1 Electrically short monopoles

A short monopole of height \(h\ll\lambda\) over an ideal ground plane corresponds to a short dipole of total length \(2h\). Thus,

\[

R_r=\frac{1}{2}\left[20\pi^2\left(\frac{2h}{\lambda}\right)^2\right]

=40\pi^2\left(\frac{h}{\lambda}\right)^2.

\]

This expression assumes an approximately triangular current, an ideal conducting plane and feed current defined at the monopole base.

4.2 Real ground

Soil has finite conductivity and nonzero permittivity. Ground currents then produce loss and alter the elevation pattern. For a good conductor, the approximate surface impedance is

\[

Z_s\approx(1+j)\sqrt{\frac{\omega\mu}{2\sigma}},

\]

where \(\sigma\) is conductivity. This approximation requires \(\sigma\gg\omega\epsilon\).

HF and medium-frequency monopoles commonly use buried or elevated radial wires. More and longer radials reduce ground loss, although practical improvement eventually becomes incremental. At VHF and above, a conductive platform or several quarter-wave radial rods can form an effective ground plane. Accurate terrestrial-wave predictions require Sommerfeld or numerical ground models rather than ideal image theory.

5. Electrically Small Loop Antennas

5.1 Uniform-current approximation

A loop is electrically small when its circumference is much less than a wavelength. Its current can then be treated as approximately uniform. For an \(N\)-turn loop of area \(A\), define the magnetic dipole moment

\[

m=NI_0A.

\]

For a loop in the \(xy\)-plane, expansion of the vector-potential integral gives the far-zone component

\[

\boxed{A_\phi=\frac{j\mu kN I_0A}{4\pi r}e^{-jkr}\sin\theta}.

\]

The factor is proportional to \(jk\), not \(k^2\). It has the dimensions required for magnetic vector potential. In the far zone,

\[

E_\phi=-j\omega A_\phi

=\frac{\eta k^2NI_0A}{4\pi r}e^{-jkr}\sin\theta,

\]

and

\[

H_\theta=-\frac{E_\phi}{\eta}.

\]

The pattern is proportional to \(\sin\theta\): maximum radiation lies in the loop plane, while nulls occur normal to the loop plane. Its polarization is orthogonal to that of a short electric dipole oriented along the loop axis.

5.2 Radiation resistance

Integrating the far-zone power gives

\[

P_{\mathrm{rad}}=rac{\eta k^4N^2I_0^2A^2}{12\pi}.

\]

Therefore,

\[

\boxed{R_r=\frac{\eta k^4N^2A^2}{6\pi}

=20k^4N^2A^2

=320\pi^4\left(\frac{NA}{\lambda^2}\right)^2}.

\]

For a one-turn circular loop of radius \(a\), \(A=\pi a^2\), giving

\[

\boxed{R_r=20\pi^2(ka)^4}.

\]

These formulas apply only when current is approximately uniform. Resonant or large loops require a nonuniform-current analysis.

Worked Example 2: Small-loop efficiency

A one-turn circular loop has radius \(a=0.10\ \mathrm{m}\) and operates at \(10\ \mathrm{MHz}\), for which \(\lambda\approx30\ \mathrm{m}\). Then

\[

ka=\frac{2\pi(0.10)}{30}=0.02094.

\]

Thus,

\[

R_r=20\pi^2(0.02094)^4

\approx3.80\times10^{-5}\ \Omega.

\]

If conductor and connection losses are \(0.20\ \Omega\),

\[

\eta_r=\frac{3.80\times10^{-5}}{0.200038}

\approx1.90\times10^{-4}=0.019\%.

\]

The calculation illustrates why small transmitting loops require thick low-resistance conductors, high-quality capacitors and careful joints. High circulating current and capacitor voltage may occur even at moderate input power.

5.3 Receiving loops and practical construction

For a uniform incident magnetic field normal to the loop, Faraday’s law gives the open-circuit voltage

\[

V_{\mathrm{oc}}=-j\omega\mu N A H_n,

\]

provided the loop is electrically small. Loop antennas are useful as direction finders because of their axial nulls. Shielding can reduce unwanted electric-field pickup, but a shield must contain a gap to prevent it from acting as a shorted turn.

At resonance, a tuning capacitor cancels the loop’s inductive reactance. The resulting bandwidth is narrow when the radiation resistance is very small. Loss, proximity effects, capacitor breakdown and environmental detuning are major design constraints.

6. Large and Resonant Loops

When loop circumference approaches one wavelength, the current is no longer uniform. A one-wavelength loop commonly has maximum radiation approximately normal to its plane, unlike an electrically small loop. Its input resistance and pattern depend strongly on shape, feed location and conductor diameter.

Square, circular and polygonal loops are used in HF and VHF arrays. Numerical full-wave analysis is usually preferable because small-loop magnetic-dipole formulas are invalid in this regime.

7. Travelling-Wave Wire Antennas

A travelling-wave antenna supports a current whose phase progresses along the conductor rather than forming a dominant standing wave. Consider a straight wire extending from \(z=0\) to \(L\), carrying

\[

I(z)=I_0e^{-j\beta z}.

\]

The radiation integral becomes

\[

F(\theta)=I_0\int_0^L e^{-j(\beta-k\cos\theta)z}\,dz

=I_0\frac{1-e^{-j(\beta-k\cos\theta)L}}

{j(\beta-k\cos\theta)}.

\]

The far field is

\[

E_\theta=\frac{j\eta k e^{-jkr}\sin\theta}{4\pi r}F(\theta).

\]

A large contribution occurs when the phase mismatch \(\beta-k\cos\theta\) is small. For a wave whose phase constant is near \(k\), radiation is concentrated toward the forward endfire region, although the element factor forces an exact null on the wire axis.

A reflected current produces standing-wave components and changes both pattern and input impedance. Travelling-wave antennas therefore employ a termination approximating their characteristic impedance. The termination absorbs residual power, improving bandwidth and pattern stability at the cost of efficiency.

Examples include terminated long wires, V antennas, rhombic antennas and Beverage receiving antennas. A Beverage antenna relies on a wire close to lossy earth and is primarily a receiving structure; its behaviour should not be inferred from an isolated-wire model alone.

8. Feeding and Baluns

A dipole is balanced: its two terminals ideally carry equal and opposite currents relative to the symmetry plane. Coaxial cable is unbalanced because the outer conductor is connected to the surrounding environment. Direct connection can drive common-mode current on the outside of the coaxial shield, causing pattern distortion, feed-line radiation, interference and impedance instability.

A balun provides a balanced-to-unbalanced transition. Common implementations include:

• Current or choke balun: presents high impedance to common-mode current while allowing differential current to pass.
• Ferrite-core current balun: provides broadband common-mode impedance when appropriate ferrite material is used.
• Quarter-wave sleeve balun: useful over a narrower microwave or VHF band.
• Transformer balun: may provide both balance and impedance transformation.

Balun selection must consider common-mode impedance, bandwidth, ferrite loss, voltage, current, temperature rise and core saturation. A voltage balun alone does not necessarily enforce equal branch currents when the antenna is environmentally asymmetric.

Balanced open-wire line offers low loss at high standing-wave ratio but requires a suitable balanced tuner and physical clearance. Microstrip-to-slotline and coaxial sleeve transitions are common in microwave wire-like radiators.

9. Mutual Impedance and Coupling

When antennas are close enough to interact, current on one produces voltage at another. For two antennas,

\[

\begin{bmatrix}V_1\\V_2\end{bmatrix}

=

\begin{bmatrix}Z_{11}&Z_{12}\\Z_{21}&Z_{22}\end{bmatrix}

\begin{bmatrix}I_1\\I_2\end{bmatrix}.

\]

Here, \(Z_{11}\) and \(Z_{22}\) are self-impedances and \(Z_{12}\), \(Z_{21}\) are mutual impedances. In a linear reciprocal medium, with consistent port definitions,

\[

Z_{12}=Z_{21}.

\]

If antenna 2 is terminated by \(Z_L\), the input impedance seen at antenna 1 is

\[

Z_{\mathrm{in},1}=Z_{11}-\frac{Z_{12}Z_{21}}{Z_{22}+Z_L}.

\]

Mutual impedance is generally complex and depends on separation, orientation, frequency and surrounding structures. Its real part describes exchanged or jointly radiated power; its imaginary part represents coupled stored energy.

Coupling affects array scan impedance, pattern, bandwidth, diversity performance and multiple-input multiple-output correlation. Pattern multiplication is valid only when the element pattern used already accounts appropriately for coupling and platform effects. Embedded-element patterns are therefore preferred in rigorous array analysis.

10. Practical Implementation from HF to Microwave

At HF, structural size, soil loss, corona, lightning protection and matching-network loss dominate design. Copper tubing, stranded wire and aluminium elements are common. Loading components should be placed and rated with awareness of current and voltage maxima.

At VHF and UHF, support structures, connectors and feed-line radiation become comparable in importance to the intended radiator. Tubular dipoles, folded dipoles, ground-plane monopoles and loop arrays are widely used.

At microwave frequencies, ordinary wires may have diameters and mounting discontinuities that are electrically significant. Printed dipoles, bond-wire antennas, planar loops, sleeve monopoles and integrated baluns are common. Surface roughness, substrate loss and connector launch geometry must be included in full-wave models.

Manufacturing tolerances should be analysed statistically when the antenna has high quality factor. Computer optimization and machine-learning surrogates can accelerate geometric exploration, but designs must still satisfy Maxwell-equation-based simulation, material limits and measured validation.

Electrically small antennas face fundamental bandwidth constraints. For an antenna enclosed by a sphere of radius \(a\), the minimum achievable quality factor grows rapidly as \(ka\) becomes small. Matching cannot remove this fundamental stored-energy limitation.

11. Illustrative Case Study: HF Quarter-Wave Monopole Installation

Background

An emergency communications facility requires an omnidirectional antenna at \(14.2\ \mathrm{MHz}\). A vertical monopole is selected because a horizontal dipole cannot be installed at sufficient height. This is an illustrative engineering case.

Problem

The design must provide acceptable efficiency, connect to a \(50\ \Omega\) coaxial line, suppress feed-line radiation and tolerate outdoor operation.

Analysis

The free-space wavelength is

\[

\lambda=\frac{299.8}{14.2}=21.11\ \mathrm{m}.

\]

An ideal quarter wavelength is

\[

h=\frac{\lambda}{4}=5.28\ \mathrm{m}.

\]

Allowing approximately 4% shortening for finite diameter and end effects gives an initial construction length of about \(5.07\ \mathrm{m}\), followed by measurement and trimming.

The ideal radiation resistance is \(36.5\ \Omega\). Suppose measured and modelled conductor-plus-ground loss is \(3.5\ \Omega\), giving a resonant resistance of about \(40\ \Omega\). Then

\[

\eta_r=\frac{36.5}{36.5+3.5}=0.9125.

\]

For a \(50\ \Omega\) line,

\[

\Gamma=\frac{40-50}{40+50}=-0.111,

\qquad

\mathrm{VSWR}=\frac{1+|\Gamma|}{1-|\Gamma|}=1.25.

\]

The mismatch loss is small, so a matching network is unnecessary unless system specifications require a lower VSWR.

Approach and Solution

The design uses multiple radial wires, a weatherproof base insulator, a coaxial current choke near the feed point and a static-discharge path. Radial number and length are increased until further measured improvement becomes uneconomic. An antenna analyser is used to trim the vertical element at its installed location rather than in isolation.

Assuming ideal monopole directivity \(3.28\), estimated realised gain is

\[

G_{\mathrm{realised}}\approx3.28(0.9125)(1-|\Gamma|^2)

\approx2.96,

\]

or approximately \(4.7\ \mathrm{dBi}\), before additional site and pattern effects.

Outcome

The design achieves a low VSWR without a lossy tuner, estimated radiation efficiency above 90%, and reduced common-mode feed-line current. Actual performance must still be confirmed through field-strength, impedance and common-mode-current measurements.

Learning Points

1. Ground loss is part of antenna efficiency, not merely a matching issue.

2. A choke controls unintended feed-line radiation.

3. Installed resonance differs from free-space prediction.

4. Low VSWR alone does not prove high efficiency.

5. Site measurements are indispensable for HF monopoles.

12. Safety, Regulation and Professional Practice

Antenna engineers must comply with spectrum licensing, permitted power, occupied bandwidth and electromagnetic-exposure regulations. High-power wire antennas can develop hazardous RF voltages at element ends and tuning capacitors. Structures require mechanical load assessment, lightning protection, grounding and safe separation from power lines.

Interference investigations should consider common-mode feed-line currents, harmonics, intermodulation and inadequate receiver immunity. Simulation assumptions, measured uncertainties and efficiency estimates should be documented honestly; a favourable impedance match must not be presented as evidence of radiated performance.

Chapter Summary

Wire-antenna radiation follows directly from the spatial distribution of conductor current. For an electrically short dipole with triangular current, the effective length is half its physical length and the radiation resistance is \(20\pi^2(L/\lambda)^2\). Resonant dipoles support approximately sinusoidal current and provide much greater radiation resistance.

An ideal monopole and its image form an equivalent dipole. For equal feed current, the monopole radiates half the corresponding dipole’s total power while concentrating radiation into one hemisphere. Real-ground loss can substantially reduce efficiency.

An electrically small loop behaves as a magnetic dipole. Its vector potential is proportional to \(jkNIA\), and its radiation resistance is \(320\pi^4(NA/\lambda^2)^2\). Because this resistance can be extremely small, conductor and tuning losses are critical.

Travelling-wave antennas obtain directional broadband behaviour through progressive-phase current and suitable termination. Baluns suppress unwanted common-mode current, while mutual impedance explains how neighbouring antennas alter input impedance and pattern. Reliable implementation requires full-wave modelling, installed measurements, safety controls and regulatory compliance.

Key Terms

Effective length: Current-weighted length that determines the far field of an electrically short antenna.

Radiation resistance: Equivalent resistance representing power radiated by an antenna.

Loss resistance: Equivalent resistance representing conductor, dielectric and ground dissipation.

Radiation efficiency: Ratio of radiated power to accepted antenna power.

Hertzian dipole: Infinitesimal current element with assumed uniform current.

Short dipole: Electrically short centre-fed conductor having an approximately triangular current.

Half-wave dipole: Resonant dipole approximately one-half wavelength long.

Monopole: Single conductor operated relative to a ground plane or counterpoise.

Image theory: Replacement of an ideal conducting boundary by equivalent image sources.

Ground radial: Conductive wire used to reduce ground-system impedance and loss.

Magnetic dipole moment: Product \(NIA\) describing an electrically small loop.

Small loop: Loop whose circumference is sufficiently small for nearly uniform current.

Travelling-wave antenna: Antenna supporting predominantly progressive-phase current.

Mutual impedance: Complex ratio describing voltage induced at one antenna by current on another.

Balun: Device or transition connecting balanced and unbalanced systems.

Common-mode current: Current flowing in the same direction on conductors intended to carry differential signals.

Embedded-element pattern: Pattern of one array element with all other ports terminated in specified loads.

Standing-wave ratio: Ratio of maximum to minimum voltage magnitude on a transmission line.

Review Questions

Short Answer Questions

1. Define transmitting effective length for an electrically short dipole.

2. Why does a triangular-current short dipole have lower radiation resistance than a uniform-current element of the same length?

3. State the image-theory relationship between a monopole and a dipole.

4. What condition permits the uniform-current approximation for a loop?

5. Why is a balun required when feeding a dipole with coaxial cable?

Long Answer Questions

1. Derive the far field and radiation resistance of a triangular-current short dipole.

2. Develop the sinusoidal-current model of a finite dipole and discuss its limitations.

3. Derive the vector potential, fields and radiation resistance of an electrically small loop.

4. Explain the operation, pattern formation and efficiency trade-offs of terminated travelling-wave antennas.

5. Discuss how finite ground conductivity affects monopole impedance, efficiency and elevation pattern.

Analytical / Application Questions

1. A \(0.4\ \mathrm{m}\) short dipole operates at \(30\ \mathrm{MHz}\). Calculate its triangular-current radiation resistance and efficiency if its loss resistance is \(1.5\ \Omega\).

2. Design an initial quarter-wave monopole for \(100\ \mathrm{MHz}\), estimate its ideal radiation resistance, and propose a practical feed and ground-plane arrangement.

3. Two coupled antennas have known \(Z_{11}\), \(Z_{22}\), \(Z_{12}\) and load \(Z_L\). Determine the driven-port input impedance and explain how changing \(Z_L\) modifies it.

Multiple Choice Questions with Answers

Question 1

What is the radiation resistance of a triangular-current short dipole with total length 0.05λ?

A. 0.123 Ω

B. 0.493 Ω

C. 1.97 Ω

D. 7.90 Ω

Correct Answer: B. 0.493 Ω

Question 2

For equal feed current, an ideal monopole compared with its corresponding isolated dipole radiates:

A. Half the total power into one hemisphere

B. Twice the total power into one hemisphere

C. The same total power into one hemisphere

D. Half the total power equally into two hemispheres

Correct Answer: A. Half the total power into one hemisphere

Question 3

What is the radiation resistance of a one-turn electrically small circular loop of radius a?

A. 20π²(ka)²

B. 80π²(ka)²

C. 20π²(ka)⁴

D. 320π⁴(ka)⁴

Correct Answer: C. 20π²(ka)⁴

Question 4

For the 0.10 m radius loop at 10 MHz described in the chapter, the radiation resistance is approximately:

A. 3.8 × 10⁻⁵ Ω

B. 3.8 × 10⁻² Ω

C. 0.20 Ω

D. 3.8 Ω

Correct Answer: A. 3.8 × 10⁻⁵ Ω

Question 5

Which expression gives the far-zone vector potential of an electrically small loop?

A. Aφ = jμkNIA e⁻ʲᵏʳ sinθ/(4πr)

B. Aφ = μk²NIA e⁻ʲᵏʳ sinθ/(4πr)

C. Aφ = ηNIA e⁻ʲᵏʳ cosθ/(4πr²)

D. Aφ = jμNIA e⁻ʲᵏʳ/(4πkr²)

Correct Answer: A. Aφ = jμkNIA e⁻ʲᵏʳ sinθ/(4πr)

Question 6

The approximate radiation resistance of an ideal quarter-wave monopole is:

A. 18.3 Ω

B. 36.5 Ω

C. 73.1 Ω

D. 146.2 Ω

Correct Answer: B. 36.5 Ω

Question 7

What is the principal purpose of a current balun at a coax-fed dipole?

A. To increase the dipole's physical length

B. To suppress common-mode current on the feed line

C. To double the antenna radiation resistance

D. To create circular polarization

Correct Answer: B. To suppress common-mode current on the feed line

Question 8

For two reciprocal coupled antennas, which relationship applies when port definitions are consistent?

A. Z12 = −Z21

B. Z12 = Z21

C. Z12 = 1/Z21

D. Z12 = Z11 + Z22

Correct Answer: B. Z12 = Z21

Question 9

A small loop has radiation resistance 3.8 × 10⁻⁵ Ω and loss resistance 0.20 Ω. Its radiation efficiency is approximately:

A. 0.019%

B. 0.19%

C. 1.9%

D. 19%

Correct Answer: A. 0.019%

Question 10

Why is a termination used on a travelling-wave wire antenna?

A. To increase reflected current and form a standing wave

B. To eliminate the need for a feed line

C. To absorb residual power and reduce reflections

D. To make the conductor electrically shorter

Correct Answer: C. To absorb residual power and reduce reflections