10.2 Quarter-Wave Verticals, Ground Systems, and Radiation Efficiency

10.2 Quarter-Wave Verticals, Ground Systems, and Radiation Efficiency

While horizontal wire antennas are staple performers for regional and transcontinental communications, long-distance intercontinental DX on the High Frequency (HF) bands requires radiation concentrated at low vertical takeoff angles. The quarter-wave ($\lambda/4$) vertical monopole is the preeminent omnidirectional antenna for achieving low-angle radiation without requiring astronomical tower heights. However, because a vertical monopole is an unbalanced system operating against the earth, its radiation efficiency, input impedance, and overall performance are governed directly by the quality of its ground return system.

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The Quarter-Wave Monopole and Image Antenna Theory

A quarter-wave vertical antenna consists of a straight vertical radiating element of electrical length $\lambda/4$, fed at its base against an underlying conductive plane.

                             Quarter-Wave Vertical Monopole

             Open Top: Voltage Antinode (V_max), Current Zero (I = 0)
                                       |
                                       |
                                       | Physical Radiator
                                       | Length h ≈ λ/4
                                       |
                                       |
             Base Feed: Current Maximum; Finite RF Feed Voltage
     ==================================o==================================
     ///// Conductive Ground Plane ///////////////////////////////////////
                                       : (Image Antenna)
                                       : Forms lower half of
                                       : virtual half-wave dipole
                                       :
                                       :

The Image Antenna Principle

The physical operation of a vertical monopole is explained through the Method of Images in electrodynamics:

1. Boundary conditions at the surface of an infinite, perfectly conducting planar ground require that the tangential component of the electric field must equal zero ($E_{\text{tan}} = 0$).
2. This boundary condition is mathematically identical to replacing the conducting ground plane with a virtual image antenna positioned beneath the surface.
3. Because the radiator is oriented perpendicularly to the ground plane, the vertical current in the real monopole induces an in-phase vertical current in the subterranean image.
4. Together, the real $\lambda/4$ vertical radiator and its virtual $\lambda/4$ image synthesize a complete half-wave ($\lambda/2$) vertical dipole operating in free space!

Current, Voltage, and Theoretical Input Resistance

The standing wave distribution on a quarter-wave monopole corresponds exactly to one half of a center-fed dipole:

• Base Feedpoint: Current is maximum and voltage is relatively low compared with the open top. The driven terminal voltage is finite: $V_{\text{feed}}=I_{\text{base}}Z_{\text{in}}$.
• Open Top: Forms a voltage antinode ($V_{\text{max}}$) and a current node ($I = 0$).

Over an infinite, perfectly conducting ground plane, the monopole radiates only into the upper hemisphere ($2\pi$ steradians), whereas a dipole radiates into the full sphere ($4\pi$ steradians). For an identical feedpoint current $I_{\text{base}}$, the total power radiated by the monopole is exactly half that of a dipole in free space:

$$P_{\text{monopole}} = \frac{1}{2} P_{\text{dipole}} = \frac{1}{2} I_{\text{base}}^2 R_{\text{dipole}} = I_{\text{base}}^2 \left(\frac{R_{\text{dipole}}}{2}\right)$$

Consequently, the theoretical radiation resistance of a resonant quarter-wave vertical over perfect ground is exactly half the radiation resistance of a resonant half-wave dipole ($73.13\ \Omega / 2$):

$$R_r = 36.6\ \Omega \approx 36.8\ \Omega$$

Elevation Pattern and DX

An ideal quarter-wave monopole over an infinite perfect conductor is omnidirectional in azimuth, has maximum radiation toward the horizon and a null along its vertical axis. Real ground loss and reflection modify this pattern, including its low-angle strength.

Low-angle radiation is often useful for long single-hop skywave paths, so a vertical with an effective ground system can be a practical DX antenna. It is not automatically superior on every path. Horizontal-dipole height and terrain also determine elevation lobes, while a low dipole can be advantageous for high-angle regional communication.

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Ground Systems and Radiation Efficiency

While the theoretical radiation resistance over perfect ground is $36.8\ \Omega$, real earth is far from a perfect conductor. Soil consists of minerals, moisture, and organic material exhibiting poor electrical conductivity (typically $\sigma \approx 10^{-4}\text{ S/m}$ for dry sand to $10^{-2}\text{ S/m}$ for rich loam) and dielectric permittivity ($\varepsilon_r \approx 5 - 15$).

                                Earth Return Currents

                                  Vertical Radiator
                                         |
                                         |
                                         |  Displacement Currents (E-Field Lines)
                                         |    )   )   )   )   )
                                         |   (   (   (   (   (
     ====================================o====================================
           <======= (J_return) ======>  Base  <====== (J_return) =======>
     High Current Concentration in Lossy Earth -> Severe I^2 * R_g Dissipation!

The Closed Electrical Circuit of an Antenna

Charge-current continuity involves both conduction current and the changing electric field. The displacement-current density is $\mathbf J_D=\partial\mathbf D/\partial t$ (or $\varepsilon\,\partial\mathbf E/\partial t$ in a uniform linear dielectric), and current through a surface is $I_D=\int\mathbf J_D\cdot d\mathbf A$. In a practical monopole, soil and the radial system participate in the RF return. This does not mean radiated energy returns to the transmitter: radiation carries energy away. The equivalent input resistance is:

$$R_{\text{in}} = R_r + R_g + R_c$$

where:

• $R_r$ is the radiation resistance (the virtual resistance that accounts for energy genuinely launched into space as radio waves).
• $R_g$ is the ground loss resistance (ohmic dissipation caused by return currents flowing through lossy soil).
• $R_c$ is the conductor resistance (RF ohmic resistance of the aluminum tubing, loading coils, and connecting leads due to skin effect).

Radiation Efficiency Formulation

The radiation efficiency ($\eta$) of an antenna is the fraction of net power accepted at its feedpoint that is radiated, rather than dissipated in antenna and ground losses:

$$\eta = \frac{P_{\text{radiated}}}{P_{\text{input}}} \times 100\% = \frac{I_{\text{base}}^2 R_r}{I_{\text{base}}^2 (R_r + R_g + R_c)} \times 100\% = \frac{R_r}{R_r + R_g + R_c} \times 100\%$$

Because $R_c$ is usually small ($< 0.5\ \Omega$) in full-size aluminum radiators, efficiency is governed almost entirely by the ratio of radiation resistance ($R_r$) to ground resistance ($R_g$):

$$\eta \approx \frac{R_r}{R_r + R_g} \times 100\%$$

Current Concentration and Radial Density

RF return currents and electric fields can produce substantial loss near the antenna base. Their distribution depends on the radiator, soil and ground-system geometry. Radial wires provide lower-loss conducting paths and reduce these near-field losses; one universal 1/r law does not describe every radial installation.

                           Buried Radial Screen Layout

                                  /   |   \
                                /     |     \
                              /       |       \
                            /         |         \
                          +-----------o-----------+
                        /   \       Base        /   \
                      /       \               /       \
                    /           \           /           \
                  /               \       /               \
                                    \   /
                            (30 to 120 Buried Radials)

Buried Radial Systems

A classic buried ground system consists of bare copper wires laid in a radial spoke pattern centered at the antenna base:

• Wire Placement: Radials are buried shallowly—typically $2\text{ to }10\text{ cm}$ below the surface—merely to prevent mechanical damage from lawn mowers and turf aeration. Because skin depth in conductive soil at HF is on the order of meters, shallow burial intercepts displacement currents at the surface. Adding radials generally reduces ground loss, with diminishing improvement as the system becomes denser. Radial count alone does not specify a resistance or efficiency: soil conductivity, frequency, wire length, burial and antenna dimensions also matter.

Elevated Resonant Radials

For suburban stations where laying 60 buried radials is impossible, elevated resonant radials provide an extraordinarily efficient alternative.

                           Elevated Resonant Ground Plane

                                  Vertical Monopole
                                         |
                                         |
                                         o  Base Feedpoint
                                       / | \
          Elevated Radial 1          /   |   \          Elevated Radial 2
     ===============================+    |    +===============================
     (Height h > 0.05λ above earth)      |      (Height h > 0.05λ above earth)
                                   Support Post
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Lossy Earth Soil ~~~~~~~~~~~~~~~~~~~~~~~~~~~~

1. Lower-Loss Return: Elevating the radial system can reduce lossy interaction with nearby soil, but does not eliminate the electric field in the ground.
2. Coupled Tuning: The radiator and radials interact with each other and earth. Their lengths, height and symmetry affect resonance and current distribution; separately cutting each wire to a nominal quarter wavelength does not establish zero loss.
3. Practical Performance: Three or four well-arranged elevated radials can perform well and, in suitable installations, compare with many ground-based radials. This result is not guaranteed by a fixed height or count. Check the actual geometry, balance and surroundings, and keep high-voltage radial ends inaccessible.

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Shortened Vertical Antennas and Loading Methods

On the low HF bands ($80\text{ m}$ and $160\text{ m}$), a full-size quarter-wave vertical requires physical heights of $20\text{ meters}$ ($66\text{ ft}$) and $40\text{ meters}$ ($130\text{ ft}$), respectively. Physical space and zoning restrictions often mandate electrically shortened radiators.

The Radiation Resistance Collapse

For an electrically short unloaded monopole ($h\ll\lambda$), the current is approximately triangular: maximum at the base and zero at the top. Over a perfect ground plane its base-referred radiation resistance is

$$R_r\approx40\pi^2\left(\frac{h}{\lambda}\right)^2\ \Omega \approx395\left(\frac{h}{\lambda}\right)^2\ \Omega.$$

This short-antenna approximation does not accurately describe every height up to $\lambda/4$.

Furthermore, because the antenna is shorter than resonance, it exhibits intense capacitive reactance ($-jX_C$) that must be canceled by inserting inductive reactance ($+jX_L$) to re-establish resonance.

                        Loading Methods for Shortened Verticals

         Base Loading                   Center Loading                 Top-Hat Loading
              |                              |                          === Capacitive Hat ===
              |                              |                                    |
              |                              |                                    |
              |                             [#] Loading Coil                      |
              |                              |                                    |
             [#] Loading Coil                |                                    |
              |                              |                                    |
       =======o=======                =======o=======                      =======o=======
         (Full coil current)           (Changes current profile)         (Capacitive end loading)

Comparative Analysis of Loading Techniques

1. Base Loading

• Mechanism: A series loading inductor is inserted at the bottom feedpoint of the mast.
• Loss: The base coil carries the full feed current and adds series resistance $r_{\text{coil}}=|X_L|/Q_0$. If radiation resistance is small, this resistance can consume a large fraction of accepted power. For example, $R_r=1\ \Omega$ and coil loss resistance $4\ \Omega$, neglecting other losses, give $\eta=1/(1+4)=20\%$. Actual loss is not fixed by the label “base loading.”
• Advantages: Mechanically simple; easily accessible for motor-driven roller inductors or switched taps.

2. Center Loading

• Mechanism: The loading inductor is positioned midway up the vertical radiator.
• Advantages: Moving loading upward can maintain more current over the lower part of the radiator and increase effective radiating height. The coil current may be lower, but its required inductance and loss also change. Calculate the resulting current distribution and loss rather than assuming a universal twofold resistance increase or 3–5 dB efficiency gain.
• Disadvantages: Heavy inductor aloft requires stronger mechanical mast guy wires.

3. Top Loading / Capacitive Hat

• Mechanism: Large metallic horizontal spokes, wires, or a perimeter ring (a capacitive top hat) are attached to the upper extremity of the mast.
• Current Profile: A capacitive hat supports charge at the top and increases current in the upper vertical section. A sufficiently large hat can make the vertical current approach a uniform distribution.
• Ideal Short-Radiator Limit: Relative to a triangular profile with effective height $h/2$, a uniform profile has effective height $h$. For the same base current, this doubles field strength and gives four times the radiation resistance:

$$R_r\approx160\pi^2\left(\frac{h}{\lambda}\right)^2\ \Omega \approx1580\left(\frac{h}{\lambda}\right)^2\ \Omega.$$

• Practical Trade-Off: A finite hat gives an intermediate current distribution and may still need a loading coil. Top loading can reduce required inductance and improve efficiency, but hat size, conductor/coil losses, ground and mechanical constraints determine the result.

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Ground Plane (GP) Antennas with Drooping Radials

A Ground Plane (GP) antenna is an elevated vertical monopole where the ground plane is formed by 3 or 4 insulated elevated radials extending outward from the base of the center radiator.

                              Drooping Ground Plane Antenna

                                     Vertical Monopole
                                      Length l ≈ λ/4
                                            |
                                            |
                                            o Feedpoint Terminals
                                          /   \
                           45° Droop    /       \    45° Droop
                         =============/           \=============
                         Radial 1                       Radial 2
                         (l ≈ λ/4)                      (l ≈ λ/4)

                       Drooping radials can move the input toward 50 ohms.

The 45° Droop Impedance Transformation

When the elevated radials are held strictly horizontal ($90^\circ$ relative to the vertical radiator):

• The antenna exhibits a radiation resistance of $R_{\text{in}} \approx 36.6\ \Omega$.
• Feeding a $36.6\ \Omega$ resistive load with standard $50\ \Omega$ coaxial cable produces a mismatch:

$$\text{SWR} = \frac{50\ \Omega}{36.6\ \Omega} \approx 1.37:1$$

Drooping the radials commonly raises feedpoint resistance toward a convenient coaxial match:

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Step-by-Step Worked Calculation: Ground Radial Efficiency Analysis

Compare two illustrative ground systems at $7.150\text{ MHz}$ with the same $100\text{ W}$ mean power accepted at the antenna feedpoint. Assume $R_r=36.8\ \Omega$ and $R_c=0.2\ \Omega$ in both cases:

• Configuration A: Ground-rod installation, assumed equivalent $R_g=38.0\ \Omega$.
• Configuration B: A 60-radial ground system, assumed equivalent $R_g=3.0\ \Omega$.

These resistance values are inputs to the example, not predictions from radial count or soil conductivity alone.

                              Worked Efficiency Comparison

     Configuration A (Single Ground Rod)            Configuration B (60 Buried Radials)
     ====================================           ===================================
     R_r = 36.8 Ω                                   R_r = 36.8 Ω
     R_g = 38.0 Ω (Assumed Ground Loss)             R_g = 3.0 Ω (Assumed Ground Loss)
     R_c =  0.2 Ω                                   R_c =  0.2 Ω
     ------------------------------------           -----------------------------------
     R_in = 75.0 Ω                                  R_in = 40.0 Ω
     Radiation Efficiency η ≈ 49.1%                 Radiation Efficiency η ≈ 92.0%

Step 1: Calculate Total Feedpoint Input Resistance ($R_{\text{in}}$)

• Configuration A:

$$R_{\text{in(A)}} = R_r + R_g + R_c = 36.8\ \Omega + 38.0\ \Omega + 0.2\ \Omega = 75.0\ \Omega$$

• Configuration B:

$$R_{\text{in(B)}} = R_r + R_g + R_c = 36.8\ \Omega + 3.0\ \Omega + 0.2\ \Omega = 40.0\ \Omega$$

Step 2: Calculate Radiation Efficiency ($\eta$)

$$\eta = \frac{R_r}{R_{\text{in}}} \times 100\%$$

• Configuration A:

$$\eta_A = \frac{36.8\ \Omega}{75.0\ \Omega} \times 100\% \approx 49.07\%$$

• Configuration B:

$$\eta_B = \frac{36.8\ \Omega}{40.0\ \Omega} \times 100\% = 92.00\%$$

Step 3: Calculate Radiated Power vs. Ground Heat Loss

Mean accepted power is $100\text{ W}$ in both configurations. Feedline and matching losses are outside this example.

• Configuration A:

$$P_{\text{rad(A)}}=100\times\frac{36.8}{75}=49.07\text{ W}$$ $$P_{\text{loss(A)}}=100-49.07=50.93\text{ W}$$

Ground loss is $100(38/75)=50.67\text{ W}$; conductor loss is $100(0.2/75)=0.267\text{ W}$.

• Configuration B:

$$P_{\text{rad(B)}}=100\times\frac{36.8}{40}=92.00\text{ W}$$ $$P_{\text{loss(B)}}=100-92=8.00\text{ W}$$

Ground loss is $100(3/40)=7.50\text{ W}$; conductor loss is $100(0.2/40)=0.50\text{ W}$. PEP alone would not specify these mean heating powers.

Step 4: Quantify the Signal Advantage in Decibels

Comparing the radiated field strength:

$$\Delta\text{dB}=10\log_{10}\left(\frac{P_{\text{rad(B)}}}{P_{\text{rad(A)}}}\right)=10\log_{10}\left(\frac{92}{49.0667}\right)\approx+2.73\text{ dB}$$

For unchanged radiation pattern, path and accepted power, the example gives about $2.73\text{ dB}$ more signal in a given direction. With the standard $6\text{ dB}$ per S-unit, this is $2.73/6\approx0.46$ S-unit, rather than nearly a full S-unit. Actual meter indications depend on calibration.

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Comparison Table: Vertical Ground Systems & Loading Configurations