Monopole & Ground-Plane Antenna Calculator

Step 1

Band & wire

Pick the band and the conductor. The wire's thickness relative to the wavelength sets how much shorter than λ/2 a resonant dipole is, and its conductivity sets the skin-effect loss. Both carry into every solve that follows.

Band, wire and feed

Band preset

Design frequency f₀

MHz

Band low edge

MHz

Band high edge

MHz

System impedance

Conductor preset

Conductor diameter

Material

Conductivity

MS/m

Length units

Wire mesh

Every change here re-solves the antenna. The band edges set the frequency sweep and the bandwidth goals in step 4.

At f₀

Resonant length against wire thickness MoM against closed form

Validation this solver against textbook results

Method. Galerkin Method of Moments on the mixed-potential electric-field integral equation, triangle basis functions spanning adjacent segments (so wire junctions conserve current), the reduced thin-wire kernel with analytic integration of the 1/R part on near segments, and a delta-gap source at the feed node. Results converge as the mesh is refined; the normal mesh is within about 1% of the converged resonance.

Step 3

Analyze

A frequency sweep of the Method of Moments model: every point is a full solve of the wires, the ground and the matching network. The dashed vertical line marks f₀ and the shaded span is your band; dashed horizontal lines are your goals.

Sweep points

Return loss at the port

Input impedance

Current along the wires

Smith chart port reflection, band in accent

Method of Moments far field

Radiation and losses

Gain is 4πU/P_in, so it already includes conductor loss and, over real earth, the power the ground absorbs. Over ground the pattern is the direct wave plus the ground reflection with the Fresnel coefficients of the soil, so it covers the upper half-space only.

3D radiation pattern at f₀

Radiation pattern at f₀

Gain against frequency peak gain, dBi

Where the power goes at f₀ share of available power

Export

The Touchstone file is the port S11 over the sweep. The NEC-2 deck (GW, GE, GN, EX, FR, RP cards) reproduces this geometry, ground and source for 4nec2, xnec2c or EZNEC import.

Method. Thin-wire Galerkin Method of Moments on the mixed-potential electric-field integral equation. Each wire is cut into segments; triangle basis functions span each pair of adjacent segments, and at a junction of n wires n − 1 of them carry current from one reference wire into each of the others, so current is conserved. The reduced kernel G = e^−jkR/(4πR) with R = √(|r − r′|² + a²) is integrated analytically for its 1/R part on near segments and by Gauss–Legendre quadrature elsewhere. A delta-gap source of 1 V at the feed node gives the input current and Z_in = V/I. Conductor loss adds the skin-effect resistance R_s/(2πa) per metre to the impedance matrix. A perfect ground is an exact image. Over real earth the image is scaled by the normal-incidence reflection coefficient of the soil for the impedance (a reflection-coefficient approximation, good above about 0.1 λ) and by the full Fresnel coefficients for the far field. Limits: the coax feed line, the balun and any common-mode current on the feed line are not modelled; matching networks are ideal and lossless.

About this monopole and ground-plane antenna calculator

This free calculator from NextAntennas designs quarter-wave monopoles and ground-plane antennas with three to eight radials, then analyzes and optimizes them in the browser with a thin-wire Method of Moments solver. It finds the resonant length of the vertical and the radials, solves for the radial droop angle that gives a direct 50 Ω coax feed, and shows the impedance, SWR, current, gain, efficiency and takeoff angle over free space, a perfect ground or real earth. Switch the antenna to Dipole in step 2 to design half-wave, off-center-fed and inverted-V dipoles with the same solver.

How to use it

Band & wire: pick a band such as 40 m, 20 m, 2 m, 70 cm, LoRa 868/915 MHz or Wi-Fi 2.4 GHz, and the wire or tubing. The calculator shows the wavelength, the 468/f length, the skin depth and the resonant-length factor for that conductor.
Antenna synthesis: choose dipole or monopole and the type, the ground and the mounting height. The Method of Moments refinement sets the length for resonance at f₀ and sizes a 1:1 or 4:1 balun, a quarter-wave transformer or an L-network, or solves the radial droop for a direct 50 Ω feed.
Analyze: S11 and VSWR, input impedance, Smith chart, the current along every wire, gain, efficiency, takeoff angle, polar and 3D radiation patterns and a power budget, with Touchstone and NEC-2 export.
Optimize: set goals for band match, resonance, gain, efficiency or takeoff angle and run Nelder–Mead or differential evolution.
Yield: Monte Carlo analysis of cut length, conductor diameter, angle, height and soil tolerances.

What it calculates

Resonant length of a dipole or monopole for the actual conductor diameter, with the closed-form value for comparison
Input impedance, S11, VSWR and 2:1 SWR bandwidth at the antenna and at the matched port
Radial length and droop angle of a ground-plane antenna for 50 Ω
Current distribution, peak gain, directivity, radiation efficiency, takeoff angle and half-power beamwidth
Conductor (skin-effect) loss and ground loss over sea water, fresh water and five soil types
Balun ratio, quarter-wave transformer impedance and length, and L-network component values

Frequently asked questions

How long should a half-wave dipole be?

A little shorter than half a free-space wavelength: L = K·λ/2, where λ = 299.79/f (metres, f in MHz) and the shortening factor K is about 0.93–0.98. K depends mainly on how thick the conductor is compared with the wavelength. For AWG 12 copper at 146 MHz (λ/d ≈ 1000) this calculator gives K = 0.959 and a resonant length of 984.5 mm in free space. Mounting height and the ground change it slightly, which step 2 of the calculator accounts for; insulated wire must be cut a further 2–5% shorter.

What is the 468/f formula and how accurate is it?

L(ft) = 468/f(MHz) is the classic amateur-radio rule for a half-wave dipole. It corresponds to K ≈ 0.952, which is about right for HF wire with λ/d of a few hundred to a few thousand. For thin wire at HF it is typically 1–3% short, and for thick tubing at VHF and UHF it can be 1–3% long. Treat it as a starting point; the calculator solves the actual wire and shows how far your conductor departs from the rule.

What is the impedance of a dipole antenna?

An infinitely thin dipole exactly λ/2 long has Z = 73.1 + j42.5 Ω. Shortening it by a few percent removes the reactance, and a resonant dipole in free space is about 70–73 Ω resistive. Over ground the resistance swings with height between roughly 50 and 100 Ω, and an inverted-V with a 90–120° apex angle drops to about 35–55 Ω.

What is the impedance of a quarter-wave monopole and a ground-plane antenna?

On an ideal, infinite ground plane a quarter-wave monopole has half the impedance of the equivalent dipole: about 36 Ω at resonance. A ground-plane antenna with four horizontal radials is lower still, about 20–25 Ω. Drooping the radials raises the resistance; at about 40–45° it reaches 50 Ω, which this calculator can solve for directly.

Why are the radials of a ground-plane antenna drooped?

Drooping the radials makes the antenna more like a vertical dipole: the radials start to radiate in phase with the vertical, which raises the input resistance from about 22 Ω towards 50 Ω and pushes the main lobe down towards the horizon. Choose “Droop radials for Z₀” in step 2 and the calculator finds the angle that gives 50 Ω (or your system impedance) at resonance.

How many radials does a ground-plane antenna need, and how long should they be?

For an elevated ground-plane antenna three or four resonant radials, each about a quarter wavelength long, are enough; more radials make the pattern rounder in azimuth but change the impedance little. Radials lying on or buried in the ground are different: they are detuned by the soil and many (often 16–60) are needed to reduce ground loss. This calculator models elevated radials.

How high should a dipole be mounted?

Height sets the elevation pattern. Over ground, a horizontal dipole at λ/2 has its main lobe at about 30°, good for long-distance HF; at λ/4 or lower most of the power goes straight up (near-vertical incidence skywave), and more is lost in the ground. At 1 λ the takeoff angle falls to about 14°. Enter your height and soil in step 2 to see the takeoff angle, the gain and the ground loss.

Is an inverted-V worse than a flat dipole?

An inverted-V needs only one support and radiates more evenly in all directions, at the cost of about 0.5–1 dB of peak gain and a lower feed resistance (about 50 Ω at a 90–120° apex, which suits coax well). Sloping arms add end capacitance, so an inverted-V is usually cut a few percent longer than a flat dipole for the same frequency.

What is an off-center-fed (OCF) dipole and why does it use a 4:1 balun?

An OCF or Windom-style dipole is fed away from its center. The current is smaller there, so the feed resistance rises roughly as R₀/cos²(kx), where x is the distance from the center: about 95 Ω at one third of the length from an end and about 200 Ω at one fifth. A 4:1 balun brings the 200 Ω case to 50 Ω. Hams use OCF dipoles because the same feed point also has moderate impedance on several harmonic bands.

Do I need a balun on a dipole?

Yes, a 1:1 current (choke) balun at the feed point is strongly recommended. A dipole is balanced and coax is not, so without a balun current flows on the outside of the shield. The feed line then radiates, the pattern skews, the SWR changes when you touch the coax and RF can reach the station. A monopole fed against its own ground system does not need a balun, though a choke on the feed line below the radials is still good practice.

How does wire thickness affect bandwidth?

A thicker conductor stores less reactive energy around itself, so the antenna has a lower Q and a wider bandwidth: going from thin wire (λ/d ≈ 10 000) to tubing (λ/d ≈ 100) widens the 2:1 SWR bandwidth of a resonant dipole from about 7% to about 17%. Thicker elements also resonate shorter. For the 146 MHz example in the theory section, AWG 12 wire gives Q ≈ 7 and a 2:1 SWR bandwidth of about 10 MHz (7%).

What gain does a dipole or monopole have, and what is the difference between dBi and dBd?

A half-wave dipole in free space has 2.15 dBi of gain (dBi is relative to an isotropic radiator; 0 dBd is a dipole, so dBd = dBi − 2.15). A quarter-wave monopole on an ideal ground plane has 5.15 dBi, 3 dB more, because all of its power goes into the upper half-space. Over real ground the reflection adds up to about 6 dB at the takeoff angle, which is why a dipole at λ/2 over good soil shows 7–8 dBi.

How does the type of ground affect an antenna?

The ground reflects part of the wave and absorbs the rest. Sea water reflects almost perfectly and gives a vertical antenna strong low-angle radiation. Average or poor soil absorbs much of the downward power, especially for vertical polarization near the pseudo-Brewster angle, which lowers efficiency and lifts the takeoff angle. Horizontal antennas are less sensitive to soil quality than verticals.

How accurate is this calculator compared with NEC or EZNEC?

It uses the same class of method as NEC-2: a thin-wire Method of Moments solution of the electric-field integral equation. In free space and over perfect ground, resonant frequencies agree to about 1% and patterns closely. Over real earth the impedance uses a reflection-coefficient approximation, which is accurate above about 0.1 λ; NEC-2 and NEC-4 with Sommerfeld ground are more accurate for antennas very close to the soil. Every page load re-runs a validation table against textbook results.

Why does my built antenna resonate at a different frequency?

Real antennas differ from the model: insulated wire is electrically longer (cut it about 2–5% shorter than bare wire), end insulators and loops of wire add capacitance, the feed line can carry common-mode current, and masts, gutters and trees nearby detune the antenna. Cut the elements a little long, then trim to resonance with an antenna analyzer such as a NanoVNA.

Can I export the design to 4nec2, EZNEC or a VNA tool?

Yes. Step 3 exports a NEC-2 card deck (GW, GE, GN, LD, EX, FR and RP cards) with the exact geometry, wire radius, conductivity, ground and source, which opens in 4nec2, xnec2c and other NEC programs, and a Touchstone .s1p file of the port S11 for circuit simulators and VNA software.

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Engineering Theory

Dipole and Monopole Antenna Theory

From the Hertzian dipole to the integral equations behind the solver: the current and pattern of a finite dipole, impedance and resonant length, bandwidth and Q, image theory, ground-plane radials, real ground, conductor loss, matching and the Method of Moments, with a worked 2 m example.

1. Dipoles and Monopoles

A dipole is a straight conductor fed at a gap, usually at its center. Its two arms carry a standing wave of current, largest at the feed of a half-wave dipole and zero at the open tips. A monopole is one arm of a dipole fed against a ground plane, which supplies the mirror image of the missing arm. Together they are the reference antennas of radio engineering: gain is quoted against an ideal dipole (dBd), and most practical antennas, from the Yagi-Uda to the J-pole, are built from dipole or monopole elements.

This calculator covers the center-fed half-wave dipole, the off-center-fed (OCF) dipole, the inverted-V, the quarter-wave monopole on an ideal ground plane and the ground-plane antenna with three to eight radials. Closed-form equations give a starting length; a thin-wire Method of Moments solution of the actual wires, junctions, conductor loss and ground then gives every result.

2. The Hertzian Dipole and the Far Field

The building block of every wire antenna is the Hertzian (infinitesimal) dipole, a current element I of length Δl ≪ λ. Far from it, its electric field is transverse and falls as 1/r:

Hertzian dipole far field

E_{θ} = j η \frac{k I Δ l}{4 π r} \sin θ e^{- j k r}

η = √(μ₀/ε₀) ≈ 376.7 Ω · k = 2π/λ · θ measured from the element axis

Integrating the radiated power over a sphere gives its radiation resistance and directivity:

Radiation resistance and directivity

R_{r} = 80 π^{2} {(\frac{Δ l}{λ})}^{2}

D = 1.5 (1.76 dBi)

A short dipole with a triangular current has a quarter of this resistance, 20π²(L/λ)²

The tiny R_r of an electrically short antenna is why short whips are inefficient: a few ohms of radiation resistance compete with the conductor and loading-coil loss. A finite antenna is a line of such elements, each weighted by its current and phased by its position; the pattern is valid in the far field, beyond about r = 2D²/λ for an antenna of largest dimension D.

3. Current and Pattern of a Finite Dipole

On a thin center-fed dipole of length L the current is very nearly a standing sine wave that vanishes at the tips:

Sinusoidal current distribution

I (z) = I_{m} \sin [k (\frac{L}{2} - | z |)]

Summing Hertzian elements along the wire gives the far field in closed form:

Far field of a dipole of length L

E_{θ} = j η \frac{I_{m} e^{- j k r}}{2 π r} [\frac{\cos (\frac{k L}{2} \cos θ) - \cos (\frac{k L}{2})}{\sin θ}]

For L = λ/2 the bracket becomes cos(½π cos θ)/sin θ

The half-wave pattern is a doughnut with nulls off the wire ends, a half-power beamwidth of 78° and a directivity of 1.64 (2.15 dBi). Lengthening the dipole narrows the main lobe: the full-wave dipole reaches 2.41 (3.8 dBi) and the 1.25 λ “extended double Zepp” about 3.3 (5.2 dBi), the most broadside gain a single straight wire can give. Beyond that the current reverses on parts of each arm and the pattern breaks into lobes. The calculator does not assume the sine current; it computes the current from the integral equation (section 11) and shows it in step 3, so off-center feeds, bends and thick wires come out correctly.

4. Radiation Resistance and Input Impedance

The induced-EMF method integrates the field of the assumed sine current against the current itself. Referred to the current maximum, the radiation resistance of a dipole of length L is:

Radiation resistance (induced EMF)

R_{m} = \frac{η}{2 π} {C + \ln (k L) - Ci (k L) + \frac{1}{2} \sin (k L) [Si (2 k L) - 2 Si (k L)] + \frac{1}{2} \cos (k L) [C + \ln (\frac{k L}{2}) + Ci (2 k L) - 2 Ci (k L)]}

C ≈ 0.5772 (Euler's constant) · Si, Ci = sine and cosine integrals

At the feed the current is I_m sin(kL/2), so power conservation transforms the resistance to the terminals; the reactance follows from the same integral and, unlike the resistance, depends on the wire radius a:

Input impedance

R_{in} = \frac{R_{m}}{\sin^{2} (k L / 2)}

Z_{in} (L = λ / 2) = 73.1 + j 42.5 Ω

Near L = λ the sine in the denominator vanishes: the full-wave dipole is anti-resonant, with an input resistance of thousands of ohms

The same method gives the mutual impedance Z₁₂ between parallel dipoles, which is how the ground (section 9) and nearby elements change the feed impedance: Z_in = Z₁₁ + (I₂/I₁)Z₁₂. For a horizontal dipole at height h over perfect ground the image carries −I₁, so Z_in = Z₁₁ − Z₁₂(2h); at h = λ/4 that adds about 12.5 + j29.9 Ω.

5. Resonant Length and the Shortening Factor

The +j42.5 Ω of an exact half wave means a real dipole must be shortened to resonate. The fringing field at the open tips adds end capacitance, and a conductor of finite radius has a current that does not quite vanish at the tip, so the electrical length exceeds the physical one:

Resonant length

L_{dipole} = K \frac{λ_{0}}{2}

L_{monopole} = K \frac{λ_{0}}{4}

K \approx 1 - \frac{1}{4.35 \ln (λ / d) - 5.6}

Fit to this solver's resonances for 100 ≤ λ/d ≤ 10⁵; K ≈ 0.93 at λ/d = 100, 0.959 at 1000 and 0.971 at 10 000

The theory goes back to King and Hallén, whose thickness parameter Ω = 2 ln(2L/a) plays the same role as ln(λ/d) here: the shortening is inversely proportional to it. The amateur-radio rule L(ft) = 468/f(MHz) assumes K = 0.952, correct for λ/d of a few hundred. Step 1 of the calculator plots the solved resonance against λ/d with the rule and the fit. Insulation shortens a wire antenna further, typically by 2–5%, because the dielectric slows the wave along the wire; this calculator models bare conductors.

6. Bandwidth, Q and Conductor Thickness

Near resonance a dipole behaves like a series RLC circuit. Its quality factor Q, the ratio of stored to radiated energy, sets the bandwidth. The calculator's sweep effectively evaluates the Yaghjian–Best expression, which needs only the input impedance and its frequency derivative:

Q from the input impedance (Yaghjian & Best)

Q (ω) \approx \frac{ω}{2 R} \sqrt{{(R^{'})}^{2} + {(X^{'} + \frac{| X |}{ω})}^{2}}

FBW \approx \frac{s - 1}{Q \sqrt{s}}

R′, X′ = derivatives with respect to ω · s = VSWR limit (2 for the usual 2:1 bandwidth), matched at resonance

A thicker element stores less energy in the near field, so its Q falls and its bandwidth widens. A resonant dipole has Q ≈ 10 for λ/d = 10 000, about 7 for λ/d = 1000 and about 4 for tubing at λ/d = 100, and its 2:1 SWR bandwidth grows from about 7% to about 17% of f₀ over that range. For electrically small antennas, such as short loaded whips, the Chu limit bounds how low Q can go: Q ≥ 1/(ka)³ + 1/(ka) for an antenna that fits in a sphere of radius a. Halving the size of a small antenna multiplies its minimum Q by about eight.

7. Monopoles and Image Theory

A perfectly conducting plane forces the tangential electric field to zero. The same boundary condition holds if the plane is removed and every current is mirrored: vertical currents keep their direction and horizontal currents reverse. A monopole of height h over the plane therefore produces, in the upper half-space, exactly the field of a dipole of length 2h, while only half of the dipole's gap voltage drives it:

Image theory

Z_{monopole} = \frac{1}{2} Z_{dipole}

D_{monopole} = 2 D_{dipole}

About 36.5 + j21.3 Ω for h = λ/4 exactly, 36 Ω at resonance, and 5.15 dBi

The doubled directivity is not extra power: the same radiated power is confined to half the sphere. The calculator overlays every monopole's impedance with half that of its equivalent free-space dipole, so you can see how far a real ground system departs from the ideal.

8. Ground-Plane Antennas and Radial Droop

An elevated ground-plane antenna replaces the infinite plane with a few quarter-wave radials. Their currents flow outward from the hub and largely cancel each other's radiation, but they are a poor mirror. With horizontal radials the structure has an input resistance of only about 22 Ω, and its peak gain is a little below a dipole's. Drooping the radials by an angle δ turns the antenna progressively into a vertical dipole bent at the bottom: the radials' vertical current components radiate in phase with the vertical element, the resistance rises, and the main lobe moves toward the horizon.

Four radials, AWG 12 copper at 146 MHz in free space; lengths re-resonated by the solver at each droop
Radial droop δ	Vertical and radials	R_in at resonance	Peak gain
0° (horizontal)	510.6 mm	22.4 Ω	1.53 dBi
41.5° (solved for 50 Ω)	485.6 mm	50.0 Ω	1.85 dBi

Three or four radials are enough for an elevated antenna; more make the azimuth pattern rounder without changing the impedance much. Radials on or in the soil behave differently: the ground detunes them and absorbs power, and dozens are needed to reduce ground loss. The calculator models elevated radials and a choke on the coax below the hub is still advisable, since the outside of the shield otherwise acts as one more radial.

9. Real Ground, Height and Takeoff Angle

Real earth is a lossy dielectric. At frequency f it has the complex relative permittivity

Complex permittivity of the soil

ε_{c} = ε_{r} - j \frac{σ}{2 π f ε_{0}}

Average soil (εr 13, σ 0.005 S/m) is mainly a conductor at 3.5 MHz (σ/ωε₀ ≈ 26) but mainly a dielectric at 146 MHz (≈ 0.6)

The wave reflected from the ground at elevation angle ψ is weighted by the Fresnel coefficients for horizontal and vertical polarization:

Fresnel reflection coefficients

Γ_{h} = \frac{\sin ψ - \sqrt{ε_{c} - \cos^{2} ψ}}{\sin ψ + \sqrt{ε_{c} - \cos^{2} ψ}}

Γ_{v} = \frac{ε_{c} \sin ψ - \sqrt{ε_{c} - \cos^{2} ψ}}{ε_{c} \sin ψ + \sqrt{ε_{c} - \cos^{2} ψ}}

For horizontal polarization Γ_h stays close to −1 at low angles over almost any soil, so a horizontal dipole at height h sees an array of itself and an inverted image. The ground factor multiplies the free-space pattern:

Ground factor and takeoff angle, horizontal dipole

| A F | = 2 | \sin (k h \sin ψ) |

ψ_{\max} = \arcsin \frac{λ}{4 h}

30° at h = λ/2, 14.5° at h = λ, straight up for h ≤ λ/4 · up to 6 dB of gain over free space at the peak

Vertical polarization behaves differently. |Γ_v| dips to a minimum at the pseudo-Brewster angle, a few degrees to about 15° over typical soil, and Γ_v → −1 at grazing, so a vertical over real ground has no radiation exactly at the horizon and loses much of its downward power to the soil. Sea water, with σ ≈ 5 S/m, pushes the pseudo-Brewster angle below one degree, which is why verticals by the sea work so well at low angles.

For the far field the calculator applies the full Fresnel coefficients to the image of every segment. For the input impedance it scales each image by the normal-incidence coefficient (1 − √ε_c)/(1 + √ε_c), the reflection-coefficient approximation of MININEC-class programs, good above about 0.1 λ. Power accepted at the feed that is neither radiated into the upper half-space nor dissipated in the wire is reported as ground loss.

10. Pocklington and Hallén Integral Equations

The sine current is an assumption. The exact current follows from the boundary condition that the total tangential electric field vanishes on a perfect conductor. For a straight thin wire along z this gives Pocklington's equation (1897):

Pocklington's integral equation

(\frac{d^{2}}{d z^{2}} + k^{2}) \int_{- L / 2}^{L / 2} I (z^{'}) G (z, z^{'}) d z^{'} = - j ω ε E_{z}^{inc} (z)

G = e^−jkR/(4πR) · reduced kernel R = √((z − z′)² + a²)

Hallén (1938) integrated the differential operator analytically, trading it for two constants fixed by I(±L/2) = 0, which gives an equation that converges faster numerically:

Hallén's integral equation, delta-gap feed V

\int_{- L / 2}^{L / 2} I (z^{'}) G (z, z^{'}) d z^{'} = - \frac{j}{η} [C_{1} \cos (k z) + \frac{V}{2} \sin (k | z |)]

Both are exact statements of the physics for a perfect, straight wire, but neither handles bends, junctions or loads directly. For general wire geometry, NEC and this calculator use the electric-field integral equation in mixed-potential form, which treats every segment the same way whatever its orientation.

11. The Method of Moments in This Calculator

Along any wire with unit tangent t̂, the scattered field of the unknown current I(l) plus the field of the source must equal the drop across the wire's surface impedance:

Mixed-potential electric-field integral equation

\hat{t} \cdot (j ω A + \nabla Φ) + Z^{'} I = \hat{t} \cdot E^{inc}, A = μ \int I \hat{t}' G d l^{'}, Φ = \frac{1}{j ω ε} \int (- \frac{d I}{d l^{'}}) G d l^{'}

Each wire is cut into segments of length Δ. The current is expanded in triangle (rooftop) basis functions f_n, each rising from zero to one across one segment and back to zero across the next, so the current is continuous and its derivative, the charge, is piecewise constant. Testing with the same functions (Galerkin's method) and integrating the scalar-potential term by parts produces a symmetric matrix equation [Z][I] = [V]:

Galerkin impedance matrix

Z_{m n} = j ω μ \iint f_{m} \cdot f_{n} G d l d l^{'} + \frac{1}{j ω ε} \iint (\nabla \cdot f_{m}) (\nabla^{'} \cdot f_{n}) G d l d l^{'} + Z^{'} \int f_{m} \cdot f_{n} d l

V_m = 1 V for the basis functions that cross the feed gap and 0 elsewhere · Z_in = 1/I_feed

Kernel. The reduced thin-wire kernel puts the source current on the wire axis and the observation point on the surface, R = √(|r − r′|² + a²). Its 1/R part is integrated analytically on nearby segments, which keeps self-terms accurate for any radius, and the smooth remainder (e^−jkR − 1)/R and distant segments use Gauss–Legendre quadrature.
Junctions. Where n wires meet, n − 1 triangle functions carry current from a reference wire into each of the others, so Kirchhoff's current law holds exactly at the hub of a ground-plane antenna or the apex of an inverted-V.
Feed. A delta-gap source at the feed node excites the basis functions that cross it. At a perfect ground plane a half-triangle carries current into its image, which is how the monopole's base feed works.
Ground. A perfect ground adds the field of every segment's image to the matrix exactly; real earth scales the image as described in section 9.
Convergence. The normal mesh uses 20 segments per half-wave. Refining it to 64 moves the resonance by less than 0.2%. The thin-wire kernel needs Δ to be several times a; the calculator lengthens segments to at least 2.5 radii and warns when that limits accuracy.
Checks. In free space the radiated power integrated over the sphere plus the conductor loss equals ½Re(VI*) to about 0.001%, a test of the whole chain from matrix to far field. The validation table in step 1 repeats this and other textbook comparisons each time the page loads.

12. Conductor Loss and Efficiency

At radio frequencies current crowds into a skin of depth δ at the conductor surface. For a round wire of radius a ≫ δ the resistance per metre is:

Skin effect

δ = \frac{1}{\sqrt{π f μ_{0} σ}}

R_{s} = \sqrt{\frac{π f μ_{0}}{σ}}

R^{'} = \frac{R_{s}}{2 π a}

Copper at 146 MHz: δ = 5.5 µm, R′ ≈ 0.49 Ω/m for AWG 12

The dissipated power is ½R′∫|I|²dl and the radiation efficiency is e_rad = P_rad/(P_rad + P_loss). Because the loss scales with |I|² and a resonant half-wave dipole has about 70 Ω of radiation resistance, a copper dipole is better than 99% efficient. Efficiency matters for short antennas: a monopole of height h ≪ λ/4 has R_r ≈ 40π²(h/λ)², only a few ohms, so the wire, the loading coil and the ground system can take most of the power. Steel and stainless conductors, with conductivity 40 times lower than copper, lose about six times as much for the same radius.

13. Directivity, Gain and Effective Aperture

The far field follows from the solved current by summing every segment's contribution with its phase:

Radiation vector and intensity

N (θ, φ) = \sum \int I (l) \hat{t} e^{j k \hat{r} \cdot r^{'}} d l

U = \frac{ω^{2} μ^{2}}{32 π^{2} η} ({| N_{θ} |}^{2} + {| N_{φ} |}^{2})

Directivity, gain, realized gain and aperture

D = \frac{4 π U_{\max}}{P_{rad}}

G = e_{rad} D = \frac{4 π U_{\max}}{P_{in}}

G_{realized} = G (1 - {| Γ |}^{2})

A_{e} = \frac{λ^{2} G}{4 π}

dBd = dBi − 2.15 · a λ/2 dipole has A_e ≈ 0.13 λ², larger than its physical area by orders of magnitude

The calculator locates the peak of U by a grid search refined to a few hundredths of a degree, so the gain and the takeoff angle it reports are not tied to the plotting grid. Over ground, gain is relative to an isotropic radiator in free space, as in NEC, so the ground reflection appears as gain.

14. Baluns and Matching

A dipole is a balanced load; coax is unbalanced. Without a balun, part of the feed current returns along the outside of the shield, which then radiates, skews the pattern and makes the SWR depend on the feed-line route. A 1:1 current (choke) balun at the feed blocks this common-mode current. An off-center feed sees a higher resistance, roughly R₀/cos²(kx) at distance x from the center, about 200 Ω at one fifth of the length from an end, and a 4:1 balun brings it to 50 Ω.

Quarter-wave transformer

Z_{t} = \sqrt{Z_{0} R_{in}}

Z_{port} = Z_{t} \frac{Z_{in} + j Z_{t} \tan β ℓ}{Z_{t} + j Z_{in} \tan β ℓ}

ℓ = V F \frac{λ_{0}}{4}

A 36 Ω monopole needs Z_t ≈ 42 Ω; a 112 Ω dipole needs 75 Ω coax as the transformer

An L-network of one series and one shunt reactance matches any load at a single frequency. When R_L > Z₀ the shunt element sits at the antenna:

L-network design (R_L > Z₀)

B = \frac{X_{L} \pm \sqrt{R_{L} / Z_{0}} \sqrt{R_{L}^{2} + X_{L}^{2} - Z_{0} R_{L}}}{R_{L}^{2} + X_{L}^{2}}

X = \frac{1}{B} + \frac{X_{L} Z_{0}}{R_{L}} - \frac{Z_{0}}{B R_{L}}

Otherwise the series element sits at the antenna. The calculator prefers the low-pass solution (series L, shunt C) and reports values in nH and pF.

The ground-plane antenna offers a third route: solve the radial droop so the antenna itself presents Z₀ at resonance, with no matching network at all (section 8).

15. Worked Example: 2 m Dipole and Ground Plane

AWG 12 copper wire (d = 2.053 mm) at f₀ = 146 MHz, λ₀ = 2053.4 mm, λ/d = 1000, K = 0.959, with the calculator's default normal mesh:

Antenna	Length	Z_in at f₀	2:1 SWR band (50 Ω)	Peak gain	Efficiency	Takeoff
Half-wave dipole, free space	984.5 mm	71.9 Ω	10.2 MHz	2.12 dBi	99.7%	—
Same, λ/2 over average earth	993.8 mm	70.6 Ω	10.3 MHz	7.16 dBi	72.6%	28.3°
λ/4 monopole on ideal ground plane	492.2 mm	35.9 Ω	15.4 MHz	5.13 dBi	99.7%	0°
Ground plane, 4 radials at 41.5°	485.6 mm each	50.0 Ω	15.6 MHz	1.85 dBi	99.7%	—

Four points stand out. The 468/f rule would cut the dipole at 977 mm, 0.8% short, so it would resonate about 1.1 MHz high. Over average earth the same dipole must be about 1% longer, and the soil absorbs over a quarter of the power at VHF, where the ground acts more as a lossy dielectric than as a mirror. The monopole has the same Q as the dipole it images (7.3 for both); its 2:1 band against 50 Ω is wider because its resistance, 36 Ω at resonance, rises toward 50 Ω above it. And the drooped ground plane gives a perfect 50 Ω match with no network, at the cost of about 3.3 dB of peak gain against the ideal-ground monopole.

16. Validity, Limitations and References

The thin-wire Method of Moments is the standard tool for wire antennas and agrees with NEC-2 to about 1% in resonant frequency when the wire is much thinner than a segment and a segment much shorter than a wavelength. Keep in mind:

The thin-wire kernel needs segments several times longer than the wire radius. Very thick tubing at UHF breaks this, and the calculator warns when it happens.
Real-ground impedance uses the reflection-coefficient approximation. For antennas below about 0.1 λ, or radials on or in the ground, a Sommerfeld-integral code such as NEC-4 is more accurate.
The feed line, the balun, connectors and insulators are not modelled, and wire insulation is not included. Common-mode current on an unbalanced feed can change the pattern and the impedance of a real dipole.
Matching networks are ideal and lossless; real coils and capacitors add loss.
Masts, guy wires, gutters and buildings detune real antennas, especially at VHF and UHF.

Cut the elements slightly long and trim to resonance with an antenna analyzer.

References

Balanis, C.A., Antenna Theory: Analysis and Design, 4th ed., Wiley, 2016, ch. 4, 8.
Harrington, R.F., Field Computation by Moment Methods, Macmillan, 1968.
Burke, G.J. & Poggio, A.J., “Numerical Electromagnetics Code (NEC) — Method of Moments,” Lawrence Livermore National Laboratory, UCID-18834, 1981.
Gibson, W.C., The Method of Moments in Electromagnetics, 3rd ed., CRC Press, 2021.
Pocklington, H.C., “Electrical Oscillations in Wires,” Proc. Cambridge Philosophical Society, vol. 9, 1897.
Hallén, E., “Theoretical Investigations into the Transmitting and Receiving Qualities of Antennae,” Nova Acta Regiae Soc. Sci. Upsaliensis, 1938.
King, R.W.P., The Theory of Linear Antennas, Harvard University Press, 1956.
Yaghjian, A.D. & Best, S.R., “Impedance, Bandwidth, and Q of Antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, 2005.
Chu, L.J., “Physical Limitations of Omni-Directional Antennas,” J. Appl. Phys., vol. 19, 1948.
Rockway, J.W. & Logan, J.C., MININEC Broadcast Professional, EM Scientific, 1996.
Stutzman, W.L. & Thiele, G.A., Antenna Theory and Design, 3rd ed., Wiley, 2012.
Pozar, D.M., Microwave Engineering, 4th ed., Wiley, 2011, ch. 5.
Straw, R.D. (ed.), The ARRL Antenna Book, ARRL.

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At f₀

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About this monopole and ground-plane antenna calculator

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What it calculates

Frequently asked questions

How long should a half-wave dipole be?

What is the 468/f formula and how accurate is it?

What is the impedance of a dipole antenna?

What is the impedance of a quarter-wave monopole and a ground-plane antenna?

Why are the radials of a ground-plane antenna drooped?

How many radials does a ground-plane antenna need, and how long should they be?

How high should a dipole be mounted?

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What is an off-center-fed (OCF) dipole and why does it use a 4:1 balun?

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Dipole and Monopole Antenna Theory

1. Dipoles and Monopoles

2. The Hertzian Dipole and the Far Field

3. Current and Pattern of a Finite Dipole

4. Radiation Resistance and Input Impedance

5. Resonant Length and the Shortening Factor

6. Bandwidth, Q and Conductor Thickness

7. Monopoles and Image Theory

8. Ground-Plane Antennas and Radial Droop

9. Real Ground, Height and Takeoff Angle

10. Pocklington and Hallén Integral Equations

11. The Method of Moments in This Calculator

12. Conductor Loss and Efficiency

13. Directivity, Gain and Effective Aperture

14. Baluns and Matching

15. Worked Example: 2 m Dipole and Ground Plane

16. Validity, Limitations and References

References

Dimensions

Return loss at the port

Input impedance

Current along the wires

3D radiation pattern at f₀

Radiation pattern at f₀

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