Two-Port Networks & S-Parameters

01 / 10

Why S21 is not unconditional gain

Four valid files cover the same 2.45 GHz node feed. Why can their traces not be cascaded yet?

The switch, filter, feed, and connector files look compatible because each has four complex traces over a common band. Their contracts disagree: the filter is normalized to 75 Ω, the connector ports are reversed, one record ends at a fixture plane, and the operating conditions are not all declared. Smooth plots do not repair those mismatches.

Switch50 Ω · P1→P2Package planes; bias and power state declared.
Filter75 Ω · P1→P2Renormalization required before a 50 Ω chain review.
Feed50 Ω · P1→P2Board launch included; finite simulated grid.
Connector50 Ω · P2→P1Port reversal is reversible; fixture state is still missing.

Think about itIf |S21| is −1.0 dB at 2.45 GHz, has the network delivered exactly 79.4% of available source power to the load?

Answer

Not from S21 alone. |S21|² is the outgoing Port 2 wave power divided by incident Port 1 wave power when a2 = 0 under the file’s reference conditions. Available source power, input mismatch, output mismatch, and multiple reflections require a transducer-power calculation and the actual terminations.

Common misconceptionS21 is gain, full stop.

S21 is a conditional complex transmission ratio. Calling it gain without the matched-port condition, planes, normalization, and operating state hides the assumptions that make the number true.

Sequence note

Modules 03.2 and 03.3 are now available. This lesson still restates the minimum reflection, structure, direction, and reference-plane context it needs so its data contract can be audited independently.

02 / 10

Why open-short network definitions get awkward

Why replace perfectly good voltage and current parameters with waves?

We do not replace voltage and current. We choose boundary variables that can be created and measured repeatably. Z-parameters define each column with the other port open. Y-parameters use a short. At microwave frequencies, an intended open has capacitance and radiation; an intended short has inductance and a displaced plane. Both become distributed structures.

$V=ZI\qquad I=YV\qquad b=Sa$ Z and Y use port voltage/current boundary conditions. S uses incident/outgoing waves at declared reference planes and impedances.

A matched termination, calibration plane, and directional receiver are usually more practical than a perfect broadband open or short. That practical advantage does not make S-parameters inherently more accurate: calibration, dynamic range, fixtures, drift, noise, and metadata still govern accuracy.

Common misconceptionS-parameters are more accurate than voltage/current parameters.

They are often more measurable at high frequency. Accuracy belongs to the full measurement and model contract, not to the letter used for the parameter family.

03 / 10

Define incident and reflected power waves

What exactly are a and b, and when do their squared magnitudes carry watts?

Pin RMS phasors, exp(+jωt), and current into every physical port. For a positive real reference impedance Z0 at one port, Kurokawa’s power-wave form reduces to the following pair. With this normalization, |a|² and |b|² have power units.

$a = \frac{V + Z_{0} I}{2\sqrt{Z_{0}}} \qquad b = \frac{V - Z_{0} I}{2\sqrt{Z_{0}}}$ $V = \sqrt{Z_{0}}(a + b) \qquad I = \frac{a - b}{\sqrt{Z_{0}}} \qquad P_{\mathrm{net,in}} = |a|^{2} - |b|^{2}$ Equal real references are used throughout the executable lesson model. Each port may have its own positive real Z0 in the general multiport definition.

a1Incident at Port 1Travels toward the DUT from the source-side plane.
b1Outgoing at Port 1Travels away from the DUT toward the source.
a2Incident at Port 2Travels toward the DUT from the load-side plane.
b2Outgoing at Port 2Travels away from the DUT toward the load.

For a complex reference impedance Zref, the power-wave definition uses a = (V + Zref I)/(2√Re{Zref}) and b = (V − Zref* I)/(2√Re{Zref}). That conjugate matters. “Voltage waves” and “power waves” can then differ, so a file must identify its convention.

Go deeperWhy current direction seems to flip at Port 2

Network currents are defined into the physical two-port at both terminals. In the ABCD chain convention used here, the right-side state is [V2, −I2]ᵀ so adjacent networks multiply in left-to-right physical order. The minus sign is bookkeeping for the chosen current arrows, not a new physical law.

04 / 10

Read every element of the S-matrix

Which incident wave is held at zero while each matrix element is defined?

Pinned convention: RMS phasors, exp(+jωt), and current into each physical port. “Incident” means toward the two-port; it does not mean left-to-right at both ports.

$\begin{bmatrix}b_1\\b_2\end{bmatrix}=\begin{bmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{bmatrix}\begin{bmatrix}a_1\\a_2\end{bmatrix}$ $S_{ij}=\frac{b_i}{a_j}\qquad\text{with every other incident wave }a_k=0$ “Zero incident wave” means the other port is terminated in its reference impedance, not physically absent.

Rows are outgoing waves; columns are incident excitations
Element	Definition	Engineering reading	Held condition
S11	b1/a1	Port 1 input reflection	a2 = 0
S21	b2/a1	Forward transmission, Port 1 to Port 2	a2 = 0
S12	b1/a2	Reverse transmission, Port 2 to Port 1	a1 = 0
S22	b2/a2	Port 2 output-side reflection	a1 = 0

Common misconceptionS11 is the device’s reflection coefficient regardless of what is connected to Port 2.

S11 is the Port 1 reflection with a2 = 0. A mismatched connected load sends a nonzero a2 back toward the DUT, so b1 = S11a1 + S12a2 and the observed input reflection can change.

Common misconceptionPort reversal changes only the labels, never the interpretation.

Reversing a two-port swaps S11↔S22 and S21↔S12. The excitation direction, matched-port condition, and physical plane attached to each matrix row and column all change. A reversible transform is still an engineering action that belongs in the conditioning log.

05 / 10

Magnitude, phase, and group delay

What does a smooth magnitude plot conceal about time and causality-sensitive behavior?

Magnitude reports a ratio; phase reports relative timing at one frequency. Group delay describes the local slope of unwrapped phase. Under exp(+jωt), a pure delay has S21 = e^−jωτ, so the minus sign in the derivative returns a positive τ.

$\begin{aligned}|S_{ij}|_{\mathrm{dB}}&=20\log_{10}|S_{ij}|\\\phi_{ij}&=\operatorname{atan2}(\operatorname{Im}\{S_{ij}\},\operatorname{Re}\{S_{ij}\})\end{aligned}$ $\tau_{g,ij}=-\frac{\mathrm d\phi_{ij,\mathrm{unwrap}}}{\mathrm d\omega}$ Phase must be in radians for the derivative. The model uses central differences inside the fixed grid and one-sided differences at its endpoints.

The factor 20 does not turn a power ratio into an amplitude convention by fiat. With normalized power waves, power is proportional to squared magnitude, so 10log10(|b|²/|a|²) equals 20log10(|b|/|a|).

Checked example · matched delay

A 75 ps matched pad has 75 ps S21 group delay.

Its analytic phase is −2πf(75 ps). Differentiating with respect to ω gives −75 ps; applying τg = −dφ/dω returns +75 ps. The deterministic test verifies every point on the 25 MHz grid.

Near a deep magnitude null, phase is numerically undefined: a tiny Cartesian perturbation can swing angle by many degrees. The bench suppresses group delay at the null and at adjacent finite-difference points instead of displaying a persuasive but meaningless spike.

Common misconceptionUnwrapped phase removes all ambiguity from group delay.

Unwrapping removes chosen 2π jumps; it cannot create reliable phase where the complex magnitude approaches zero, repair inadequate frequency spacing, or justify an extrapolation beyond the measured band.

06 / 10

Test reciprocity, symmetry, passivity, and losslessness

Which matrix test supports each label—and which labels do not imply the others?

Reciprocity: $S_{12} = S_{21}$
Port symmetry: $S_{11} = S_{22}$
Losslessness: $S^{H}S = I$
Passivity: $\sigma _{\mathrm{max}}(S) \le 1$

The executable check reports numeric residuals against a base tolerance of 1×10⁻⁹, scaled by the relevant matrix, Gram-matrix, or singular-value magnitude with a floor of one. For passivity it computes the largest singular value through the two eigenvalues of SᴴS. Tolerance is part of the result, not an invisible “close enough.”

Common misconceptionReciprocal means lossless.

A matched attenuator has S12 = S21 and can be symmetric while converting incident power to heat. Reciprocity constrains direction exchange; losslessness constrains energy conservation.

Common misconceptionIf every individual |Sij| is below one, no matrix passivity check is needed.

Each coefficient can be below one while a coherent combination of incident waves produces too much outgoing power. With equal real power-wave references, passivity is a matrix condition for every incident vector, so test the maximum singular value. Unequal or complex normalizations require the corresponding power metric.

Common misconceptionIf a sampled S-matrix is non-passive, the device is unstable.

An active gain block is intentionally non-passive. Stability is a feedback and termination question requiring additional analysis over relevant frequencies and conditions. A passivity failure can also expose bad de-embedding or inconsistent data; it is not a complete stability verdict.

07 / 10

Reference impedance and reference plane are metadata

If the hardware is unchanged, why do all four S-parameters change after renormalization or a plane shift?

S describes waves relative to chosen coordinates. Renormalization changes the impedance used to split V and I into a and b. A reference-plane shift changes the propagation assigned to the record. Neither operation edits the hardware, and neither authorizes overwriting the native file.

Checked renormalization · preserve native record

A 50 Ω series impedance viewed at 50 Ω and 75 Ω

$\mathrm{ABCD}=\begin{bmatrix}1&50\,\Omega\\0&1\end{bmatrix}$ The physical two-port stays fixed.
$S_{50}=\begin{bmatrix}1/3&2/3\\2/3&1/3\end{bmatrix}$ At 50 Ω, |S11| = 0.3333 and |S21| = 0.6667.
$S_{75}=\begin{bmatrix}1/4&3/4\\3/4&1/4\end{bmatrix}$ At 75 Ω, |S11| = 0.2500 and |S21| = 0.7500.

The bench converts S→ABCD using the native real Zref, then ABCD→S at the analysis Zref. Conversion stops if S21 is too close to zero for this representation. The original 50 Ω record and its metadata remain the authority.

$S_{11}' = S_{11}e^{-j 2\beta l_{1}} \qquad S_{22}' = S_{22}e^{-j 2\beta l_{2}}$ $S_{21}' = S_{21}e^{-j\beta (l_{1}+l_{2})} \qquad S_{12}' = S_{12}e^{-j\beta (l_{1}+l_{2})}$ Positive li adds a matched lossless line between the old and new Port i planes under exp(+jωt). Negative li removes it. The bench pins εeff = 3.4 and applies equal shifts.

Common misconceptionReference impedance is just a label; changing it cannot change S.

Zref defines the wave coordinates, so a renormalized S-matrix must change. The device behavior is invariant only when transformed consistently into a physical representation and back.

Common misconceptionReference-plane motion changes only phase in every real fixture.

The bench deliberately adds a lossless teaching line, so magnitudes stay fixed. A real port extension can add conductor/dielectric loss, dispersion, mismatch, or radiation and therefore change magnitude as well as phase. Moving a calibrated plane through a fixture is a measurement/de-embedding operation deferred to Path 08.4.

08 / 10

A Touchstone file is data plus a contract

Can you interpret one numeric row without auditing the header and the measurement record?

Touchstone 2.1 is the current ratified IBIS specification. Its keywords make port count, frequency count, optional two-port ordering, reference impedances, and the end of a 2.x file explicit. The option line declares frequency unit, network-parameter type, complex format, and a reference value. The specification states that, apart from the literal [Version] value, a 2.1 file is identical to its 2.0 counterpart; Version 1.x files rely more heavily on defaults and filename conventions.

Synthetic header · audit exercise · not measured data

Read this before plotting it

! Illustrative node-filter export; analytic values, not a measurement
[Version] 2.1
# MHz S RI R 75
[Number of Ports] 2
[Number of Frequencies] 3
[Reference] 75 75
[Two-Port Data Order] 12_21
[Network Data]
2300  0.05 0.02   0.64 -0.41   0.63 -0.42   0.04 0.01
2450  0.04 0.03   0.72 -0.45   0.71 -0.46   0.03 0.02
2600  0.06 0.02   0.62 -0.50   0.61 -0.51   0.05 0.02
[End]

Header audit before numeric interpretation
Field	What the file says	Audit result
Version	2.1	Current ratified syntax; [End] present.
Option line	MHz · S · RI · R 75	Frequency in MHz; real/imaginary pairs; 75 Ω normalization.
Port/order	2 ports · 12_21	Each row is S11, S12, S21, S22—not the common 21_12 order.
Grid	3 increasing frequencies	No interpolation policy; too sparse for a defensible group-delay result.
Planes/fixture	Absent from the file/comments	Reject for cascade until supplied externally.
Conditions	Absent	Bias, power, temperature, and terminations are unknown.

In Touchstone, MA means magnitude/angle, DB means 20log10(magnitude)/angle, and RI means real/imaginary; angles are degrees. A missing 1.x option-line field can invoke a default, but an engineering archive should not force a reviewer to rely on memory. Preserve the file and add an external provenance record when the format cannot hold the needed fixture, plane, uncertainty, or operating-condition detail.

Common misconceptionA smooth Touchstone plot proves the metadata and de-embedding are correct.

Plot smoothness can coexist with swapped columns, wrong units, wrong Zref, displaced planes, an included fixture, or a physically impossible transform. Audit syntax and provenance before judging the trace shape.

09 / 10

S-matrices do not multiply to cascade

What state vector passes naturally across the internal connection between two networks?

With current into both physical ports, this lesson pins [V1, I1]ᵀ = [A B; C D][V2, −I2]ᵀ. Adjacent two-port ABCD matrices then multiply in physical left-to-right order. Convert compatible S records to ABCD, multiply, and convert the product back to S at the intended reference impedance.

$T_{\mathrm{total}} = T_{1} T_{2} \ldots T_{\mathrm{n}}$ $S_1\to T_1\qquad S_2\to T_2\qquad T_1T_2\to S_{\mathrm{total}}$ The conversion used by the model requires a non-negligible S21. A deep transmission null is a representation limit to handle explicitly, not a reason to divide by a tiny number.

Checked counterexample · two matched 6 dB pads

Direct multiplication puts the 12 dB result in the wrong matrix cells.

$S=\begin{bmatrix}0&t\\t&0\end{bmatrix},\quad t=10^{-6/20}=0.501187$ One reciprocal matched pad.
$\text{Correct cascade}=\begin{bmatrix}0&t^2\\t^2&0\end{bmatrix}$ S21 = 0.251189 = −12.00 dB; both reflections remain zero.
$\text{Direct }S\cdot S=\begin{bmatrix}t^2&0\\0&t^2\end{bmatrix}$ It falsely reports zero through transmission and −12 dB reflections.

Common misconceptionNetwork matrices are matrices, so ordinary S-matrix multiplication must cascade them.

Matrix multiplication only has physical meaning when the vector being passed between blocks matches the connection variables. S maps all incident waves to all outgoing waves; the internal incident wave depends on the adjacent network. Chain variables encode that connection.

10 / 10

Audit the node feed-network chain

Which records can be accepted, which can be conditioned, and which must stop the cascade?

AcceptSwitch · 50 ΩDirection, planes, bias, power, and grid match the chain contract.
ConditionFilter · 75→50 ΩRenormalize a derived copy and preserve the native 75 Ω record.
AcceptFeed · 50 ΩBoard-launch inclusion and simulated plane endpoints are explicit.
RejectConnector · reversedReversal is solvable, but missing fixture/plane state blocks composition.

Cascade only the records whose contracts can be made compatible without inventing information. Archive the ordered source list, every reversible transform, model version, fixture version, frequency-grid policy, property checks, and exclusions. The result is then reviewable rather than merely reproducible on one engineer’s laptop.

Interactive decision bench

Two-Port Sanity Bench

Inspect one fixed synthetic record, condition only what the metadata permits, then compare a physical cascade with the tempting direct-multiplication mistake.

Accept under the declared contract

The synthetic record already matches the requested direction, reference impedance, and reference planes. The finite-band and fixture limits still apply.

Fixture contract

Evidence: Simulated
Model: two-port-sanity/1.0.0
Native Zref: 50 Ω
Analysis Zref: 50 Ω
Incoming order: normal
Equal shift: 0 mm per port
Cascade: 1 physical section
Direction: Port 1 to Port 2
Grid: 2.300 to 2.600 GHz in 25 MHz steps
Planes: Port planes are the modeled filter package leads
Fixture: Idealized connector launches included in the declared planes
De-embedding: No de-embedding applied; planes are already at the modeled DUT boundaries
Interpolation: None; evaluate only on the fixed fixture grid
Plane-shift model: Matched lossless line; equal port extensions; phase velocity c/sqrt(3.4)
Conditions: Linear small-signal model; 25 °C nominal; output terminated in native Zref; no power sweep

Conditioning log

preservedorientation
Kept the declared Port 1 input and Port 2 output orientation.
preservednormalization
Kept the native 50 ohm normalization.
preservedreference-plane
Kept both native package-boundary reference planes.
cascadedcascade
Composed 1 identical section in physical order through ABCD matrices.

Magnitude only; the dashed or dotted trace styles remain distinguishable without color. The vertical cursor marks the committed frequency.

Committed cursor

2.450 GHz network state

Conditioned S-matrix at 2.450 GHz
	a1	a2
b1	S11`-0.0064 − j0.0697`-23.10 dB ∠ -95.2°	S12`-0.9162 + j0.0837`-0.72 dB ∠ 174.8°
b2	S21`-0.9162 + j0.0837`-0.72 dB ∠ 174.8°	S22`-0.0064 − j0.0697`-23.10 dB ∠ -95.2°

Port 1 incident100.0%

Reflected0.5%

Transmitted to match84.6%

Unaccounted / dissipated14.9%

Power fractions assume a unit incident wave at Port 1 and a matched Port 2 at the committed real reference impedance. A negative remainder signals energy delivery or an incompatible contract, not “negative loss.”

Matrix tests

Properties at the cursor

Claim	Test	Residual	Tolerance	Status
Reciprocity	$\|S_{12}-S_{21}\|$	5.56e-16	1.00e-9	Pass
Port symmetry	$\|S_{11}-S_{22}\|$	1.40e-17	1.00e-9	Pass
Losslessness	$\\|S^{\mathrm H}S-I\\|_{\mathrm F}$	2.10e-1	1.00e-9	Flag
Passivity	$\max(\sigma)\le1$	0	1.00e-9	Pass

Maximum singular value: 0.922659. Passing these sampled checks is evidence within this grid, not a proof outside it.

Checked counterexample

ABCD cascade versus direct S multiplication

Two-port count in this comparison: 2. Frobenius difference at the cursor: 1.702e+0.

Physical-order ABCD cascade of 2, converted back to S
	a1	a2
b1	S11`-0.0222 − j0.1265`-17.83 dB ∠ -100.0°	S12`0.8285 − j0.1519`-1.49 dB ∠ -10.4°
b2	S21`0.8285 − j0.1519`-1.49 dB ∠ -10.4°	S22`-0.0222 − j0.1265`-17.83 dB ∠ -100.0°

Direct S-matrix product — intentionally wrong
	a1	a2
b1	S11`0.8276 − j0.1525`-1.50 dB ∠ -10.4°	S12`0.0233 + j0.1267`-17.80 dB ∠ 79.6°
b2	S21`0.0233 + j0.1267`-17.80 dB ∠ 79.6°	S22`0.8276 − j0.1525`-1.50 dB ∠ -10.4°

Auditable output

Full conditioned frequency table

Each cell reports magnitude in dB, wrapped and unwrapped phase in degrees, and group delay in picoseconds. “Withheld” marks an unsafe result near a deep null.

2.45 GHz reciprocal filter; model two-port-sanity/1.0.0; fixtures two-port-fixtures/1.0.0
Frequency	S11	S12	S21	S22
2.300 GHz	-16.29 dB∠ -65.0°φu -65.0°τg 380.3 ps	-6.57 dB∠ -155.0°φu -155.0°τg 380.3 ps	-6.57 dB∠ -155.0°φu -155.0°τg 380.3 ps	-16.29 dB∠ -65.0°φu -65.0°τg 380.3 ps
2.325 GHz	-17.11 dB∠ -68.4°φu -68.4°τg 406.0 ps	-3.63 dB∠ -158.4°φu -158.4°τg 406.0 ps	-3.63 dB∠ -158.4°φu -158.4°τg 406.0 ps	-17.11 dB∠ -68.4°φu -68.4°τg 406.0 ps
2.350 GHz	-18.02 dB∠ -72.3°φu -72.3°τg 466.7 ps	-1.69 dB∠ -162.3°φu -162.3°τg 466.7 ps	-1.69 dB∠ -162.3°φu -162.3°τg 466.7 ps	-18.02 dB∠ -72.3°φu -72.3°τg 466.7 ps
2.375 GHz	-19.04 dB∠ -76.8°φu -76.8°τg 547.0 ps	-0.91 dB∠ -166.8°φu -166.8°τg 547.0 ps	-0.91 dB∠ -166.8°φu -166.8°τg 547.0 ps	-19.04 dB∠ -76.8°φu -76.8°τg 547.0 ps
2.400 GHz	-20.20 dB∠ -82.1°φu -82.1°τg 641.7 ps	-0.74 dB∠ -172.1°φu -172.1°τg 641.7 ps	-0.74 dB∠ -172.1°φu -172.1°τg 641.7 ps	-20.20 dB∠ -82.1°φu -82.1°τg 641.7 ps
2.425 GHz	-21.53 dB∠ -88.4°φu -88.4°τg 726.5 ps	-0.72 dB∠ -178.4°φu -178.4°τg 726.5 ps	-0.72 dB∠ -178.4°φu -178.4°τg 726.5 ps	-21.53 dB∠ -88.4°φu -88.4°τg 726.5 ps
2.450 GHz (current cursor)	-23.10 dB∠ -95.2°φu -95.2°τg 762.0 ps	-0.72 dB∠ 174.8°φu -185.2°τg 762.0 ps	-0.72 dB∠ 174.8°φu -185.2°τg 762.0 ps	-23.10 dB∠ -95.2°φu -95.2°τg 762.0 ps
2.475 GHz	-21.53 dB∠ -102.1°φu -102.1°τg 726.5 ps	-0.72 dB∠ 167.9°φu -192.1°τg 726.5 ps	-0.72 dB∠ 167.9°φu -192.1°τg 726.5 ps	-21.53 dB∠ -102.1°φu -102.1°τg 726.5 ps
2.500 GHz	-20.20 dB∠ -108.3°φu -108.3°τg 641.7 ps	-0.74 dB∠ 161.7°φu -198.3°τg 641.7 ps	-0.74 dB∠ 161.7°φu -198.3°τg 641.7 ps	-20.20 dB∠ -108.3°φu -108.3°τg 641.7 ps
2.525 GHz	-19.04 dB∠ -113.6°φu -113.6°τg 547.0 ps	-0.91 dB∠ 156.4°φu -203.6°τg 547.0 ps	-0.91 dB∠ 156.4°φu -203.6°τg 547.0 ps	-19.04 dB∠ -113.6°φu -113.6°τg 547.0 ps
2.550 GHz	-18.02 dB∠ -118.1°φu -118.1°τg 466.7 ps	-1.69 dB∠ 151.9°φu -208.1°τg 466.7 ps	-1.69 dB∠ 151.9°φu -208.1°τg 466.7 ps	-18.02 dB∠ -118.1°φu -118.1°τg 466.7 ps
2.575 GHz	-17.11 dB∠ -122.0°φu -122.0°τg 406.0 ps	-3.63 dB∠ 148.0°φu -212.0°τg 406.0 ps	-3.63 dB∠ 148.0°φu -212.0°τg 406.0 ps	-17.11 dB∠ -122.0°φu -122.0°τg 406.0 ps
2.600 GHz	-16.29 dB∠ -125.5°φu -125.5°τg 380.3 ps	-6.57 dB∠ 144.5°φu -215.5°τg 380.3 ps	-6.57 dB∠ 144.5°φu -215.5°τg 380.3 ps	-16.29 dB∠ -125.5°φu -125.5°τg 380.3 ps

Server-rendered fallbackOpen the default 2.450 GHz matrix, checks, and complete table

Default reciprocal filter × 1 at 50 Ω

This independent HTML record remains available without running the client interaction. It uses model two-port-sanity/1.0.0 and fixture set two-port-fixtures/1.0.0. Decision: Accept under the declared contract.

Default synthetic filter metadata
Evidence / model	Simulated · two-port-sanity/1.0.0 · two-port-fixtures/1.0.0
Native contract	50 Ω · Port 1 input, Port 2 output · Port 1 to Port 2
Frequency	2.300 to 2.600 GHz in 25 MHz steps · units GHz · None; evaluate only on the fixed fixture grid
Planes / shift	Port planes are the modeled filter package leads · Matched lossless line; equal port extensions; phase velocity c/sqrt(3.4)
Fixture / de-embedding	Idealized connector launches included in the declared planes · No de-embedding applied; planes are already at the modeled DUT boundaries
Conditions	Linear small-signal model; 25 °C nominal; output terminated in native Zref; no power sweep

Conditioned S-matrix at 2.450 GHz; rows are b1/b2 and columns are a1/a2
Output	a1	a2
b1	`S11 = -0.0064 − j0.0697` -23.10 dB ∠ -95.2°	`S12 = -0.9162 + j0.0837` -0.72 dB ∠ 174.8°
b2	`S21 = -0.9162 + j0.0837` -0.72 dB ∠ 174.8°	`S22 = -0.0064 − j0.0697` -23.10 dB ∠ -95.2°

Reciprocity: Pass · residual 5.56e-16
Port symmetry: Pass · residual 1.40e-17
Losslessness: Flag · residual 2.10e-1
Passivity: Pass · residual 0.00e+0

Default conditioning log

orientation · preserved: Kept the declared Port 1 input and Port 2 output orientation.
normalization · preserved: Kept the native 50 ohm normalization.
reference-plane · preserved: Kept both native package-boundary reference planes.
cascade · cascaded: Composed 1 identical section in physical order through ABCD matrices.

Canonical fixture checks

Independent analytic anchors at 2.450 GHz
Fixture	Reflection	Transmission	Power / delay check
30 mm matched line	S11 = S22 = 0.0	S21 = S12 = -0.21 dB ∠ -162.7°	Uses exp(−γl), l = 30 mm, α = 0.8 Np/m, vp = c/√3.4.
6 dB matched attenuator	S11 = S22 = 0.0	\|S21\| = 0.501187	25.1189% transmitted power · τg 75.0 ps.

Dataset audit checklist

Evidence source, fixture/model version, and original provenance preserved.
Frequency unit, range, monotonic order, sample count, and interpolation policy declared.
Touchstone version, network type, RI/MA/DB representation, and two-port column order verified.
Physical port orientation, propagation direction, native Zref, and analysis Zref reconciled.
Reference planes, fixture inclusion, and de-embedding state named without claiming calibration fidelity.
Linear/small-signal state, bias, power, temperature, and terminations recorded.
Reciprocity, symmetry, passivity, and losslessness tested only over the supplied grid.
Every reversal, renormalization, plane shift, exclusion, and cascade-order decision logged.

Full fixed frequency grid; magnitude and phase
Frequency	S11	S12	S21	S22
2.300 GHz	-16.29 dB ∠ -65.0°	-6.57 dB ∠ -155.0°	-6.57 dB ∠ -155.0°	-16.29 dB ∠ -65.0°
2.325 GHz	-17.11 dB ∠ -68.4°	-3.63 dB ∠ -158.4°	-3.63 dB ∠ -158.4°	-17.11 dB ∠ -68.4°
2.350 GHz	-18.02 dB ∠ -72.3°	-1.69 dB ∠ -162.3°	-1.69 dB ∠ -162.3°	-18.02 dB ∠ -72.3°
2.375 GHz	-19.04 dB ∠ -76.8°	-0.91 dB ∠ -166.8°	-0.91 dB ∠ -166.8°	-19.04 dB ∠ -76.8°
2.400 GHz	-20.20 dB ∠ -82.1°	-0.74 dB ∠ -172.1°	-0.74 dB ∠ -172.1°	-20.20 dB ∠ -82.1°
2.425 GHz	-21.53 dB ∠ -88.4°	-0.72 dB ∠ -178.4°	-0.72 dB ∠ -178.4°	-21.53 dB ∠ -88.4°
2.450 GHz	-23.10 dB ∠ -95.2°	-0.72 dB ∠ 174.8°	-0.72 dB ∠ 174.8°	-23.10 dB ∠ -95.2°
2.475 GHz	-21.53 dB ∠ -102.1°	-0.72 dB ∠ 167.9°	-0.72 dB ∠ 167.9°	-21.53 dB ∠ -102.1°
2.500 GHz	-20.20 dB ∠ -108.3°	-0.74 dB ∠ 161.7°	-0.74 dB ∠ 161.7°	-20.20 dB ∠ -108.3°
2.525 GHz	-19.04 dB ∠ -113.6°	-0.91 dB ∠ 156.4°	-0.91 dB ∠ 156.4°	-19.04 dB ∠ -113.6°
2.550 GHz	-18.02 dB ∠ -118.1°	-1.69 dB ∠ 151.9°	-1.69 dB ∠ 151.9°	-18.02 dB ∠ -118.1°
2.575 GHz	-17.11 dB ∠ -122.0°	-3.63 dB ∠ 148.0°	-3.63 dB ∠ 148.0°	-17.11 dB ∠ -122.0°
2.600 GHz	-16.29 dB ∠ -125.5°	-6.57 dB ∠ 144.5°	-6.57 dB ∠ 144.5°	-16.29 dB ∠ -125.5°

Five-claim synthesis · correct the review meeting

Replace shorthand with conditions and evidence.

01
“S21 is −2 dB, so efficiency is 63%.”
Correct to: |S21|² = 63.1% of the incident Port 1 wave power reaches the outgoing Port 2 wave only for a2 = 0 under the declared wave and reference conditions.
02
“S11 is −10 dB, so insertion loss is 10 dB.”
Correct to: |S11|² = 10% reflected power for the matched-other-port condition. Insertion loss is a transmission statement, not return loss.
03
“The network is reciprocal, therefore lossless.”
Correct to: reciprocity tests S12 = S21. Losslessness requires SᴴS = I; a reciprocal pad can dissipate power.
04
“The plot is smooth, so the file is trustworthy.”
Correct to: smoothness does not verify units, data order, normalization, planes, fixture state, de-embedding, or operating conditions.
05
“Multiply the S-matrices to get the chain.”
Correct to: first condition compatible records, convert to a chain representation, multiply in physical order, then convert back and re-run checks.

Think about itWhat should happen if a model update changes one fixture formula but leaves the plotted curve looking almost identical?

Answer

Increment the fixture version, rerun deterministic tests, and regenerate the archived result. Provenance must expose a model change even when visual inspection cannot.

Ungraded review

Check your understanding

Answer each question in your own words, then reveal the model answer.

01What is the complete meaning of S21 in a two-port S-matrix?
Model answer
S21 = b2/a1 with a2 = 0 under the declared wave definition, port order, reference impedances, reference planes, frequency, and operating conditions. It is forward transmission under a matched Port 2 condition—not unconditional gain or an efficiency by itself.
02For equal, positive real Z0 and RMS phasors, how are incident and reflected power waves defined?
Model answer
With current defined into each physical port, a = (V + Z0 I)/(2√Z0) and b = (V − Z0 I)/(2√Z0). Then V = √Z0(a+b), I = (a−b)/√Z0, and net power entering the port is |a|²−|b|².
03Which equalities test reciprocity and port symmetry, and why are they not losslessness tests?
Model answer
Reciprocity tests S12 = S21. Port symmetry tests S11 = S22 under the same port definitions. Losslessness instead requires SᴴS = I for equal real references. A reciprocal or symmetric attenuator can still dissipate power.
04What metadata must be compatible before two S-parameter records are composed?
Model answer
Frequency grid or an explicit interpolation policy, parameter and complex-data format, port count and order, propagation direction, reference impedance per port, reference planes, fixture/de-embedding state, and operating conditions such as bias, power, temperature, and terminations.
05Why does shifting a reflection reference plane by length l produce twice the one-way phase?
Model answer
The incident wave travels from the new plane to the DUT and the reflected wave returns. Under exp(+jωt), adding a matched line therefore multiplies the reflection coefficient by exp(−j2βl). Transmission uses the sum of the two port-extension distances.
06Why are two S-matrices not multiplied directly to cascade two networks?
Model answer
The internal incident and reflected waves are coupled by the connection between the networks; they are not the independent state vector used by ordinary matrix multiplication. Convert each compatible record to a chain representation such as ABCD, multiply in physical order, then convert back to S.

Sources and further study

Accessed 5 September 2026. Equations, diagrams, synthetic headers, fixtures, and checked examples are original to this lesson and pinned to the stated conventions. No source figure or device measurement is reproduced.

Wave and network theory

K. Kurokawa, “Power Waves and the Scattering Matrix,” IEEE Transactions on Microwave Theory and Techniques, vol. 13, no. 2, 1965, DOI 10.1109/TMTT.1965.1125964. Primary source for power waves and complex reference impedances.
David M. Pozar, Microwave Engineering, 4th ed., Chapters 4–5. Stable textbook treatment of network analysis, S-parameters, and matching.
Keysight Technologies, Using User-Defined Models, Advanced Design System 2011, network-parameter conversion equations. Vendor documentation supporting the ABCD/S convention audit.

File and measurement contracts

IBIS Open Forum, Touchstone File Format Specification, Version 2.1, ratified 26 January 2024. Normative syntax, option-line fields, two-port data order, per-port references, and 2.x keywords.
IEEE, IEEE 370-2020, Electrical Characterization of Printed Circuit Board and Related Interconnects at Frequencies up to 50 GHz, active standard, with errata dated 21 January 2022. Normative context for fixtures, reference planes, and interconnect quality.
Keysight Technologies, Network Analyzer Basics. Measurement-oriented explanation of incident, reflected, and transmitted waves, error correction, and S-parameter display.

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