Antennas: Turning a Circuit Into Radiated Waves

From a Guided Wave to a Free Wave

In 1888 Heinrich Hertz did something that looked like a parlour trick. He sparked a current across a tiny gap in a wire on one side of his lab, and across the room a second loop of wire — touching nothing — sparked back. Energy had crossed empty air with no wire to carry it. Hertz had built the first transmitting and receiving antennas, and in doing so confirmed Maxwell's equations: a changing current does not just sit on the metal, it sheds part of itself into space as an electromagnetic wave.

That is the whole job of an antenna: it is a transducer sitting at the boundary between two worlds. On one side is a *guided* wave — a current and voltage trapped on a transmission line, coax or microstrip, where energy obediently follows the copper. On the other side is a *free-space* wave that has detached from any conductor and is sailing outward at the speed of light. Transmitting, the antenna converts guided power into radiated power; receiving, it does the exact reverse, catching a passing wave and funnelling it back down the line. By reciprocity, the same antenna does both equally well.

Input Impedance: The Load Your Transmitter Actually Sees

Here is the idea that ties this rung back to everything you learned about matching. To the transmitter, an antenna is not a mysterious wave-launcher — it is simply a load, a complex number of ohms at the connector. That number is the antenna input impedance, written Z_in = R + jX. The transmitter cannot see space; it only sees this impedance, exactly the way an amplifier only sees the resistor you bolt to its output.

The real part R is the most beautiful part of the story. It splits into two: R = R_rad + R_loss. The loss resistance R_loss is ordinary ohmic heating in the metal — wasted. But R_rad, the radiation resistance, is the equivalent resistor whose 'heat' is actually power that left as a wave and never came back. A half-wave dipole has R_rad ≈ 73 Ω. Nothing is glowing hot at 73 Ω; that 73 ohms *is the sky*. The reactive part jX represents near-field energy that sloshes back and forth without escaping — energy stored, not radiated — and at resonance, by design, X goes to zero.

Two Workhorses: The Dipole and the Patch

Of the hundreds of antenna shapes, two appear again and again, and they teach almost everything. The first is the half-wave [[ee-dipole-antenna|dipole]]: two straight rods fed in the middle, the rabbit-ears descendant of Hertz's original. The second is the [[ee-patch-antenna|patch]]: a flat metal rectangle floating above a ground plane on a PCB, the antenna hiding inside your phone, your Wi-Fi router, and almost every GPS receiver. What both share is the rule that governs *all* antennas: their physical size is set by the wavelength λ of the signal.

Why half a wavelength? Picture standing waves on a string. A dipole is a piece of wire that has been opened up and stretched flat, and like a plucked guitar string it has a natural resonance when its length equals half a wavelength. At that length the current is maximum at the centre (where you feed it) and zero at the tips — a perfect half-sine standing wave. Feed it there and the antenna rings sympathetically, its reactance vanishes, and R_rad lands near that famous 73 Ω. Make it too short and it looks capacitive and weakly radiating; too long and it goes inductive. Resonance is a length, and that length is λ/2.

The patch works by the same standing-wave logic but on a flat resonator. The patch length is roughly half a wavelength *inside the dielectric* of the board — and that 'inside the dielectric' caveat shrinks it. A wave slows down in a material of relative permittivity ε_r, so the effective wavelength is λ / √ε_r. On a board with ε_r ≈ 4, every dimension shrinks by a factor of two, which is precisely how a 2.4 GHz patch fits comfortably inside a wristwatch. The radiating edges are the two open ends of the patch, where the fringing fields spill into space.

   HALF-WAVE DIPOLE                       PATCH ANTENNA
   (current = half-sine)                  (side view)

   tip          tip                          patch  (L ~ lambda/2 / sqrt(eps_r))
    |            |                       ___________________
    |  I=0       |  I=0                  |                 |  <- fringing fields
    |            |                       |_________________|     radiate here
    +---<feed>---+   <- I = max          ====================  <- ground plane
    |            |                              |
    |            |                            <feed via>

   total length L = lambda / 2            board: dielectric eps_r slows the wave
   R_rad ~ 73 ohm at resonance           Z_in often 100-300 ohm; inset feed tunes it

Where the Energy Goes: Pattern, Directivity, Gain, Beamwidth

An antenna almost never radiates equally in all directions. Picture a bare lightbulb versus a flashlight: same bulb, but the flashlight's reflector pours the light into a beam. Antennas do the same with radio power, and the radiation pattern is the map of where it goes — a plot of radiated strength versus angle. It has a main lobe (the beam you aim with), side lobes (smaller spills of energy off to the sides), and nulls (directions where, by interference, almost nothing comes out).

Directivity D puts a number on this concentration. It asks: in the antenna's best direction, how much stronger is the signal than if the *same total power* were smeared evenly over a sphere? An ideal point-source 'isotropic' radiator has D = 1 (0 dBi) by definition. A half-wave dipole has D ≈ 1.64, about 2.15 dBi — modestly focused. A dish antenna can reach 40 dBi, ten thousand times the isotropic level in its beam. Directivity is pure geometry: it is set entirely by the shape of the pattern, nothing else.

Antenna gain G is directivity *minus the losses*. A real antenna is never lossless — some power heats the metal and the dielectric instead of leaving. Gain bundles directivity and efficiency into one number: G = η · D, where η (typically 0.5 to 0.95) is the radiation efficiency. Gain is the number that actually appears in your link budget, because it tells you the real on-the-air boost in the chosen direction. A '+9 dBi patch' means: in its main beam, the wave is nine decibels — about eight times — stronger than an isotropic antenna fed the same power would manage. Gain costs you nothing in power; you buy it purely by *focusing* the power you already have.

Worked Example: Sizing a Dipole

Let's build a half-wave dipole for the FM broadcast band — say a station at 100 MHz. Everything flows from one equation, c = f · λ, the bridge between frequency and length that you met on the wavelength rung.

1. Find the wavelength. λ = c / f = (3 × 10⁸ m/s) / (100 × 10⁶ Hz) = 3.0 metres. At 100 MHz one full wave is as long as a small car.
2. Take half of it. A half-wave dipole is λ/2 = 1.5 m long in total — two rods of 0.75 m each, fed in the middle.
3. Apply the velocity factor. Real wire is slightly slow and has end effects, so the resonant length is about 95% of the ideal: L ≈ 0.95 × 1.5 m ≈ 1.43 m. Engineers shorthand this as the '468 / f(MHz) feet' rule, which gives the same answer.
4. Read off the electrical numbers. At resonance the reactance X ≈ 0, the radiation resistance R_rad ≈ 73 Ω, directivity ≈ 2.15 dBi, and the main lobe is a doughnut broadside to the rods with a beamwidth of about 78° in the elevation plane.
5. Match it. Your 50 Ω feedline meets 73 Ω: a VSWR of about 1.46, which is usually acceptable, but a small matching network or a folded dipole (which steps the impedance up to ~300 Ω) can clean it up. This is the matching step from the earlier rung doing real work.

  DIPOLE SIZING, 100 MHz
  -----------------------------------------
  lambda     = c / f = 3e8 / 100e6 = 3.0 m
  L (ideal)  = lambda / 2          = 1.50 m
  L (real)   = 0.95 * 1.50         = 1.43 m   <- cut to this

  Scaling to other bands (real length ~ 0.95 * lambda/2):
    FM   100 MHz  ->  L ~ 1.43 m
    GPS  1.575 GHz->  L ~ 9.0 cm
    WiFi 2.45 GHz ->  L ~ 5.8 cm
    5G   28 GHz   ->  L ~ 5.1 mm  (now a patch on a chip makes more sense)

  Resonant Z_in ~ 73 + j0 ohm ;  feed = 50 ohm  ->  VSWR ~ 1.46

Why This Sets Up the Link Budget

You now hold the one number that the next rung lives on: gain. A radio link is an accounting exercise — you start with the power your amplifier produces, *add* the gain of the transmitting antenna, *subtract* the enormous spreading loss of free space, then *add* the gain of the receiving antenna, and see whether what's left clears the noise floor. Every term is in decibels, so the whole budget is just addition and subtraction, and two of its terms are antenna gains you now know how to compute.

So the chain is complete. Wavelength told you how big to build it. Impedance and matching told you how to feed it without wasting power. The input impedance was the bridge between those circuit ideas and this radiating object. And the pattern, directivity, and gain tell you, finally, where the energy lands. A circuit has become a wave in flight — and a wave in flight, in the next rung, becomes a working radio link.