Learning Objectives

1. I can define and calculate fundamental antenna properties — gain, directivity, effective aperture, beamwidth, and sidelobe level — apply the reciprocity principle that links an antenna's transmit and receive behavior, and use the Friis transmission equation to predict received power in a link.

Where we were

Lesson 1 called the antenna a transducer: the device that hands a guided wave off to free space and takes it back again. That is the right picture, but it is not yet a number. Today we build the vocabulary that turns the picture into engineering — power density, radiation intensity, directivity, gain, efficiency, beamwidth, and effective aperture — and we walk the physics chain from Maxwell’s equations down to the plane wave that makes those definitions legal in the first place. Then we spend the vocabulary: reciprocity ties the transmit side to the receive side, and the Friis equation turns two antennas, a frequency, and a range into a received power in dBm. By the end of the lesson you can close a link budget on the back of an envelope.

Maxwell’s equations (source-free, linear media)

Every antenna analysis in this course starts from Maxwell’s equations. Antennas are a boundary-condition problem: currents on the antenna set tangential fields, those fields must match a solution that radiates outward. To see why they radiate, we walk the chain once.

The two curl equations are the ones that make waves. A changing \(\mathbf{H}\) in time creates a spatial curl in \(\mathbf{E}\), and a changing \(\mathbf{E}\) in time creates a spatial curl in \(\mathbf{H}\). Time and space are locked together — you cannot change one without changing the other. That coupling is the entire story of propagation.

The Telegrapher’s Equations

On a two-conductor transmission line, integrating Maxwell over the cross-section collapses the fields onto voltage \(v(z,t)\) and current \(i(z,t)\), with the geometry surviving only as a per-unit-length inductance \(L\) [H/m] and a per-unit-length capacitance \(C\) [F/m]:

Both give you the wave equation

Same structure as the curl pair: a spatial derivative on one side, a time derivative on the other. Differentiate one, substitute the other, and both \(v\) and \(i\) satisfy the 1-D wave equation

The propagation speed drops out: \(u = 1 / \sqrt{L C}\). On an air-filled line this is essentially \(c\).

In free space, the same “curl-of-curl” trick applied to Maxwell gives

The exact same equation applies to \(\mathbf{H}\). This is the equation whose solutions the rest of the course lives in.

A plane-wave solution

The canonical solution, propagating in \(+\hat{z}\):

• \(\omega = 2 \pi f\) is the time frequency — how fast the field oscillates at a fixed point.

• \(k = 2 \pi / \lambda\) is the space frequency (wave number) — how fast the field oscillates at a fixed instant in time.

• \(\omega / k = c\) ties them together: the wave crest moves through space at \(c\).

Impedance of free space and the far-field structure

Plugging the plane-wave \(\mathbf{E}\) back into Faraday’s law forces \(\mathbf{H}\) to be transverse, perpendicular to \(\mathbf{E}\), and

Far from an antenna, the radiated field looks locally like a plane wave — transverse E and H, ratio \(\eta_{0}\) — with a \(1/r\) amplitude fall-off that we’ll derive from the radiation integrals in L6. That \(1/r\) (equivalently \(1/r^{2}\) in power) is why the antenna’s directional properties even matter — they set how much power lands on your receiver in a given direction.

The Poynting vector

With the wave equation in hand, we can define the working vocabulary for the rest of Module 1.

Before we can say how much power an antenna sends in a given direction, we need the quantity that measures power flow itself. The Poynting vector is

the instantaneous power per unit area carried by the field, pointing in the direction of propagation.

The time-average Poynting vector

For a time-harmonic field we almost always want the time-average,

and in the far field — where \(\mathbf{E}\) and \(\mathbf{H}\) are transverse and locally plane-wave-like with \(|\mathbf{E}| / |\mathbf{H}| = \eta_{0}\) — its magnitude collapses to

Power density in the far field

Because the far-field amplitude falls as \(1/r\), this power density falls as \(1/r^{2}\) — the inverse-square law. This \(S_{\text{rad}}\) is exactly the quantity that appears in the radiation-intensity definition next.

Radiation intensity

Power radiated per unit solid angle, in a given direction:

where \(S_{\text{rad}} = |\mathbf{E}|^{2} / (2 \eta_{0})\) is the time-average Poynting magnitude in the far field. The \(r^{2}\) cancels the \(1/r^{2}\) in \(S_{\text{rad}}\), so \(U\) depends only on direction, not distance. Total radiated power is

Directivity

How well the antenna concentrates power in a given direction relative to an isotropic radiator:

\(D\) is dimensionless (or in dBi when converted to decibels). “Directivity” — no losses, purely geometric.

Directivity — the pencil-beam approximation

A useful rule of thumb: for a pencil beam with half-power beamwidths \(\theta_{1}\) and \(\theta_{2}\) (in radians),

That 41,253 is the lossless geometric bound — it assumes every watt lands inside the main beam. Real horns and dishes leak into sidelobes and spill past the reflector, so in practice you substitute a constant of 26,000 to 32,400, which is the same estimate docked the 1 to 2 dB that the sidelobes and spillover actually cost.

Gain and efficiency

Gain is directivity with losses folded in:

Efficiency accounts for ohmic and dielectric losses in the antenna structure. Mismatch loss at the antenna terminals is usually broken out separately as realized gain \(G_{\text{re}} = (1 - |\Gamma|^{2}) G\).

When your instrument or datasheet reports gain in dBi, it is almost always the realized gain in the peak direction.

Γ, VSWR, and Power Reflected

Recall the transmission-line result you already own.

Power as fractions of the incident power

Power splits as fractions of the incident power:

Quick Reference — Γ vs VSWR vs Power

Γ, VSWR, and Power — the verdict table

Beamwidth, boresight, and lobes

Read on a radiation pattern:

• Boresight — the direction of peak radiation. Antennas are usually aimed along boresight.

• Half-power beamwidth (HPBW) — angular width where the pattern drops by 3 dB (to \(1/2\) in power).

• First-null beamwidth (FNBW) — angular width between the first nulls on either side of the main lobe. Roughly \(2\times\) HPBW for most simple patterns.

• Main lobe — the lobe containing boresight.

• Sidelobes — every other lobe. Reported by sidelobe level (SLL) in dB below the main lobe peak.

• Back lobe — the lobe centered at 180° from boresight. The front-to-back ratio (F/B) is main-lobe peak divided by back-lobe peak.

Beamwidth and sidelobe level are the two knobs you’ll trade against each other in every array-tapering problem in Module 3.

Reciprocity — the same antenna, both ways

Everything so far described an antenna transmitting: it takes power from a source and shapes it into a pattern. But the same piece of metal also receives, and you might worry it behaves differently in the two roles. It does not.

Reciprocity principle. For any antenna made of ordinary (linear, passive, isotropic) materials, its receiving behavior is identical to its transmitting behavior. The pattern, the directivity and gain, the input impedance, and the polarization are the same function of angle whether the antenna is launching a wave or catching one. An antenna that radiates best toward the horizon is also most sensitive to signals arriving from the horizon.

This is not an accident of any particular geometry — it follows from the symmetry of Maxwell’s equations themselves (the Lorentz reciprocity theorem). Swap the source and the observation point in a reciprocal medium and the fields trade places unchanged.

Why we care

Two payoffs. First, you can characterize an antenna in whichever mode is convenient — a measurement range almost always puts the antenna under test in receive — and the numbers carry straight over to transmit. Second, it ties the two sides of the antenna together: the transmit quantity gain and the receive quantity effective aperture, which we define next, are not independent. Reciprocity locks them to a single universal ratio.

The one exception. Reciprocity fails when the medium itself is non-reciprocal — a magnetically biased ferrite, or a magnetized plasma like the ionosphere. That is exactly what makes a circulator or isolator work: pass energy one way, block the other. Antennas radiating into air are reciprocal, so we use it freely.

Effective aperture

Flip the antenna around to receive. A passing wave carries power density \(S_{\text{inc}}\) (W/m²); the antenna delivers some power \(P_{\text{rx}}\) to its load. The ratio has units of area and is the effective aperture (you will hear “capture area” in the wild — avoid it; the course term is effective aperture):

Effective aperture — capturing an incident wave

The antenna is a receiving circuit

So how big is that effective aperture — and where does the famous \(\lambda^{2}/4\pi\) come from? We can derive it.

An incident field \(E\) aligned with the antenna induces an open-circuit voltage \(V_{\text{oc}} = E\ell_e\), where \(\ell_e\) is the antenna’s effective length. The antenna then behaves like a Thévenin source with internal radiation resistance \(R_{\text{rad}}\); a conjugate-matched load \(R_L = R_{\text{rad}}\) draws the maximum available power:

Where the 8 comes from

Where the 8 comes from: at match the load sees \(V_{\text{oc}}/2\) across it (that is the factor of 4), and time-averaging a sinusoid adds another 2 — hence 8.

Example: a short dipole

Do it for the simplest antenna — an infinitesimal (Hertzian) dipole with uniform current. For that idealized element the effective length is the physical length, \(\ell_e = \ell\), and the radiation resistance is \(R_{\text{rad}} = 80\pi^{2}(\ell/\lambda)^{2}\).

Divide, and watch the ℓ’s cancel

The incident power density is \(S = E^{2}/2\eta_{0}\) with \(\eta_0 = 120\pi\ \Omega\). Divide, and watch the \(\ell\)’s cancel:

Recognize the number

That 1.5 is exactly the short dipole’s directivity \(D\) (we derive it from the radiation integral in Lesson 6 — borrow it here). So \(A_e = D\lambda^{2}/4\pi\); folding in ohmic losses (\(G = \eta_{\text{rad}}D\)), and assuming a lossless match with the reactance tuned out and the polarization aligned,

It’s universal. We computed only one antenna — but reciprocity guarantees the ratio \(A_e/G\) is the same constant for every antenna, from a short dipole to a giant dish. (Equivalently, a thermodynamic argument — an antenna in a blackbody cavity must absorb and re-radiate equally in each direction — pins down the same constant.) So the boxed relation is universal.

Physical vs. effective aperture

An antenna with a real opening of area \(A_{\text{phys}}\) (a horn, a dish) never uses all of it perfectly — illumination taper, spillover, and feed blockage waste some. The aperture efficiency \(\eta_{\text{ap}}\) (typically 0.5–0.7 for a dish) captures this:

Physical vs. effective aperture — dish, horn, and dipole

Worked example — gain and captured power of a 1.2 m dish

A subtlety worth pinning down

People often say “effective aperture shrinks with frequency,” but that is only true if you hold gain fixed — then \(A_{e} = (\lambda^{2}/4\pi)G\) falls as \(\lambda^{2}\). For a fixed physical dish, the metal fixes \(A_{e} = \eta_{\text{ap}} A_{\text{phys}}\), and it does not change with frequency — to first order: \(\eta_{\text{ap}}\) itself drifts with illumination taper and surface tolerance (the Ruze relation), which is why real dishes stop gaining at the top of their band. Instead the gain climbs as \(f^{2}\), because the same aperture spans many more wavelengths. Both statements are the same universal relation read in opposite directions.

The Friis transmission equation

Effective area is what a receiver catches; gain is what a transmitter concentrates. Put one at each end of a link and you can predict the received power directly.

A transmitter feeds \(P_t\) into an antenna of gain \(G_t\), so at range \(R\) the power density on boresight is the isotropic value \(P_t/4\pi R^{2}\) boosted by \(G_t\):

The Friis transmission equation — combining the two

The receive antenna, with effective aperture \(A_e = G_r \lambda^{2}/4\pi\), collects \(P_r = S_{\text{inc}}A_e\). Combining the two:

— the Friis transmission equation, the backbone of every link budget.

Three things to read off it

• \(P_t G_t\) is the EIRP (effective isotropic radiated power): the whole transmit side collapses to one number — the power an isotropic radiator would need to match this antenna on boresight.

• The \((\lambda / 4\pi R)^{2}\) factor is the free-space path loss. In dB, \(\text{FSPL} = 20 \log_{10}\!\left( \dfrac{4 \pi R}{\lambda} \right)\) — it climbs with both frequency and range, and it is by far the largest term in most links.

• Everything is multiplicative, so in decibels the link budget is just addition: \(P_r[\text{dBm}] = P_t + G_t + G_r - \text{FSPL}\).

Worked example — a 2.4 GHz satellite uplink budget

Summary — wave and power quantities

Summary — directivity, gain, and pattern

Summary — aperture and reciprocity

Summary — the link

Where this is going

The rest of Module 1 takes each term you just defined and makes it a real engineering quantity. Lesson 3 attaches a direction to \(\mathbf{E}\) — polarization — and asks how every parameter here drifts with frequency, which is bandwidth. Lesson 4 walks into the antenna terminals: input impedance, \(S_{11}\), and the matching networks that turn the \(\Gamma\) and VSWR algebra of today into hardware. Lesson 5 asks where the plane-wave picture is even legal, which is the \(2D^2/\lambda\) far-field boundary and the reason antenna ranges are as long as they are. Lesson 6 then goes back and earns the results we borrowed today, deriving the \(1/r\) far field — and the \(D = 1.5\) — from an arbitrary current distribution.

Before next lesson

The link budget is the payoff you will keep reusing. Every array in Module 3 exists to raise \(G_t\) or \(G_r\) in that one equation, and every radar problem later in the course is Friis applied twice. Before the next class, read the assigned sections on polarization and bandwidth, and come ready to explain what “vertical polarization” means in terms of the plane-wave solution we wrote today.