Learning Objectives

1. I can determine the polarization of an antenna and describe the bandwidth characteristics of common antenna types.

Where we were

Lesson 2 handed you the terms of a link budget — directivity, gain, effective aperture, Friis. Every one of them quietly assumed the two antennas were polarization matched and operating at a single frequency. Today you pay for both assumptions: polarization decides how much of an arriving wave an antenna can see at all, and bandwidth decides over what span of frequencies any of the datasheet numbers still hold. They are also the two most misquoted specs on an antenna datasheet, and the last part of this lesson prices each misquote in dB.

Part 1: Polarization

What polarization is

Fix a point in space and watch the tip of the electric field vector \(\mathbf{E}(t)\) trace a curve over one period. That curve is the polarization.

If the tip moves back and forth along a line, the wave is linearly polarized. If it traces a circle, the wave is circularly polarized. If it traces an ellipse (the general case), the wave is elliptically polarized. Those are the only three shapes on offer:

Building any polarization from two linear components

You can write any plane wave propagating along \(+\hat{z}\) as the sum of two orthogonal linear components:

Three numbers fix the polarization completely

Three numbers fix the polarization completely: the amplitudes \(E_{x}\) and \(E_{y}\), and the relative phase \(\delta\) between them.

Right or left hand?

IEEE convention — point your right thumb along the direction of propagation. If the E-vector rotates in the direction your fingers curl, it’s right-hand polarized. Left-hand is the mirror image.

Axial ratio

For the general elliptical case, the axial ratio is the ratio of the major axis of the ellipse to the minor axis:

The polarization ellipse

Those are the two lengths marked on the ellipse below, along with the tilt of the ellipse in the transverse plane:

A “CP antenna” is really a nearly-CP antenna

In dB: \(\text{AR}_{\text{dB}} = 20 \log_{10}(\text{AR})\).

• \(\text{AR} = 1\) (0 dB) — pure circular polarization.

• \(\text{AR} \to \infty\) (\(\infty\) dB) — pure linear polarization.

• Everything else is elliptical, with \(1 < \text{AR} < \infty\).

A “CP antenna” is really a nearly-CP antenna. Real specs read something like “AR ≤ 3 dB across the operating band.” Read that carefully: it does not mean the antenna is never worse than 3 dB down. It means the power a linear antenna recovers swings over a 3 dB range as you rotate it — between −1.8 and −4.8 dB, rather than sitting at the flat −3 dB an ideal circular wave would give. Part 3 derives those two numbers.

Worked example — axial ratio of a measured wave

Worked example — right-hand elliptical

Polarization loss factor (PLF)

When a wave with polarization \(\hat{\rho}_{\text{w}}\) arrives at an antenna with polarization \(\hat{\rho}_{\text{a}}\), the fraction of incident power the antenna captures is the polarization loss factor:

The unit vectors are complex when either wave is not purely linear — that dot product hides the phase relationship between \(E_{x}\) and \(E_{y}\).

PLF cheat sheet

Five cases you should know cold:

Every row here assumes perfect circular polarization. Datasheets spec real “CP” hardware at AR ≤ 3 dB, which moves several of these numbers substantially; Part 3 prices the difference.

Two linear antennas tilted by angle ψ

For two linear antennas tilted by angle \(\psi\) relative to each other, \(\text{PLF} = \cos^{2}\psi\) — this is why aligning your handheld radio matters. (The tilt gets its own symbol \(\psi\) because \(\theta\) is already spoken for as the polar angle.)

Polarization loss factor against tilt angle

The curve is gentle near alignment and brutal near 90°:

Worked example — pricing two everyday mismatches (GPS and a linear whip)

Worked example — pricing two everyday mismatches (two linear antennas)

Why satellite and GPS links use circular polarization

Faraday rotation in the ionosphere rotates the plane of a linearly polarized wave by an unpredictable angle as it passes through — so a purely linear satellite downlink would fade in and out as ionospheric conditions changed. Circular polarization, by contrast, is invariant under rotation about the propagation axis (a rotated circle is still a circle), so CP dodges the problem entirely. Cost: a fixed 3 dB penalty when either end is only linear.

The same argument applies whenever the receiver’s orientation isn’t fixed — cubesats tumble, handhelds rotate, aircraft bank.

Part 2: Bandwidth

An antenna has multiple bandwidths

An antenna has multiple bandwidths, one for each parameter you care about:

• Impedance bandwidth — range of frequencies over which the antenna is well matched to its feed (usually VSWR ≤ 2, i.e. \(|\Gamma| \le 1/3\), or return loss (RL) ≥ 9.5 dB). This is the most common definition and the one meant by an unqualified “bandwidth.”

• Pattern bandwidth — range where the radiation pattern (gain, beamwidth, sidelobe level) stays within spec.

• Polarization bandwidth — range where axial ratio stays below a threshold, typically 3 dB.

The three don’t have to coincide

The three don’t have to coincide. You can match a patch antenna over a wider band than it produces good CP, so its polarization bandwidth is narrower than its impedance bandwidth. Stack the three bands against frequency and the antenna is only usable where all of them overlap:

Quote the widest bar and you have quoted the impedance bandwidth

Quote the widest bar and you have quoted the impedance bandwidth. Quote the overlap and you have quoted something a system can actually use.

Fractional and ratio bandwidth

Report bandwidth as a fraction of the center frequency, not just Hz:

100 MHz means very different things at 500 MHz and at 50 GHz.

For wideband antennas the more useful figure is the ratio bandwidth (RBW):

Narrowband, broadband, ultra-wideband

Rough categories:

• Narrowband — FBW \(\lesssim 5\%\). Resonant antennas: patches, small loops, dielectric resonator antennas.

• Broadband — FBW 5%–50%. Dipoles, slots, horns, most practical designs.

• Ultra-wideband (UWB) — RBW \(\ge 2{:}1\), which is the same bar as FBW \(\ge 67\%\) with the definitions above. Log-periodic, spiral, Vivaldi/TSA, biconical.

That 2:1 bar is this course’s working definition of “ultra-wideband,” and it is a demanding one. The FCC’s regulatory definition is a separate and much looser test: a UWB emission is one whose fractional bandwidth is at least 0.20 or whose absolute bandwidth is at least 500 MHz. An antenna can satisfy the FCC and still be nowhere near 2:1.

Size limits bandwidth

There is no free lunch. The Chu-Harrington bound ties minimum antenna Q to the smallest sphere that encloses the antenna:

where \(a\) is the sphere radius and \(k = 2\pi/\lambda\). The bandwidth form is the half-power one: \(\text{FBW} \approx 1/Q\) is the fraction of the center frequency over which a single resonance stays within 3 dB. At the tighter bar we usually quote a match to, VSWR \(\le s\), the same \(Q\) buys less band — \(\text{FBW} \approx (s-1)/(Q\sqrt{s})\), or \(\approx 0.71/Q\) at \(s = 2\). Same physics, a factor of about 1.4 apart, and worth knowing which one a datasheet has used.

Minimum Q against electrical size

The knee sits at ka = 1

The knee sits at \(ka = 1\) — an antenna that just fills a sphere of radius \(\lambda/2\pi\), where \(Q \approx 2\). To the left of it the curve turns nearly vertical: that \(1/(ka)^{3}\) term means halving the size octuples the minimum \(Q\). Small antennas (small \(ka\)) inevitably have high \(Q\) and narrow bandwidth.

You can beat the bound — but only by adding loss

You can beat the bound — but only by adding loss. Resistive loading damps the resonance and widens the band, and the wasted power comes straight out of gain. The bound constrains the lossless case; it never promised you could not build a bad antenna with a wide match.

Resonant antennas

An antenna’s bandwidth depends less on its material than on what the current does when it reaches the end of the structure.

Resonant antennas. The current travels out along the structure, reflects off the open end, and travels back. Forward and reflected waves add to a standing wave, and the antenna radiates efficiently only when its length fits that standing-wave pattern — near \(\ell \approx \lambda/2\). Move away in frequency and the pattern no longer fits: the input reactance climbs, the match collapses, and the band ends. One resonance buys one narrow band. Patches, half-wave dipoles, and slots all work this way.

Traveling-wave and self-scaling antennas

Traveling-wave antennas. Here the current is gone before it can come back — radiated away as it travels, or dumped into a resistive termination at the far end. With little or no reflected wave there is no standing wave and no sharp resonance, and with no resonance to fall out of, the useful band is wide. Vivaldi (tapered-slot) antennas, axial-mode helices, and terminated long-wire and rhombic antennas are the usual examples.

Self-scaling antennas. A third route to a wide band: build the structure as a scaled copy of itself, so that at every frequency some section of it is the right electrical size. That section is the active region, and it slides along the antenna as frequency changes. Log-periodics and spirals are the classic cases.

Key point

Module 2 takes these families apart one at a time, starting with the simple resonant antennas in L7.

Bandwidth by Antenna Type

Bandwidth by Antenna Type — the wideband families

Part 3: Reading a datasheet — the two most misquoted specs

This lesson opened with a claim — that polarization and bandwidth are the two most misquoted specs on an antenna datasheet. Here is the evidence. None of what follows is a lie on the vendor’s part. Each is a defensible reading of a real measurement, which is precisely why these survive all the way to a design review.

Misquote 1 — whose right hand?

The handedness convention introduced above is the IEEE one: point your right thumb along the direction of propagation. There is a second convention in wide use.

Same physical wave, opposite name

The two label the same physical wave with opposite handedness. A part sold as “RHCP” under the optics convention is IEEE LHCP. Order its mate on the strength of the label and you have designed in a sense mismatch — which, per the PLF table above, is the one polarization error that takes the whole link rather than a few dB of it.

There is no clever fix, only a procedural one: on any CP datasheet, confirm which way the observer is facing before you commit to the other end of the link.

Misquote 2 — “circularly polarized” at what axial ratio?

The PLF table above is the idealized one — it assumes perfect circular polarization, AR = 0 dB. Real hardware is sold as “circular” at AR ≤ 3 dB, and parts marketed as circular at 6 dB are not hard to find. It is worth knowing exactly what that costs.

The polarization vector of a real CP antenna

Work in the basis of the ellipse’s own major and minor axes. An antenna with axial ratio \(A\) (linear, \(\ge 1\)) has polarization unit vector

PLF as a function of orientation

A linearly polarized wave arriving at angle \(\psi\) from the major axis is \(\hat{\rho}_{\text{w}} = \cos\psi \hat{u}_{\text{maj}} + \sin\psi \hat{u}_{\text{min}}\). Then

This single expression contains three rows of the earlier table. Set \(A = 1\) (perfect CP) and it collapses to \(1/2\) for every \(\psi\) — the fixed −3 dB, independent of orientation. Let \(A \to \infty\) (perfectly linear) and it becomes \(\cos^{2}\psi\), the two-linear-antennas rule.

The captured power swings between two limits

In between, the captured power swings with orientation between

Their ratio is what matters

the powers in the major and minor axes respectively — note that \(p_{\max} + p_{\min} = 1\), as it must be. Their ratio is what matters:

Key point

The same treatment prices the other idealization

The same treatment prices the other idealization. Two antennas of equal axial ratio, opposite sense, major axes aligned, gives \(\hat{\rho}_{\text{b}} = (A \hat{u}_{\text{maj}} - j \hat{u}_{\text{min}})/\sqrt{A^{2}+1}\) and therefore

What 3 dB of axial ratio actually costs

At \(A = 1\) this is zero — the \(-\infty\) dB in the ideal table. At a real 3 dB axial ratio it is −9.6 dB. Collecting the numbers:

The lesson: the “infinite” cross-polarization rejection of opposite-sense CP is worth about 10 dB in practice, and at 6 dB axial ratio it is worth 4.5 dB — essentially no isolation at all. Budget accordingly.

Misquote 3 — “4% bandwidth” of what?

Recall that an antenna has impedance, pattern, and polarization bandwidths, and that they need not coincide. A single quoted bandwidth figure is nearly always the impedance bandwidth — generally the widest of the three, and so the most flattering.

The usable band is the intersection

Two further qualifications usually go unstated:

• Threshold. “Impedance bandwidth” means nothing until you state the VSWR bar. The same antenna is wider at VSWR ≤ 3 than at ≤ 1.5.

• Scan angle. Datasheets typically quote axial ratio at boresight only. It degrades off-axis, so a patch holding AR ≤ 3 dB on boresight may be well outside it at ±45° — exactly where a wide-beam GPS or telemetry antenna has to work.

The usable band is the intersection of all three bandwidths, over the scan angles you actually operate at. A CP patch matched over 4% may hold AR ≤ 3 dB over barely 1% of it, and only near boresight.

How to read the spec properly

Summary — polarization

Summary — the misquotes

Summary — bandwidth

Where this is going

Lesson 4 takes the impedance side of the story apart. Bandwidth was defined here as the span over which the match holds; next lesson you see what sets that match in the first place — input impedance, \(S_{11}\), feed lines, and baluns — and how the return-loss curve defines the impedance bandwidth we have been quoting. Everything narrowband about a resonant antenna shows up there as a reactance sweeping through zero.

Polarization keeps coming back. In Module 3 an array inherits the polarization of its elements, so a linearly polarized element in a circularly polarized array is a mistake you pay for at the system level; in Module 4 radars exploit polarization deliberately, alternating H and V pulses to read a target’s scattering matrix. Before Lesson 4, read the assigned sections on input impedance, feed lines, and baluns, and come ready to say what a “50 Ω antenna” actually means and why a dipole fed by coax needs a balun.