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Antenna design models

Connect wavelength, radiation, impedance matching, aperture, arrays, and link budgets with twenty solved examples.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Fields and radiation

Definitions & inputs. c speed in vacuum, f frequency, λ wavelength, U radiation intensity, Pr total radiated power.

  1. Outgoing power spreads over spherical area.

    λ=c/f,S(r,θ,ϕ)=U(θ,ϕ)/r2\lambda=c/f,\quad S(r,\theta,\phi)=U(\theta,\phi)/r^2

  2. Normalize a pattern to the isotropic intensity at the same total radiated power.

    Pr=∫4πU dΩ,D=4πU/PrP_r=\int_{4\pi}U\,d\Omega,\quad D=4\pi U/P_r

  3. Radiation efficiency reduces gain relative to directivity.

    G=ηrDG=\eta_r D

Interpretation. Realized gain additionally includes mismatch loss; specify which gain is used.

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2. Impedance and matching

Definitions & inputs. ZA load antenna impedance, Z0 real line impedance, Γ voltage reflection coefficient.

  1. Enforce voltage and current continuity at the termination.

    Γ=(ZA−Z0)/(ZA+Z0)\Gamma=(Z_A-Z_0)/(Z_A+Z_0)

  2. Standing-wave maxima and minima follow interference of forward and reflected waves.

    VSWR=(1+∣Γ∣)/(1−∣Γ∣)VSWR=(1+|\Gamma|)/(1-|\Gamma|)

  3. Reflected power is the squared amplitude ratio; return loss uses a positive dB convention.

    ηm=1−∣Γ∣2,RL=−20log⁡10∣Γ∣\eta_m=1-|\Gamma|^2,\quad RL=-20\log_{10}|\Gamma|

Interpretation. A well-matched antenna may still have poor radiation efficiency.

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3. Apertures and links

Definitions & inputs. Ae effective collecting area, G power gain, r separation, Pt transmitted accepted power.

  1. Reciprocity relates receiving aperture and transmitting gain.

    Ae=Gλ2/(4π)A_e=G\lambda^2/(4\pi)

  2. Transmit gain concentrates the far-field power density.

    S=PtGt/(4πr2)S=P_tG_t/(4\pi r^2)

  3. Multiply incident density by receiving effective aperture.

    Prx=PtGtGr(λ/(4πr))2P_{rx}=P_tG_tG_r(\lambda/(4\pi r))^2

Interpretation. Near field, obstruction, fading, atmosphere and polarization mismatch require extra terms.

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4. Geometry and arrays

Definitions & inputs. Dap largest aperture size, d element spacing, θ angle from broadside, N elements.

  1. Aperture path curvature sets a conventional far-field distance estimate.

    rFF≳2Dap2/λr_{FF}\gtrsim2D_{ap}^2/\lambda

  2. Superpose element fields including propagation and feed phase.

    AF(θ)=∑n=0N−1ejn(kdsin⁡θ+β)AF(\theta)=\sum_{n=0}^{N-1}e^{jn(kd\sin\theta+\beta)}

  3. Choose the feed progression so fields add at the steering angle.

    β=−kdsin⁡θ0,k=2π/λ\beta=-kd\sin\theta_0,\quad k=2\pi/\lambda

Interpretation. Element pattern and coupling modify the array result; scan-dependent grating lobes must be checked.

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Graphical worked example

Real reference impedance, single-mode line; matching does not establish radiation efficiency. X axis: Reflection magnitude |Γ| (dimensionless). Y axis: Voltage standing-wave ratio (dimensionless).
Real reference impedance, single-mode line; matching does not establish radiation efficiency. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Free-space wavelength

Definitions & inputs. f=1 GHz, c=299792458 m/s.

  1. Choose the governing model and isolate the requested quantity.

    λ=c/f\lambda=c/f

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    λ=299792458/109\lambda=299792458/10^9

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2997925 m\mathrm{Result}=0.2997925\ {\rm m}

Interpretation. Wavelength in a dielectric differs from the free-space value.

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Example 02. Nominal half-wave dipole

Definitions & inputs. f=100 MHz in free space.

  1. Choose the governing model and isolate the requested quantity.

    L≃λ/2L\simeq\lambda/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    L≃299792458/(2⋅108)L\simeq299792458/(2\cdot10^8)

  3. Evaluate the expression; the result uses the units shown.

    Result=1.498962 m\mathrm{Result}=1.498962\ {\rm m}

Interpretation. Resonant physical length is affected by radius, end effects, and mounting.

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Example 03. Nominal quarter-wave monopole

Definitions & inputs. f=300 MHz over an ideal infinite ground plane.

  1. Choose the governing model and isolate the requested quantity.

    L≃λ/4L\simeq\lambda/4

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    L≃299792458/(4⋅3×108)L\simeq299792458/(4\cdot3\times10^8)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.249827 m\mathrm{Result}=0.249827\ {\rm m}

Interpretation. A finite ground plane changes pattern and resonance.

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Example 04. Gain from efficiency

Definitions & inputs. Directivity 8, radiation efficiency .75.

  1. Choose the governing model and isolate the requested quantity.

    G=ηDG=\eta D

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    G=0.75(8)G=0.75(8)

  3. Evaluate the expression; the result uses the units shown.

    Result=6 \mathrm{Result}=6\ {}

Interpretation. This gain excludes impedance mismatch.

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Example 05. Gain in dBi

Definitions & inputs. Linear gain 6.

  1. Choose the governing model and isolate the requested quantity.

    GdBi=10log⁡10GG_{dBi}=10\log_{10}G

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    GdBi=10log⁡106G_{dBi}=10\log_{10}6

  3. Evaluate the expression; the result uses the units shown.

    Result=7.781513 dBi\mathrm{Result}=7.781513\ {\rm dBi}

Interpretation. Isotropic gain is the reference.

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Example 06. Reflection coefficient

Definitions & inputs. Real antenna resistance 75 Ω on 50 Ω line.

  1. Choose the governing model and isolate the requested quantity.

    Γ=(RA−Z0)/(RA+Z0)\Gamma=(R_A-Z_0)/(R_A+Z_0)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Γ=(75−50)/(75+50)\Gamma=(75-50)/(75+50)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2 \mathrm{Result}=0.2\ {}

Interpretation. The reflection phase is zero in this resistive example.

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Example 07. VSWR

Definitions & inputs. Reflection magnitude .2.

  1. Choose the governing model and isolate the requested quantity.

    VSWR=(1+∣Γ∣)/(1−∣Γ∣)VSWR=(1+|\Gamma|)/(1-|\Gamma|)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    VSWR=1.2/0.8VSWR=1.2/0.8

  3. Evaluate the expression; the result uses the units shown.

    Result=1.5 \mathrm{Result}=1.5\ {}

Interpretation. A ratio of one indicates perfect impedance match.

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Example 08. Return loss

Definitions & inputs. Reflection magnitude .2.

  1. Choose the governing model and isolate the requested quantity.

    RL=−20log⁡10∣Γ∣RL=-20\log_{10}|\Gamma|

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    RL=−20log⁡100.2RL=-20\log_{10}0.2

  3. Evaluate the expression; the result uses the units shown.

    Result=13.9794 dB\mathrm{Result}=13.9794\ {\rm dB}

Interpretation. Larger return loss means less reflection.

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Example 09. Mismatch efficiency

Definitions & inputs. Reflection magnitude .2.

  1. Choose the governing model and isolate the requested quantity.

    ηm=1−∣Γ∣2\eta_m=1-|\Gamma|^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ηm=1−0.22\eta_m=1-0.2^2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.96 \mathrm{Result}=0.96\ {}

Interpretation. Four percent of incident power reflects at this interface.

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Example 10. Radiation efficiency

Definitions & inputs. Radiation resistance 60 Ω, loss resistance 15 Ω.

  1. Choose the governing model and isolate the requested quantity.

    ηr=Rr/(Rr+Rl)\eta_r=R_r/(R_r+R_l)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ηr=60/75\eta_r=60/75

  3. Evaluate the expression; the result uses the units shown.

    Result=0.8 \mathrm{Result}=0.8\ {}

Interpretation. Both resistances use the same feed-current convention.

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Example 11. Short-dipole resistance

Definitions & inputs. Triangular-current short dipole, total length L=.05λ.

  1. Choose the governing model and isolate the requested quantity.

    Rr≃20π2(L/λ)2R_r\simeq20\pi^2(L/\lambda)^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Rr≃20π2(0.05)2R_r\simeq20\pi^2(0.05)^2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.4934802 Ω\mathrm{Result}=0.4934802\ \Omega

Interpretation. The uniform-current Hertzian dipole has a different coefficient.

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Example 12. Effective aperture

Definitions & inputs. Gain 10, wavelength .1 m.

  1. Choose the governing model and isolate the requested quantity.

    Ae=Gλ2/(4π)A_e=G\lambda^2/(4\pi)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Ae=10(0.1)2/(4π)A_e=10(0.1)^2/(4\pi)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.007957747 m2\mathrm{Result}=0.007957747\ {\rm m}^2

Interpretation. Receiving polarization and impedance are matched.

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Example 13. Dish gain

Definitions & inputs. Diameter 1 m, wavelength .03 m, aperture efficiency .6.

  1. Choose the governing model and isolate the requested quantity.

    G=ηa(πD/λ)2G=\eta_a(\pi D/\lambda)^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    G=0.6(π/0.03)2G=0.6(\pi/0.03)^2

  3. Evaluate the expression; the result uses the units shown.

    Result=6579.736 \mathrm{Result}=6579.736\ {}

Interpretation. Aperture efficiency includes illumination and other aperture losses.

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Example 14. Dish approximate beamwidth

Definitions & inputs. D=1 m, λ=.03 m; illustrative 70° coefficient.

  1. Choose the governing model and isolate the requested quantity.

    θHPBW≃70λ/D\theta_{HPBW}\simeq70\lambda/D

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    θHPBW≃70(0.03)\theta_{HPBW}\simeq70(0.03)

  3. Evaluate the expression; the result uses the units shown.

    Result=2.1 deg\mathrm{Result}=2.1\ {\rm deg}

Interpretation. The coefficient depends on illumination; this is not a universal dish specification.

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Example 15. Far-field distance estimate

Definitions & inputs. Aperture maximum size 1 m, λ=.03 m.

  1. Choose the governing model and isolate the requested quantity.

    rFF≃2D2/λr_{FF}\simeq2D^2/\lambda

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    rFF≃2/0.03r_{FF}\simeq2/0.03

  3. Evaluate the expression; the result uses the units shown.

    Result=66.66667 m\mathrm{Result}=66.66667\ {\rm m}

Interpretation. Also require distance large compared with wavelength and antenna dimensions.

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Example 16. Isotropic power density

Definitions & inputs. Radiated power 10 W, distance 100 m.

  1. Choose the governing model and isolate the requested quantity.

    S=P/(4πr2)S=P/(4\pi r^2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    S=10/(4π1002)S=10/(4\pi100^2)

  3. Evaluate the expression; the result uses the units shown.

    Result=7.957747×10−5 W m−2\mathrm{Result}=7.957747\times10^{-5}\ {\rm W\,m}^{-2}

Interpretation. The isotropic source is a normalization model.

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Example 17. EIRP

Definitions & inputs. Transmit accepted power 2 W, transmit gain 10.

  1. Choose the governing model and isolate the requested quantity.

    EIRP=PtGtEIRP=P_tG_t

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    EIRP=2(10)EIRP=2(10)

  3. Evaluate the expression; the result uses the units shown.

    Result=20 W\mathrm{Result}=20\ {\rm W}

Interpretation. Feed cable loss must be applied before accepted antenna power.

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Example 18. Friis received power

Definitions & inputs. Pt=1 W, Gt=Gr=1, λ=.1 m, r=100 m.

  1. Choose the governing model and isolate the requested quantity.

    Pr=PtGtGr(λ/(4πr))2P_r=P_tG_tG_r(\lambda/(4\pi r))^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Pr=[0.1/(4π100)]2P_r=[0.1/(4\pi100)]^2

  3. Evaluate the expression; the result uses the units shown.

    Result=6.332574×10−9 W\mathrm{Result}=6.332574\times10^{-9}\ {\rm W}

Interpretation. This is a matched, unobstructed free-space link.

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Example 19. Polarization coupling

Definitions & inputs. Two linear polarizations separated by 30°.

  1. Choose the governing model and isolate the requested quantity.

    ηp=cos⁡2ψ\eta_p=\cos^2\psi

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ηp=cos⁡230∘\eta_p=\cos^2 30^\circ

  3. Evaluate the expression; the result uses the units shown.

    Result=0.75 \mathrm{Result}=0.75\ {}

Interpretation. Circular/elliptical states need a vector polarization calculation.

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Example 20. Array steering phase

Definitions & inputs. Half-wavelength element spacing, beam θ0=30° from broadside.

  1. Choose the governing model and isolate the requested quantity.

    β=−2π(d/λ)sin⁡θ0\beta=-2\pi(d/\lambda)\sin\theta_0

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    β=−2π(0.5)sin⁡30∘\beta=-2\pi(0.5)\sin30^\circ

  3. Evaluate the expression; the result uses the units shown.

    Result=−1.570796 rad\mathrm{Result}=-1.570796\ {\rm rad}

Interpretation. Adjacent element phase is negative under the stated array convention.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.