Conjugate Match, 50 Ω Match and Power Transfer
Conjugate Match, 50 Ω Match and Power Transfer
A match is meaningful only when its two networks, reference plane, viewing direction, wave definition and power quantity are stated.
A load can be the complex conjugate of the source-side impedance at one junction while differing from the 50 Ω reference used by a coaxial line or VNA. Conversely, a one-port can measure exactly 50 + j0 Ω without representing the optimum large-signal load of a transmitter’s active device. These statements answer different engineering questions.
Four Quantities to Name First
| Quantity | Definition | Question it answers |
|---|---|---|
| Source impedance, ZS | The impedance looking back into a specified linear source equivalent at a named plane | Which conjugate load maximises power absorbed from that source model? |
| Load impedance, ZL | The impedance looking into the load at the same plane | What voltage-current relationship does the load present? |
| Characteristic impedance, Z0 | The forward-wave voltage-to-current relationship of a uniform transmission-line mode | What termination suppresses reflection on that physical line? |
| Reference impedance, Zref | The impedance used to normalise an S-parameter port, commonly a real 50 Ω | Relative to what impedance are the measured or simulated waves defined? |
In a nominal 50 Ω coaxial system, Z0 and Zref are often both treated as real 50 Ω. They remain different concepts. A real line has frequency-dependent loss and may have a complex characteristic impedance, while VNA data can be renormalised to another declared reference impedance.
Minimum complete statement: “At the tuner input connector, looking toward the tuner, the impedance is 50 + j0 Ω at 14.2 MHz, referred to the VNA’s calibrated 50 Ω plane.”
The Complex Conjugate Condition
For a linear Thevenin source with positive resistance:
ZS = RS + jXS
ZL = ZS* = RS − jXS
The second equation is the conjugate-match condition at that junction. It cancels the total reactance and makes the load resistance equal to the source resistance. For a fixed linear source equivalent, it maximises average power absorbed by the load.
If Vth is the RMS open-circuit Thevenin voltage, the source’s available power is:
Pavs = |Vth|² / (4RS)
Available power is a property of the defined source equivalent. It is the power that would be delivered to a conjugate load connected at that plane. It is not automatically the DC input power, saturated transmitter output, antenna accepted power or radiated power.
A real 50 Ω match is one special conjugate match when the effective source-side impedance at the same plane is 50 + j0 Ω. The number 50 by itself does not establish that condition.
The 50 Ω Line-Match Condition
For a uniform line with a real 50 Ω characteristic impedance, the load-plane voltage reflection coefficient is:
ΓL = (ZL − 50) / (ZL + 50)
VSWR = (1 + |ΓL|) / (1 − |ΓL|)
ZL = 50 + j0 Ω gives ΓL = 0, so there is no load reflection on that ideal 50 Ω line. Any other passive load gives a nonzero reflection and standing-wave pattern. This comparison says nothing by itself about the impedance looking back into the complete terminated source-side network.
For real Zref and the corresponding power-wave normalisation, the net power accepted at a passive one-port is:
Paccepted = Pincident − Preflected = Pincident(1 − |Γ|²)
That is a port-wave balance. It is not the complete source-to-load result when a mismatched source, multiple reflections, tuner loss or feedline attenuation is present.
Available, Delivered, Accepted and Radiated Power
| Power term | Reference boundary | What can reduce it |
|---|---|---|
| Available source power | Defined source port with a hypothetical conjugate load | It is the source-model reference, not an after-loss result |
| Delivered load power | Actual load in the complete connected system | Source/load mismatch and intervening network loss |
| Antenna accepted power | Net power crossing the antenna feedpoint | Reflection at that plane |
| Radiated power | Power leaving as radiation | Antenna conductor, dielectric, ground and common-mode losses |
Transducer power gain is GT = Pdelivered/Pavs. It includes the effect of source and load terminations as well as the intervening network. Antenna radiation efficiency is a different ratio, Pradiated/Paccepted. A low input SWR establishes neither ratio.
What S11 Measures
A VNA reports incident and reflected waves at its calibrated reference plane. For a one-port, S11 is the complex ratio b1/a1 under the instrument’s declared normalisation. With a conventional real 50 Ω reference, it maps to impedance through:
Z = 50(1 + S11) / (1 − S11)
S11 = 0 therefore means a 50 Ω reference-impedance match at the calibrated plane. It does not measure a transmitter’s large-signal Thevenin impedance and does not prove a remote antenna is 50 Ω unless every intervening cable and fixture has been calibrated out or accurately de-embedded.
For complex characteristic or reference impedances, travelling-wave, pseudo-wave and Kurokawa power-wave definitions are not interchangeable. NIST’s waveguide-circuit work shows why the wave definition and reference impedance must accompany the data. The familiar Smith-chart formula is safest when its real reference impedance and wave convention are stated.
A Lossless Quarter-Wave Example
Consider a lossless quarter-wave line with real Z0 = 50 Ω and:
ZL = 100 + j50 Ω
Zin = Z0² / ZL = 20 − j10 Ω
The load is not matched to the line. Its load-plane |Γ|² is 0.20 and its VSWR is about 2.62:1. Those numbers describe the 50 Ω line mismatch.
Now terminate the source side with ZS = 20 + j10 Ω. The transformed line-and-load input Zin = 20 − j10 Ω is its conjugate, so the source plane is conjugately matched. Looking back through the same ideal line from the load plane gives:
Zback = 50² / (20 + j10) = 100 − j50 Ω = ZL*
The two networks are also conjugately matched at the load plane, even though the load is not 50 Ω and a standing wave exists on the line. This is a property of the stated linear, lossless reciprocal transformation. Loss, non-reciprocal elements, control loops or nonlinear operation remove the automatic equivalence.
Do not interpret |Γ|² = 0.20 as “20% of transmitter power is lost.” It is the reflected fraction of one incident wave at that 50 Ω load plane. Net source-to-load delivery depends on the source reflection, repeated interactions and all network losses.
Tuners Create Different Reference Planes
A typical station tuner is adjusted so that its input presents a load near 50 + j0 Ω to the radio’s external port. The impedance at the tuner output is generally not 50 Ω; it is whatever value allows the tuner to transform the connected feed system to the input target.
| Plane | Useful question | What a 50 Ω result means |
|---|---|---|
| Transmitter output / tuner input | Does the radio see its intended external load? | The tuner-plus-feed-system input is matched to the nominal radio interface |
| Tuner output / feedline input | What impedance does the line and antenna present to the tuner? | Only a special case; the tuner output does not need to be 50 Ω |
| Antenna feedpoint | What is the installed antenna-system impedance? | ΓL is zero only if the antenna equals the line’s reference impedance there |
A shack tuner can give the transmitter a low SWR while high SWR remains on the line beyond the tuner. A remote tuner placed at the antenna end can instead terminate the long coax near its Z0, leaving the tuner’s short output connection to carry the transformed voltage and current. Neither layout is universally superior; line loss, length, SWR, voltage, current, common mode, weatherproofing and maintenance decide.
A Transmitter Port Is an Operating Interface
The optimum load of a driven RF power transistor is normally established by nonlinear load-pull, bias, compression, harmonic terminations, efficiency, linearity and device-stress limits. It is not obtained by measuring the transistor’s small-signal output impedance and attaching its conjugate.
The transmitter’s output network transforms the device’s required load to the specified external interface, commonly nominal 50 Ω. Control and protection may reduce power when the external load leaves the allowed region. For station design, follow the transmitter manufacturer’s load and SWR limits; do not infer them from the maximum-power-transfer theorem.
Choose the Match That Solves the Actual Problem
- For minimum load reflection on a uniform coax: terminate that line in its characteristic impedance at the load plane.
- For maximum power from a fixed linear source equivalent: present its complex-conjugate load at the same plane.
- For a transmitter: present the manufacturer’s permitted external load over the required power, modulation and frequency range.
- For a long mismatched line: calculate matched-line attenuation plus mismatch/re-reflection effects and check peak voltage and current.
- For antenna performance: measure or bound matching-network, feedline, conductor, dielectric, ground and common-mode losses separately from SWR.
A Measurement and Reporting Workflow
- Draw the system boundary. Mark transmitter, tuner input and output, line, choke, transformer and antenna feedpoint.
- Name the plane and direction. State where the port is and which network is being viewed.
- State Z0 and Zref. Do not use “50 Ω system” as a substitute for both.
- Calibrate or de-embed. Move the VNA plane to the connector of interest or characterise every intervening fixture.
- Record complex data. Save resistance, reactance and complex S11, not only minimum SWR.
- Identify the power quantity. Available, incident, reflected, accepted, delivered and radiated power are not synonyms.
- Include loss. Measure or model the tuner, line, transformer, choke and antenna dissipation at the operating state.
- Check transmitter behaviour. Record power foldback, protection, distortion and device limits under the tested load.
- Report uncertainty. Include calibration, connector repeatability, cable movement, drift and power-measurement mismatch uncertainty.
Engineering conclusion: a 50 Ω match sets the reflection condition relative to a line or measurement reference. A conjugate match sets a source-load complex-impedance relationship for a defined linear power-transfer problem. They coincide only under a stated special case, and neither one alone establishes feed-system efficiency or radiated performance.
Primary engineering references
- K. Kurokawa — Power Waves and the Scattering Matrix
- NIST — A General Waveguide Circuit Theory
- NIST — Travelling-wave and power-wave definitions with complex impedances
- Keysight — Reflection, impedance, VSWR and S11 measurements
- Keysight — RF power transfer, generator/load mismatch and measurement uncertainty
- Keysight — Large-signal load-pull and power-amplifier operating tradeoffs
- ARRL — Smith Chart and Another Look at Reflections resources
Mini-FAQ
- Is a 50 Ω load always a conjugate match? No. It is conjugately matched only when the effective source-side impedance at the same plane is 50 + j0 Ω.
- Does high SWR prove there is no conjugate match? No. It proves a mismatch to the line reference. An ideal lossless reciprocal network can still produce a conjugate source-load relationship across the larger system.
- What does S11 = 0 mean? The one-port is matched to the VNA’s declared reference impedance at its calibrated plane. It does not identify a transmitter’s large-signal optimum load.
- Does a tuner make the antenna 50 Ω? It can make the combined tuner-plus-feed-system input appear as 50 Ω while the tuner output, feedline and antenna feedpoint have different impedances.
- What is available power? It is the maximum power a defined linear source equivalent can deliver to a conjugate load at the stated source plane.
- Is accepted antenna power the same as radiated power? No. Conductor, dielectric, ground and common-mode losses can dissipate part of the net power accepted at the feedpoint.
- Does |Γ|² equal transmitter power lost? Not generally. It is a port-wave reflected fraction under a stated normalisation; complete delivery also depends on source mismatch, repeated reflections and network loss.
- Where should a tuner be placed? Choose from line loss and stress, SWR, common mode, accessibility, weatherproofing and maintenance. A shack tuner and remote tuner solve different reference-plane problems.