Two-Port Impedance Model

• Two-port impedance model is defined as a network representation relating two port voltages and currents via a 2x2 matrix that captures driving-point and mutual characteristics.
• Modal and equivalent-circuit parameterizations characterize asymmetry, modal mixing, and reciprocity using eigenstate decomposition and positive-real synthesis techniques.
• The model’s applications span microwave circuits, grid-tied converters, and time-varying devices while addressing practical concerns in extraction, calibration, and statistical analysis.

Searching arXiv for recent and relevant papers on two-port impedance modeling and closely related formulations. A two-port impedance model represents a device, subsystem, or interaction by relating two port voltages and two port currents, conventionally as

$\begin{bmatrix}V_1\V_2\end{bmatrix} = \begin{bmatrix}Z_{11}&Z_{12}\Z_{21}&Z_{22}\end{bmatrix} \begin{bmatrix}I_1\I_2\end{bmatrix}.$

In this form, $Z_{11}$ and $Z_{22}$ are driving-point impedances, while $Z_{12}$ and $Z_{21}$ are transfer or mutual impedances. In recent literature, however, the topic spans more than fixed frequency-domain $Z$-matrices: it includes hybrid immittance descriptions, measured operating-point-specific models, modal equivalent circuits, and energetic port-Hamiltonian formulations when a stationary impedance matrix is not mathematically faithful (Hernández-Escobar et al., 2020, Arbustini et al., 29 Sep 2025, Jeltsema, 2022).

1. Canonical meaning of a two-port impedance description

The standard reciprocal two-port is written either in impedance form,

$$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$

or in admittance form,

$$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$

Reciprocity is expressed by $Z_{12}=Z_{21}$ or $Y_{12}=Y_{21}$, while non-symmetry means $Z_{11}$0 or $Z_{11}$1. This distinction is central in microwave and antenna structures, especially reciprocal but non-symmetric networks that require three complex degrees of freedom for complete characterization (Hernández-Escobar et al., 2020).

In wave-oriented formulations, the same two-port may be represented by scattering parameters. For measured microwave cavities and related systems, the impedance–scattering relation is

$Z_{11}$2

with $Z_{11}$3 the diagonal matrix of channel characteristic impedances; for equal real reference impedance $Z_{11}$4, the inverse relation can be written as

$Z_{11}$5

Accordingly, a two-port impedance model is often one coordinate choice inside a larger immittance/scattering framework rather than an isolated formalism (Yeh et al., 2013, Prod'homme et al., 2024).

2. Modal and equivalent-circuit parameterizations

A general equivalent circuit for lossy reciprocal non-symmetric two-ports can be derived by eigenstate decomposition. In that construction, the complete model is specified by three complex parameters,

$Z_{11}$6

where $Z_{11}$7 are modal immittances and $Z_{11}$8 encodes eigenvector shape, asymmetry, and modal mixing. The corresponding admittance decomposition is

$Z_{11}$9

with each term a rank-one modal contribution. When $Z_{22}$0, the formulation reduces to the usual even/odd lattice interpretation; when $Z_{22}$1 is real, the eigenstates are orthogonal; when $Z_{22}$2 is complex, the modes are non-orthogonal. A key property is that for passive reciprocal networks the real parts of the modal immittances are non-negative, avoiding negative-real-part branch artifacts that can arise in simpler $Z_{22}$3 or $Z_{22}$4 fits (Hernández-Escobar et al., 2020).

An allied synthesis viewpoint treats the passive linear subsystem as a black-box multiport impedance $Z_{22}$5, fits it as a positive-real rational matrix, and then realizes it exactly by a finite lumped circuit. In the two-port case,

$Z_{22}$6

and the off-diagonal terms are realized by multiport transformer structure, especially Belevitch transformers. The positive-real property is decisive: a finite passive lumped circuit exists exactly realizing $Z_{22}$7 only if $Z_{22}$8 is positive real (Solgun et al., 2015).

3. Energetic completion beyond ordinary $Z_{22}$9-parameters

A notable limitation of the conventional two-port impedance picture appears in explicitly time-varying devices. For a time-varying capacitor, the classical constitutive law

$Z_{12}$0

is mathematically valid, but incomplete as a physical one-port model. Using the storage function

$Z_{12}$1

its derivative is

$Z_{12}$2

The extra term has no associated port in the one-port description, so the missing power must be exchanged with the mechanism that changes the capacitance (Jeltsema, 2022).

The complete formulation promotes $Z_{12}$3 to a state and introduces a second, mechanical-like modulation port: $Z_{12}$4

$Z_{12}$5

The two ports are then $Z_{12}$6 and $Z_{12}$7, with power $Z_{12}$8 and $Z_{12}$9, and the balance law becomes

$Z_{21}$0

In this completed representation, the device is a nonlinear, time-invariant, lossless two-port state-space system.

This model is not a conventional LTI impedance matrix. The paper is explicit that no ordinary frequency-domain impedance matrix exists globally, because the relation depends multiplicatively on the time-varying state/input $Z_{21}$1 or $Z_{21}$2. The closest faithful description is port-Hamiltonian: $Z_{21}$3 Accordingly, a local incremental linearization around an operating trajectory may yield a linear time-varying small-signal two-port matrix, but not a fixed impedance matrix (Jeltsema, 2022).

4. Measurement, extraction, and calibration of two-port models

A frequency-domain two-port impedance representation is used for a converse magnetoelectric magnetometer with two electrical interfaces: a resonator/excitation side and a pickup-coil side. Its measured model is

$Z_{21}$4

where $Z_{21}$5 is the composite cantilever impedance, $Z_{21}$6 the coil impedance, $Z_{21}$7 the forward transfer impedance, and $Z_{21}$8 the reverse transfer impedance. The parameters are obtained by measuring S-parameters with a Bode 100 Vector Network Analyzer under fixed AC excitation amplitude of $Z_{21}$9, $Z$0 resolution, averaging factor $Z$1, and $Z$2 measurement bandwidth, and then converting them to Z-parameters in MATLAB. Under open-circuit output conditions,

$Z$3

while under short-circuit conditions,

$Z$4

The model is explicitly an LTI approximation around a specific operating condition and does not account for dependence of impedance on excitation amplitude and external magnetic field (Arbustini et al., 29 Sep 2025).

A complementary extraction route starts directly from Maxwell’s equations in a finite-element setting. For an $Z$5-branch system, one excites one branch at a time with current $Z$6, computes branch voltages from terminal-averaged compensated scalar-potential differences, and assembles the impedance matrix column by column via

$Z$7

For two ports, this yields $Z$8 from excitation of port 1 and $Z$9 from excitation of port 2. The method supports arbitrary conductor geometry and inhomogeneous permittivities and permeabilities, and includes a mandatory low-frequency stabilization scheme (Stysch et al., 2020).

Metrological and bench-measurement variants often use terminal-pair or modal formulations rather than a direct $$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$0 $$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$1-matrix. In a two terminal-pair digital impedance bridge, each standard is modeled as a two-port $$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$2 network with transadmittance $$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$3, high-to-shield stray admittance $$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$4, and low-to-shield stray admittance $$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$5, and the corrected ratio model is

$$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$6

$$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$7

In the twin-wires method with a two-port VNA, the measured transmission includes modal contamination,

$$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$8

so hybrid isolation and splitter magnitude balance enter directly into the extracted impedance error budget (Callegaro et al., 2014, Liang-Sheng et al., 2014).

5. Statistical and interconnected two-port models

In complicated wave-scattering systems, the two-port impedance matrix is treated statistically. The random coupling model uses the normalized impedance

$$[Z]=\begin{bmatrix} z_{11} & z_{12} \ z_{12} & z_{22} \end{bmatrix},$$9

where $$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$0 contains nonuniversal average response due to coupling and short trajectories. For a reciprocal two-port, the impedance variance ratio

$$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$1

is predicted to be a universal function of the loss parameter

$$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$2

whereas the corresponding scattering variance ratio is generally nonuniversal unless $$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$3 (Yeh et al., 2013).

A geometrically refined average impedance for a realistic two-port chaotic cavity is

$$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$4

where $$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$5 is the radiation impedance and the matrix $$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$6 sums short ray trajectories connecting port $$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$7 to port $$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$8: $$[Y]=\begin{bmatrix} y_{11} & y_{12} \ y_{12} & y_{22} \end{bmatrix}.$$9 In the two-port case this includes not only $Z_{12}=Z_{21}$0 and $Z_{12}=Z_{21}$1, but also the off-diagonal inter-port terms $Z_{12}=Z_{21}$2 and $Z_{12}=Z_{21}$3, including the direct trajectory between the two ports without wall bounces (Yeh et al., 2010).

For assemblies of interconnected subnetworks, a closed-form equivalent scattering matrix is

$Z_{12}=Z_{21}$4

and the equivalent two-port impedance matrix can then be recovered, for equal real reference impedance $Z_{12}=Z_{21}$5, by

$Z_{12}=Z_{21}$6

The same work argues that scattering parameters have a fundamental advantage for connection schemes that require some interconnections to be treated as delayless, lossless, reflectionless, and reciprocal two-port scattering systems (Prod'homme et al., 2024).

6. Applications, limits, and recurrent misunderstandings

In grid-tied voltage-source converters, the fundamental small-signal model is inherently coupled. In the $Z_{12}=Z_{21}$7 frame,

$Z_{12}=Z_{21}$8

and, after transformation, the sequence-domain model is likewise a $Z_{12}=Z_{21}$9 impedance matrix. The paper shows that an accurate SISO sequence model derived by closed-loop equivalence gives identical stability conclusions to the full MIMO model, whereas a reduced SISO model based on the strong-grid assumption may lead to wrong results if the PLL bandwidth is large (Zhang et al., 2017).

In physical human–robot interaction, two-port analysis is used in a hybrid form rather than a pure $Y_{12}=Y_{21}$0-matrix form. For series damped elastic actuation under velocity-sourced impedance control, the controlled actuator is modeled as a two-port hybrid immittance between the human side and the virtual-environment side. The paper provides necessary and sufficient conditions for two-port passivity and proves the necessity of a dissipative element parallel to the series elastic component and the necessity of a virtual coupler with dissipation for absolute stability and two-port passivity. In a different nonreciprocal setting, a three-electrode Hall device becomes a two-port when one electrode is chosen as common ground; the resulting two-port admittance and scattering description predicts anti-reciprocity at

$Y_{12}=Y_{21}$1

and self-matching in the low-$Y_{12}=Y_{21}$2 regime for simple rational values of $Y_{12}=Y_{21}$3, especially $Y_{12}=Y_{21}$4 (Mengilli et al., 2020, Bosco et al., 2016).

Several recurrent misunderstandings are directly addressed in the literature. One is that a constitutive equation may be mathematically correct but physically incomplete: for the time-varying capacitor, $Y_{12}=Y_{21}$5 is valid, but not a complete one-port model. Another is that any two-port can be assigned a fixed global $Y_{12}=Y_{21}$6-matrix: the cME magnetometer model is explicitly an operating-point-specific LTI approximation, and the time-varying capacitor admits at most local small-signal linearization around a trajectory. A third is that every port-based quantum or nanoscale device already has a derived two-port matrix model: the low-frequency double-quantum-dot paper provides a one-port admittance seen from the gate electrode,

$Y_{12}=Y_{21}$7

and states that a genuine two-port extension would require additional terminal definitions and electrostatic coupling relations beyond the derived model (Arbustini et al., 29 Sep 2025, Jeltsema, 2022, Esterli et al., 2018).