Current Distribution
How Current Shape Sets the Far Field
Every antenna is, at its core, a controlled current distribution in space. Maxwell's equations make the relationship exact: the radiated electric field at any far-field angle is the superposition of the fields from each infinitesimal current element, weighted by that element's amplitude, phase, and position. Mathematically the far-field pattern is the spatial Fourier transform of the current along the structure. Two antennas with identical physical outlines but different current tapers will radiate completely different patterns, which is precisely why aperture and array engineers spend so much effort shaping I(z) rather than just the metal.
For a thin straight wire the current behaves like that on an open-circuited transmission line. The reflection at each open end forces a current null there, and the standing wave that results is closely sinusoidal. A center-fed half-wave dipole therefore carries a current maximum at its feed point and tapers smoothly to zero at the tips. Integrating the radiated power against this distribution yields the textbook radiation resistance of 73 Ω and a directivity of 2.15 dBi. A full-wave (one wavelength) dipole, by contrast, develops a current reversal near its center, which splits the pattern and drives the input resistance into the kilohm range, a vivid demonstration that the current shape, not the antenna length alone, governs behavior.
The sinusoidal model is an approximation valid for wires thinner than roughly 0.01λ. As conductors become thicker, are bent, or sit near other metal, mutual coupling and end-capacitance distort the distribution. Engineers then turn to numerical electromagnetics: the method of moments solves for the true segment currents by enforcing the boundary condition that the tangential field vanish on the conductor, and the pattern, gain, and impedance all fall out of that computed current.
Governing Relations for Wire and Aperture Currents
I(z) = I0 sin[β(L/2 − |z|)], β = 2π / λ
Half-wave dipole approximation:
I(z) ≈ I0 cos(βz), −λ/4 ≤ z ≤ λ/4
Far-field pattern (line-source transform):
E(θ) ∝ ∫ I(z) · e jβz cosθ dz
Radiation resistance from the current integral:
Rr = 2Prad / |I0|2 → 73 Ω for an ideal λ/2 dipole
Where I0 = peak (feed) current, β = phase constant, θ = angle from the wire axis, Prad = total radiated power. Sidelobe level is set by the current taper: uniform → −13 dB, cosine taper → ≈ −23 dB.
Current Taper vs. Pattern Performance
| Current Distribution | Typical Structure | First Sidelobe | Beamwidth Factor | Aperture Efficiency |
|---|---|---|---|---|
| Uniform | End-fed slot array, leaky-wave | −13.2 dB | 1.0 (narrowest) | 100% |
| Cosine | Open-ended waveguide aperture | −23 dB | 1.19 | 81% |
| Cosine² (cos²) | Horn, reflector feed | −32 dB | 1.44 | 67% |
| Sinusoidal (cos βz) | Half-wave dipole | None (broad) | 1.05 | n/a, Rr=73 Ω |
| Taylor / Chebyshev | Phased-array aperture | −25 to −40 dB (design) | 1.1 to 1.4 | 70 to 95% |
Frequently Asked Questions
Why is the current distribution on a half-wave dipole assumed to be sinusoidal?
A thin straight wire acts like an open-circuited transmission line, so its current forms a standing wave that must vanish at the open ends. For a center-fed λ/2 dipole this gives I(z) = I0 cos(βz), peaking at the feed and zero at the tips. The model is accurate within a few percent for wires thinner than about 0.01λ and produces the classic 73 Ω radiation resistance. Thick elements, biconicals, or wires near other metal need a full method-of-moments solution to capture the true shape and reactance.
How does current distribution determine the antenna radiation pattern?
The far-field is the spatial Fourier transform of the current. Each current element radiates, and the pattern is their vector sum, weighted by amplitude, phase, and position: E(θ) ∝ ∫ I(z) ejβz cosθ dz. A uniform taper yields the narrowest beam with −13 dB sidelobes, while a tapered current (cosine, Taylor) lowers sidelobes at the cost of a wider beam and reduced aperture efficiency. Shaping the current taper across an array aperture is the primary tool for sidelobe and beamwidth control.
What numerical method is used to compute current distribution on a real antenna?
The method of moments (MoM) is the standard for wire and surface antennas. It segments the conductor, expands the unknown current in basis functions, and enforces that the tangential electric field vanish on the surface, turning the integral equation into a matrix equation Z·I = V solved for the segment currents. NEC and commercial MoM solvers then derive pattern, gain, and impedance from the computed I(z). Electrically large or dielectric-loaded structures instead use FEM or FDTD solvers to find the equivalent current.