Why Back-to-Back EFHW Measurements Keep Fooling People

Connect two high-ratio EFHW transformers nose-to-nose, put a 50 Ω VNA port at each outer end and the ideal turns ratios cancel. That is a useful experiment. It measures the complete cascade under one small-signal fixture condition—not one transformer uniquely, not a transformer on an arbitrary antenna load and certainly not radiated EFHW efficiency.

Draw the Actual Fixture Before Reading S21

A common back-to-back arrangement is:

The drawing is incomplete until it also shows the low-side connection, connector shells, coax shields, instrument chassis, enclosure, any primary shunt capacitors, the physical high-impedance link and nearby conductive surfaces. The two VNA ports often share a chassis reference through their cables and instrument. At kilohm-order impedance, a few picofarads of fixture capacitance and a short piece of exposed wire can materially change the result.

Record these boundaries with the trace:

• the exact transformer topology and which winding or autotransformer terminals form each port;
• the VNA reference impedance and calibration planes;
• the connection used at the high-impedance junction;
• all compensation capacitors, chokes, shields and enclosures;
• orientation and spacing of both units and cables;
• frequency span, sweep power, IF bandwidth, averaging and trace uncertainty; and
• temperature before and after the measurement.

Change any of those and you have measured a different two-port.

S21 Is a Cascade Result

A full two-port measurement produces four complex quantities: S11, S21, S12 and S22. With the unused port terminated in its reference impedance, S11 and S22 describe reflections and S21 and S12 describe forward and reverse transmission. A single S21 magnitude plot discards phase, reverse transmission and both port matches.

For two cascaded networks A and B whose connected ports use the same reference impedance, the forward transmission contains an internal-reflection term:

The denominator is why the two individual S21 values cannot generally be multiplied while ignoring match. Waves can reflect repeatedly at the high-impedance junction. Depending on phase, that interaction can raise or lower the cascade transmission at a particular frequency.

Keysight’s S-parameter design guidance defines the four two-port terms and the distinction between reciprocal and lossless networks. Its VNA de-embedding note uses signal-flow and scattering-transfer matrices for exactly this reason: cascades must be treated as complete complex networks.

Reciprocal Does Not Mean Symmetric

A passive transformer made from ordinary reciprocal materials will normally satisfy S21 = S12 when measured with consistent reference definitions. That does not require S11 = S22. Reciprocity describes transmission in opposite directions; symmetry requires the two ports to look alike.

A high-ratio transformer is intentionally unlike at its ports. One side is designed around a lower impedance and the other around a higher impedance. Reversing the second unit can cancel the ideal turns ratio, but it does not cancel winding capacitance, leakage inductance, magnetizing admittance, connection geometry or port mismatch.

Measure all four S-parameters in both physical orientations. Agreement between S21 and S12 is a useful reciprocity check. It is not permission to replace S11 with S22 or to assume the two finished units are identical.

When Dividing the Paired Loss by Two Is Defensible

For a passive matched cascade, insertion loss is commonly written:

Even mirror symmetry plus one measured 2× cascade does not uniquely determine both fixture halves without additional constraints. Keysight’s automatic-fixture-removal guidance uses time-domain information and structural assumptions to resolve that ambiguity. If those assumptions are unavailable, report the pair result as a pair result.

Calibration Stops Where Its Standards Stop

A full two-port SOLT or TRL calibration can remove systematic VNA and cable errors up to its defined planes. It does not automatically remove everything connected after those planes. If the calibration ends at two coax connectors, the transformer launches, leads, enclosure and high-impedance bridge remain in the measurement.

Port extension corrects delay; it is not a general substitute for fixture de-embedding. A defensible fixture treatment uses characterized S-parameters, suitable calibration standards, a validated electromagnetic model or a method such as TRL that moves the reference planes to the device terminals.

NIST’s fixture de-embedding work compares general fixture extraction with open-short-load and thru-reflect-line methods. IEEE 370-2020 likewise requires fixture design, data-quality checks and consistency for high-frequency interconnect characterization. The frequencies differ from many EFHW tests, but the metrology principle does not: unremoved fixture behavior belongs to the result.

The High-Impedance Junction Is Not a Perfect Load

A nominal 49:1 transformation maps 50 Ω to 2450 Ω only in the ideal ratio calculation. The high-side port of a real transformer has complex magnetizing admittance, leakage inductance, winding capacitance, conductor loss and core loss. Joining two high sides creates a new network whose impedance can be very different from a 2450 Ω resistor.

That makes a back-to-back test neither automatically gentle nor automatically representative. At milliwatt VNA power the absolute voltage and flux are low, but the high-impedance link can still emphasize stray capacitance and self-resonance. At operating power, voltage, flux, winding current and temperature depend on the exact complex load and topology.

A resistor is useful for a declared comparison because it is repeatable. It is not a universal EFHW substitute. Test a matrix of complex loads spanning the modeled or measured antenna-terminal impedance, and keep the fixture’s voltage, current and dissipation within rated limits.

An EFHW Feed Is a Two-Terminal Port

An EFHW may have one conspicuous long radiator, but its feed is still electrically a two-terminal port: voltage is defined between two terminals and current leaves through one and returns through the other. The second terminal may connect to a dedicated counterpoise, transformer enclosure, coax shield or another deliberate conductor.

In an installed antenna, displacement current to the environment and current on the coax exterior, mast, equipment or bonding can extend the return path. Those conductors can create additional common-mode behavior and change the radiating system. They do not make the input “not two-terminal”; they make the boundary and modal description more important.

A coaxial VNA reports the modes presented at its calibrated ports. It does not automatically observe current that leaves through the fixture exterior or converts into radiation. Use a characterized clamp-current probe, a shielded fixture, mixed-mode measurements where applicable and orientation/separation checks to reveal those paths. Do not assume a visually symmetric bench arrangement has zero common-mode excitation.

Reactive Current Is Conditional, Not the Villain

Reactive voltage and current exchange stored energy with inductance and capacitance. They do not themselves consume average real power, but they can increase RMS winding current, flux or peak voltage, which then increases conductor, dielectric or core loss. How much depends on the network and load.

A back-to-back fixture does not universally eliminate reactive current. Its internal match and parasitics may reduce it at one frequency and increase it near another. Measure or calculate branch voltage, current and dissipated power from a validated equivalent circuit rather than assigning all transformer heating to a generic “reactive current” mechanism.

The Fair-Rite broadband-transformer note separates magnetizing impedance, leakage, winding capacitance and loss in the transformer equivalent circuit. The TDK ferrite data book makes core loss a function of material, frequency, flux density and temperature and adds winding skin/proximity loss to the thermal result.

The Shunt Capacitor Belongs to the Measured Network

A capacitor across the low-impedance side can compensate transformer reactance and alter S11, S21, internal current and loss. It may improve or degrade power transfer depending on frequency and load. It is not merely cosmetic, and it cannot be credited with improved efficiency from SWR alone.

If both back-to-back units include capacitors, the cascade contains two compensation networks. Record their values, tolerances and positions, then measure configurations with and without them while keeping the same calibrated planes. A resonance that improves S21 may represent useful compensation, internal re-reflection, reduced mismatch, increased dissipation or a combination. Full complex data and a power balance are needed to separate them.

Small-Signal S-Parameters Have a Clear Boundary

A VNA sweep is highly valuable when the network remains linear and time-invariant at the tested power and temperature. It can reveal bandwidth, mismatch, resonances, reciprocity, unit-to-unit spread and small-signal cascade loss.

It cannot by itself certify operating-power loss, thermal equilibrium, insulation margin or ferrite linearity. Core loss and permeability move with frequency, flux density and temperature; winding resistance changes with temperature and skin/proximity effect. High-power evaluation therefore needs an appropriately rated, enclosed fixture, calibrated forward/reflected and delivered-power planes, remote temperature measurement, declared duty cycle and an uncertainty budget. A low-power trace and a high-power thermal test answer complementary questions.

Component Insertion Loss Is Not Antenna Radiation Efficiency

With normalized power waves and port 2 matched to its reference impedance:

That component efficiency belongs to the measured network, load and planes. It does not include an antenna unless an antenna is actually inside the declared boundary.

Antenna radiation efficiency is radiated power divided by net power accepted at the antenna port. Total efficiency also incorporates mismatch. NIST’s antenna-efficiency research measures radiated power and uncertainty to establish those quantities. A back-to-back transformer pair has no installed radiator or ground environment, so its S21 cannot establish EFHW radiation efficiency, realized gain, pattern or coax contribution.

For the installed system, keep a power ledger:

• incident and reflected power at the declared feed-line plane;
• feed-line loss under the measured mismatch;
• matching-assembly accepted and delivered power;
• radiator, ground, nearby-object and exterior-path loss;
• total radiated power and pattern; and
• measurement or model uncertainty at every transition.

A Measurement Ladder That Builds Evidence

The strongest conclusion is often modest and useful: “This pair measured X dB with these four S-parameters, these calibration planes and this fixture at this power and temperature.” From there, every broader claim needs another measured link.

Primary Technical References

• Keysight: S-Parameter Techniques for Network Design
• Keysight: De-Embedding and Embedding S-Parameter Networks
• NIST: De-Embedding and Unterminating Microwave Fixtures
• IEEE 370-2020: Electrical Characterization of Fixtures and Interconnects
• Fair-Rite: Use of Ferrites in Broadband Transformers
• TDK: Ferrite Core Loss, Flux, Temperature and Winding Loss
• NIST: Radiation and Total Efficiency of Antennas

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