Maxwell's Equations in RF and Antenna Design

Fig 1 — Electric field (blue) and magnetic field (orange) patterns in four structures governed by Maxwell's equations: coaxial line, rectangular waveguide (TE₁₀), microstrip and half-wave dipole

Introduction to Maxwell's Equations in RF Engineering

Over nine articles we have built Maxwell's equations from first principles (Parts 1–7), derived electromagnetic waves (Part 8) and worked out what happens at interfaces (Part 9). This final article shows how the equations become the everyday tools of RF engineering. Each of the main RF structures — transmission lines, waveguides, microstrip circuits and antennas — is simply a particular solution of Maxwell's equations with particular boundary conditions:

Structure	What Maxwell's equations give	Engineering parameters
Free space	Plane and spherical waves	Path loss, power density, wave impedance η₀
Transmission line	TEM wave guided between two conductors	Z₀, velocity factor, Γ, VSWR, loss
Waveguide	TE and TM modes inside a hollow conductor	Cut-off frequency, guide wavelength, mode count
Microstrip	Quasi-TEM wave across a dielectric–air interface	ε_eff, Z₀ versus W/h, dispersion
Antenna	Radiation from time-varying currents	Pattern, directivity, gain, polarisation, input impedance

Electromagnetic Wave Propagation in RF Systems

An RF link is a chain of wave-guiding and wave-launching structures: a power amplifier drives a transmission line, the line feeds an antenna, the antenna converts guided waves into free-space waves, the waves spread and are partly absorbed, reflected and diffracted on the way, and a receiving antenna converts a tiny fraction back into a guided wave. Far from the transmitting antenna, the radiated field is a spherical wave that looks locally like a plane wave (Part 8), with power density spreading as 1/r². Combined with the antenna gains, that spreading gives the Friis transmission equation:

Friis transmission equation and free-space path loss

\frac{P_{r}}{P_{t}} = G_{t} G_{r} {(\frac{λ}{4 π d})}^{2} FSPL = 20 \log_{10} (\frac{4 π d}{λ}) dB

Valid in the far field of both antennas with matched polarisation and no obstructions. Real links add absorption (rain, foliage, walls), multipath reflection, diffraction and polarisation mismatch — each of which is again a Maxwell's-equations problem.

Applications in Transmission Lines

Electric and Magnetic Fields in Transmission Lines

A transmission line has two conductors: a signal conductor and a return. The voltage between them corresponds to an electric field from one conductor to the other, terminating on surface charges (Gauss's law, Part 2). The current corresponds to a magnetic field circling the signal conductor, terminated by surface currents (Ampère's law, Part 5). In a coaxial line the electric field is purely radial and the magnetic field purely circumferential, both confined between the conductors.

TEM Wave Propagation

Because two-conductor lines in a uniform dielectric can support fields with no component along the direction of travel, they carry a TEM mode: 𝐄 and 𝐇 lie entirely in the cross-section. In a TEM mode the transverse fields satisfy the same equations as static fields, which is why the line's voltage and current can be defined uniquely and why its static capacitance and inductance per unit length fully describe it. The TEM mode has no cut-off frequency: it propagates from DC upward, at velocity v = c/√ε_r.

Characteristic Impedance

Characteristic impedance — lossless and general line

Z_{0} = \sqrt{\frac{L^{'}}{C^{'}}} Z_{0} = \sqrt{\frac{R^{'} + j ω L^{'}}{G^{'} + j ω C^{'}}} Z_{0, coax} = \frac{η_{0}}{2 π \sqrt{ε_{r}}} \ln \frac{b}{a}

Z₀ is the ratio of voltage to current of a single travelling wave. It is fixed entirely by cross-section geometry and materials — not by length. In Parts 2 and 5 we derived C′ and L′ for coax from Gauss's and Ampère's laws and obtained Z₀ ≈ 47 Ω for an RG-58-type cable.

Reflection and Standing Waves

At a load Z_L ≠ Z₀, the voltage-to-current ratio of the wave cannot match the load, so part of the wave reflects — exactly as a plane wave reflects at an impedance step in Part 9. Incident and reflected waves superpose into a standing wave with maxima and minima λ/4 apart:

Reflection coefficient, VSWR, return loss and mismatch loss

Γ = \frac{Z_{L} - Z_{0}}{Z_{L} + Z_{0}} VSWR = \frac{1 + | Γ |}{1 - | Γ |} RL = - 20 \log_{10} | Γ | ML = - 10 \log_{10} (1 - {| Γ |}^{2})

The Smith chart is a graphical map of Γ in the complex plane, used to design matching networks that bring Γ to zero at the frequencies of interest.

Applications in Waveguides

Electromagnetic Field Distribution

A hollow metal waveguide has only one conductor, so it cannot carry a TEM wave (there is no second conductor for the electric field lines to end on, and no closed path enclosing a current for 𝐇 to circle in the cross-section). Instead, the fields must satisfy the Helmholtz equation inside the guide with the PEC boundary conditions of Part 9 — tangential 𝐄 = 0 and normal 𝐁 = 0 on the walls. Only certain standing-wave patterns across the cross-section fit; each is a mode. In the dominant TE₁₀ mode of a rectangular guide with broad wall a, the electric field runs between the broad walls and varies as sin(πx/a): zero at the side walls (as the boundary condition demands) and maximum at the centre.

TE and TM Modes

TE (transverse electric) modes, TE_mn: the electric field has no component along the guide (E_z = 0); the magnetic field does (H_z ≠ 0).
TM (transverse magnetic) modes, TM_mn: H_z = 0, E_z ≠ 0. In a rectangular guide the lowest is TM₁₁.
The indices m and n count half-wave variations across the broad and narrow dimensions. TE₁₀ — one half-wave across a, none across b — has the lowest cut-off of all modes when a > b.

Cutoff Frequency

Cut-off frequency of the TE_mn / TM_mn mode in a rectangular guide (a × b)

f_{c, m n} = \frac{c}{2 \sqrt{μ_{r} ε_{r}}} \sqrt{{(\frac{m}{a})}^{2} + {(\frac{n}{b})}^{2}} f_{c, TE 10} = \frac{c}{2 a}

Below f_c a mode cannot propagate: its fields decay exponentially along the guide (evanescent). This makes a waveguide a natural high-pass filter, and a short waveguide section below cut-off is used as a "waveguide-below-cutoff" vent in shielded enclosures.

Waveguide Propagation

Propagation above cut-off

β = β_{0} \sqrt{1 - {(\frac{f_{c}}{f})}^{2}} λ_{g} = \frac{λ_{0}}{\sqrt{1 - {(f_{c} / f)}^{2}}} v_{p} = \frac{c}{\sqrt{1 - {(f_{c} / f)}^{2}}} v_{g} = c \sqrt{1 - {(f_{c} / f)}^{2}}

(Air-filled guide.) The guide wavelength is longer than in free space and the phase velocity exceeds c, but energy travels at the group velocity, below c, with v_p·v_g = c². Waveguides are used where low loss and high power matter: radar, satellite ground stations, high-power transmitters and mmWave test systems.

Applications in Microstrip Circuits

Electric and Magnetic Fields in Microstrip Lines

A microstrip line is a copper strip of width W on a dielectric substrate of thickness h and permittivity ε_r, over a ground plane. Most of the electric field runs straight down from the strip to the ground through the substrate, but some fringes through the air above and beside the strip. The magnetic field circles the strip, and the return current concentrates in the ground plane directly underneath it — the boundary conditions of Part 9 at work.

Effective Dielectric Constant

Because the wave travels partly in air and partly in the substrate, it behaves as if it were in a uniform medium of effective permittivity ε_eff between 1 and ε_r. A widely used closed-form approximation (Hammerstad) for W/h ≥ 1 is:

Microstrip effective permittivity and characteristic impedance (W/h ≥ 1)

ε_{eff} = \frac{ε_{r} + 1}{2} + \frac{ε_{r} - 1}{2} \frac{1}{\sqrt{1 + 12 h / W}} Z_{0} = \frac{120 π}{\sqrt{ε_{eff}} [W / h + 1.393 + 0.667 \ln (W / h + 1.444)]}

Wider strips put more field in the substrate (higher ε_eff, lower Z₀). Guided wavelength λ_g = λ₀/√ε_eff sets the length of every λ/4 and λ/2 element on the board.

Quasi-TEM Propagation

A pure TEM wave requires a single uniform dielectric — in a mixed air/substrate cross-section, the boundary conditions cannot be satisfied with zero longitudinal field components, because the wave "wants" to travel at different speeds in the two media. The microstrip mode therefore has small E_z and H_z components. At low frequencies these are negligible and the line behaves as if it were TEM — hence quasi-TEM. As frequency rises, more of the field is pulled into the substrate, ε_eff increases (dispersion), and eventually higher-order and surface-wave modes appear, limiting the useful frequency range of a given substrate thickness.

Microstrip Filters and Power Dividers

Stub and stepped-impedance filters use short sections of high- and low-Z₀ line to approximate series inductors and shunt capacitors in a lumped low-pass prototype.
Coupled-line band-pass filters use the electric and magnetic coupling between parallel λ/4 resonators — the even- and odd-mode fields of two coupled strips.
The Wilkinson power divider splits power equally using two λ/4 lines of impedance √2·Z₀ (70.7 Ω in a 50 Ω system) and a 2Z₀ (100 Ω) isolation resistor between the outputs.
Branch-line and rat-race couplers combine λ/4 and 3λ/4 sections to create 90° and 180° hybrid couplers for balanced amplifiers, mixers and beam-forming networks.

Applications in Antenna Engineering

Electromagnetic Radiation

Radiation is produced by time-varying currents — accelerating charges. A static charge has only a static field; a steady current only a static magnetic field; but when current changes, the changes in field propagate outward at c and, at distances beyond about a wavelength, detach from the source as a wave carrying energy away permanently. Maxwell's equations are solved for a given current distribution 𝐉 by introducing the magnetic vector potential of Part 3, computed with a "retarded" phase factor e^{−jβR} that accounts for the travel time from each piece of current to the observation point:

Retarded vector potential and far field of a short (Hertzian) dipole

𝐀 = \frac{μ}{4 π} \int_{V} \frac{𝐉 e^{- j β R}}{R} d V E_{θ} \approx j η \frac{β I Δ l}{4 π r} \sin θ e^{- j β r} H_{φ} = \frac{E_{θ}}{η}

The radiated field falls off as 1/r (so power as 1/r²), is proportional to frequency (through β), and has E/H = η — a local plane wave. The short dipole's radiation resistance is R_rad = 80π²(Δl/λ)²: tiny for antennas much shorter than λ, which is why electrically small antennas are inefficient.

Current Distribution on Antennas

The radiated field depends entirely on how current is distributed over the antenna. On a thin centre-fed dipole, current must fall to zero at the open ends (no conductor to carry it further), and the antenna behaves like an open-circuited transmission line folded out: the current forms a standing wave. For a half-wave dipole, I(z) = I₀ cos(βz), maximum at the feed and zero at the tips. Integrating the Hertzian-dipole field over this distribution gives the half-wave dipole's pattern, cos(π/2·cos θ)/sin θ, its radiation resistance of about 73 Ω and directivity of 1.64 (2.15 dBi). For patch antennas, the current flows on the patch and ground plane and the radiation can be modelled as coming from the fringing fields at the two radiating edges (equivalent magnetic currents, Part 3).

Near-Field and Far-Field Regions

Region	Distance (antenna size D)	Character
Reactive near field	r < 0.62√(D³/λ) (≈ λ/2π for small antennas)	Stored energy dominates; E/H ≠ 377 Ω; fields fall as 1/r² and 1/r³; strong coupling to nearby objects
Radiating near field (Fresnel)	0.62√(D³/λ) < r < 2D²/λ	Radiation dominates but the pattern still changes with distance
Far field (Fraunhofer)	r > 2D²/λ (and r ≫ λ, r ≫ D)	Plane-wave-like; pattern independent of r; E/H = η₀; Friis equation valid

NFC, RFID, wireless charging and near-field probes deliberately work in the reactive near field, where the magnetic field of a coil couples by Faraday's law (Part 4) rather than by radiation. Antenna patterns, by contrast, must be measured in the far field — or in the near field with a scanner whose data are mathematically transformed to the far field.

Antenna Radiation Patterns

The radiation pattern plots the far-field intensity versus direction, usually as E-plane and H-plane cuts in dB. Key features are the main lobe and its half-power beamwidth, side lobes, nulls and front-to-back ratio. A half-wave dipole has a doughnut pattern with nulls along its axis; a patch radiates a broad beam away from its ground plane; a phased array steers a narrow beam by controlling the phase of each element's current — all predicted by integrating the fields of the current distribution.

Polarization and Gain

Directivity, gain and polarisation mismatch

D = \frac{4 π U_{max}}{P_{rad}} G = e_{rad} D A_{e} = \frac{λ^{2} G}{4 π} PLF = \cos^{2} ψ

U is radiation intensity (W/sr); e_rad is radiation efficiency (losses in conductors and dielectrics); A_e is effective aperture; ψ is the angle between two linear polarisations. Polarisation is the direction of the far-field 𝐄 vector — linear, circular (GNSS, satellite) or elliptical. A linearly polarised antenna receiving a circularly polarised wave loses 3 dB; crossed linear polarisations ideally lose all signal.

Electromagnetic Simulation and Maxwell's Equations

Only simple geometries have closed-form solutions. Real antennas, connectors, packages and PCBs are analysed with computational electromagnetics: the structure is divided into many small cells, Maxwell's equations are written for each, and the resulting very large system of equations is solved numerically.

Finite Element Method (FEM)

FEM solves the frequency-domain (Helmholtz) form of Maxwell's equations on an unstructured mesh, usually of tetrahedra. Within each element the field is approximated by simple basis functions; enforcing the equations in a weighted-average sense across all elements produces a sparse matrix equation. Unstructured tetrahedra follow curved and intricate geometry accurately, and adaptive meshing refines the mesh where the fields change fastest until S-parameters converge. FEM is especially strong for resonant, enclosed and 3D structures: connectors, filters, cavities, packages and antennas with complex feeds.

Finite-Difference Time-Domain (FDTD)

FDTD discretises the two curl equations directly in space and time on a staggered grid — the Yee cell (1966), with 𝐄 components on cell edges and 𝐇 components on face centres. 𝐄 and 𝐇 are updated alternately in a leapfrog scheme, so a pulse can be launched and its propagation simulated step by step. One run with a broadband pulse gives the response over a wide frequency range via a Fourier transform. Two rules govern accuracy and stability:

FDTD cell size and Courant stability limit (uniform cubic cells)

Δ \leq \frac{λ_{min}}{10 \dots 20} Δ t \leq \frac{Δ}{c \sqrt{3}}

λ_min is the shortest wavelength in the model, inside the densest dielectric. For a 1 mm cell, Δt ≤ 1.93 ps; at 10 GHz in FR-4 (λ ≈ 14.3 mm) a λ/20 cell is 0.71 mm.

Related time-domain methods include the finite integration technique (FIT) used in CST Studio Suite's time-domain solver. A third major family, the method of moments (MoM), solves for currents on conducting surfaces and wires only — very efficient for wire antennas and planar circuits (FEKO, Keysight Momentum, Sonnet).

Applications in HFSS and Other EM Solvers

Tool	Main method	Typical uses
Ansys HFSS	FEM (frequency domain), plus integral-equation and SBR+ solvers	Antennas, connectors, filters, packages, radomes, antenna placement
CST Studio Suite	FIT time domain, FEM frequency domain, MoM	Broadband antennas, EMC/EMI, signal integrity, transient problems
Altair FEKO	MoM, MLFMM, hybrid methods	Wire and electrically large antennas, installed performance on vehicles and aircraft
Keysight Momentum / Sonnet	Planar MoM	Microstrip and stripline circuits, PCB and MMIC layouts
openEMS	FDTD (open source)	Antennas and microstrip structures on a budget

Whatever the tool, the engineer supplies the geometry, the material constants (ε_r, tan δ, σ, μ_r — the constitutive relations of Part 7), the excitation ports and the boundary conditions (PEC, radiation/PML or symmetry planes — Part 9). Understanding the underlying field theory is what lets you set these up correctly, choose sensible mesh settings, and recognise when a result is non-physical.

Practical Examples in RF Engineering

Mismatch: a 75 Ω Load on a 50 Ω Line

Problem: A 50 Ω coaxial line feeds a 75 Ω load. Find Γ, VSWR, return loss and mismatch loss.

Γ = \frac{75 - 50}{75 + 50} = 0.2 VSWR = \frac{1.2}{0.8} = 1.5 RL = - 20 \log 0.2 = 14.0 dB ML = - 10 \log (0.96) = 0.18 dB

Answer: Γ = 0.2 | VSWR = 1.5 | RL = 14.0 dB | ML = 0.18 dB (4% of power reflected)

A VSWR of 1.5 costs only 0.18 dB of delivered power, but the reflected wave can still upset a sensitive amplifier's stability or create ripple in a long cable run. A λ/4 section of √(50 × 75) = 61.2 Ω line would match it at one frequency.

WR-90 Waveguide at 10 GHz

Problem: An air-filled WR-90 waveguide has a = 22.86 mm and b = 10.16 mm. Find the cut-off frequencies of TE₁₀, TE₂₀ and TE₀₁, the single-mode band, and the guide wavelength, phase velocity and group velocity of TE₁₀ at 10 GHz.

f_c = c/(2a), c/a, c/(2b)

f_{c, 10} = \frac{2.998 \times 10^{8}}{2 \times 0.02286} = 6.557 GHz f_{c, 20} = 13.11 GHz f_{c, 01} = 14.75 GHz

At 10 GHz: √(1 − (6.557/10)²) = 0.755; λ₀ = 29.98 mm

λ_{g} = \frac{29.98}{0.755} = 39.7 mm v_{p} = \frac{c}{0.755} = 3.97 \times 10^{8} m/s v_{g} = 0.755 c = 2.26 \times 10^{8} m/s

Answer: TE₁₀ 6.56 GHz, TE₂₀ 13.1 GHz, TE₀₁ 14.8 GHz | single-mode 6.56–13.1 GHz | λ_g = 39.7 mm, v_p = 1.32c, v_g = 0.755c

The recommended operating band of WR-90 is 8.2–12.4 GHz — comfortably above TE₁₀ cut-off (where loss and dispersion rise sharply) and below TE₂₀. Check: v_p·v_g = c².

A 50 Ω Microstrip Line on FR-4

Problem: A microstrip trace 3.0 mm wide sits on 1.6 mm FR-4 (ε_r = 4.4). Estimate ε_eff and Z₀, and the length of a quarter-wave section at 2.4 GHz.

W/h = 1.875

ε_{eff} = 2.7 + \frac{1.7}{\sqrt{1 + 12 / 1.875}} = 2.7 + \frac{1.7}{2.720} = 3.32

Z₀ = 376.7 / (√3.32 × [1.875 + 1.393 + 0.667 ln(3.319)])

Z_{0} = \frac{376.7}{1.823 \times 4.068} = 50.8 Ω \frac{λ_{g}}{4} = \frac{124.9}{4 \times 1.823} = 17.1 mm

Answer: ε_eff ≈ 3.32 | Z₀ ≈ 50.8 Ω | λ_g/4 ≈ 17.1 mm at 2.4 GHz

About two-thirds of the way from air (1) to FR-4 (4.4) — most of the field is in the substrate. In practice FR-4's ε_r varies by ±5–10% between suppliers and with frequency, so critical lengths are tuned after measurement or the design moves to a controlled-ε_r RF laminate.

A 915 MHz Half-Wave Dipole

Problem: Design a half-wave dipole for 915 MHz (ISM band). Find the free-space wavelength, the ideal half-wave length, a practical cut length, its radiation resistance and gain, and a sensible minimum distance for far-field pattern measurement.

λ = c/f

λ = \frac{2.998 \times 10^{8}}{915 \times 10^{6}} = 327.6 mm \frac{λ}{2} = 163.8 mm L_{cut} \approx 0.48 λ \approx 157 mm

Far-field distance with D = λ/2: 2D²/λ = λ/2 — so the "r ≫ λ" condition dominates

\frac{2 D^{2}}{λ} = 164 mm r_{min} \approx 3 λ \approx 1 m

Answer: λ = 327.6 mm; ideal λ/2 = 163.8 mm; cut ≈ 157 mm | R_rad ≈ 73 Ω, G ≈ 2.15 dBi | measure at ≥ ~1 m

A real dipole is cut a few percent short because its finite thickness and end effects make it electrically longer; the shortening also removes the ≈ +42 Ω reactance of an exact half-wave dipole so the input is resistive. For a small antenna the Fraunhofer formula 2D²/λ gives a distance less than λ, so the other far-field conditions (r ≫ λ) set the limit.

A 2.4 GHz Free-Space Link Budget

Problem: A 2.4 GHz transmitter delivers 20 dBm (100 mW) to a dipole (2.15 dBi). The receiver, 100 m away in free space, also uses a dipole. Find the free-space path loss and received power. Is the link viable with a receiver sensitivity of −90 dBm?

λ = 0.1249 m; FSPL = 20 log(4πd/λ)

FSPL = 20 \log \frac{4 π \times 100}{0.1249} = 20 \log (10060) = 80.1 dB

P_r = P_t + G_t + G_r − FSPL

P_{r} = 20 + 2.15 + 2.15 - 80.1 = - 55.8 dBm

Answer: FSPL = 80.1 dB | P_r ≈ −55.8 dBm — 34 dB of margin above −90 dBm

That margin is what absorbs real-world losses: walls (3–15 dB each), body loss, multipath fading (often 10–20 dB), cable and mismatch losses. Every term in this budget — gain, path loss, wall loss, fading — is ultimately a solution of Maxwell's equations.

Importance of Maxwell's Equations in Modern Wireless Communications

5G and mmWaveAt 28–39 GHz, wavelengths are about a centimetre: packaging, PCB vias and even the phone case become part of the antenna. Design is impossible without full-wave simulation.

Massive MIMO and beamformingPhased arrays steer beams by controlling the phase of each element's current — superposition of radiated fields, directly from Maxwell's equations.

Wi-Fi, Bluetooth and IoTCompact chip and PCB antennas must work next to batteries, displays and hands; near-field interactions and detuning are field problems.

Satellite and GNSSCircular polarisation, high-gain reflectors and phased-array user terminals rely on careful control of radiated field polarisation and patterns.

Automotive radar77 GHz radar sensors depend on antenna-in-package design, radome transparency (Part 9) and low-loss substrates.

EMC, SAR and coexistenceRegulatory compliance — emissions, immunity, specific absorption rate in the body — is assessed through field measurements and simulations.

As frequencies rise and devices shrink, the gap between "circuit" and "field" disappears. Signal-integrity engineers designing 112 Gb/s serial links, power-electronics engineers fighting switching noise and RF engineers designing antennas all end up solving the same four equations.

Summary and Conclusion

Application	Key result	Built on
Transmission lines	Z₀ = √(L′/C′); Γ = (Z_L − Z₀)/(Z_L + Z₀)	Gauss, Ampère, TEM waves
Waveguides	f_c = c/2a (TE₁₀); λ_g = λ₀/√(1 − (f_c/f)²)	Helmholtz equation + PEC walls
Microstrip	1 < ε_eff < ε_r; λ_g = λ₀/√ε_eff	Boundary conditions at substrate–air
Antennas	E ∝ βIΔl sin θ / r; G = e·D	Retarded potentials from time-varying currents
Links	P_r/P_t = G_tG_r(λ/4πd)²	Far-field plane waves
Simulation	FEM, FDTD, MoM	Discretised Maxwell's equations

This completes our ten-part series. Starting from the idea of a field, we developed the four laws one at a time, assembled them into Maxwell's equations, derived electromagnetic waves, worked out the rules at material boundaries, and finally saw how those few equations underpin every transmission line, waveguide, circuit board and antenna in modern wireless systems.

Part	Article	Core idea
1	Introduction to Electromagnetic Fields	Charges make 𝐄; currents make 𝐁
2	Understanding Electric Flux and Gauss's Law	∇·𝐃 = ρ_v
3	Magnetic Flux and Gauss's Law for Magnetism	∇·𝐁 = 0
4	Faraday's Law of Electromagnetic Induction	∇ × 𝐄 = −∂𝐁/∂t
5	Ampere's Circuital Law	∇ × 𝐇 = 𝐉
6	Maxwell's Displacement Current	+ ∂𝐃/∂t
7	The Four Maxwell's Equations Explained	The complete system + constitutive relations
8	Deriving the Electromagnetic Wave Equation	∇²𝐄 = με ∂²𝐄/∂t²; c, η₀
9	Maxwell's Equations and Electromagnetic Boundary Conditions	Interfaces, Γ, matching, shielding
10	Applications of Maxwell's Equations in RF Engineering and Antennas	Lines, guides, microstrip, antennas, simulation

To keep going, put the theory to work: use the Smith chart to design a matching network for the 75 Ω load in Example 1, or run the link budget in Example 5 with your own antenna gains and distances.

Put It Into Practice

Use the NEXT ANTENNAS calculators to apply the ideas from this article to real designs.

Antenna Design Patch Antenna Calculator Width, length, effective permittivity and inset feed of a rectangular microstrip patch for your frequency and substrate. Open calculator → Transmission Lines Microstrip Calculator Trace width for a target impedance, effective permittivity and guided wavelength on any PCB substrate. Open calculator → Impedance Matching Smith Chart Visualiser Plot impedances, reflection coefficient and VSWR, and build matching networks step by step. Open calculator → All Tools All RF Calculators Browse every NEXT ANTENNAS calculator for RF, microwave and antenna design in one place. Open calculator →

Applications of Maxwell's Equations in RF Engineering and Antennas