ch5: why antennas are sized in λ

date: May 17 2026

this chapter: why are antenna sizes in multiples of λ/2, and can they be different? what happens off-resonance? what physical limit you hit when you try to make an antenna much smaller than λ.

1. An antenna is a transducer

Antennas are weirdly specific about their size. A 5 cm piece of wire is a great antenna at 3 GHz and an inert chunk of metal at 300 MHz. The dependence on frequency isn’t smooth either, it’s a series of sharp resonances at integer multiples of half a wavelength along the wire. The whole story comes down to standing waves on the conductor.

The antenna sits between two fundamentally different EM regimes: guided current on a metal wire and free-space radiation. The two regimes have different boundary conditions, and at the antenna’s terminals the only way to satisfy both at once is to set up a standing wave on the conductor whose half-wavelength on the wire matches the wavelength radiated into free space, $\lambda = c/f$ . Sizes that match this number couple constructively and radiate strongly. Sizes that don’t either reflect energy back into the source or radiate it weakly.

2. Standing waves and the natural λ/2 resonance

A straight wire with open ends is an open-circuited transmission line. Current is the flow of charge, and charge has nowhere to go past the wire’s ends, so the current must be zero at both tips. Voltage, by contrast, is maximum at the tips and minimum at the center for the fundamental mode.

The simplest standing-wave pattern that puts current nulls at both tips is one with a single half-cycle along the length:

I(z) \;=\; I_0 \cos\!\left(\frac{\pi z}{L}\right), \qquad -L/2 \le z \le L/2

For this standing wave to oscillate sinusoidally in time at frequency $f$ , the wavelength on the wire must equal $2L$ , which means $L = \lambda/2$ . That is the fundamental resonance of a dipole.

Higher harmonics fit additional half-cycles: $L = \lambda$ , $3\lambda/2$ , $2\lambda$ , and so on. Each higher mode produces a different current distribution (with sign reversals along the wire) and so a different radiation pattern.

The transmission-line picture

A dipole can be thought of as a parallel-wire transmission line, shorted at the feedpoint, with its two arms unfolded into a straight line. An open-circuited transmission line of length $\lambda/4$ presents zero input impedance (it transforms the open-circuit infinite impedance through a quarter-wave to short). Unfolding gives a $\lambda/2$ dipole with low feedpoint impedance and a clean current standing wave. Other lengths give other impedance values, most of them reactive and inconvenient.

So $L = \lambda/2$ isn’t really a design choice. The wire’s open ends force current nulls there, the smallest sinusoidal current pattern that obeys that boundary condition fits exactly one half-wavelength along the wire, and that’s the smallest mode the boundary conditions allow. Everything else is harmonics on top.

3. What breaks off-resonance

The dipole still radiates if its length is something other than $\lambda/2$ , but the geometry of the current distribution and the resulting radiation pattern change with $L/\lambda$ .

Shorter than λ/2

When $L < \lambda/2$ , the natural sinusoidal current would still want a half-cycle, but the wire is too short to support it; the current ends up approximately triangular, peaking at the feedpoint and tapering to zero at the tips. The radiation pattern stays roughly the same shape (a broadside donut), but two things degrade:

Radiation resistance collapses as $(L/\lambda)^2$ . From Chapter 4, $R_\text{rad} \approx 80\pi^2(L/\lambda)^2 \approx 790(L/\lambda)^2 \,\Omega$ . A $\lambda/10$ dipole has $R_\text{rad} \approx 8\,\Omega$ , compared to $73\,\Omega$ at $\lambda/2$ .
Input reactance becomes large and capacitive. The feedpoint impedance has a small real part and a large negative imaginary part, requiring an inductor in series to tune it for matching. The matching network adds loss and narrows bandwidth.

Longer than λ/2

When $L > \lambda/2$ , the natural standing-wave pattern fits more than one half-cycle along the wire, and the current changes sign one or more times along the length. The pattern and the feedpoint impedance evolve differently as $L$ grows:

At $L = \lambda$ , the broadside pattern is actually a narrower donut than at $\lambda/2$ , with directivity around 3.82 dBi - higher gain, single main lobe, no significant side lobes. The catch lives in the impedance: $I(0) = I_0 \sin(k L / 2) = \sin(\pi) = 0$ , so the current is zero at the feedpoint, the input impedance is effectively infinite, and the antenna is hard to feed without a transformer that adds loss and narrows bandwidth.
Maximum single-lobe gain occurs around $L \approx 1.25\lambda$ at about 5 dBi, with input impedance still high and reactive.
At $L = 3\lambda/2$ , the main lobe splits into two beams off broadside (peaks near ±48° from broadside). Feedpoint current is back to maximum, so the impedance is moderate and feedable.
At $L = 2\lambda$ , broadside is a deep null and the pattern is four lobes off-axis.

Two separate things must line up for an antenna to be both feedable and useful:

Feedpoint current. The input impedance is low and resistive when $I(0)$ is at a current maximum. This happens at odd multiples of $\lambda/2$ ( $L = \lambda/2, 3\lambda/2, 5\lambda/2, \ldots$ ) and is poor at integer multiples of $\lambda$ .
Pattern shape. A clean single broadside lobe exists for $L$ between about $\lambda/4$ and $1.25\lambda$ . Beyond that, the pattern splits.

$L = \lambda/2$ is the only length where both conditions are excellent: maximum feedpoint current (so the cleanest impedance match at ~73 Ω) and a clean broadside donut. Other lengths win on one and lose on the other.

Dipole radiation pattern vs L/λ

A center-fed dipole with an assumed sinusoidal current distribution. The slider sweeps the length in units of λ. The left panel shows the current along the wire, the right panel shows the resulting radiation pattern in polar form (broadside is right, dipole axis is up/down).

L / λ 0.50 G ≈ 2.15 dBi

Left: current $I(z) = I_0 \sin(k(L/2 - |z|))$ along the wire. Right: normalized power pattern $|F(\theta)|^2$ on a polar plot, where $F(\theta) = [\cos(kL\cos\theta/2) - \cos(kL/2)] / \sin\theta$ . Pattern stays single-lobe up to $L \approx 1.25\lambda$ ; it splits into off-broadside beams at $L = 3\lambda/2$ and develops a broadside null at $L = 2\lambda$ .

What feedpoint impedance actually is

Feedpoint impedance is the ratio of voltage to current at the antenna’s terminals, as seen by whatever is driving it:

Z_\text{in} \;=\; \frac{V_\text{feed}}{I_\text{feed}} \;=\; R + j X

The real part $R$ is power leaving (radiation plus a small ohmic loss). The imaginary part $X$ is energy sloshing back and forth between the source and the reactive near field without ever leaving the antenna. The behavior of an antenna at the source side is fully captured by these two numbers at the operating frequency.

What “matched” means

A source with internal impedance $Z_s$ delivers maximum power to a load when $Z_\text{load} = Z_s^*$ (complex conjugate match). For a purely resistive 50 Ω source feeding a real-valued antenna impedance, the match is just $R = 50\,\Omega$ . Cable impedances landed at 50 Ω (or 75 Ω for video and TV) by industry convention: 50 Ω is roughly the compromise between minimum loss and maximum power-handling for air-dielectric coax. A half-wave dipole at $R \approx 73 \,\Omega$ matches 75 Ω cable essentially exactly and 50 Ω cable to within a 1.5:1 mismatch ratio - good enough that hobbyists routinely use 50 Ω feed without worrying.

What reflection looks like

When source and load do not match, part of the wave reflects back. The voltage reflection coefficient is

\Gamma \;=\; \frac{Z_\text{load} - Z_s}{Z_\text{load} + Z_s}

The fraction of incident power that reflects is $|\Gamma|^2$ ; the fraction that gets into the antenna is $1 - |\Gamma|^2$ . Some reference points for a 50 Ω source:

$Z_\text{load} = 73\,\Omega$ (half-wave dipole): $|\Gamma| \approx 0.19$ , reflected power 3.7%, antenna gets 96.3% (-0.16 dB mismatch loss). Fine without matching.
$Z_\text{load} = 300\,\Omega$ (folded dipole or rough estimate at $3\lambda/2$ ): $|\Gamma| \approx 0.71$ , reflected power 51%, antenna gets 49% (-3.1 dB). Needs a transformer or stub.
$Z_\text{load} = 5000\,\Omega$ (typical full-wave dipole at $L = \lambda$ ): $|\Gamma| \approx 0.98$ , reflected power 96%, antenna gets 4% (-14 dB). Effectively unfeedable from 50 Ω without a tightly tuned matching network.
$Z_\text{load} = \infty$ (ideal full-wave dipole, lossless): $|\Gamma| = 1$ , all power reflects back. The source sees an open circuit.

The mismatch loss is one-way; in radar, where the same antenna is used for TX and RX, mismatch hits twice. A bad match also creates standing waves on the feed cable, which raises peak voltage and can damage the source. The standing-wave ratio (VSWR) $= (1 + |\Gamma|)/(1 - |\Gamma|)$ is the conventional metric: 1:1 is perfect, 2:1 is the usual acceptable ceiling, and above 3:1 things get problematic.

The source impedance is itself a design choice

The 50 Ω convention is not a fixed property of the source - it is engineered. A matching network (L-section, π-section, or a commercial antenna tuner) inserted between source and antenna transforms whatever impedance the antenna presents into whatever the source wants to see. Mobile phones use tunable matching circuits with varactor diodes or RF-MEMS switches that adapt the source-side impedance as the user’s hand detunes the antenna, keeping a single short antenna roughly matched across the 700 MHz to 6 GHz cellular range. The cost is tenths of a dB of loss in the tuner components and a switching time of microseconds to milliseconds per band change.

Engineers usually visualise the match on a Smith chart: every possible load impedance maps to a point inside a unit disc parametrised by $\Gamma$ , and constant-VSWR contours are concentric circles around the centre. A matching network reads as a path that walks the load point toward the centre; each series capacitor, shunt inductor, or transmission-line stub traces a specific arc. The match is also frequency-dependent. As you sweep, $Z_\text{load}(f)$ draws a curve on the chart, and the VSWR-vs-frequency plot is the V-shape that curve cuts through the constant-VSWR circles. A narrowband resonant antenna draws a tight loop near $f_0$ ; a wideband design (Vivaldi, ridged horn, tapered slot) stays near the centre over a decade or more.

4. Other standard sizes (it is still λ/2)

The $\lambda/2$ scale shows up in nearly every standard antenna geometry, often with one factor of two hidden somewhere.

antenna	resonant dimension	why
λ/4 monopole over ground plane	λ/4	The ground plane mirrors the antenna; image theory gives a virtual $\lambda/2$ dipole. Half the physical size, same electrical behavior. Used in nearly every Wi-Fi and Bluetooth device.
Half-wave folded dipole	λ/2	Two parallel $\lambda/2$ conductors connected at the ends. Same length, ~4× higher feedpoint impedance (~300 Ω, matches old TV twin-lead). Wider bandwidth than a plain dipole.
Patch antenna	≈ λ/(2√ε_r)	A resonant cavity between a conducting patch and a ground plane. The radiating edge is roughly $\lambda/2$ in the dielectric, so the physical size is reduced by $\sqrt{\epsilon_r}$ .
Small loop	circumference ≪ λ	Behaves as a magnetic dipole. Radiation resistance is even smaller than a short electric dipole (scales as $(C/\lambda)^4$ ). Used where size matters more than efficiency (AM radios, RFID).
Resonant loop	circumference ≈ λ	A loop of total path length $\lambda$ has a sinusoidal current standing wave that supports efficient radiation. Higher gain than a half-wave dipole.
Horn antenna	aperture > λ²	Gain scales with aperture area (in units of $\lambda^2$ ). Aperture size is a free parameter, but the waveguide section feeding the horn must support the propagating mode: rectangular waveguide width > $\lambda/2$ for the dominant TE10 mode.
Parabolic dish	aperture ≫ λ	Same as horn: gain ∝ aperture area / $\lambda^2$ . The dish itself is an aperture antenna; the actual radiator is a feed (usually a horn or dipole) at the focus.

Apertures (horns, dishes) are the exception that proves the rule: their aperture can be any size, but they always contain a smaller radiator somewhere (the horn throat, the dish feed) sized in $\lambda/2$ units to do the actual coupling between guided wave and free space. The aperture’s job is to shape the beam, not to do the transducing.

5. Cheating: making it smaller

An antenna shorter than $\lambda/2$ can be tuned back to resonance with a series inductor that cancels its capacitive reactance. The antenna will then accept power from a matched source and radiate it. But the bandwidth around the new resonance shrinks.

The reason is a hard limit, due to Chu (1948) and refined by Harrington and others: for any antenna that fits inside a sphere of radius $a$ , the minimum achievable quality factor at frequency $\omega$ (with $k = \omega/c$ ) is

Q_\text{min} \;=\; \frac{1}{(ka)^3} + \frac{1}{ka}

The fractional bandwidth scales as $1/Q$ , so

\frac{\Delta f}{f_0} \;\le\; \frac{1}{Q_\text{min}} \;\approx\; (ka)^3 \quad \text{when } ka \ll 1

Halving the antenna’s physical size at the same frequency cuts the maximum achievable bandwidth by a factor of 8. A $\lambda/20$ antenna ( $ka \approx 0.16$ ) is bounded at roughly 0.4% fractional bandwidth, no matter what design tricks you bring. This is a theorem about energy stored vs energy radiated; no antenna geometry beats it.

What this means in practice

$L \gtrsim \lambda/2$ : the antenna lives on the right side of the Chu bound, far from the cliff. Bandwidth is plentiful and matching is easy.
$L \sim \lambda/4$ to $\lambda/10$ : the antenna can be tuned with a matching network, but bandwidth drops to single-digit percent and matching network loss starts eating efficiency. The Wi-Fi “rubber duck” sits here.
$L < \lambda/20$ : bandwidth is sub-percent. Used for narrowband applications (RFID, ISM keyfobs); ill-suited to anything that needs to sweep frequency, like FMCW radar.

The bound assumes a lossless conductor. Real designs can flatten the VSWR curve by adding ohmic loss (a resistor in the feed network, lossy dielectrics in the matching elements), which lowers $Q$ but at the cost of dissipating input power as heat rather than radiating it. The total bandwidth-efficiency budget is unchanged; you only get to trade one against the other. Practical electrically small antennas sit 2 to 5× above $Q_\text{min}$ once feed losses, finite ground planes, and imperfect conductors are folded in, so the achievable bandwidth is tighter than the Chu formula alone suggests.

For FMCW radar this matters concretely: the antenna needs to pass the full chirp bandwidth without distortion. A 1 GHz chirp at 6 GHz centre is 17% fractional bandwidth, which is fine for an electrically large antenna but completely out of reach for a small one with $Q_\text{min} \approx 20$ . Below that limit, the chirp edges get attenuated and phase-distorted, and the FMCW range FFT picks up the edge attenuation as ghost targets at the band edges.

next: ch6: arrays and beamforming →