10.4 Antenna Gain, Aperture Antennas & Horns, ERP/EIRP, Radiation Patterns

10.4 Antenna Gain, Aperture Antennas & Horns, ERP/EIRP, Radiation Patterns

Radio communication across the electromagnetic spectrum depends fundamentally on launching radio frequency energy into space in controlled spatial distributions. An antenna does not amplify power; rather, it acts as an electromagnetic transducer that concentrates transmitter power into preferred spatial directions. On High Frequency (HF) and Very High Frequency (VHF) bands, directional control is achieved primarily through wire arrangements and parasitic elements. At Ultra High Frequency (UHF) and microwave frequencies, where wavelengths contract to centimeters and millimeters, radio amateurs utilize aperture antennas, including horn antennas and parabolic reflectors. Mastering antenna gain references, radiation patterns, aperture physics, and Effective Isotropic Radiated Power (EIRP) is essential for advanced amateur communications, satellite tracking, and Earth-Moon-Earth (EME) weak-signal operations.

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Antenna Gain References: dBi vs. dBd and ERP vs. EIRP

Antenna gain expresses the power density radiated in the direction of maximum beam intensity relative to the power density produced by a reference antenna supplied with identical input power. Gain incorporates both directional concentration (directivity) and electrical efficiency:

$$G = \eta_{\text{rad}} \times D_{\text{antenna}}$$

                              Gain Reference Comparison

           Isotropic Radiator (dBi)                  Half-Wave Dipole (dBd)
        Theoretical Dimensionless Point              Resonant Wire in Free Space
           Radiates Uniform Sphere                   Toroidal Figure-Eight Pattern

                    . : .
                  :  /|\  :                                (   ( | )   )
                 : -- O -- :                             (     ( | )     )
                  :  \|/  :                                (   ( | )   )
                    ' : '
             Directivity = 1.0 (0 dBi)                Directivity = 1.64 (2.15 dBi)

1. The Isotropic Radiator ($0\text{ dBi}$) and Half-Wave Dipole ($0\text{ dBd}$)

• Isotropic Radiator ($\text{dBi}$): A hypothetical, lossless point source radiating uniformly across all $4\pi$ steradians. At distance $r$, power flux density is $S_{\text{iso}} = P_{\text{tx}} / (4\pi r^2)$. It possesses a reference gain of $0\text{ dBi}$.
• Half-Wave Dipole ($\text{dBd}$): A real resonant wire dipole in free space. Because a dipole produces toroidal radiation with zero emission along its wire axis, broadside power density is $1.6406$ times greater ($+2.15\text{ dB}$) than an isotropic radiator:

$$G_{\text{dBi}} = G_{\text{dBd}} + 2.15\text{ dB}$$ $$G_{\text{dBd}} = G_{\text{dBi}} - 2.15\text{ dB}$$

2. Effective Radiated Power (ERP) and Effective Isotropic Radiated Power (EIRP)

Delivered transmitter power ($P_{\text{del}}$) equals transmitter output power minus transmission line and connector attenuation. Directional radiated power is quantified in two formats:

• Effective Radiated Power (ERP): Power referenced to a resonant half-wave dipole:

$$\text{ERP} = P_{\text{del}} \times 10^{G_{\text{dBd}} / 10}$$

• Effective Isotropic Radiated Power (EIRP): Power referenced to an isotropic source:

$$\text{EIRP} = P_{\text{del}} \times 10^{G_{\text{dBi}} / 10} = \text{ERP} \times 1.6406$$

Under the Australian Radiocommunications (Amateur Stations) Class Licence, Advanced licensees may operate up to $400\text{ W}$ PEP transmitter output. In designated bands with shared services—such as $472-479\text{ kHz}$ ($630\text{ m}$)—statutory limits are explicitly framed as radiated limits ($5\text{ W}$ PEP EIRP).

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Radiation Patterns, Lobes, and Beamwidth

An antenna's spatial radiation pattern plots electric field intensity or relative power density in spherical coordinates, sliced into the E-plane (containing the electric field vector) and H-plane (containing the magnetic field vector).

                             Directive Radiation Pattern

                                       Main Lobe
                                       /=======\
                         Side Lobe    /         \    Side Lobe
                           (==)      |  +0 dB    |      (==)
                              \      |           |      /
                               \=====(   Feed    )=====/ 
                                      \         /
                                       \=======/
                                       Back Lobe

Lobes, Front-to-Back, and Front-to-Rear Ratios

• Main Lobe: The primary beam containing peak power density.
• Side Lobes: Unwanted secondary radiation peaks caused by aperture edge currents and phase cancellation limits.
• Back Lobe: Residual radiation directed $180^\circ$ opposite the main beam.
• Front-to-Back ($F/B$) Ratio: Ratio of main lobe peak power to power radiated at $180^\circ$ azimuth.
• Front-to-Rear Ratio: Ratio of main lobe peak to the highest side or back lobe across the entire rear hemisphere ($90^\circ$ to $270^\circ$).

Half-Power Beamwidth (HPBW)

The Half-Power Beamwidth ($\theta_{-3\text{dB}}$) is the angular separation between pattern points where radiated power density drops to half ($-3\text{ dB}$) of the main beam maximum:

$$\text{HPBW}=|\theta_2-\theta_1|\quad\text{(both edge points are at half power)}$$

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Aperture Antennas: Fundamental Radiation Physics

Unlike wire antennas that radiate via oscillating conduction currents along thin metallic elements, aperture antennas launch electromagnetic waves through a physical opening or surface into free space.

Waveguide -> Gradual horn flare -> Radiating aperture -> Far-field beam
              larger aperture      amplitude and phase
                                    shape the beam

Waveguide-to-Space Transition

An open waveguide can radiate, but its abrupt termination presents a radiation load and produces a broad beam. A horn gradually enlarges the guide into a radiating aperture, improving the transition to free space and increasing directivity.

Free-space intrinsic wave impedance is:

$$\eta_0=\sqrt{\frac{\mu_0}{\varepsilon_0}}\approx376.73\ \Omega$$

This does not make the aperture a simple junction between two uniform transmission lines. Its reflection cannot be found just by substituting free-space and guided-wave impedances into a two-resistance reflection formula.

Horn aperture fields are not generally uniform in phase. A short horn with a wide flare has appreciable phase error because paths to the aperture edge are longer than paths to its centre. Aperture dimensions and horn length therefore trade compactness against gain, beamwidth and sidelobes.

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Microwave Horn Antenna Topologies

                             Horn Antenna Topologies

          Pyramidal Horn             E-Plane Sectoral            H-Plane Sectoral
        Flared in E and H              Flared in E only            Flared in H only

             /=====\                      /=====\                     /===========\
            /       \                     |     |                    /             \
           /         \                    /=====\                   /===============\
          |===========|                  /       \                  |               |
          |           |                 /=========\                 |               |
          \===========/                 \=========/                 \===============/

1. Pyramidal Horn: Flared simultaneously along both the E-plane and H-plane walls. Fed by rectangular waveguide, it produces symmetric pencil beams and moderate gain ($12\text{ to }25\text{ dBi}$). It is the global standard for antenna test range calibrations.
2. E-Plane Sectoral Horn: Flared only along the electric field vector dimension (narrow waveguide wall), keeping the H-plane dimension constant. Produces a fan-shaped beam that is narrow in elevation (E-plane) and broad in azimuth (H-plane).
3. H-Plane Sectoral Horn: Flared only along the magnetic field vector dimension (broad waveguide wall). Produces a beam that is narrow in azimuth (H-plane) and broad in elevation (E-plane).
4. Conical Horn: Flared from a circular waveguide feed. Ideal for circular polarization and corrugated feed horns, delivering extremely low cross-polarization and symmetric radiation patterns.

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Parabolic Reflector (Dish) Antennas

For forward gains exceeding $30\text{ to }50\text{ dBi}$, microwave stations employ parabolic reflector antennas. A feed horn positioned at the focal point illuminates a paraboloidal metallic dish, collimating spherical wavefronts into planar wavefronts.

                              Dish Feed Architectures

             Prime-Focus Feed                             Cassegrain Feed
        Feed Horn at Focal Point                     Convex Subreflector at Focus

                 \                                            \
                  \                                            \       Subreflector
                   \                                            \          (\)
                    +---[ Feed Horn ]                            +----------+
                   /                                            /      [ Feed Horn ]
                  /                                            /       (At dish apex)
                 /                                            /

Feed Architectures: Prime-Focus vs. Cassegrain and Gregorian

• Prime-Focus: The feed horn sits directly at the focus in front of the dish. While structurally simple, feed support struts and transmission lines obstruct the radiating aperture, producing blockage loss and high side lobes.
• Cassegrain Configuration: Employs a convex hyperboloidal subreflector placed near the focus. The feed horn mounts at the vertex of the main dish. Shorter waveguide runs reduce feeder loss and its contribution to receiver noise temperature.
• Gregorian Configuration: Employs a concave ellipsoidal subreflector placed past the focal point, providing similar electrical advantages.

Aperture Illumination Efficiency and Edge Taper

Total power gain depends on the effective aperture area ($A_e$) relative to physical aperture area ($A_p = \pi D^2 / 4$):

$$A_e = \eta_a A_p$$

The aperture efficiency ($\eta_a$, typically $55\%-70\%$) balances two opposing effects:

1. Aperture Illumination Taper: Uniform illumination maximizes aperture utilization but causes severe spillover loss.
2. Spillover Loss: Feed energy missing the reflector edge adds thermal noise from the warm earth ($290\text{ K}$). An edge taper of $-10\text{ dB}$ to $-12\text{ dB}$ at the dish rim provides optimal compromise.

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Quantitative Formulas for Parabolic Dishes

For a circular parabolic dish of diameter $D$ at wavelength $\lambda$:

1. Parabolic Dish Gain

$$G = \frac{4\pi}{\lambda^2} A_e = \eta_a \left(\frac{\pi D}{\lambda}\right)^2$$ $$G_{\text{dBi}} = 10 \log_{10}\left[\eta_a \left(\frac{\pi D}{\lambda}\right)^2\right] = 20 \log_{10}\left(\frac{\pi D}{\lambda}\right) + 10 \log_{10}(\eta_a)$$

2. Half-Power Beamwidth (HPBW)

$$\theta_{-3\text{dB}} \approx 70^\circ \frac{\lambda}{D}$$

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Amateur Microwave Allocations & EME Communications

Amateurs deploy aperture antennas across microwave allocations:

• $1.2\text{ GHz}$ ($23\text{ cm}$) & $2.4\text{ GHz}$ ($13\text{ cm}$): High-speed digital links, Wi-Fi experimentation, and amateur satellites (QO-100 uplink).
• $5.7\text{ GHz}$ ($6\text{ cm}$) & $10\text{ GHz}$ ($3\text{ cm}$): Terrestrial line-of-sight and rain-scatter DX records exceeding $500\text{ km}$.
• $24\text{ GHz}$ ($1.2\text{ cm}$) & $47\text{ GHz}$: Precision millimeter-wave propagation.
• Earth-Moon-Earth (EME / Moonbounce): Bouncing microwave signals off the lunar surface requires overcoming round-trip path attenuation exceeding $250 - 270\text{ dB}$. Amateurs deploy dishes with gains $\ge 35\text{ dBi}$ and HPBW under $2^\circ$, necessitating precision azimuth/elevation tracking rotators.

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Step-by-Step Worked Calculation: 10 GHz EME Parabolic Dish Design

An Australian Advanced amateur designs a $10.368\text{ GHz}$ ($3\text{ cm}$) Earth-Moon-Earth station. The installation features a prime-focus dish of diameter $D = 2.40\text{ m}$ illuminated by a circular horn feed with an aperture efficiency of $\eta_a = 60\%$ ($0.60$). A $10.368\text{ GHz}$ solid-state power amplifier delivers $P_{\text{tx}} = 50\text{ W}$ PEP into a low-loss waveguide feed with $0.80\text{ dB}$ attenuation.

                              Worked Problem Layout

     [ 50 W SSPA ] ===( Waveguide Loss = 0.8 dB )===> [ Feed Horn ] ===> [ 2.4 m Dish ]
      f = 10.368 GHz                                                       η_a = 60%

Step 1: Calculate Wavelength

$$\lambda = \frac{c}{f} = \frac{2.9979 \times 10^8\text{ m/s}}{10.368 \times 10^9\text{ Hz}} \approx 0.028915\text{ m} = 28.915\text{ mm}$$

Step 2: Determine Physical and Effective Aperture Area

$$A_p = \frac{\pi D^2}{4} = \frac{\pi (2.40)^2}{4} \approx 4.5239\text{ m}^2$$ $$A_e = \eta_a A_p = 0.60 \times 4.5239\text{ m}^2 \approx 2.7143\text{ m}^2$$

Step 3: Calculate Dish Power Gain and Gain in dBi

$$G = \frac{4\pi}{\lambda^2} A_e = \frac{4\pi}{(0.028915)^2} \times 2.7143 \approx 40,796$$ $$G_{\text{dBi}} = 10 \log_{10}(40,796) \approx 46.11\text{ dBi}$$

Step 4: Calculate Half-Power Beamwidth

$$\theta_{-3\text{dB}} \approx 70^\circ \times \frac{\lambda}{D} = 70^\circ \times \frac{0.028915\text{ m}}{2.40\text{ m}} \approx 0.843^\circ$$

Step 5: Calculate Net EIRP Along Main Beam

• Delivered transmitter power:

$$P_{\text{del}} = 50\text{ W} \times 10^{-0.80 / 10} = 50 \times 0.8318 \approx 41.59\text{ W}$$

• Effective Isotropic Radiated Power:

$$\text{EIRP} = P_{\text{del}} \times G = 41.59\text{ W} \times 40,796 \approx 1,696,705\text{ W} \approx 1.70\text{ MW EIRP}$$ $$\text{EIRP (dBW)} = 10 \log_{10}(41.59) + 46.11\text{ dBi} = 16.19\text{ dBW} + 46.11\text{ dBi} = 62.30\text{ dBW}$$

The $2.4\text{ m}$ parabolic reflector concentrates $50\text{ W}$ of RF power into a razor-sharp $0.84^\circ$ pencil beam, generating $1.70\text{ Megawatts}$ EIRP to conquer lunar path loss.

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Comparison Table: Directional and Aperture Antenna Architectures