Physics-Consistent Multiport-Network Model

• The topic is a physics-consistent multiport-network model that represents electromagnetic systems as matrix networks with ports, capturing mutual coupling and reconfigurable load effects.
• It overcomes conventional RIS abstractions by accurately modeling the nonlinear interplay of phase and amplitude, structural scattering, and load interactions.
• The framework supports optimization, calibration, and experimental validation in RIS, BD-RIS, and layered metasurfaces to ensure physically realizable and predictive designs.

A physics-consistent multiport-network model is a network-theoretic representation of an electromagnetic or wireless system in which transmitters, receivers, reconfigurable intelligent surface (RIS) elements, stacked metasurface layers, feeds, and, when needed, environmental scatterers are represented as ports of a common multiport. Reconfigurability enters through tunable load impedances, load reflection coefficients, or tunable multiport terminations, while the observable channel is obtained by terminating and eliminating internal ports. In contemporary RIS and programmable-metasurface research, this modeling class is used as an electromagnetically consistent alternative to heuristic “phase-shift-array” abstractions because it preserves mutual coupling, structural scattering, loading, mismatch, reciprocity, passivity, and the generally nonlinear dependence of the end-to-end response on the tunable loads (Renzo et al., 2024, Nossek et al., 2023).

1. Foundational viewpoint

Multiport network theory (MNT) treats the electromagnetic system as a matrix-valued network rather than as a diagonal array of independent reflectors. In this viewpoint, each physical entity can be represented as a port, and the quantities of interest are impedance, admittance, and scattering matrices. A basic linear multiport relation is written as

$$\mathbf{v}=\mathbf{Z}\mathbf{i},\qquad \mathbf{i}=\mathbf{Y}\mathbf{v},\qquad \mathbf{b}=\mathbf{S}\mathbf{a},$$

with $\mathbf{Y}=\mathbf{Z}^{-1}$ when the inverse exists. The choice between $\mathbf{Z}$, $\mathbf{Y}$, and $\mathbf{S}$ is not merely representational; the literature explicitly treats each as making different physical properties and optimization tasks more transparent (Renzo et al., 2024).

This viewpoint arose in response to a central deficiency of conventional RIS channel abstractions. The standard model assumes that each element can impose an arbitrary phase shift while keeping unit amplitude, but physically consistent analyses show that even with isotropic radiators, perfect matching, no intra-array coupling, no losses, and reactive one-port terminations, changing a reactive load changes both phase and amplitude. In that sense, the multiport model is not an optional refinement layered on top of the conventional abstraction; it is the modeling level at which the actual port physics is retained (Nossek et al., 2023).

2. Mathematical representation and channel reduction

In a standard RIS-parametrized setting with $N_\mathrm{T}$ transmit antennas, $N_\mathrm{R}$ receive antennas, and $N_\mathrm{S}$ RIS elements, the full system is represented as an $N\times N$ network with $N=N_\mathrm{A}+N_\mathrm{S}$ and $\mathbf{Y}=\mathbf{Z}^{-1}$0. The network can be described by a scattering matrix $\mathbf{Y}=\mathbf{Z}^{-1}$1 or an impedance matrix $\mathbf{Y}=\mathbf{Z}^{-1}$2, related by

$\mathbf{Y}=\mathbf{Z}^{-1}$3

RIS elements are modeled as auxiliary ports terminated by tunable loads. Eliminating those ports yields the measurable antenna-port impedance

$\mathbf{Y}=\mathbf{Z}^{-1}$4

followed by the measurable scattering matrix

$\mathbf{Y}=\mathbf{Z}^{-1}$5

and the end-to-end channel block

$\mathbf{Y}=\mathbf{Z}^{-1}$6

For conventional diagonal RIS (D-RIS), the key structural constraint is that the load matrix $\mathbf{Y}=\mathbf{Z}^{-1}$7 is diagonal because each element is terminated independently (Hougne, 2024).

An equivalent scattering-domain representation writes the channel as

$\mathbf{Y}=\mathbf{Z}^{-1}$8

In this form, $\mathbf{Y}=\mathbf{Z}^{-1}$9 is the direct antenna-to-antenna scattering block, $\mathbf{Z}$0 and $\mathbf{Z}$1 describe antenna-to-RIS coupling, and $\mathbf{Z}$2 captures RIS self-interaction and mutual coupling. The inverse term is the locus of multi-bounce interactions between load states and the static environment (Hougne, 2 Jul 2025).

3. Electromagnetic consistency and corrective role

The defining content of “physics-consistent” is that the model respects the actual coupling physics between ports. Off-diagonal impedance entries $\mathbf{Z}$3 represent mutual impedances, and off-diagonal scattering terms represent corresponding interaction channels; hence the natural network representation is generally non-diagonal. A diagonal-only model assumes element independence and is acknowledged in the literature as, at best, an approximation that can fail when elements are closely spaced or strongly interacting (Renzo et al., 2024).

The same corrective role appears in the treatment of phase control. For a single reactive RIS element with load $\mathbf{Z}$4, the normalized transfer factor becomes

$\mathbf{Z}$5

so that

$\mathbf{Z}$6

The amplitude is therefore not preserved except in the trivial case $\mathbf{Z}$7. The one-element and two-element toy examples show that the amplitude-phase interrelation is already present before adding realistic antennas, mismatch, losses, or intra-array coupling (Nossek et al., 2023).

A second correction concerns structural scattering. Multiport S-parameter formulations explicitly show that a realistic RIS reradiates even when the programmable load state corresponds to a matched or “off” condition. This structural scattering produces an unwanted specular reflection and modifies the direct block seen in the end-to-end model. Accordingly, an RIS “OFF” state is not equivalent to zero scattering, and design procedures that ignore this term can mischaracterize both desired and parasitic paths (Abrardo et al., 2023).

A third correction concerns physical feasibility. Reciprocity is associated with symmetry conditions such as $\mathbf{Z}$8 and $\mathbf{Z}$9 under the relevant conventions, while passivity is expressed through conditions such as $\mathbf{Y}$0 or $\mathbf{Y}$1 under passive, matched normalization assumptions. These constraints are not ancillary bookkeeping; they delimit the admissible search space for optimization and exclude channel responses that do not correspond to realizable passive hardware (Renzo et al., 2024).

The literature also emphasizes a subtler point: even if off-diagonal mutual coupling is negligible, nonlinear load dependence need not disappear. In the Neumann-series interpretation of the scattering model, zero off-diagonal coupling alone is not sufficient for the series to collapse if the diagonal of $\mathbf{Y}$2 remains nonzero, because local multiple reflections between the loads and the environment can still persist (Hammami et al., 3 Aug 2025).

4. Architectural classes and generalizations

The D-RIS/BD-RIS distinction is central to recent work. In D-RIS, each auxiliary RIS port is terminated by its own independent load, so $\mathbf{Y}$3 is diagonal. In beyond-diagonal RIS (BD-RIS), the auxiliary RIS ports are terminated by a load circuit that couples those ports to each other. A key result is that the load circuit of a BD-RIS is itself a multiport network with ports $\mathbf{Y}$4 terminated by individually tunable diagonal loads $\mathbf{Y}$5, while the internal load-circuit matrix $\mathbf{Y}$6 can be fully populated. Eliminating $\mathbf{Y}$7 gives an effective non-diagonal termination

$\mathbf{Y}$8

The conceptual consequence is that a BD-RIS-parametrized environment can be reinterpreted as a cascade of the radio environment and the load circuit, followed by independent diagonal terminations. The mapping is summarized as

$\mathbf{Y}$9

which implies that the BD-RIS problem can be handled within the conventional D-RIS formalism once the environment is replaced by the environment-plus-load-circuit cascade (Hougne, 2024).

This reinterpretation has immediate architectural implications. It implies that no BD-RIS-specific optimization formalism is required at the level of the physics-consistent model, and it raises a fairness caveat for comparative studies: at fixed number of RIS elements $\mathbf{S}$0, a BD-RIS often has more tunable lumped elements $\mathbf{S}$1 than a D-RIS, so comparisons at fixed $\mathbf{S}$2 may not compare like hardware complexity with like hardware complexity (Hougne, 2024).

Multiport formulations also support reciprocal and non-reciprocal BD-RIS. In full-duplex settings, reciprocal BD-RIS corresponds to a symmetric RIS scattering matrix, whereas non-reciprocal BD-RIS allows a non-symmetric unitary scattering matrix for a lossless surface. Under the physically consistent channel model, non-reciprocity enables direction-dependent behavior and can relax the restrictive alignment conditions required for reciprocal surfaces to maximize uplink and downlink signal-of-interest power simultaneously (Li et al., 2024).

The same architectural logic extends to stacked intelligent metasurfaces (SIMs). A SIM is modeled as a controllable electromagnetic object embedded in a larger multiport network, with transmitter ports, SIM internal ports, and receiver ports. Its termination is a reconfigurable multiport network $\mathbf{S}$3, giving the end-to-end law

$\mathbf{S}$4

For layered transmissive RIS structures, an alternative but still physics-consistent formulation uses transfer scattering parameters, replacing recursive $\mathbf{S}$5-parameter cascades with a product

$\mathbf{S}$6

together with pseudo-unitary and reciprocity constraints in the $\mathbf{S}$7-domain (Pettanice et al., 15 Mar 2026, Yahya et al., 12 Mar 2025).

5. Estimation, calibration, and optimization

Because the tunable loads are the actual reconfiguration knobs, multiport models are directly suitable for optimization. The literature surveys Neumann-series-based optimization, direct optimization without Neumann approximation, scattering-parameter-based optimization, and gradient-descent-based parameter estimation, including experimentally grounded settings with limited measurements (Renzo et al., 2024).

A recurrent practical difficulty is that the RIS design and the radio environment may be partially or completely unknown. For experimentally relevant multi-bit RISs, a hybrid estimation procedure addresses this by combining closed-form and gradient-descent steps. The procedure first estimates the direct TX–RX block from a reference configuration, then uses single-element state perturbations and the rank-one structure of the corresponding channel differences to estimate RIS-to-antenna and antenna-to-RIS coupling directions via SVD, and finally uses gradient descent on additional random configurations to estimate the remaining scaling factors, the mutual-coupling block, and the load-state vector. The closed-form part uses only $\mathbf{S}$8 measurements, and the resulting parameter ambiguities—scaling, rotation, and translation of the estimated load-state constellation—are treated as operationally irrelevant because the objective is accurate prediction of the configuration-to-channel mapping rather than unique physical identification (Hougne, 2 Jul 2025).

For 1-bit-tunable RIS elements under mutual coupling, comparative optimization studies based on statistical MNT ensembles report that coordinate descent with random initialization yields the most favorable trade-off in terms of performance, execution time and memory usage except when mutual coupling is negligible. The same work introduces a statistical ensemble construction for passive reciprocal scattering matrices, with a single hyper-parameter $\mathbf{S}$9 scaling the off-diagonal entries of $N_\mathrm{T}$0 and therefore tuning the strength of mutual coupling (Hammami et al., 3 Aug 2025).

Computational tractability is an active theme. For arbitrarily complex interconnections of multi-port subsystems, a closed-form supersystem-plus-connection-system formula gives the connected scattering matrix

$N_\mathrm{T}$1

and Woodbury-identity updates permit efficient reevaluation when only a small subsystem changes. The same Woodbury logic is exploited in RIS and SIM discrete optimization to accelerate single-element or single-cell updates without recomputing dense inverses from scratch (Prod'homme et al., 2024).

6. Experimental validation, extensions, and recurring caveats

Physics-consistent multiport-network models have been validated both numerically and experimentally. The overview literature emphasizes agreement with full-wave electromagnetic simulations and measurements and notes that a compact MNT model can sometimes be calibrated with a single full-wave simulation even in rich-scattering environments with metallic walls, dielectric objects, and strong structural scattering (Renzo et al., 2024). SIM work validates predicted model behavior against FEKO plus SPICE co-simulation, using the multiport model as the optimization engine and the full-wave/circuit co-simulation as the physical benchmark (Pettanice et al., 15 Mar 2026).

The strongest experimental evidence comes from parameter-estimation studies. For an eight-element, 8-bit-programmable RIS in an unknown rich-scattering environment, the MC-aware MNT model reached a peak prediction accuracy of $N_\mathrm{T}$2 dB, whereas the MC-unaware cascaded model reached $N_\mathrm{T}$3 dB; yet the optimized end-to-end communication performance of the two models was very similar for the tested KPIs. This establishes an important methodological caveat: substantial gains in forward-model fidelity do not necessarily translate into proportionally large gains in the final communication objective (Hougne, 2 Jul 2025).

For a 19-GHz dynamic metasurface antenna with 7 feeds and 96 elements coupled through a chaotic cavity, an experimentally estimated proxy MNT model predicted the reflected field at the feeds with accuracy $N_\mathrm{T}$4 dB and the radiated field with accuracy $N_\mathrm{T}$5 dB, while an MC-unaware benchmark achieved only $N_\mathrm{T}$6 dB and $N_\mathrm{T}$7 dB, respectively. The same study motivates “auxiliary calibration feeds” because parameter estimation quality depends strongly on the number of accessible feeds, even when normal operation is intended to use only a subset of them (Tapie et al., 27 Dec 2025).

The framework now extends beyond static single-frequency RISs. Time-Floquet RISs are modeled by harmonic-lifted multiport scattering systems in which the static subsystem is block diagonal across harmonics while the time-varying load subsystem produces cross-harmonic coupling. In that setting, segmented estimation followed by cross-harmonic ambiguity alignment is essential; omitting alignment can reduce accuracy by up to about $N_\mathrm{T}$8 dB in the most informative measurement mode (Hougne, 12 May 2026). Closely related multifrequency models for time-modulated scatterers extend structural scattering, mutual coupling, non-digital modulation, and non-periodic configurations to harmonic-domain S-parameter systems (Kuznetsov et al., 17 Jul 2025).

The same modeling principles have also been applied to reconfigurable pinching-antenna systems, where a pinching antenna is represented as a multiport scattering network coupled to a waveguide and free-space channel, and to SIM-aided near-field sensing, where a SIM is optimized to implement a physically realizable subspace projection that preserves near-field wavefront curvature while reducing the number of RF chains (Wang et al., 6 Sep 2025, Abrardo et al., 22 May 2026).

Two recurring caveats cut across the literature. First, the model parameters of proxy multiport descriptions are often non-unique; operational correctness rather than exact physical identifiability is the relevant criterion in calibration studies. Second, hardware-aware comparisons matter: whether one fixes the number of RIS elements, the number of tunable loads, the degree of non-reciprocity, or the number of accessible calibration ports can materially change the meaning of “performance improvement.” These caveats do not weaken the multiport approach; they delineate the conditions under which its physical fidelity should be interpreted.