Antenna Parameters That Describe Different Things

Failure: good S11 is reported as a “good antenna”

The VNA traces agree. Why does one node still lose its link?

In our illustrative engineering case, a battery-powered condition-monitoring node near 2.45 GHz must work in free space, on a metal machine, in a plastic enclosure, near a hand, and in two mounting orientations. The gateway may eventually use two antennas. We have no measurements of those products or states yet.

Start with a narrower, reproducible question. Candidate A and Candidate B have the same signed S11 at the same feed reference. A accepts 90% of the incident power. So does B. That agreement says nothing about how much accepted power is dissipated, or whether the gateway lies in a pattern null.

In the first B comparison, the dipole-like shape stays fixed: the 3.979400 dB drop is an efficiency effect. In the second, B keeps its 20% efficiency and changes only to a distorted scalar shape whose null is at +x. Its total radiated power stays 0.18 mW. Neither variant predicts the effect of an actual enclosure or hand.

The first decision is therefore to name the missing metric. We will trace the energy before judging where the antenna sends it.

From guided current to radiated fields

What changes when a guided signal reaches the antenna?

A guided mode confines its field around a transmission structure. At an antenna, spatially distributed, time-varying currents couple that guided excitation to radiating field modes. Contributions from different parts of the structure combine with direction-dependent phase and orientation. They can reinforce in one direction and cancel in another.

For a passive, linear antenna in reciprocal materials, this coupling works in reverse: an incident spatial mode can excite the port. Reciprocity does not require perfect efficiency. It requires compatible frequency, geometry, medium, direction, and polarization conventions.

1. R2 · guided portIncident and reflected traveling waves. Current excites the antenna and its return path.
2. Current distributionConductor and displacement currents, stored fields, material loss. Schematic orientation only; no current solver here.
3. S0 · spatial modesFar-field direction and polarization. Radiated power is distributed over a sphere.

The x, y, z in this shape are components of a unit direction vector, not distances. A spherical pattern plot places a value in each direction; it does not draw the physical boundary of radio coverage. A nearby enclosure can change the current distribution, but this lesson supplies no model of that change.

Informative foundation: Staelin, Chapter 10, §§10.2–10.3, on radiation and antenna circuits. Our q functions below are original disclosed teaching shapes. Next, count the energy crossing each boundary.

Build the port-to-space power ledger

Of the incident milliwatt, which part actually escapes into space?

Freeze the interface: R2 is the antenna-feed plane with real reference impedance Z0 = 50 Ω and a matched source. Incident traveling-wave power is defined here. Upstream R1-to-R2 feed loss is absent in this local fixture. S0 names spatial radiation. R3 remains the detector/decision boundary; it is not a receive antenna port.

The complex reflection coefficient Γ has magnitude and phase. Our signed scalar S11 value gives only the magnitude. Positive return loss has the opposite sign.

At −10 dB signed S11, |Γ| = 0.316227766; the reflected power fraction is its square, 0.1. It is not 31.6%. Subtract reflected power from incident power to get accepted power, then divide accepted power between radiation and dissipation.

These shorthand numerical identities mean Pinc = Pref + Pdiss + Prad and Pacc = Pdiss + Prad, with every term expressed in mW. A power-conservation residual identifies a bookkeeping error before a link-budget error.

Carry the same denominator into the next section. That is what distinguishes the two efficiencies.

Radiation efficiency and total efficiency answer different questions

Does “50% efficient” refer to accepted power or incident power?

Radiation efficiency asks how much of the energy that entered the antenna was radiated. Total efficiency also accounts for energy that never entered because it reflected. Both ratios are useful; neither can replace the other without the mismatch information.

For A, 0.45/0.9 = 50% radiation efficiency, while 0.45/1 = 45% total efficiency. For B’s efficiency-only variant, the same 90% acceptance factor multiplies 20%, giving 18%. Equal mismatch loss does not imply equal material loss.

The practical consequence is to choose a denominator first, then measure the needed numerator. Now hold the radiated total fixed and ask where it goes.

Directivity, gain, and realized gain

How can an antenna have gain without adding energy?

A sphere contains 4π steradians of solid angle. Radiation intensity U is power per solid angle, in W/sr. In the far field, U = r²Srad for outward cycle-average radial power density Srad in W/m². An isotropic radiator distributes its total power equally, with intensity Prad/(4π).

Directivity compares intensity in one direction with that sphere-average intensity. We write D(θ,φ) for the directional value and state explicitly when discussing its peak. Gain instead references accepted antenna power; realized gain references incident power at R2.

For A at +x, D = 1.5, G = 0.5 × 1.5 = 0.75, and Greal = 0.9 × 0.75 = 0.675. Apply 10 log₁₀ to these power ratios: 1.760912591 dBi, −1.249387366 dBi, and −1.706962272 dBi. Negative gain in dBi is perfectly possible: dissipation can outweigh directional concentration.

A’s +x EIRP is 0.675 mW (−1.706962 dBm); its radiated total is 0.45 mW. A directional equivalent and a sphere-integrated total can have different numbers. With the chosen normalization, ∫G dΩ = 4πηrad and ∫Greal dΩ = 4πηtot.

The ledger settles energy conservation. The next task is to keep a pattern’s geometry and normalization attached to the result.

Read patterns, cuts, beams, sidelobes, and nulls

What is missing when a plot is simply labeled “antenna pattern”?

At minimum, the plot needs frequency, specimen/configuration, coordinate frame, angular convention, cut or full-sphere context, plotted quantity, normalization, and polarization basis. Without them, even a carefully drawn curve can answer the wrong engineering question.

θ = 0° is +z, θ = 90° is the equator, and θ = 180° is −z. φ increases from +x toward +y. At either pole, all φ refer to the same physical direction. Elevation above the xy plane would be 90° − θ, so an elevation label cannot silently replace θ.

For a concrete sidelobe reading exercise, take a separate illustrative total-gain cut at 2.450 GHz, φ = 0°, specimen θ = 0°–180°, with a 5 dBi main peak and a −7 dBi lesser peak. Polarization decomposition is unknown. The relative sidelobe level is −12 dB, or 12 dB suppression. Those hypothetical numbers illustrate the subtraction; they are not outputs of q and are not measurement evidence for this node.

With the angular basis secured, polarization becomes a vector relationship rather than an unlabeled antenna property.

Polarization is a vector relationship

Can two antennas point at each other and still reject the wanted wave?

Yes. At a fixed far-field location, polarization describes how the electric-field vector moves in the plane transverse to propagation. A line, a circle, and an ellipse describe different motions of that vector tip. They are not three different scalar power-pattern shapes.

With δ = 0° the components rise and fall together, giving a line. Equal nonzero amplitudes with δ = ±90° give a circle. Ax = 2 V/m, Ay = 1 V/m, δ = +90° gives an ellipse with major/minor amplitude ratio 2: its axial ratio is 20 log₁₀2 = 6.020599913 dB. A circle has axial ratio 1 (0 dB); a line has a zero minor axis and an unbounded axial ratio.

General ellipticity does not require quadrature. Both relative amplitude and phase determine the major/minor axes and tilt. Ax/Ay is the axial ratio only when the chosen axes are the ellipse’s principal axes, as in the quadrature example.

For unit complex polarization vectors expressed in a consistent incoming-wave receive basis, coupling is |eRXᴴ ewave|², where H denotes conjugate transpose. A linear vector (1,0) and a circular vector (1,±j)/√2 give 1/2. Orthogonal circular basis vectors give zero. Receive-vector conventions must account for propagation direction; copying a transmit handedness label into a dot product without that mapping can reverse the answer.

Cross-polarization is the component along the declared orthogonal basis, not a universal “bad field.” A co = eθ / cross = eφ decomposition at one direction does not automatically describe the whole sphere. At a radiation null the vector vanishes and its polarization state is undefined.

Polarization adds a receive coupling factor. To turn field density into available port power, we also need effective aperture.

Effective aperture and reciprocal receive behavior

How much power can this antenna collect from a known incident field?

Effective aperture converts incident plane-wave power density into available receive power. It is an electrical capture area, not necessarily the metal’s physical area. For the same passive linear reciprocal antenna and environment, transmit gain connects to that area under conjugate matching and aligned incident polarization.

Here c = 299792458 m/s exactly and f = 2.450e9 Hz, so λ = 0.12236426857142857 m. Using A’s +x gain G = 0.75 gives Ae = 0.000893635959967125 m², or 8.936359600 cm². This is already reduced by its 50% radiation efficiency.

Accepted transmit power is a net flow into an excited antenna. Available receive power is the maximum that the incoming wave can deliver under conjugate matching. Delivered receive power depends on the connected load. None is EIRP; none is detector power at R3.

A short bridge to receive noise

The receive pattern weights the external noise environment: looking at warm ground and looking at cold sky need not give the same antenna noise temperature. Antenna dissipation also contributes thermal noise; receiver noise must be referred to a compatible plane. Gain alone specifies neither noise temperature nor G/T. Do not apply receiver noise figure to a pattern or infer sensitivity from S11. Revisit Noise; the full conditional link belongs to 06.5.

Everything so far was conditional on one frequency. A bandwidth statement must say which of these conditions it preserves.

An antenna has several bandwidths

Can a 100 MHz S11 span establish a 100 MHz usable antenna band?

Only if match is the entire requirement—and for a real link it usually is not. Impedance bandwidth limits reflection. Efficiency bandwidth limits dissipative conversion. Realized-gain bandwidth constrains a directional, loss-inclusive result. Pattern bandwidth can constrain beamwidth, null location, or coverage. Polarization bandwidth can constrain axial ratio, handedness, or cross-polarization within a stated region.

Define a usable band as the frequencies at which all required characteristics meet their own stated limits in the specified configurations. The intersection depends on the application; there is no universal antenna bandwidth independent of the requirement.

The intervals 2.40–2.45 and 2.45–2.50 GHz are unknown. No interpolation establishes a continuous guaranteed band. Pattern and polarization compliance are also unknown at all three points. This exercise uses a separate synthetic specimen/version; it does not silently change Candidate A’s fixed −10 dB S11 at 2.450 GHz to −15 dB.

In the ledger above, select Evidence required. The interpretation changes while the antenna inputs stay fixed. This is deliberate: a better answer can require new evidence rather than a different number in the same model.

Informative context: Antenna Basics, §3.11, makes bandwidth conditional on the characteristic and criterion. The thresholds in our exercise are hypothetical, not normative limits.

Write a complete antenna requirement for the node

What could a test engineer measure to accept or reject the claim?

Write the quantity, threshold, feed/reference plane, band, direction or coverage region, polarization basis, mechanical/environmental state, and population statistic together. Then name the evidence method, uncertainty, and decision rule. If a field is not applicable—direction for a port match, for example—say so rather than leaving it ambiguous.

Below are three hypothetical requirements for a proposed N1 verification set. They use chosen thresholds; none claims those thresholds have been achieved. The mechanical plate, enclosure, phantom, feed fixture, and mounts need controlled drawings before test execution.

1. Requirement · REQ-MATCH

   A feed requirement

   For five identified node prototypes of build N1, at R2 (real 50 Ω, fixture de-embedded), require signed S11 ≤ −10 dB at every 1 MHz test point from 2.400 to 2.500 GHz. Test each unit in free space, plastic enclosure, on the specified metal-machine plate, and with the specified hand phantom, in mounts O1 (+z upright) and O2 (90° rotation about +y). Direction and polarization are not applicable to this port measurand; retain the specimen x/y/z registration. Report the worst point per unit/state and calibrated VNA method, fixture drawing, and expanded uncertainty (k = 2). Accept only if the upper uncertainty bound meets the limit; compliance between points remains unclaimed.

2. Requirement · REQ-EFF

   A radiation requirement

   For the same five N1 units and all four mechanical states and both O1/O2 mounts, require radiation efficiency ≥ 40% at 2.400, 2.450, and 2.500 GHz. Reference accepted power to R2 (real 50 Ω); integrate both orthogonal transverse polarizations over the full S0 sphere registered to specimen x/y/z. Report each unit/state/frequency and the worst result, using a calibrated full-sphere radiated-power measurement divided by accepted feed power. Include cable-removal/de-embedding treatment, angular sampling and integration error, and expanded uncertainty (k = 2). The lower uncertainty bound must meet 40%; unsampled frequencies remain unknown.

3. Requirement · REQ-COVERAGE

   A coverage requirement

   For the same five N1 units, four mechanical states, and O1/O2 mounts, require co-polar realized gain ≥ −6 dBi over S0 coverage θ = 60°–120°, φ = 0°–360°, at 2.400, 2.450, and 2.500 GHz. Use specimen-fixed θ/φ, co = eθ and cross = eφ in this pole-free region; reference incident feed power to real 50 Ω R2 and include antenna dissipation and mismatch once. Use calibrated dual-polarization chamber gain measurements on a 5° angular grid, report the minimum per unit/state/frequency plus angular interpolation uncertainty and expanded uncertainty (k = 2). Accept the continuous-region claim only when the lower bound, including justified between-point uncertainty, stays above −6 dBi; otherwise report grid-only results and request denser data.

The antenna and channel evidence map

This local snapshot, p06-evidence-map-v1, is the start of the path’s evidence map. It contains every field needed to interpret its rows; no previous visit or saved workbook is assumed. Requirement rows, illustrative calculated results, and missing measurement evidence remain visibly distinct.

For the family-selection handoff, carry the coverage region, match and efficiency requirements, enclosure/ground constraints, orientations, and unresolved evidence. Do not carry a scalar q preset as a validated antenna geometry. The next lesson will ask which physical antenna families can plausibly satisfy this contract.