Reciprocity, passivity and RF network measurements

Reciprocity, physical port-exchange symmetry and passivity answer different questions. For a reciprocal two-port, the transfer impedances are equal wherever a Z representation exists. Equal driving-point impedances are an additional symmetry property, and neither identity asserts that power cannot be dissipated.

A real resistance is reciprocal and passive but lossy. A biased nonreciprocal passive component can have unequal transfer terms; passivity is a statement about net absorbed power under the chosen current references, not about the transpose of the parameter matrix.

Write the terminal relation V=ZI. With both currents entering, the cycle-averaged absorbed power for peak phasors is one half the real part of I†ZI. Any matrix can be separated into Hermitian and skew-Hermitian parts; only the Hermitian part contributes to average real power.

Consequently every passive linear two-port whose Z representation exists must have H_Z=(Z+Z†)/2 positive semidefinite for each real frequency. For a 2×2 Hermitian matrix check both diagonal entries and its determinant. Pointwise nonnegative absorption alone is not a full proof of broadband causal positive-real behavior.

An admittance representation I=YV gives the equivalent identity P_abs=(1/2)Re(V†YV). To see positivity without trusting a black-box matrix test, build a reciprocal π network with G₁=0.010 S to ground at the input, G₂=0.005 S at the output, and G_b=0.020 S between the two nodes.

Kirchhoff gives Y11=G₁+G_b=0.030 S, Y22=G₂+G_b=0.025 S and Y12=Y21=−0.020 S. For peak voltages V1=1 V, V2=0.4 V, currents are I1=0.022 A and I2=−0.010 A. The absorbed average power is 9 mW, exactly the sum of the three resistor losses.

A useful counterexample to the belief that nonreciprocal means active is the ideal gyrator. For one allowed orientation, V1=R_g I2 and V2=−R_g I1, producing an antisymmetric real Z matrix. Its off-diagonal entries have opposite signs, so this device is not reciprocal.

Nevertheless Z†=−Z, its Hermitian part is zero and the time-average real absorbed power vanishes for arbitrary current phasors. A lossless gyrator can exchange power between ports while conserving the net balance. Orientation changes the signs without changing that conclusion.

Scattering measurements describe waves traveling toward and away from a selected port reference plane. For the moment take the same real positive reference impedance Z₀ at both ports, and define incident a=(V+Z₀I)/(2√Z₀), outgoing b=(V−Z₀I)/(2√Z₀), with I entering the DUT.

These are peak-phalor-derived wave amplitudes, not the RMS power waves used in some VNA textbooks. Their net port power is P_abs=(a†a−b†b)/2, not a†a−b†b. The factor one half is a consequence of the phasor-amplitude convention and must be preserved in numerical exercises.

For a linear DUT define b=Sa. Eliminate V and I from the wave definitions to obtain S=(Z−Z₀I)(Z+Z₀I)⁻¹, when the inverse exists and both ports share the same real Z₀. Equivalently S=(I−Z₀Y)(I+Z₀Y)⁻¹ if the admittance form exists.

The peak-wave power balance becomes P_abs=(a†[I−S†S]a)/2. Passivity under every incident wave requires I−S†S positive semidefinite. For reciprocal devices with properly power-normalized real references S12=S21, but a lossy reciprocal device need not have a unitary S matrix.

At one port an incident power P_inc produces reflected power |S11|²P_inc. Return loss is −20log10|S11| in dB and should not be confused with insertion loss −20log10|S21|. For |S11|=0.3 and P_inc=1 mW, reflection is 0.09 mW and the accepted power is 0.91 mW.

For two measured ports with a2=0, outgoing powers sum to (|S11|²+|S21|²)P_inc; a passive device has a sum no larger than the incident power. The difference may be dissipated or radiated through channels excluded from the two-port. A 6 dB matched attenuator is not lossless merely because it is reciprocal.

A VNA measures reflection and transmission at its calibrated reference planes. Cables, adapters, launches and fixture parasitics can lie between these planes and the actual DUT. A calibration procedure corrects specified systematic measurement errors, whereas de-embedding explicitly removes a modeled or characterized fixture.

Keep the frequency vector, port numbering, reference impedances, phase convention and Touchstone units together. An S matrix that slightly violates a passivity inequality near measurement noise does not by itself prove gain. State uncertainty, bandwidth and fixture assumptions before interpreting the result as a material or device property.

1. Explain why a reciprocal passive network may dissipate power and why port symmetry is a separate property.
2. Derive the Hermitian Z power criterion and give all 2×2 positive-semidefinite conditions.
3. Using G₁=0.010 S, G₂=0.005 S and G_b=0.020 S, derive the π-network Y matrix.
4. With V1=1 V and V2=0.4 V peak, calculate terminal currents and the independent branch-loss power.
5. Show that a gyrator with Z12=+R_g and Z21=−R_g is nonreciprocal but lossless.
6. From the peak wave definitions with real Z₀, derive the factor 1/2 in the absorbed-power identity.
7. Derive S from Z for equal real Z₀ and state the invertibility assumption.
8. Given |S11|=0.3 and incident power 1 mW, compute reflected power and return loss.
9. A two-port has S11=0, |S21|=0.5 with a2=0. Explain whether it can be reciprocal and compute the fraction dissipated in a passive matched device.
10. Describe which properties must be specified before interpreting measured Touchstone S data as intrinsic DUT behavior.

The following RF course will develop practical S-parameter measurements, VNA calibration and uncertainty analysis.