RLC Equivalent Circuit Model

• RLC equivalent circuit models are reduced-order representations that map physical resonances to lumped resistor, inductor, and capacitor elements, capturing critical observables like resonance frequency and damping.
• The models are derived directly from physical principles, compressing complex electromagnetic or quantum interactions into interpretable circuit parameters via systematic reduction techniques.
• Applications span on-chip circuit QED, graphene plasmon absorbers, reflective metasurfaces, and accelerator RF structures, providing practical insights into quality factors, group delay, and loss mechanisms.

Searching arXiv for the supplied topic and cited papers to ground the article in current arXiv records. I’ll look up the listed arXiv records and closely related RLC-equivalent-circuit papers. An RLC equivalent circuit model is a representation in which the response of a resonant physical system is mapped onto lumped resistors, inductors, capacitors, and, when required by geometry, coupling capacitors, mutual inductors, transmission-line sections, or composite RC/RL subnetworks. In the cited literature, the same modeling principle is used for on-chip circuit QED, graphene plasmonic absorbers, reflective metasurfaces, slow-light analogues, resonant-tunneling diodes, accelerator RF structures, and finite-frequency quantum electronic circuits. Across these domains, the model serves two related purposes: it compresses a higher-dimensional field, Hamiltonian, or transport problem into a low-order network, and it preserves the observables of primary interest, such as resonance frequencies, linewidths, reflection, transmission, absorption, decoherence signatures, and frequency-dependent admittance (Mátyás et al., 2011, Barzegar-Parizi et al., 2019, Peri et al., 2023, Nousiou et al., 9 Apr 2025, Cromières et al., 2017, Ourednik et al., 2024, Pelz et al., 8 Jul 2025).

1. Formal definition and canonical relations

At its most basic level, the RLC equivalent circuit model identifies a resonant mode with an inductive-capacitive pair and represents dissipation by a resistor. The common resonance condition is

$$\omega_0 = 2\pi f_0,\qquad \omega_0^2 = \frac{1}{LC},\qquad C=\frac{1}{\omega_0^2 L}.$$

For a series resonator, the quality factor is

$Q_0=\frac{\omega_0 L}{R_s},$

whereas for a parallel resonator it is

$Q_0=\frac{R_p}{\omega_0 L}.$

The bandwidth relation is

$\Delta f = \frac{f_0}{Q_0}.$

In accelerator-RF notation, the cavity shunt impedance is also written

$$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$

These identities provide the minimal algebraic bridge between measured or simulated RF quantities and circuit elements (Pelz et al., 8 Jul 2025).

When the model is used as a one-port seen by an incident wave, the central object is the input admittance or input impedance. For the lumped-element-loaded metasurface,

$$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$

and the reflection coefficient is

$$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$

For the graphene absorber, the corresponding total admittance is

$$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$

with perfect absorption when $Y_{\rm in}=\eta_0^{-1}$ and real. These formulations show that the RLC abstraction is not restricted to isolated resonators; it also functions as an interface model between structured surfaces and free-space or transmission-line environments (Nousiou et al., 9 Apr 2025, Barzegar-Parizi et al., 2019).

2. Derivation pathways from physics to circuit parameters

A defining feature of the modern RLC equivalent circuit literature is that the circuit is often derived rather than postulated. In on-chip quantum electrodynamics, the circuit is obtained from a general interaction Hamiltonian. The resonator parameters are mapped as

$$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$

while the qubit branch in the dispersive or small-signal limit $Q_0=\frac{\omega_0 L}{R_s},$0 obeys

$Q_0=\frac{\omega_0 L}{R_s},$1

and the coupling capacitance is

$Q_0=\frac{\omega_0 L}{R_s},$2

Here the circuit element values are direct encodings of resonator frequency, qubit frequency, linewidth, dephasing time, and vacuum Rabi coupling (Mátyás et al., 2011).

In the unified linear-response theory of quantum electronic circuits, the starting point is the susceptibility $Q_0=\frac{\omega_0 L}{R_s},$3. The gate admittance is written

$Q_0=\frac{\omega_0 L}{R_s},$4

with

$Q_0=\frac{\omega_0 L}{R_s},$5

For a closed $Q_0=\frac{\omega_0 L}{R_s},$6-level system, Foster’s theorem yields a set of parallel lossless LC resonators, one per transition. For an open system with relaxation and dephasing, the admittance decomposes into

$Q_0=\frac{\omega_0 L}{R_s},$7

where $Q_0=\frac{\omega_0 L}{R_s},$8 is a quantum RLC branch with linewidth $Q_0=\frac{\omega_0 L}{R_s},$9, $Q_0=\frac{R_p}{\omega_0 L}.$0 is a series RC “Sisyphus” branch, and $Q_0=\frac{R_p}{\omega_0 L}.$1 is a composite “Hermes” branch that becomes important when decoherence is comparable to or faster than the transition frequency (Peri et al., 2023).

A third route appears in the resonant-tunneling-diode literature, where the circuit follows from charge continuity, terminal-current relations, and a linearization around a DC operating point. The resulting large-signal admittance is

$Q_0=\frac{R_p}{\omega_0 L}.$2

and the last term is reinterpreted as the admittance of a series $Q_0=\frac{R_p}{\omega_0 L}.$3 branch with

$Q_0=\frac{R_p}{\omega_0 L}.$4

This gives a universal RLRC topology whose configuration remains fixed while the parameters vary with bias and drive amplitude (Ourednik et al., 2024).

3. Network topologies and architectural variants

The phrase “RLC equivalent circuit model” does not denote a single circuit shape. The supplied literature exhibits a family of architectures chosen to preserve the dominant coupling, symmetry, and boundary conditions of the original problem.

Domain	Equivalent topology	Source
On-chip circuit QED	Resonator RLC branch coupled by $Q_0=\frac{R_p}{\omega_0 L}.$5 to a qubit RLC branch with decoherence resistors	(Mátyás et al., 2011)
Graphene absorber	Surface admittance as an infinite parallel array of series RLC branches, above a grounded dielectric TL section	(Barzegar-Parizi et al., 2019)
Reflective metasurface	Shunt surface admittance $Q_0=\frac{R_p}{\omega_0 L}.$6 in parallel with a grounded substrate TL section	(Nousiou et al., 9 Apr 2025)
Slow-light analogue	Ladder of cells with series $Q_0=\frac{R_p}{\omega_0 L}.$7 and two parallel series-RLC branches per cell	(Cromières et al., 2017)
RTD dynamics	Parallel capacitor, DC conductance, and series $Q_0=\frac{R_p}{\omega_0 L}.$8 branch	(Ourednik et al., 2024)
Accelerator RF structure	TL blocks, resonant series or parallel RLC branches, and capacitive or inductive inter-block coupling	(Pelz et al., 8 Jul 2025)
Multi-level quantum circuit	Per-transition H, $Q_0=\frac{R_p}{\omega_0 L}.$9, and L branches in parallel	(Peri et al., 2023)

Two structural patterns recur. The first is modal decomposition. In the graphene absorber, the patterned array is represented by

$\Delta f = \frac{f_0}{Q_0}.$0

so that each series RLC branch corresponds to a graphene plasmon polariton mode. In practice, the infinite sum is truncated to a two-pole model when only the first two modes lie in the frequency window of interest (Barzegar-Parizi et al., 2019).

The second is hierarchical composition. In the metasurface work, a 1×1 cell is extended to 2×1 and 2×2 cells, enabling dual-polarization and azimuth-dependent reflection. In the accelerator-RF methodology, one “cuts” a 3D electromagnetic structure into natural sub-structures, models each sub-structure by a sub-circuit, and then reconstructs the full response by cascade or parallel combination. This suggests that the RLC model often functions as a reduced-order network language rather than a single isolated resonator (Nousiou et al., 9 Apr 2025, Pelz et al., 8 Jul 2025).

4. Dissipation, decoherence, and non-ideal effects

The resistor in an equivalent circuit is not a uniform concept across applications. It can represent conductor loss, radiation loss, linewidth, dephasing, dynamic dissipation, or effective conductance. In the on-chip circuit-QED model, resonator loss $\Delta f = \frac{f_0}{Q_0}.$1 is represented by the resistor $\Delta f = \frac{f_0}{Q_0}.$2, while qubit decoherence $\Delta f = \frac{f_0}{Q_0}.$3 is represented by a parallel resistor $\Delta f = \frac{f_0}{Q_0}.$4 across $\Delta f = \frac{f_0}{Q_0}.$5 and, equivalently, a series resistor $\Delta f = \frac{f_0}{Q_0}.$6 with $\Delta f = \frac{f_0}{Q_0}.$7. The same summary explicitly states that qubit relaxation $\Delta f = \frac{f_0}{Q_0}.$8 is neglected in the purely linear model but could be added as a further branch (Mátyás et al., 2011).

In the slow-light ladder, the single-branch linewidth is

$\Delta f = \frac{f_0}{Q_0}.$9

and the two slightly detuned resonators produce two absorption peaks with a transparency window between them when $$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$0. Here resistance controls both absorptive depth and dispersive steepness, and thus directly affects the group delay obtainable from a finite cascade (Cromières et al., 2017).

In the unified quantum-circuit theory, dissipation is split explicitly into several mechanisms. The total admittance is decomposed into a reactive part and a dissipative part through $$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$1 and $$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$2, but the open-system theory goes further by assigning distinct circuit sub-branches to coherent Hamiltonian response, energy-relaxation-driven Sisyphus processes, and the Hermes correction associated with consistent perturbation of the Lindblad jump operators. This formalism makes non-unitary effects part of the circuit topology itself rather than an after-the-fact linewidth fit (Peri et al., 2023).

The RTD model uses conductances rather than a single loss resistor. The DC large-signal conductance $$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$3 sets the low-frequency branch, while the difference $$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$4 is encoded by the series $$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$5 path. The paper emphasizes that the topology is universal, but the parameter set $$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$6 depends on operating point and drive amplitude (Ourednik et al., 2024).

A physically complementary viewpoint appears in accelerator RF design: strong electric-field regions suggest capacitive elements, strong magnetic-field regions suggest inductive elements, and loss is represented by a resistor in series with $$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$7 for a series-branch resonator or in parallel for a parallel-branch resonator. This field-based interpretation anchors the circuit parameters in stored electric and magnetic energy rather than purely numerical fitting (Pelz et al., 8 Jul 2025).

5. Phenomena captured across application domains

The range of phenomena reproduced by RLC equivalent circuits is wider than the simplicity of the networks might suggest. In on-chip circuit QED, the model reproduces the vacuum-Rabi anticrossing in resonator transmission, with splitting $$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$8 at zero detuning, and the dispersive Lamb shift

$$R_s \equiv \frac{V^2}{P_{\rm loss}} = \frac{Q_0}{\omega_0 C} = Q_0 \omega_0 L.$$9

The coupling rate itself appears directly as

$$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$0

Within the stated dispersive and linear regime, the circuit retains the principal observables of cavity–qubit interaction (Mátyás et al., 2011).

In graphene absorbers, each resonance band is associated with a selected graphene plasmon polariton. For a given branch,

$$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$1

and under the Drude-dominated conductivity assumption the closed-form elements are

$$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$2

The design strategy is to excite selectively the first two modes, impose phase-matching and amplitude-matching at the target frequencies, and thereby realize dual-band perfect absorption (Barzegar-Parizi et al., 2019).

In the ladder analogue of slow light, the single-cell transfer function is

$$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$3

and for $$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$4 cascaded cells,

$$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$5

The group delay becomes

$$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$6

so the delay scales linearly with the number of cells. The transparency window is therefore not merely a spectral curiosity; it is the region of maximal $$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$7 and hence of delayed pulse propagation (Cromières et al., 2017).

In quantum electronic circuits, the unified linear-response theory organizes operation into adiabatic and resonant regimes, and into coherent and incoherent regimes. The two-level charge-qubit example shows that at $$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$8 the response reduces to a quantum-plus-tunnelling capacitance with Sisyphus resistance, whereas at $$Y_{\rm in}(\omega)=Y_{\rm surf}(\omega)+Y_{\rm sub}(\omega),\qquad Z_{\rm in}(\omega)=\frac{1}{Y_{\rm in}(\omega)},$$9 one obtains Rabi-peak splitting and a conductance peak at $$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$0. The Majorana-qubit example further shows that parity-dependent tunnel splittings generate parity-dependent admittances, enabling readout through $$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$1 (Peri et al., 2023).

For reflective metasurfaces, the circuit model is generalized beyond normal incidence. Angle-dependent transmission-line impedances, lumped-element corrections $$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$2, and azimuthal mixing formulas for $$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$3 allow the circuit to track dual-polarization behavior under arbitrary obliquity and azimuthal rotation, up to the stated accuracy limits (Nousiou et al., 9 Apr 2025).

6. Validation, accuracy, and regime of validity

The strength of an equivalent-circuit model is determined by how closely it reproduces experiment or full-wave calculation within its intended regime. In the circuit-QED example based on parameters from Fragner et al. and Armat et al., the resonator values are $$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$4 GHz and $$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$5, with $$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$6 MHz, $$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$7 MHz, and $$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$8. The mapped values include

$$\Gamma(\omega)=\frac{Z_{\rm in}(\omega)-Z_0}{Z_{\rm in}(\omega)+Z_0}.$$9

and AC simulation of the full RLC network reproduces both the vacuum-Rabi anticrossing and the qubit Lamb shift with excellent quantitative agreement to the measured data (Mátyás et al., 2011).

In the graphene ribbon-array design for dual bands at $$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$0 THz, the reported parameters are

$$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$1

and the circuit-model absorption agrees with HFSS to better than $$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$2, with $$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$3 for the lower band and $$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$4 for the upper band. For the disk array, a design for $$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$5 THz yields $$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$6 THz with $$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$7–$$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$8 absorption in both bands and good agreement between the circuit model and HFSS (Barzegar-Parizi et al., 2019).

For the lumped-element-loaded metasurface, the reported resonance-frequency error $$Y_{\rm in}(\omega)=Y_{\rm sur}(\omega)-jY_s\cot(\beta_s h),$$9 relative to full-wave simulation is $Y_{\rm in}=\eta_0^{-1}$0 for the 1×1 cell and $Y_{\rm in}=\eta_0^{-1}$1 for the 2×1 cell at $Y_{\rm in}=\eta_0^{-1}$2, $Y_{\rm in}=\eta_0^{-1}$3 and $Y_{\rm in}=\eta_0^{-1}$4 at $Y_{\rm in}=\eta_0^{-1}$5, and $Y_{\rm in}=\eta_0^{-1}$6 and $Y_{\rm in}=\eta_0^{-1}$7 at $Y_{\rm in}=\eta_0^{-1}$8. At $Y_{\rm in}=\eta_0^{-1}$9, the 1×1 error reaches $$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$0 and the 2×1 ECM breaks down. Over $$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$1–$$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$2 GHz, the absorption-magnitude root-mean-square error satisfies

$$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$3

The abstract summarizes this as robustness up to $$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$4 degrees (Nousiou et al., 9 Apr 2025).

In the accelerator-RF example of a coax-fed two-cavity particle gun, the optimized equivalent circuit reproduces $$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$5 to within $$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$6 over the $$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$7 GHz band. The broader methodology states explicitly that such circuits are valid over a limited band around the lowest few modes, and that each lumped $$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$8 or $$C_r=\frac{1}{Z_0\omega_r},\qquad L_r=\frac{Z_0}{\omega_r},\qquad R_r=\frac{1}{\gamma C_r}=\frac{Z_0\omega_r}{\gamma},$$9 should be small compared to $Q_0=\frac{\omega_0 L}{R_s},$00 at $Q_0=\frac{\omega_0 L}{R_s},$01 (Pelz et al., 8 Jul 2025).

The slow-light ladder makes its own assumptions explicit: neglecting reflections and backward currents requires either strong total loss or a matched load. Under the fitted values

$Q_0=\frac{\omega_0 L}{R_s},$02

the measured Bode plot shows two deep absorption lines of about $Q_0=\frac{\omega_0 L}{R_s},$03 dB, a $Q_0=\frac{\omega_0 L}{R_s},$04 dB residual dip in the window, and a phase slope corresponding to $Q_0=\frac{\omega_0 L}{R_s},$05; a $Q_0=\frac{\omega_0 L}{R_s},$06 Gaussian pulse is delayed by about $Q_0=\frac{\omega_0 L}{R_s},$07 and lengthened to about $Q_0=\frac{\omega_0 L}{R_s},$08 (Cromières et al., 2017).

Taken together, these results delimit a general principle: an RLC equivalent circuit model is exact neither by definition nor by aspiration. It is a regime-dependent reduction whose credibility comes from explicit derivation, physically interpretable parameters, and quantitative agreement with the target observable over a specified band, drive level, coherence regime, or angular range.