10.1 Resonant Dipoles, Inverted-Vs, and Multiband Wire Antennas

10.1 Resonant Dipoles, Inverted-Vs, and Multiband Wire Antennas

In radio frequency communications, the wire antenna is the classic foundation upon which all modern electromagnetic radiator theory is built. Simple in construction yet mathematically elegant, wire antennas utilize standing electrical waves along conductive spans to convert guided radio frequency energy from a transmitter into propagating electromagnetic wavefronts in free space. For the Australian Advanced amateur radio operator, understanding the precise current and voltage distributions, complex feedpoint impedance variations over real earth, and broadband impedance matching architectures of wire systems is crucial for designing reliable, high-efficiency HF transmitting installations.

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The Resonant Half-Wave Dipole

The fundamental building block of wire antenna design is the center-fed resonant half-wave dipole (often designated $\lambda/2$ dipole). Consisting of a straight electrical conductor severed at its physical center to accept an RF transmission line, the dipole functions as an open-ended resonant transmission line folded outward into free space.

Open end A              Center feed              Open end B
   o-------------------o A     B o-------------------o
 I near zero           Current maximum          I near zero
 Voltage maximum       Relative voltage minimum Voltage maximum

A and B are separate feed terminals connected to the feedline.
The RF voltage across A-B is finite when power is supplied.

Current and Voltage Distributions

When excited at its fundamental resonant frequency, the standing wave pattern along the dipole conductor is governed by the boundary conditions imposed by its physical geometry:

1. Current Distribution: Because electrons cannot leave the conductor tips into insulating air, the RF current must drop strictly to zero at the wire extremities ($I = 0$). By wave reflection, an incident current wave and reflected current wave interfere to establish a sinusoidal standing wave. At a distance of $\lambda/4$ inward from each open tip (the physical center of the half-wave span), the currents constructively interfere to form a current antinode ($I_{\text{max}}$). The current distribution as a function of position $x$ measured from the center ($-\lambda/4 \le x \le +\lambda/4$) is:

$$I(x) = I_0 \cos\left(\frac{2\pi x}{\lambda}\right)$$

1. Voltage Distribution: Charge accumulates near the open ends, where RF voltage is high. The two ends have opposite instantaneous polarities. Near the center, voltage is relatively low while current is high; this is not a zero voltage across the driven feed terminals.

The antenna's input impedance is the voltage across its feed terminals divided by the current entering them:

• Center Feed: A resonant half-wave dipole has a low, approximately resistive input, commonly quoted as about $73\ \Omega$ in the free-space model. A finite current therefore requires a finite feed voltage.
• End Feed: Feeding near an end gives a high but finite impedance, commonly thousands of ohms. That driven input is different from an undriven open tip, where the ideal current boundary is zero.

Physical vs. Electrical Length Formula

In theoretical free space, a half-wavelength is defined strictly by the speed of light ($c \approx 299,792,458\text{ m/s}$):

$$l_{\text{free-space}} = \frac{\lambda}{2} = \frac{c}{2f} \approx \frac{150}{f_{\text{MHz}}}\text{ meters}$$

However, a practical wire antenna suspended in the atmosphere must be physically cut shorter than the theoretical free-space half-wavelength to achieve zero reactance ($X_{\text{in}} = 0$) at the desired design frequency. This shortening is dictated by two electromagnetic effects:

1. Conductor Diameter-to-Length Ratio ($l/d$): Radio waves travel along a metallic wire slightly slower than in vacuum ($v = k \cdot c$). For practical copper antenna wires (such as $1.6\text{ mm}$ to $2.5\text{ mm}$ diameter, AWG 14 to AWG 10), the ratio of length to diameter ($l/d$) is large, establishing a velocity factor of $k \approx 0.95$ to $0.98$.
2. End-Effect Capacitance: Stray capacitive coupling exists between the open ends of the wire conductor and surrounding space (trees, soil, support insulators). This capacitive fringe field stores electrical energy, effectively adding electrical length to the wire extremities.

Combining the nominal velocity factor ($k \approx 0.95$) and end-effect leads to the classic physical length formula for a practical half-wave wire dipole:

$$l = \frac{142.5}{f_{\text{MHz}}}\text{ meters} \quad \left(\text{or } l = \frac{468}{f_{\text{MHz}}}\text{ feet}\right)$$

Free-Space Feedpoint Impedance

For an infinitesimally thin half-wave dipole situated in free space at an exact length of $l = \lambda/2$, Maxwell's equations yield a complex feedpoint impedance of:

$$Z_{\text{in}} = 73.13 + j42.5\ \Omega$$

Notice the $+j42.5\ \Omega$ inductive reactance! An exact theoretical half-wave dipole is slightly inductive. Trimming the physical length downward by approximately $2\%$ to $3\%$ cancels this inductive reactance against the capacitive end-effect, establishing true series resonance:

$$Z_{\text{resonant}} = 73 + j0\ \Omega$$

Feedpoint Impedance Variation with Height Above Ground

In real terrestrial installations, an antenna does not radiate in free space; it radiates above conducting earth. The ground acts as an electrical boundary that reflects incident electromagnetic radiation, creating an apparent image antenna beneath the surface.

                         Horizontal Dipole at Height h
                       ================================= (+I)
                                       |
                                       | h
                                       |
     ~~~~~~~~~~~~~~~~~~~~ Real Earth Surface ~~~~~~~~~~~~~~~~~~~~
                                       |
                                       | h
                                       |
                       ================================= (-I)
                           Ground Image (180° Inverted)

The real antenna couples electromagnetically with its subterranean image via mutual impedance ($Z_{12}$). The total input impedance seen at the dipole terminals is the sum of its self-impedance ($Z_{11}$) and the mutual impedance ($Z_{12}$):

$$Z_{\text{in}} = Z_{11} - Z_{12}$$

The image is separated from the real horizontal dipole by $2h$. Changing height changes their mutual impedance, so both the feedpoint resistance and reactance vary. The impedance follows the complete current distribution and ground properties, rather than a single reflected-ray phase calculation:

• Very low heights: The opposite image current reduces radiation resistance over an ideal conducting ground. Real soil also introduces loss, so measured input resistance includes more than radiation resistance.
• Heights of a few tenths of a wavelength: Radiation resistance rises and can exceed the free-space value of about $73\ \Omega$. A quarter-wavelength mounting height does not by itself establish a $70\ \Omega$ input or zero reactance.
• Increasing height: Resistance oscillates above and below its free-space value as coupling with the image changes. The oscillations diminish as the antenna is raised farther from the ground.

Use these trends to understand installation effects, then measure and tune the antenna at its working height. Height alone does not specify its exact impedance.

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The Inverted-V Antenna

The Inverted-V is an extremely popular practical variant of the half-wave dipole. Instead of requiring two tall support masts at opposite ends of the span, the inverted-V utilizes a single central mast to elevate the feedpoint apex, allowing the two antenna legs to slope symmetrically downward toward the ground.

                         Center support / feedpoint
                                   /\
                         Leg A    /  \    Leg B
                                 /    \
                    Insulator   o      o   Insulator
                                |      |
                         Ends kept above accessible areas
                         Tie-off ropes extend to anchors

Included apex angle is between the two sloping wire legs.
The wire ends are insulated; they are not grounded electrodes.

Apex Angle and Impedance Transformation

The included apex angle $\theta_{\text{apex}}$ is commonly around $90^\circ$–$120^\circ$ or wider. Bringing the legs closer together generally lowers feedpoint resistance compared with a flat dipole. A $50\ \Omega$ feed can therefore be a useful choice, but angle alone does not determine the exact resistance or reactance.

Very sharp angles increase field cancellation between the legs and can reduce radiation resistance and bandwidth. Keep the intended geometry when tuning.

Radiation Pattern and Mechanical Advantages

• H-Plane Fill-In: A horizontal dipole exhibits sharp radiation nulls off its ends (along the wire axis). Because the legs of an inverted-V slope downward, their currents possess both horizontal and vertical vector components. The vertical component radiates an omnidirectional wave that partially fills in the end nulls, transforming the classic figure-eight pattern into an oval, semi-omnidirectional radiation pattern.
• Reduced Wire Span: For straight equal legs of total wire length $L$ and included apex angle $\theta_{\text{apex}}$, the horizontal distance between wire ends is $L\sin(\theta_{\text{apex}}/2)$. Anchor positions also depend on the tie-off ropes and are not fixed by this wire-span calculation.
• End Insulation and Access: The wire ends and matching components can carry high RF voltage. Use suitable insulators and arrange the installation to prevent contact during transmission. A fixed height of 2.5 or 3 meters alone does not certify safety or compliance; assess accessible conductors and RF exposure for the actual station.

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The Folded Dipole

A folded dipole consists of two parallel, closely spaced electrical conductors connected together at their extreme ends, with one conductor opened at the center to accept the feedline.

        +---------------------------------------+
        |                                       |
        +---------------o       o---------------+
                        A       B
                        |       |
                     [ 4:1 balun ]
                        |       |
                       75-ohm coax

A-B: about 300 ohms in this example.
After the 4:1 transformation: about 75 ohms, not 50 ohms.

The 4:1 Impedance Step-Up Factor

The operating principle of the folded dipole can be analyzed by decomposing its currents into two orthogonal modes: an antenna mode (radiating) and a transmission-line mode (non-radiating):

1. The two parallel wires are separated by a distance $d$ that is tiny compared to the operating wavelength ($d \ll \lambda$). Consequently, the electromagnetic fields radiated by the two conductors add in-phase in the far field.
2. When fed with RF energy, the total antenna current $I_{\text{total}}$ required to radiate a given power is divided equally between the two identical conductors:

$$I_1 = I_2 = \frac{I_{\text{total}}}{2}$$

1. The feedline connects only to Conductor 2, meaning the input current drawn from the generator is $I_{\text{in}} = I_{\text{total}} / 2$.
2. Because power radiated is $P = I_{\text{total}}^2 R_0 = I_{\text{in}}^2 R_{\text{in}}$, substituting $I_{\text{total}} = 2 I_{\text{in}}$ yields:

$$R_{\text{in}} = \frac{(2 I_{\text{in}})^2 R_0}{I_{\text{in}}^2} = 4 \times R_0$$

Since a standard half-wave dipole exhibits a radiation resistance of $R_0 \approx 72\ \Omega$ in free space, the input resistance of an identical two-wire folded dipole is stepped up by a factor of $4$:

$$R_{\text{in}} = 4 \times 72\ \Omega \approx 288\ \Omega \text{ to } 300\ \Omega$$

If the two conductors have different diameters ($d_1$ for the continuous wire, $d_2$ for the fed wire), the impedance step-up ratio $r$ can be custom engineered via the formula:

$$r = \left(1 + \frac{\ln(2s / d_2)}{\ln(2s / d_1)}\right)^2$$

Impedance Bandwidth

The close parallel conductors increase effective conductor diameter, which can broaden impedance bandwidth. The approximate modal circuit also combines the radiating and transmission-line responses:

$$Z_{\text{in}}\approx(4Z_A)\parallel(2Z_{\text{TL}}), \qquad Z_{\text{TL}}=jZ_0\tan(\beta L/2).$$

Here $Z_A$ is the equivalent straight-dipole impedance and $L$ is the folded dipole's span. The two shorted line halves form a series $2Z_{\text{TL}}$ branch in this model. Near half-wave resonance that branch is high impedance. Just above it, its capacitive reactance gives positive susceptance, which can offset the radiating branch's inductive response. Actual SWR bandwidth still depends on dimensions and the feed impedance; folding alone does not guarantee coverage of an entire band.

Feeding the Folded Dipole

Because its natural feedpoint impedance is $\approx 300\ \Omega$, a folded dipole can be fed directly with $300\ \Omega$ balanced twin-lead or ladder line. To interface with modern $50\ \Omega$ unbalanced coaxial cable, a 4:1 Guanella current balun is installed directly at the center feedpoint, transforming $300\ \Omega$ down to $75\ \Omega$ (or transforming $200-240\ \Omega$ down to $50-60\ \Omega$ when mounted at heights where ground coupling depresses base resistance).

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The Off-Center Fed Dipole (OCFD / Windom)

The Off-Center Fed Dipole (OCFD), historically known as the Windom antenna, is an asymmetric multiband wire antenna designed to provide low SWR on multiple harmonically related amateur bands using a single feedline.

               Off-Center-Fed Dipole: example arrangement

Long arm (about 2L/3) --------------o       o---------- Short arm
                                    A       B           (about L/3)
                                    |       |
                                 [ 4:1 current
                                   transformer ]
                                    |       |
                                   50-ohm coax
                                 with suitable choking

A-B are separate feed terminals. A 200-ohm load would transform
to 50 ohms here; measure the actual impedance on each working band.

Harmonic Current Distributions and the One-Third Feedpoint

A center-fed half-wave wire has a current maximum at the feed on its fundamental and odd harmonics, but a high-impedance current minimum near the center on even harmonics. Matching those even-harmonic impedances to coax can be difficult.

Moving the feed near one-third of the span avoids center-feed nulls on the fundamental, second, fourth and eighth harmonics. This underlies common $80/40/20/10\text{ m}$ OCF designs. It does not avoid every harmonic null: in the sinusoidal-current model, the one-third point is a current null on the third and sixth harmonics. Thus a basic $80\text{ m}$ one-third-fed wire does not automatically include $15\text{ m}$; changed dimensions or additional elements may be needed.

Matching and Common-Mode Choking

Practical OCF designs often use nominal input impedances around $200$–$300\ \Omega$, but neither this value nor a low SWR is fixed across all working bands. Select the impedance transformer from the actual design and measured load. An ideal 4:1 unit transforms $200\ \Omega$ to $50\ \Omega$; it does not guarantee SWR below $2:1$ on every band.

The asymmetric antenna can excite common-mode feedline current. Use a current-balancing transformer and adequate choking for the intended bands. Choke location and impedance depend on the antenna and RF return path; there is no universal $3$–$5\text{ m}$ placement rule.

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Multiband Trap Dipoles

A trap dipole achieves multiband operation along a single collinear wire by inserting parallel-resonant LC tuned circuits—known as traps—at calculated intervals.

                    Symmetrical two-band trap dipole

Outer wire --[parallel LC]-- Inner wire --o A   B o-- Inner wire --[parallel LC]-- Outer wire

A-B are separate center-feed terminals connected to a balanced
feed arrangement, commonly coax through a suitable current balun.
At the higher band the traps have high impedance, so the inner
span is the principal radiator. Below trap resonance, current also
flows in the outer sections and the traps provide inductive loading.

Parallel Resonance Operation Mechanics

A parallel LC circuit exhibits an impedance given by:

$$Z_{\text{trap}} = \frac{j\omega L}{1 - \omega^2 L C}$$

1. Higher-Band Operation: Choose the trap resonance near the higher operating frequency, $f_{\text{high}}=1/(2\pi\sqrt{LC})$. At parallel resonance the ideal inductive and capacitive susceptances cancel, giving infinite impedance in the lossless model. A real trap has a finite high impedance determined by loss and loading. In the intended design this greatly reduces current in the outer sections, leaving the inner span as the principal high-band radiator.
2. Operation at Lower Band ($f_{\text{low}}$): At a lower operating frequency ($f_{\text{low}} < f_{\text{high}}$), the operating frequency is below the trap's self-resonance ($ω^2 LC < 1$). The trap exhibits a net inductive reactance ($+jX_L$). The trap ceases to act as an insulator; instead, it functions as an inductive loading coil. RF current passes through the trap into the outer wire extensions.

Physical Shortening and Bandwidth Trade-Offs

Because the traps behave as series inductors on the lower bands:

• Physical Shortening: The inductive loading electrically lengthens the wire, meaning the physical span required for low-band resonance is substantially shorter than a full-size half-wave dipole.
• Narrowed Bandwidth: Inductive loading concentrates magnetic energy, lowering radiation resistance and raising the loaded $Q$ of the antenna. Consequently, the operating bandwidth (2:1 SWR curve) on the lowest band is significantly narrower than that of a full-sized wire dipole, often requiring an ATU to cover an entire band such as $80\text{ m}$.

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The End-Fed Half-Wave (EFHW) Antenna

The End-Fed Half-Wave (EFHW) antenna has become one of the most widely deployed portable and permanent HF antennas. Rather than driving the antenna at its center current antinode, the EFHW is driven directly at one extreme end.

50-ohm coax  ===> [ 49:1 matching transformer ] ----> Half-wave wire
                                |
                         RF return connection
                                |
                    Counterpoise / defined return path

The coax port and high-impedance antenna port are distinct.

Voltage-Fed High Impedance Physics

An EFHW feeds the wire near an end, where the voltage-to-current ratio is high. Its driven terminal still carries finite current and has an RF return path. Typical practical end-fed inputs are resistances of a few thousand ohms with reactance that depends on tuning and installation:

$$Z_{\text{in}} \approx 2000\ \Omega \text{ to } 4000\ \Omega$$

Because the antenna is an electrical half-wavelength at its fundamental frequency ($f_0$), it also naturally supports integer multiples of half-wavelengths ($n \cdot \lambda/2$) at all harmonic frequencies:

• Fundamental ($f_0$, e.g., $7.1\text{ MHz}$): $1 \times \lambda/2$, end is a voltage antinode ($Z_{\text{in}} \approx 3000\ \Omega$).
• 2nd Harmonic ($2f_0$, e.g., $14.2\text{ MHz}$): $2 \times \lambda/2 = 1\lambda$, end is a voltage antinode ($Z_{\text{in}} \approx 3000\ \Omega$).
• 3rd Harmonic ($3f_0$, e.g., $21.3\text{ MHz}$): $3 \times \lambda/2$, end is a voltage antinode ($Z_{\text{in}} \approx 3000\ \Omega$).
• 4th Harmonic ($4f_0$, e.g., $28.4\text{ MHz}$): $4 \times \lambda/2$, end is a voltage antinode ($Z_{\text{in}} \approx 3000\ \Omega$).

An EFHW cut for the $40\text{ m}$ band operates naturally across $40\text{ m}, 20\text{ m}, 15\text{ m},\text{ and } 10\text{ m}$ without any internal traps or moving parts!

Broadband 49:1 and 64:1 Matching Transformers

To transform the high end impedance down to the $50\ \Omega$ characteristic impedance of coaxial cable, a broadband RF autotransformer wound on a high-permeability ferrite toroidal core (such as Fair-Rite Mix 43) is employed:

• 49:1 Impedance Ratio: Wound with a $2:14$ turns ratio ($1:7$ primary-to-secondary turns ratio). The impedance transformation scales as the square of turns ratio ($N^2$):

$$\frac{Z_{\text{sec}}}{Z_{\text{pri}}} = \left(\frac{14}{2}\right)^2 = 7^2 = 49:1 \implies Z_{\text{in}} = 50\ \Omega \times 49 = 2450\ \Omega$$

• 64:1 Impedance Ratio: Wound with a $2:16$ turns ratio ($1:8$ turns ratio), yielding $8^2 = 64:1$, matching $50\ \Omega$ coax to $3200\ \Omega$.

A small capacitor across the primary can improve the upper-band match by compensating the complete transformer response. For example, the cited construction uses $100\text{ pF}$ rated $2\text{ kV}$. Its value and voltage rating belong to that design; choose and check the compensation with the intended transformer and load rather than assuming universal leakage-inductance cancellation.

Counterpoise Requirements

An EFHW requires an RF return path. For a given resistive input of $3000\ \Omega$ and $100\text{ W}$ accepted power, $I_{\text{feed}}=\sqrt{100/3000}\approx0.18\text{ A}$; this current is finite. The return may involve a counterpoise, the coax exterior and capacitance to the surroundings.

Provide the return arrangement specified by the antenna design and use appropriate common-mode choking to control current on the remaining feedline. Counterpoise length and choke position affect tuning and current distribution; a fixed distance cannot guarantee the same result on all bands.

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Step-by-Step Worked Calculation: 40-Meter Inverted-V Antenna Design

An Australian Advanced amateur operator desires to construct an inverted-V antenna for the 40-meter amateur allocation, optimized for center frequency $f_0 = 7.100\text{ MHz}$. The antenna apex will be supported at a height of $12\text{ meters}$ on a fiberglass mast, with the two wire legs sloping downward at an included apex angle of $\theta_{\text{apex}} = 100^\circ$.

                         Apex: 12 m
                             /\
                  9.835 m   /  \   9.835 m
                           /    \
                 Wire end o      o Wire end
                          |      |
                          |      |  End height: about 5.68 m
Ground -------------------+------+-------------------
                     7.53 m from mast to each wire end

Included apex angle: 100 degrees.
Tie-off ropes and their ground anchors are not dimensioned here.

Step 1: Calculate the Free-Space Wavelength

$$\lambda = \frac{c}{f_0} = \frac{299.792 \times 10^6\text{ m/s}}{7.100 \times 10^6\text{ Hz}} \approx 42.224\text{ meters}$$

Step 2: Calculate the Baseline Resonant Dipole Length

Applying the standard half-wave physical wire formula accounting for end-effect and conductor velocity factor ($k \approx 0.95$):

$$l_{\text{dipole}} = \frac{142.5}{f_{\text{MHz}}} = \frac{142.5}{7.100} \approx 20.070\text{ meters}$$

Each leg of a standard horizontal dipole would measure $l_{\text{leg}} = 20.070 / 2 = 10.035\text{ meters}$.

Step 3: Adjust the Installed Length

The starting formula does not establish a fixed inverted-V shortening percentage. Suppose measurements in this example installation show that resonance requires a $2\%$ reduction from the baseline length:

$$l_{\text{inverted-V}} = l_{\text{dipole}} \times (1 - 0.02) = 20.070 \times 0.98 \approx 19.669\text{ meters}$$

Dividing into two symmetrical legs:

$$l_{\text{leg}} = \frac{19.669\text{ m}}{2} \approx 9.835\text{ meters per leg}$$

Step 4: Verify Ground Clearance and Anchor Distance

Using trigonometry on the right triangle formed by the mast, ground, and one wire leg:

• The half-angle at the apex is $\alpha = \theta_{\text{apex}} / 2 = 100^\circ / 2 = 50^\circ$.
• The vertical drop of the wire leg is:

$$\Delta h = l_{\text{leg}} \times \cos(50^\circ) = 9.835\text{ m} \times 0.6428 \approx 6.322\text{ meters}$$

• The physical height of the insulator ends above ground is:

$$h_{\text{end}} = h_{\text{apex}} - \Delta h = 12.000\text{ m} - 6.322\text{ m} = 5.678\text{ meters}$$

The calculated wire ends are about 5.68 meters above level ground. This geometric result does not certify RF safety: prevent access to live conductors and assess the installation’s exposure and clearances.

• The horizontal distance from the mast to the vertical projection of each wire end is:

$$d_{\text{end}}=l_{\text{leg}}\sin(50^\circ)=9.835\times0.7660\approx7.534\text{ m}$$

The horizontal wire-end span is $2\times7.534=15.068\text{ m}$, compared with about $20.07\text{ m}$ for the baseline flat dipole. Additional space for tie-off ropes depends on the chosen anchor positions.

Step 5: Measure the Feedpoint and Calculate SWR

The apex angle and height do not uniquely specify radiation resistance, ground loss or reactance. Measure the installed antenna at its feedpoint, or remove the feedline's transformation from the measurement.

For illustration, suppose the tuned feedpoint is measured as $45+j0\ \Omega$. On a $50\ \Omega$ line:

$$\text{SWR}=\frac{50}{45}\approx1.11:1.$$

This is a useful direct match. If the measured impedance includes reactance, calculate the reflection coefficient from the complete complex impedance or adjust the antenna/matching network. Do not add an assumed ground-loss resistance merely to obtain exactly $50\ \Omega$.

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Comparison Table: Practical Wire Antenna Configurations