Monopole / Whip Antenna Calculator

A monopole is a single vertical conductor of height $h$ fed against a ground plane: a car whip, a handheld's rubber duck over its own case, a quarter-wave ground-plane antenna on a mast. Over a perfectly conducting, infinite plane the ground acts as a mirror. The current on the element and its image form a centre-fed dipole of length $2h$, so the monopole radiates into the upper half-space exactly as that dipole does.

That image gives the whole calculation. The monopole has half the dipole's input impedance and twice its directivity, because the same field fills half the space:

The dipole is taken from the induced-EMF method with a sinusoidal current, as Balanis gives it, with $k = 2\pi/\lambda$, $l = 2h$, wire radius $a$ and the sine and cosine integrals Si and Ci:

$R_r$ and $X_m$ are referred to the current maximum. Dividing by $\sin^2(kh)$ refers them to the base, where the feed is. For a quarter-wave monopole the two are the same, and the result is the familiar $36.5 + j21.3\ \Omega$ and 5.16 dBi: half the half-wave dipole's $73.1 + j42.5\ \Omega$, twice its directivity of 1.64. The positive reactance is why a resonant quarter-wave whip is cut a few percent shorter than $\lambda/4$; the thicker the wire, the more it must be shortened.

Short whips and base loading

Below a quarter wave the input resistance falls quickly, towards $R_{in} \approx 40\pi^2 (h/\lambda)^2$, and the reactance becomes large and capacitive. A loading coil at the base cancels it:

The calculator reports that inductance whenever the reactance is capacitive, and zero when the element is already inductive, which calls for shortening instead.

Validity

The model assumes a thin wire: the conductor radius should not exceed 0.0159 λ, the thin-wire limit $2\pi a/\lambda \le 0.1$ that NEC-2 also uses. Near a height of half a wavelength, or any multiple of it, the base current vanishes and the modelled impedance grows without bound. Within 0.08 λ of such a height the calculator warns that the impedance is not a design value, and within 0.01 λ it does not report one at all. The ground is perfect and infinite. Radials, finite ground planes, car roofs and lossy earth are not modelled; for those, open the antenna simulator from this page. The velocity factor only shortens the quarter-wave cutting length; the impedance is computed for the element as entered.