2. Antenna Parameters: A Vocabulary

Antenna performance is described by a small set of figures of merit. Almost every datasheet for every antenna lists these numbers. Once you understand them, you can translate between any two antenna types.

2.1 Radiation pattern

The radiation pattern is a 3D map of how much power the antenna radiates in each direction. Usually we plot two 2D cuts: the E-plane (containing the electric field vector) and the H-plane (containing the magnetic field vector). On a polar plot, you can see the lobes, nulls, and asymmetries at a glance.

A typical pattern has a main lobe (where most of the power goes), some side lobes (smaller peaks in unwanted directions), a back lobe (the lobe pointing opposite the main beam), and nulls (directions where almost nothing is radiated).

       Main lobe
         ▲
        / \
       /   \
      /     \
     /       \
─── ─ ─ ─ ─ ─ ─ ─ ─ ─ ─  (axis)
     \       /
      \     /
   side lobe   side lobe
       \ /
        v
     Back lobe

A flashlight beam is a perfect physical analogy. Its main lobe is the bright cone in front; the small bit of light leaking sideways is the side lobes; the dim glow behind the bulb (because some light reflects off the inside of the housing) is the back lobe. A dim lamp with no reflector radiates almost equally everywhere, the way an isotropic antenna would, and so it has no main lobe to speak of.

2.2 Beamwidth: HPBW and FNBW

Beamwidth quantifies "how narrow is the beam." Two definitions are standard:

• Half-Power Beamwidth (HPBW): the angular width between the points where radiated power drops to half the peak (–3 dB). For a narrow beam, this is a small angle. For an omnidirectional antenna, HPBW can exceed 180° (essentially undefined for the full doughnut).
• First-Null Beamwidth (FNBW): the angular width between the first nulls on either side of the main lobe. Always larger than HPBW.

A high-gain dish at 10 GHz with a 1 m diameter has HPBW around 2°. A half-wave dipole has HPBW around 78°. A patch antenna has HPBW around 60° to 80°. A small whip antenna has HPBW essentially 360° in azimuth (omni in the horizontal plane).

2.3 Polarization: linear, circular, elliptical

Polarization is the direction of the electric field vector in the radiated wave. We talked about this in Chapter 9. From the antenna's perspective:

• Linear polarization: $\vec{E}$ stays along one axis. A vertical dipole radiates vertically polarized waves. A horizontal dipole radiates horizontally polarized waves. Most cellular and TV antennas use linear polarization.
• Circular polarization: the $\vec{E}$ vector rotates as the wave propagates. From the source's perspective, looking outward, it rotates either clockwise (right-hand circular, RHCP) or counterclockwise (left-hand circular, LHCP). GPS satellites transmit RHCP. Satellite TV often uses circular polarization so dish orientation does not matter for reception.
• Elliptical polarization: the general case, where $\vec{E}$ traces an ellipse. Linear and circular are the two special cases.

A polarization mismatch costs you. Linear-to-circular is a 3 dB loss (half the power). Crossed linear (vertical to horizontal) is theoretically infinite loss, in practice 20 to 30 dB depending on antenna purity. Right-hand circular to left-hand circular is also infinite in theory.

This matters operationally. If your handheld radio is vertical and you tilt it, you partially mismatch with a vertical base-station antenna, dropping the link. It also matters in spy-craft: receivers tuned to the wrong polarization simply cannot pick up the target signal, no matter how high the gain.

2.4 Directivity: how concentrated the beam is

Directivity is the ratio of the radiation intensity in the direction of the main lobe to the average over all directions:

$D = \frac{4\pi U_{max}}{P_{rad}}$

An isotropic radiator (a hypothetical antenna that radiates equally in all directions) has $D = 1$, conventionally written as 0 dBi. Real antennas always have $D \geq 1$, because at least a little of the radiation has to point somewhere.

For reference:

Higher directivity means a narrower beam; a narrower beam means more reach in the chosen direction at the cost of seeing nothing in other directions. A flashlight has high directivity. A naked light bulb has low directivity.

2.5 Gain and efficiency

Real antennas are lossy. Some of the input power is dissipated in conductor resistance (ohmic loss), some in dielectric losses, some leaks out the wrong way through impedance mismatch. The fraction that actually gets radiated is the antenna efficiency:

$e = \frac{P_{rad}}{P_{input}}$

Typical $e$ ranges from 0.5 to 0.95. The gain is then

$G = e \cdot D$

Gain is what you actually measure on a real antenna. Directivity is what you would have if the antenna were lossless.

A subtle point that confuses beginners: a high-gain antenna does not produce more total power. It just concentrates the power you already have into a smaller solid angle. If the transmitter outputs 1 W and the antenna has 10 dBi of gain, the equivalent isotropic radiated power (EIRP) in the boresight direction is 10 W, but the total radiated power is still less than 1 W (less because of the efficiency loss). Conservation of energy is intact; you just borrowed photons from "useless" directions and concentrated them toward the target.

2.6 Effective area and aperture efficiency

For receive antennas, the most useful figure of merit is the effective area $A_e$. If a plane wave with power density $S$ (in W/m²) hits the antenna, the antenna delivers $A_e \cdot S$ watts to a matched load. For a parabolic dish, $A_e$ is around 50–70% of the physical area (the rest is lost to spillover, blockage, edge effects, surface roughness). The fraction $A_e / A_{phys}$ is the aperture efficiency $\eta_{ap}$.

The crucial relationship between gain and effective area is

$G = \frac{4\pi A_e}{\lambda^2}$

This is one of the most useful equations in antenna engineering. It says that for a given physical aperture, higher frequency means higher gain (smaller $\lambda$). That is why satellite communications keep marching to higher frequencies (X-band, Ku-band, Ka-band, now V-band): the same dish gives more gain.

A confusion to address head-on: can effective area exceed physical area? Generally no. For an aperture antenna, $A_e \leq A_{phys}$. But for non-aperture antennas like a dipole, the concept of "physical area" is poorly defined; a half-wave dipole has effective area $A_e = 1.64 \lambda^2 / (4\pi) \approx 0.13 \lambda^2$, which dwarfs the wire's cross-sectional area. The antenna gathers energy from a region around itself larger than its physical wire, which is intuitively strange but correct.

Lighthouse analogy. A lighthouse uses a giant Fresnel lens to focus the bulb's light into a narrow beam that sweeps the horizon. The beam can be seen 30 miles offshore. The bulb itself is dim. The lens has done two things at once: it has given the bulb directivity (most light goes into the beam) and it has given an effective collecting area (a ship's eye sees the bulb through the lens, concentrated). The lens has not made the bulb brighter in any absolute sense; it has just routed the photons. A parabolic dish does the same job with radio waves.

2.7 Radiation resistance: a "real" Ohmic value with no heat

Consider feeding an antenna with an alternating current. From the feedline's perspective, the antenna draws power. By Ohm's law, $P = I^2 R$, where $R$ is some resistance. We call this resistance the radiation resistance $R_r$.

The puzzle: this is a real Ohmic value, with units of ohms, but no power is dissipated as heat. Where does it go? Out into space, as electromagnetic waves. The "loss" on the line is real (the source is doing work), but the destination is the far-field rather than a hot wire.

For standard antennas:

The total input impedance of an antenna is $Z_a = (R_r + R_{loss}) + jX$, where $R_{loss}$ is the actual Ohmic loss in the conductor and $X$ is the reactive part. For maximum efficiency, you want $R_r \gg R_{loss}$. That is why short antennas are inefficient: their $R_r$ is tiny (because a short wire is a poor radiator), so even small conductor losses dominate.

This is the deep reason FM antennas are physically big. At 100 MHz, $\lambda = 3$ m, so a quarter-wave whip is 75 cm. You cannot make it much shorter without driving radiation resistance into the milliohms, where conductor losses turn your transmitter into a heater rather than a radio.

2.8 The Friis transmission equation

Now we tie it all together. Suppose antenna A radiates total power $P_t$ with gain $G_t$ in some direction. At distance $r$ in that direction, the power density is

$S = \frac{P_t G_t}{4\pi r^2}$

The $4\pi r^2$ in the denominator is the surface area of a sphere of radius $r$. If the antenna were isotropic, the power would spread evenly over that sphere. Since it has gain $G_t$, it concentrates the power by a factor of $G_t$ in our direction.

Antenna B is at distance $r$ with effective area $A_{e,r}$. The received power is

$P_r = S \cdot A_{e,r} = \frac{P_t G_t A_{e,r}}{4\pi r^2}$

Substituting $A_{e,r} = G_r \lambda^2 / (4\pi)$:

$\boxed{P_r = P_t G_t G_r \left(\frac{\lambda}{4\pi r}\right)^2}$

This is the Friis transmission equation. It is one of the most-used equations in communications engineering, the basis of every link budget calculation. Read it as: received power equals transmitted power times the two antenna gains times a path-loss factor.

In dB form, it is even cleaner:

$P_r(\text{dBm}) = P_t(\text{dBm}) + G_t(\text{dB}) + G_r(\text{dB}) - 20\log_{10}(r) - 20\log_{10}(f) + 147.55$

where $r$ is in meters and $f$ is in Hz. The constant 147.55 absorbs the conversion from $\lambda = c/f$ and the units of $r$.

Worked example: WiFi to your phone. A router transmits 100 mW (20 dBm) at 2.4 GHz with a 2 dBi antenna. Your phone has a 0 dBi antenna 10 m away. $P_r = 20 + 2 + 0 - 20\log(10) - 20\log(2.4 \times 10^9) + 147.55 = 20 + 2 + 0 - 20 - 187.6 + 147.55 = -38$ dBm. A typical phone needs about –80 dBm minimum for a stable connection, so we have 42 dB of margin. The link is comfortable.

Worked example: deep-space, Voyager 1 to Earth. Voyager 1 is at about 24 billion km ($2.4 \times 10^{13}$ m). It transmits 22 W (43 dBm) at 8.4 GHz with a 48 dBi dish. The DSN receives with a 70 m dish (74 dBi). $P_r = 43 + 48 + 74 - 20\log(2.4 \times 10^{13}) - 20\log(8.4 \times 10^9) + 147.55 = 43 + 48 + 74 - 267.6 - 198.5 + 147.55 = -153.5$ dBm. That is $4.5 \times 10^{-19}$ W. The thermal noise of a 1 Hz bandwidth at 30 K is about $4 \times 10^{-22}$ W. So the SNR in 1 Hz is about 1000:1, which sustains a few hundred bits per second of data after coding gain. Friis applied to the most distant human-made object.

The path-loss term $20\log(r) + 20\log(f)$ tells you that doubling distance costs 6 dB and doubling frequency also costs 6 dB. This is the inverse-square law showing up in dB.

2.9 Link budget

A link budget is a spreadsheet that adds up all the gains and subtracts all the losses on a link, telling you if there is enough margin. Real link budgets include:

• Transmitter output power
• Cable losses to antenna
• Antenna gains (transmit and receive)
• Free-space path loss
• Atmospheric loss (rain, oxygen, water vapor)
• Polarization mismatch
• Pointing error
• Receiver noise figure
• Required signal-to-noise ratio
• A safety margin (typically 3–10 dB)

If the budget closes (received power minus required signal level is positive), the link works. If not, you increase $P_t$, $G_t$, $G_r$, or move the antennas closer. Every commercial wireless system on Earth lives or dies by this calculation.