# Fractional-Order Equivalent Circuit Modeling of Antennas – Part I: Modeling

François Torrès, Marc Thévenot

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François Torrès, Marc Thévenot. Fractional-Order Equivalent Circuit Modeling of Antennas – Part I: Modeling. IEEE Access, 2025, pp.1-1. 10.1109/ACCESS.2025.3640412. [hal-05401855v1](https://hal.science/hal-05401855v1)

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Received 21 November 2025, accepted 1 December 2025. Date of publication 00 xxxx 0000, date of current version 00 xxxx 0000.

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# Fractional-Order Equivalent Circuit Modeling of Antennas—Part I: Modeling

FRANÇOIS TORRÈS AND MARC THÉVENOT

XLIM Research Institute, UMR 7252, CNRS, University of Limoges, 87000 Limoges, France

Corresponding author: François Torrès (francois.torres@xlim.fr)

**ABSTRACT** This article introduces a new approach for devising accurate general equivalent circuit models of the input impedance of antennas. The major advance, compared to existing equivalent circuits models, consists in using fractional-order capacitances and inductances within RLC (resistance, inductance, and capacitance) resonators. The resulting fractional frequential dispersion faithfully models the complex dispersion laws in antennas. The addition of fractional-order mutual coupling to model the coupling between antenna modes further improves the capacity of the model to account for the distributed physical phenomena occurring in antennas. This leads to elegant, compact, powerful and versatile equivalent circuits with unprecedented accuracy for such simple models. Various examples, either based on full-wave simulations of different types of antennas or experimental measurements are provided, outlining the power and polyvalence of the proposed model. In the companion paper of this series (part II), this fractional-order equivalent circuit model will be used to build parametric macromodels of antennas capable of highly accurate parameters interpolation (and limited extrapolation), and allowing frequency extrapolation.

**INDEX TERMS** Antenna model, equivalent circuit, fractional circuit element.

## I. INTRODUCTION

Modeling the input impedance of antennas through lumped-element equivalent circuits is now customary, as circuit models provide an analytical description of the frequency response of an antenna, and allow to compensate for the lack of simple approximate models or empirical design formulas. Besides providing meaningful insights into the physical mechanisms involved in antennas, these equivalent circuits have useful applications, such as determining matching limits [1], [2], or as a surrogate for antenna design or optimization, as circuit calculations are much faster than electromagnetic (EM) full-wave computations [3], [4].

The creation of equivalent circuits for the input impedance of antennas is generally a two-step process: starting from impedance data (measured or computed), an identification phase, which primarily relies on either a physically-based or a network-oriented approach, assesses the circuit topology and gives a rough estimate of the circuit elements values. The

second step generally consists in an optimization of these circuit elements values in order to fit the impedance with minimum error. In essence, the physical approach models each antenna resonance with a resistor/inductor/capacitor (RLC) resonator (series or parallel) [5], [6]. Given that antennas inherently possess multiple resonances, it is possible to devise broadband equivalent circuits by cascading resonators cells [7], [8], [9], [10], and adding extra capacitors and inductors to account for the low and high-frequency behavior of the impedance. The rough values of circuit elements of a narrowband RLC tank model can be estimated from resonance frequencies, bandwidths and/or quality factors, [9], [11], [12], [13]. But they can also be calculated from approximate or empirical formulas relating the physical characteristics of the antennas to circuit elements: relations between circuit elements and physical dimensions or electrical parameters [14], [15], [16], or identification of cavity modes of a patch antenna to circuit resonances [17].

The network-oriented approach stems from pioneering network theory works from Foster [18], Cauer [19], Brune [20], and Schelkunoff [21], [22]. These works assert that the

The associate editor coordinating the review of this manuscript and approving it for publication was Feng Wei.
driving-point impedance of a two-terminal linear network of resistors, inductors, and capacitors, irrespective of their interconnection, can be expressed as a rational function of the complex Laplace variable, eventually broken up into a series expansion of elementary fractions involving individual poles/residues of the rational function. System identification methods, such as vector fitting [23], [24], allow for poles identification, and the equivalent lumped-element circuit can then be synthesized [25], [26].

As Foster and Cauer canonical expansions only deal with lossless structures, they can be discarded for antenna modeling, although an extension of Foster expansion to account for lossy networks is possible [27], [28], but it can lead to stability issues due to negative-valued circuit elements [8], [29]. Equivalent networks for lossy structures can be more efficiently built using more general synthesis methods, such as Brune's [30], [31], [32], Miyata's [33], [34], or Bott-Duffin's [35], [36], [37]. In addition, a specific network synthesis technique dedicated to antennas has been proposed by Baum [38], [39], based on a particular series expansion of his, the Singularity Expansion Method (SEM) [40]. But this synthesis involves complex network elements, whose interpretation and realizability are not immediate.

It is clear from the aforementioned references and other extensive literature that these equivalent circuit modeling approaches are the most noteworthy and give excellent results. However, the number of studies also clearly outlines that this modeling strategy still has room for improvement.

For physically-based approaches, one way of improving accuracy and performances of the models is to modify the circuit topology. This can be achieved with additional inductors and capacitors to series resonant cells [12], resistor-damped cells [11], Drude-Lorentz circuits [41], high-pass cells [42], [43], or the addition of transmission lines [44], [45] or transformers [46]. For network-based approaches, passivity and realizability conditions have been proposed [47], [48], and the efficiency of synthesis can be enhanced through the utilization of contemporary optimization algorithms [3], [36], or machine-learning techniques [49], [50]. In addition, novel approaches have been put forth for the synthesis of equivalent circuits. These approaches include the integration of SEM with generalized transmission lines theory [51], the derivation of equivalent circuits models from eigenmodes decomposition [52] or characteristic modes [42], [53].

However, impedances of EM structures involving distributed phenomena, such as antennas, are generally transcendental functions, thereby evading the classical network theory and rational functions expansions. Nevertheless, this theoretical challenge is not new to network theorists: sixty years ago, Carlin [54] pointed out that network synthesis of such impedances may require additional building blocks beyond the elementary ones (R, L, C). Hence the use of transmission lines, for example, to introduce a degree of “distribution” into circuit models. Furthermore, relaxing local passivity and reciprocity requirements, new building

blocks have been introduced with the utilization of negative-valued elements [55], [56], or more exotic elements such as the nullator or the norator [57]. In addition, methodologies for devising equivalent circuit models from transcendental impedances have been proposed by Fry [58] and Zinn [59] in specific cases, and some approaches using non-rational functions expansions have been developed and discussed [60], [61], [62], including hypergeometric functions [63]. But, even if a comprehensive framework remains elusive [61], these network theory developments for representing linear passive distributed impedances using either rational or non-rational series expansions bear striking similarities with fractional-order formalisms, i.e., involving $s^\alpha$, $s$ being the complex angular frequency ($s = j\omega$), and $\alpha$ an arbitrary real number (where generally $-1 \le \alpha \le 1$).

Connection between network theory and fractional-order operators is indeed a long-known fact: the $s^{-1/2}$ behavior of the impedance of infinite RC ladders [64], [65], [66], [67], network synthesis of arbitrary order impedances [68], [69], [70], [71], or constant-argument admittances [72], [73], [74]. However, all of the aforementioned work aims at synthesizing fractional-order functions with integer-order operators or circuit elements. But the utilization of fractional-order circuit elements per se in network design was only marginal until recent advances in fabrication of fractional-order capacitors [75], [76], [77], or inductors [78], which triggered significant attention for fractional-order circuits and systems [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89]. A large part of the published research is devoted to theoretical aspects such as stability [90], [91], passivity [80], [92], or realizability [82]. However, literature on the application of fractional-order circuit elements to EM distributed structures is much less extensive. Some developments of fractional transmission lines and interconnects can be noted [93], [94], [95], [96], [97], but, to the best of our knowledge, there are no published works on antennas impedance modeling with fractional-order circuits. Yet, antennas are, in our opinion, a valuable candidate for such a modeling: as fractional calculus is particularly powerful to account for nonlocality [98], the distributed nature of EM phenomena involved in antennas should be more accurately described by fractional-order circuits. Moreover, the spatial dispersion due to fringing effects on antennas is either expressed as series expansion [99] or fractional-order empirical formulas [100]. And such are the radiative losses in antennas, associated to the radiation resistance, whose frequency dependence involves transcendental functions [101], and are approximated by series expansions [102] or distributed lossy networks [45], [103].

Therefore, we propose in this article a fractional-order equivalent circuit modeling methodology for antennas and – more generally – passive resonant EM structures involving distributed phenomena. To the best of the authors’ knowledge, this is the first instance of the utilization of fractional circuit elements for modeling the impedance of antennas. As it will be shown, the utilization of fractional inductances
<figure><img src="image_3.png" /><figcaption>FIGURE 1. RLC fractional resonators: parallel (left) and series (right).</figcaption></figure>

and capacitances as circuit elements results in simple and elegant equivalent circuits, which achieve an unprecedented modeling accuracy, and contributes bridging the gap between lumped and distributed circuits.

Section II presents the fundamentals of fractional-order circuits in frequency domain, and describes the equivalent circuit synthesis methodology for antennas. Some modeling results are presented, showing the performance of fractional-order equivalent circuits compared to first-order models, but also outlining some limitations. Section III addresses these issues by introducing new fractional-order elements. Section IV validates the proposed modeling methodology through various examples of EM simulations. Finally, the fractional circuit approach is successfully applied to experimental data in Section V, thereby demonstrating the efficiency, accuracy and versatility of the fractional-order equivalent circuit models of antennas.

## II. FRACTIONAL EQUIVALENT CIRCUITS FOR ANTENNAS

### A. FRACTIONAL CIRCUIT ELEMENTS AND RESONATORS

The fractional-order circuit elements $L_β$ and $C_α$ have been introduced to generalize the classical first order inductance $L$ and capacitance $C$ [80]. In basic form, impedance $Z_L$ of the $L_β$ inductance and admittance $Y_C$ of the $C_α$ capacitance in frequency domain are simply defined as:

$$Z_L(\omega) = L(j\omega)^\beta = L\omega^\beta e^{j\beta\frac{\pi}{2}}, \quad (1)$$

and

$$Y_C(\omega) = C(j\omega)^\alpha = C\omega^\alpha e^{j\alpha\frac{\pi}{2}}, \quad (2)$$

where $L$ and $C$ are the nominal inductance and capacitance, $\omega$ the angular frequency, and $\alpha$ and $\beta$ are the fractional-order parameters (without dimension).

Equation (1) can be rewritten as:

$$Z_L(\omega) = L\omega^\beta \cos\left(\beta\frac{\pi}{2}\right) + jL\omega^\beta \sin\left(\beta\frac{\pi}{2}\right). \quad (3)$$

Despite its apparent simplicity, the above equation conceals a few subtleties. First of all, fractional order introduces a frequency-dependent real part with non-integer dispersion. Furthermore, if $\beta > 1$, the real part of the inductance impedance becomes negative, thereby rendering the element active, which has to be carefully avoided in the case of passive structures such as antennas. These elements can be used to build fractional RLC resonators, either parallel or series (Fig. 1).

Such resonators have been extensively analyzed and discussed [80], [104], [105], [106], [107], and some of their

properties will be recalled here. First, the angular resonance frequencies, i.e., the angular frequencies where the impedance is purely real ($\Im(Z) = 0$), can be calculated as:

$$\omega_{rs} = \left( \frac{\sin(\alpha\pi/2)}{LC\sin(\beta\pi/2)} \right)^{\frac{1}{\alpha+\beta}} \quad \text{for the series tank,} \quad (4)$$

and

$$\omega_{rp} = \left( \frac{\sin(\beta\pi/2)}{LC\sin(\alpha\pi/2)} \right)^{\frac{1}{\alpha+\beta}} \quad \text{for the parallel tank.} \quad (5)$$

In the special case $\alpha = \beta = 1$, (4) and (5) reduce to the first-order classical angular resonance frequency $\omega_r = 1/\sqrt{LC}$.

Compared to first-order resonators, the fractional-order parameters $\alpha$ and $\beta$ provide additional degrees of freedom, allowing for precise adjustments of the resonance frequency, dispersion, bandwidth and phase response of the impedance [105], [107]. However, unlike the first-order case, a pure imaginary angular frequency $\omega_{pi}$ can exist, i.e., a frequency – other than $\omega = 0$ – where the impedance is purely imaginary ($\Re(Z) = 0$). This means the circuit has the ability to function as a negative resistor below (or beyond) this angular frequency. As antennas are passive devices, it is obviously imperative to avoid this condition. For simple fractional LC circuits (without a resistor), it has been analytically shown [80] that satisfying $\alpha + \beta \le 2$ is a sufficient criterion to ensure passivity, but in the case of RLC tanks, the presence of a resistor fundamentally changes the conditions [90]. The pure imaginary angular frequency $\omega_{pi}$ must verify $\Re[Z(\omega_{pi})] = 0$, which reduces to

$$L + RLC\cos(\alpha\pi/2)\omega_{pi}^{\alpha} + R\cos(\beta\pi/2)\omega_{pi}^{-\beta} = 0 \quad (6)$$

for a parallel resonator, and to

$$RC + \cos(\alpha\pi/2)\omega_{pi}^{-\alpha} + LC\cos(\beta\pi/2)\omega_{pi}^{\beta} = 0 \quad (7)$$

for a series resonator. Unfortunately, these transcendental equations do not possess an analytical general solution, with the exception of certain specific cases for $\alpha$ and $\beta$ [90], [108], and have to be numerically solved on a case-by-case basis. Therefore, the resistorless $\alpha + \beta \le 2$ passivity criterion has been adopted for the moment.

### B. EQUIVALENT CIRCUIT SYNTHESIS

This section describes the methodology for devising a fractional equivalent circuit for an antenna. First, the reflection coefficient ($\Gamma_{FW}$) of the antenna to be modeled is computed using a full-wave electromagnetic software, namely CST Microwave Studio, and the input impedance of the antenna ($Z_{ant}$) is calculated as $Z_{ant} = Z_0 (1 + \Gamma_{FW}) / (1 - \Gamma_{FW})$, where $Z_0$ is the reference impedance (50Ω). The equivalent circuit topology is then determined by the number and type (series or parallel) of resonances in the full-wave impedance.

The chosen approach for establishing the equivalent circuit topology is quite conventional, and consists in associating one
<figure><img src="image_4.png" /><figcaption>FIGURE 2. Equivalent circuit topologies for antenna modeling: Parallel cells in series (left) and Series cells in parallel (right).</figcaption></figure>

fractional RLC resonator (hereafter referred to as a “cell”) per resonance in the antenna impedance. Depending on the full-wave impedance behavior, these cells can be parallel or series ones, and are cascaded in order to cover the whole bandwidth of the impedance data. Parallel cells are connected in series, and series cells are paralleled (Fig. 2).

The out-of-band behavior of the equivalent circuit also has to be considered. There is a general agreement to represent high-frequency behavior of the impedance by the addition of an inductance, which accounts for the effects of the higher-order modes of the antenna [13]. However, antennas are intrinsically multi-resonant structures, which implies that the next higher resonance de facto exists and contributes to the impedance curve at lower frequencies. Therefore, a more accurate – and more physical – solution is the addition of an auxiliary resonant cell, whose resonance lies beyond the frequency band of interest. This principle similarly pertains to the low-frequency behavior of the impedance: if the frequency band of interest starts beyond the first natural resonance of the antenna, a low-frequency resonant cell has to be added. Otherwise, a simple circuit (capacitor or/and inductor) can be inferred from the antenna type and feeding structure to represent the quasi-static behavior of the impedance [13]. These additions are obviously made at the expense of the number of circuit parameters, but the resulting model accuracy more than justifies the cost.

Once the topology is chosen, the rough values of circuit elements of each cell are estimated from resonance frequencies and bandwidths, assuming first-order (i.e., $\alpha = \beta = 1$) circuit elements. If $\omega_r$ is an angular resonance frequency, the R, L, and C values of the corresponding cell can be calculated as follows.

First, the resistor value R is given by the real part of impedance (resp. admittance for the series cells) at the resonance frequency $\omega_r$:

$$R = \mathfrak{R}[Z_{ant}(\omega_r)] \text{ for parallel cells} \quad (8)$$

or

$$R = 1/\mathfrak{R}[Y_{ant}(\omega_r)] \text{ for series cells,} \quad (9)$$

where $Y_{ant} = 1/Z_{ant}$ is the antenna admittance.

Then, L and C values are calculated from the bandwidth around the resonance. However, in the presence of multiple and possibly close and overlapping resonances, classical bandwidth determination may be difficult. Therefore, a more general approach was adopted, which consists in exploring

<figure><img src="image_5.png" /><figcaption>FIGURE 3. Bandwidth determination in the case of overlapping resonances.</figcaption></figure>

the real part of the full-wave impedance (resp. admittance) around the resonance. In this region, the real part is expected to decrease until it ideally reaches half its value at resonance, or – if not – until the slope of the real part begins to decrease (Fig. 3).

If $\omega_\Delta$ is the angular frequency where this behavior occurs (Fig. 3), the K ratio is defined as:

$$K = \mathfrak{R}[Z_{ant}(\omega_r)]/\mathfrak{R}[Z_{ant}(\omega_\Delta)] \text{ for parallel cells} \quad (10)$$

or

$$K = \mathfrak{R}[Y_{ant}(\omega_r)]/\mathfrak{R}[Y_{ant}(\omega_\Delta)] \text{ for series cells.} \quad (11)$$

Then, using $\omega_r = 1/\sqrt{LC}$, L and C values are calculated as:

$$L = R \frac{|\omega_r^2 - \omega_\Delta^2|}{\omega_\Delta \omega_r^2 \sqrt{K-1}} \quad \text{then} \quad C = \frac{1}{L \omega_r^2} \quad \text{for parallel cells} \quad (12)$$

or

$$C = \frac{1}{R} \frac{|\omega_r^2 - \omega_\Delta^2|}{\omega_\Delta \omega_r^2 \sqrt{K-1}} \quad \text{then} \quad L = \frac{1}{C \omega_r^2} \quad \text{for series cells.} \quad (13)$$

In the case off-band cells are needed, the initial values of their elements are calculated in the same manner. The off-band cell is assumed to resonate with a given amplitude and bandwidth at an harmonic frequency. These are inferred from the distribution of in-band resonance frequencies, amplitudes, and bandwidths, and elements values are given by (8)-(13).

The estimated values, along with the fractional-order parameters initially set to one, are then used as an initial guess in an extraction procedure to evaluate the circuit elements values. Extraction is performed with nonlinear optimization using MATLAB's fmincon and the sequential quadratic programming (SQP) algorithm, with the following main parameters: TolX = 1E-10; TolFun = 1E-10; TolConSQP = 1E-10; MaxSQPIter = 500; MaxIter = 1000; and MaxFunctionEvals = 5000. The extraction procedure aims at minimizing the error function

$$\mathcal{E} = 20 \log \left( \frac{1}{N_f} \sum_{1}^{N_f} |\Gamma_{FW} - \Gamma_C| \right) \quad (\text{in dB}), \quad (14)$$

where $\Gamma_{FW}$ is the complex reflection coefficient given by the full-wave simulation, $\Gamma_C$ the complex reflection coefficient
of the equivalent circuit model, and $N_f$ the number of frequency points in the impedance data.

As fractional-order cells may become active, a nonlinear constraint was added to ensure the total impedance of the circuit ($Z_{circ}$) remains a positive-real function, i.e., the real part of the impedance is always positive or null [36]. Rather than enforcing $\alpha + \beta \le 2$ within each cell or verifying the possible existence of a pure imaginary angular frequency (6)-(7), a simple check on the real part was implemented to ensure it remains positive over the whole frequency band ($\Re[Z_{circ}(\omega)] \ge 0$ $\forall \omega$). As it will be explained later, this simple constraint provided key insights that led to significant improvements in the model.

### C. FIRST FRACTIONAL EQUIVALENT CIRCUIT MODELS

This section shows the performances of the fractional-order equivalent circuits approach for some simple antennas.

#### 1) ABOUT EM SOLVERS

As a preamble, it should be noted that all full-wave EM simulations presented in this article were carried out using the time-domain solver of CST Microwave Studio (MWS). In fact, CST's frequency-domain solver (as most of the frequency-domain solvers of the commercial EM softwares) relies on rational functions (vector fitting) [109] or model order reduction approaches [110] to interpolate between the frequency points actually calculated. As previously mentioned, while these approximations can be quite effective in general, they do not fully capture the intricate and transcendental nature of the electromagnetic response. Yet, it would be ironic and pointless to apply to such results a model designed to avoid such approximations. The only way to address the approximations made by frequency-domain solvers would be to increase the number of frequency samples. This would reduce the interpolated part of the numerical results. However, the computational cost involved would be prohibitive compared to that of a time-domain solver, even with an extremely fine mesh. If further substantiation of this claim were needed, it appeared during this study that the extraction procedure systematically encountered much greater challenges in converging and minimizing error when working with results from the frequency-domain solver than with results from the time-domain solver.

With regard to the CST time-domain solver, which relies on a hexahedral cartesian meshing of the structures, precautions have been taken to circumvent numerical artifacts and guarantee the physical significance of the results. Consequently, the meshes have been refined until convergence of the results is achieved, i.e., until the results are independent of the mesh fineness. Furthermore, the time-domain window for analysis has been expanded beyond CST MWS default values to ensure proper signals damping and clean Fourier transforms.

#### 2) DIPOLE ANTENNA

First, a thin-wire center-fed dipole (each arm is 400 mm long, wire radius is 2 mm, and feeding gap is 0.5 mm) has been

first modeled and simulated using CST MWS from 0.1 GHz to 1 GHz, encompassing the first three resonances of the antenna. As dipoles are generally characterized by their series resonances, the chosen equivalent circuit uses four paralleled series cells. The first three cells model the three series resonances in the antenna impedance, and the fourth cell accounts for the next out-of-band resonance (Fig. 4). No additional capacitor is needed to model the quasi-static capacitive behavior of the dipole, as the low-frequency impedance of such a circuit is purely capacitive.

<figure><img src="image_6.png" /><figcaption>FIGURE 4. Fractional equivalent circuit of the dipole.</figcaption></figure>

In order to underline the contribution of fractional-order elements, equivalent circuit extraction has been performed both with first-order cells (by enforcing $\alpha = \beta = 1$), and fractional-order circuit elements, all else being equal. Fig. 5 presents the resulting impedances.

<figure><img src="image_7.png" /><figcaption>FIGURE 5. Equivalent circuit impedance (fractional-order and first-order) of the dipole compared to full-wave impedance.</figcaption></figure>

The concordance between full-wave and fractional-order circuit impedances is impressive, with a fitting error (14) of only -48 dB, while the first-order circuit could not achieve a fitting error any lower than -21 dB. The fractional elements undeniably help to match frequency dispersion of the dipole as closely as possible, while first-order elements face difficulties in achieving this level of accuracy. This clearly shows that classical first-order resonant cells are not suitable for equivalent circuit modeling of antennas, which explains the numerous attempts to modify the topology of first-order resonant cells [11], [12], [37], [41].

Table 1 reports the values of the different elements for the fractional-order circuit, and it is worth noting that, despite the only nonlinear constraint applied is to ensure that the total circuit impedance is a positive-real function, the $\alpha + \beta \le$
<table><caption>TABLE 1. Extracted fractional-order equivalent circuit parameters for the Dipole Antenna.</caption><thead><tr><th>Element</th><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th><th>Cell 4</th></tr></thead><tbody><tr><td>C</td><td>618.35 fF</td><td>66.855 fF</td><td>28.548 fF</td><td>25.477 pF</td></tr><tr><td>α</td><td>1.0553</td><td>1.0583</td><td>1.052</td><td>0.804</td></tr><tr><td>L</td><td>10.656 μH</td><td>4.1481 μH</td><td>2.5509 μH</td><td>48.485 pH</td></tr><tr><td>β</td><td>0.8464</td><td>0.8891</td><td>0.9102</td><td>1.1902</td></tr><tr><td>R</td><td>0.5423 Ω</td><td>1.4445 Ω</td><td>2.3244 Ω</td><td>1.1555 Ω</td></tr></tbody></table>

2 requirement to ensure the passivity of each cell has been automatically met by the extraction procedure.

### 3) MICROSTRIP PATCH ANTENNA

Next, fractional-order equivalent circuit modeling has been applied to a simple microstrip patch antenna on a finite ground plane (Fig. 6). The antenna is fed with a coaxial probe, and simulated with CST MWS from 3 to 4.5 GHz, thus covering the first two resonances.

<figure><img src="image_8.png" /><figcaption>FIGURE 6. Microstrip patch antenna geometry (not to scale) and design dimensions (in mm): W = 35.1, L = 27, h = 1.524, Wg = 100, P = 2.7, a = 1.27, D = 4.06. The substrate is Rogers RT5880, with an $ε_r$ of 2.2 and a tgδ of 0.9e-3 at 10GHz.</figcaption></figure>

The equivalent circuit model is presented in Fig. 7 and consists of three parallel cells in series, the first two modeling the two in-band resonances, while the third cell accounts for higher modes. The addition of the $C_0$ capacitor allows representing the quasi-static behavior of the impedance.

<figure><img src="image_9.png" /><figcaption>FIGURE 7. Equivalent circuit model for the microstrip patch antenna.</figcaption></figure>

As previously, circuit extraction has been performed with both first-order and fractional-order cells, and Fig. 8 presents the full-wave and circuit impedances. Once again, the fractional-order circuit outperforms the first-order model, with a fitting error (14) of -62 dB vs. -19 dB, although the

<figure><img src="image_10.png" /><figcaption>FIGURE 8. Equivalent circuit impedance (3 cells, fractional-order and first-order) of the patch antenna compared to full-wave impedance.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>FIGURE 9. $|S_{11}|$ of the equivalent circuits (fractional-order and first-order) of the patch antenna compared to full-wave result.</figcaption></figure>

<table><caption>TABLE 2. Extracted fractional equivalent circuit parameters for the Microstrip Patch Antenna.</caption><thead><tr><th>Element</th><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th></tr></thead><tbody><tr><td>C (pF)</td><td>91.95</td><td>0.6345</td><td>2.7797</td></tr><tr><td>α</td><td>0.8719</td><td>1.0173</td><td>0.9386</td></tr><tr><td>L (pH)</td><td>32.2</td><td>3290.6</td><td>725.82</td></tr><tr><td>β</td><td>1.1152</td><td>0.9733</td><td>0.9984</td></tr><tr><td>R (kΩ)</td><td>0.8301</td><td>42.577</td><td>643.01</td></tr><tr><td>C₀ (pF)</td><td>21.764</td><td></td><td></td></tr></tbody></table>

differences in the curves of Fig. 8 may seem quite negligible. However, the return losses ($|S_{11}|$) of the equivalent circuits tell a different story, as shown in Fig. 9.

The discrepancies are much more obvious, reflecting the gap between the fitting error values, and underlining the excellent performances of the fractional-order model compared to the first-order circuit. Finally, Table 2 lists the values of the fractional-order circuit elements, and once again it can be noted that the $α + β ≤ 2$ requirement in each cell has been met without explicit enforcement.

Fractional equivalent circuit modeling for antennas impedances thus appears to demonstrate exceptional efficacy, offering a compact and elegant approach to account for the complex frequency dispersion characteristics of antennas. However, in certain cases, it has been found that full-wave
impedances still elude the fractional circuit modeling, thereby necessitating further refinement.

### D. FRACTIONAL CIRCUIT MODEL LIMITATIONS

While testing the fractional-order circuit model on various antennas, it was observed that the approach was not successful in some cases. Fig. 10 presents one of such case, which will be examined in detail.

This microstrip patch antenna is deliberately unusual in that it has no right angles or symmetry, since it was intended to degrade the classical cavity modes. Time-domain EM simulations were performed using CST MWS between 0.5 GHz and 1.5 GHz. Due to the particular shape of the antenna, an exceedingly fine spatial mesh was used to ensure the convergence of results with respect to the mesh fineness. As a result, the full-wave impedance appeared to involve four parallel resonances. Therefore, the equivalent circuit resemble the one in Fig. 7, except that five cells were used, the last one being devoted to higher modes. Fractional equivalent circuit extraction could not achieve a fitting error any lower than -33 dB. Although this may be regarded as satisfactory, it is not as accurate as expected, given the previous results. As before (Fig. 8), if the impedances show almost no difference, the return loss is more revealing of the discrepancies (Fig. 11).

<figure><img src="image_12.png" /><figcaption>FIGURE 10. Example of coaxial probe-fed microstrip skewed antenna eluding fractional equivalent circuit modeling (not to scale). All dimensions are in mm, and substrate is air. The feeding probe is centered on the ground plane.</figcaption></figure>

An even more interesting insight is provided by examining the fractional parameters $\alpha$ and $\beta$ (and their sum) derived from the extraction procedure (Table 3).

<table><caption>TABLE 3. Extracted fractional parameters for the Skewed Antenna.</caption><thead><tr><th>Parameter</th><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th><th>Cell 4</th><th>Cell 5</th></tr></thead><tbody><tr><td>α</td><td>0.9634</td><td>1.0358</td><td>1.0466</td><td>1.0411</td><td>0.9747</td></tr><tr><td>β</td><td>1.117</td><td>1.0536</td><td>1.0019</td><td>1.1031</td><td>1.0585</td></tr><tr><td>α+β</td><td>2.0804</td><td>2.0894</td><td>2.0485</td><td>2.1442</td><td>2.0332</td></tr></tbody></table>

These results show that within each cell, the passivity requirement ($\alpha + \beta \le 2$) is not met, although the positive- real function criterion is satisfied ($|S_{11}| \le 0$ dB, Fig. 11). A closer

<figure><img src="image_13.png" /><figcaption>FIGURE 11. $|S_{11}|$ of the fractional equivalent circuit (5 cells) of the skewed patch antenna compared to full-wave result.</figcaption></figure>

look at the impedance of each cells ($Z_{cell}$) revealed that only the fifth cell impedance showed $\Re[Z_{cell}] < 0$ for some frequencies. However, this negative part is balanced by the sum of $\Re[Z_{cell}]$ of all the other cells, maintaining a passive global impedance. Therefore, it seems that the optimizer is trying to address two conflicting objectives: the necessity of maintaining a passive circuit impedance ($\Re[Z_{circ}] \ge 0$), and the apparent requirement for an active part within the cells. The algorithm is therefore missing additional levers to satisfy the passivity constraints while maintaining accuracy. In other words, the fractional-order circuit model is missing a key element, and the next section presents the way to enhance it through mutual coupling.

### III. FRACTIONAL EQUIVALENT CIRCUIT IMPROVEMENT

#### A. MUTUAL COUPLING

As previously outlined, there appears to be a need for active elements within the cells. The central question is as follows: of what origin is this active part? Given that antennas fundamentally are passive and lossy devices, the only physical answer lies in the existence of mutual coupling between the resonance modes. This hypothesis is consistent with the fact that the antenna in Fig. 10 was deliberately designed to degrade the cavity modes, thereby potentially fostering coupled modes. And from the circuit standpoint, taking mutual coupling into account involves active current/voltage-driven generators linked to passive reactive elements, which could provide the missing active part in our circuit model and enforce passivity ($\alpha + \beta \le 2$) within each cell. This factual concordance leads to the concept of integrating modes coupling into the equivalent circuit model through mutual coupling between the cells.

Coupled modes in electromagnetics are a long-known fact [111], but it has recently received renewed interest in the antenna domain with the theory of characteristic modes, for which coupled-mode theory provided notable improvements [43], [53], [112], [113]. However, couplings between antennas' modes involve complex mechanisms and mainly have to be studied on a case-by-case basis, which is incompatible with a general modeling methodology. Therefore,
modes coupling phenomena have been accounted for through a general equivalent circuit approach.

As previously mentioned, the natural translation of EM coupled modes to equivalent circuits shall involve mutual coupling between the resonant cells. And as the EM modes coupling mechanisms are not a priori known, equivalent circuit mutual coupling has to be both inductive and capacitive [43], [52]. Moreover, as fractional order seems to better describe the frequency dispersion of the modes, fractional order has also been adopted for mutual couplings.

Like previous fractional circuit elements, fractional-order mutual inductance [114] and capacitance [115] have been introduced to generalize their classical first-order counterparts, and the equivalent circuits of Fig. 2 are modified as shown in Fig. 12. All the previous reactive elements have been kept fractional, and inductive mutual coupling (in red) is represented by voltage generators in series with the inductances, while capacitive coupling (in green) is represented by current generators in parallel with the capacitances.

<figure><img src="image_14.png" /><figcaption>FIGURE 12. Parallel cells in series (top) and series cells in parallel (bottom) with mutual coupling.</figcaption></figure>

The frequency-dependent coupled voltage $V_{Mi}$ (s) in the $i^{th}$ cell, (s being the complex angular frequency, $s = j\omega$), is given by

$$V_{Mi}(s) = \sum_{j=1}^{n, i \neq j} Z_{Mij}(s) I_j(s) = \sum_{j=1}^{n, i \neq j} s^{\beta_{Mij}} M_{Lij} I_j(s), \quad (15)$$

where $I_j(s)$ is the current flowing through the inductance of the $j^{th}$ cell, $M_{Lij}$ the mutual inductance (frequency-independent) between the $i^{th}$ and $j^{th}$ cells, $\beta_{Mij}$ the fractional order of this mutual inductance, and $n$ the total number of cells. Similarly, the frequency-dependent coupled current

$I_{Mi}(s)$ in the $i^{th}$ cell is given by

$$I_{Mi}(s) = \sum_{j=1}^{n, i \neq j} Y_{Cij}(s) V_j(s) = \sum_{j=1}^{n, i \neq j} s^{\alpha_{Mij}} M_{Cij} V_j(s), \quad (16)$$

where $V_j(s)$ is the voltage across the capacitor of the $j^{th}$ cell, $M_{Cij}$ the mutual capacitance (frequency-independent) between the $i^{th}$ and $j^{th}$ cells, with a fractional order $\alpha_{Mij}$.

As the elementary cells are now coupled, circuit equations are no longer independent within each cell, and a system of coupled equations has to be solved. From (15) and (16), defining the mutual impedances and admittances between cell $i$ and cell $j$ as

$$Z_{Mij}(s) = s^{\beta_{Mij}} M_{Lij} \quad \text{and} \quad Y_{Cij}(s) = s^{\alpha_{Mij}} M_{Cij}, \quad (17)$$

the mutual coupling matrices $[M_L]$ and $[M_C]$ are written as

$$[M_L] = \begin{bmatrix} Z_{L1}(s) & \cdots & Z_{Mij}(s) \\ \vdots & \ddots & \vdots \\ Z_{Mji}(s) & \cdots & Z_{Ln}(s) \end{bmatrix} \quad \text{and} \quad (18)$$

$$[M_C] = \begin{bmatrix} Y_{C1}(s) & \cdots & Y_{Cij}(s) \\ \vdots & \ddots & \vdots \\ Y_{Cji}(s) & \cdots & Y_{Cn}(s) \end{bmatrix}, \quad (19)$$

where the diagonal terms of the matrices are equal to $Z_{Li}(s) = s^{\beta_i} L_i$ and $Y_{Ci}(s) = s^{\alpha_i} C_i + 1/R_i$ for parallel cells, and to $Z_{Li}(s) = R_i + s^{\beta_i} L_i$ and $Y_{Ci}(s) = s^{\alpha_i} C_i$ for series cells. As antennas generally are reciprocal devices, reciprocity conditions are assumed, i.e., $M_{Lji}(s) = M_{Lij}(s)$ and $M_{Cji}(s) = M_{Cij}(s)$. However, non-reciprocity could theoretically be handled if needed (for antennas on ferrite substrates, for instance). Finally, the equation system to be solved is

$$(Z_c) = ([I] + [M_L][M_C])^{-1} [M_L] \quad (20)$$

for parallel cells in series, and

$$(Y_c) = ([I] + [M_C][M_L])^{-1} [M_C] \quad (21)$$

for paralleled series cells. ($Z_c$) (resp. ($Y_c$)) is the vector containing the unknown impedance (resp. admittance) of each individual cell, $[I]$ is the identity matrix, and (1) is the unit vector. The total circuit impedance $Z_{circ}$ (resp. admittance $Y_{circ}$) is the sum of all the elements of ($Z_c$) (resp. ($Y_c$)). Circuit equations (20) and (21) have been implemented in MATLAB. However, it is also possible to implement these equations in circuit simulators such as Keysight's Advanced Design System (ADS) or SPICE, as detailed in the Appendix.

Parameter extraction is performed in the same way as before, except that the resonant cells are sorted by ascending resonance frequencies prior to the introduction of mutual coupling (cell #1 has the lowest resonance frequency). This maintains consistency in the mutual coupling matrices during the extraction process.

The main drawback of the introduction of mutual coupling is the considerable number of additional circuit parameters.
<figure><img src="image_15.png" /><figcaption>FIGURE 13. |S11| of the fractional equivalent circuit (5 cells, with partial mutual coupling) of the Skewed Antenna compared to full-wave result.</figcaption></figure>

<table><caption>TABLE 4. Extracted fractional circuit parameters for the Skewed Antenna.</caption><thead><tr><th>Element</th><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th><th>Cell 4</th><th>Cell 5</th></tr></thead><tbody><tr><td>C (pF)</td><td>749.26</td><td>90.559</td><td>1.2992</td><td>908.2</td><td>1.7407</td></tr><tr><td>α</td><td>0.82176</td><td>0.96053</td><td>1.0188</td><td>0.7933</td><td>0.94588</td></tr><tr><td>L (nH)</td><td>0.2596</td><td>1.9737</td><td>90.822</td><td>6.1198e-3</td><td>204e-6</td></tr><tr><td>β</td><td>1.1011</td><td>0.97272</td><td>0.90565</td><td>1.1494</td><td>1.0356</td></tr><tr><td>R (Ω)</td><td>2209.2</td><td>577.03</td><td>138.5e3</td><td>82.65</td><td>25.856</td></tr><tr><td>α+β</td><td>1.92286</td><td>1.93325</td><td>1.92445</td><td>1.9427</td><td>1.98148</td></tr><tr><td colspan="6">Coupling</td></tr><tr><td>M<sub>L</sub> (fH)</td><td>0.93062</td><td>1.3175</td><td>2.3765</td><td>1.4108</td><td></td></tr><tr><td>β<sub>M</sub></td><td>1.1463</td><td>1.0139</td><td>1.3052</td><td>0.94019</td><td></td></tr><tr><td>M<sub>C</sub> (fF)</td><td>0.99995</td><td>0.68741</td><td>93.997</td><td>0.72366</td><td></td></tr><tr><td>α<sub>M</sub></td><td>0.96831</td><td>1.2858</td><td>1.0348</td><td>1.1944</td><td></td></tr></tbody></table>

However, numerous tests have shown that, in most of the cases, the coupling between distant cells (i.e., cells whose resonance frequencies are far apart) could be neglected without loss of accuracy. Most of the coupling actually occurs between adjacent cells: for instance, cell #2 is coupled with cell #1 and #3 only. This partial coupling scheme allows the reduction of the total number of parameters without compromising global accuracy. It should be noted, however, that this scheme is only applicable if the resonances originate from a single resonant structure. In the case of composite structures with multiple resonators, the possibility exists that the resonance modes of one resonator may be coupled to the resonance modes of the other structures. Therefore, in this scenario, implementing the general coupling scheme is essential to accurately represent the behavior of the impedance. Such a case is depicted in Section V, with the equivalent circuit modeling of a U-slot antenna.

As an example, the new fractional equivalent circuit of the skewed antenna of Fig. 10 has been extracted. Fig. 13 presents comparison of the full-wave and equivalent circuit return loss, (only partial coupling has been used here). The fractional equivalent circuit including mutual coupling achieves a remarkable fitting error of -70 dB, to be compared to the -33 dB error without mutual coupling (cf. Fig. 11).

Table 4 presents the extracted circuit parameters. It can be noticed that mutual coupling is very small compared to

the internal inductance and capacitance of the cells. And as coupling deliberately introduces active parts in the cells, the cell passivity criterion ($α + β ≤ 2$) is no longer violated by the reactive fractional elements, as it was previously (Table 3). Therefore, the initial assumption of the need of mutual coupling proves to be correct. In addition, numerous tests on various antennas have shown that both inductive and capacitive fractional mutual coupling are imperative to achieve the lowest fitting error. Any attempt to use only one type of mutual coupling, as well as only first-order coupling, yielded a significantly lower accuracy.

## B. ONE STEP ASIDE

Fractional mutual coupling is definitely essential to achieve accurate circuit modeling. But this also prompts the question of why the first two examples presented (dipole and simple patch antenna) yield excellent results despite the lack of mutual coupling, all the more so since the cell passivity requirement ($α + β ≤ 2$) was met. In both cases, a careful examination of the impedance of each cell ($Z_{cell}$) revealed that, despite verifying $α + β ≤ 2$ in each cell, some of the cells' impedances were not positive-real functions, i.e., $\Re[Z_{cell}] < 0$ for some frequency range. And as long as the total impedance remained passive, this point was not checked. However, as previously mentioned, this negative part is balanced by the other cells, maintaining a passive global impedance, which was the optimization constraint to be met.

It appears that the passivity criterion previously defined ($α + β ≤ 2$) is definitely unreliable. Even if met, it may lead to active cells, but it can be violated without leading to active cells in the frequency band of interest: for the skewed antenna (Table 3), all of the five cells displayed $α + β > 2$, but only the fifth one had a negative real part in the frequency band. Without this criterion, the only way to detect active cells during the fitting procedure would be to check whether a pure imaginary angular frequency exists in each cell, i.e., determining the existence of solutions to (6) or (7). As these transcendental equations have no analytical solution, such a constraint would be computer-intensive, and ultimately useless, since it applies only to uncoupled cells.

In the case of mutual coupling, the cells are inherently active, and the very notion of passivity criterion for a cell becomes a nonsense. As no analytical solutions to circuit equations (20) and (21) exist, the only option left is to verify on the fly that the total circuit impedance is a positive-real function. Yet, the $α + β ≤ 2$ criterion has been retained, no longer as a constraint, but as an indicator of the fitting process quality.

## IV. RESULTS

The proposed fractional equivalent circuit model methodology has been applied to various antennas, and this section gives examples of the performance of this approach. To underline the improvements of the fractional-order approach, circuit parameter extraction has also been per-
formed with first-order circuit elements, including first-order mutual couplings.

## A. MICROSTRIP PATCH ANTENNA

First, the microstrip patch antenna of Fig. 6 has been modeled once more, but with the inclusion of partial mutual coupling. As previously (Fig. 7), the equivalent circuit consists of three parallel cells in series, in series with a capacitor (Fig. 14).

<figure><img src="image_16.png" /><figcaption>FIGURE 14. 3-cell equivalent circuit of the microstrip patch antenna including partial mutual coupling.</figcaption></figure>

Three different scenarios were considered: first, the exact same dimensions as in Fig. 6, then the feeding probe position was set to P = 8 mm with the same substrate height, and finally the substrate was thickened to h = 5 mm, along with probe position as in the first case (P = 2.7 mm). All the other dimensions of the antenna remained unaltered.

Equivalent circuit extraction has been performed in the three cases, and Fig. 15 compares the return losses of the three fractional circuits with full-wave results.

<figure><img src="image_17.png" /><figcaption>FIGURE 15. [S11] of the fractional-order equivalent circuits (3 cells, with partial mutual coupling) of the Microstrip Patch Antenna for the three sets of h and P parameters, compared to CST full-wave results.</figcaption></figure>

In all cases, the fractional equivalent circuits exhibit an excellent agreement with full-wave results, thereby demonstrating the relevance and the adaptability of the proposed circuit modeling approach. Table 5 details the extracted circuit parameters, both for fractional order (gray backgrounded values) and first order (white background). Given the reduced number of degrees of freedom in first-order circuits, the optimization algorithm must adopt alternative pathways to achieve optimal convergence, particularly with respect to mutual couplings. Consequently, a direct circuit element comparison between fractional and first-order values is not meaningful or relevant.

<table><caption>TABLE 5. Extracted circuit parameters for the Microstrip Patch Antenna for the three scenarios (Gray backgrounded values: Fractional-Order Model; white backgrounded values: First-Order Model).</caption><thead><tr><th rowspan="2"></th><th colspan="3">P=2.7mm h=1.5mm</th><th colspan="3">P=8mm h=1.5mm</th><th colspan="3">P=2.7mm h=5mm</th></tr><tr><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th></tr></thead><tbody><tr><td>C (pF)</td><td>4.4775</td><td>1.1507</td><td>2.8403</td><td>4.1197</td><td>0.7358</td><td>2.0226</td><td>144.37</td><td>7.9472</td><td>7.5399</td></tr><tr><td>α</td><td>58.735</td><td>0.5101</td><td>2.3716</td><td>2.8808</td><td>0.6069</td><td>0.2241</td><td>3.3035</td><td>0.0058</td><td>12.372</td></tr><tr><td>L (pH)</td><td>0.9561</td><td>0.9959</td><td>1.0940</td><td>0.9461</td><td>1.0143</td><td>1.0772</td><td>0.8761</td><td>0.9352</td><td>1.2163</td></tr><tr><td>β</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>R (Ω)</td><td>607.16</td><td>3297.6</td><td>384.83</td><td>864.85</td><td>3333.9</td><td>348.63</td><td>419.05</td><td>6191.7</td><td>305.38</td></tr><tr><td>α</td><td>36.342</td><td>3081.9</td><td>162.18</td><td>1031.2</td><td>3261.1</td><td>542.12</td><td>1170.1</td><td>1742.6</td><td>201.04</td></tr><tr><td>β</td><td>1.0255</td><td>0.9809</td><td>0.8971</td><td>1.0385</td><td>0.9608</td><td>0.9264</td><td>1.0014</td><td>0.9626</td><td>0.7502</td></tr><tr><td>R (Ω)</td><td>4443.7</td><td>11402</td><td>145.51</td><td>9396.4</td><td>14906</td><td>181.86</td><td>2121</td><td>3408.6</td><td>100.24</td></tr><tr><td>α</td><td>20.007</td><td>11927</td><td>127.88</td><td>610.04</td><td>8467.1</td><td>286.66</td><td>490.24</td><td>242.95</td><td>41.596</td></tr><tr><td>M<sub>L</sub>M<sub>C</sub></td><td>1↔2</td><td>2↔3</td><td></td><td>1↔2</td><td>2↔3</td><td></td><td>1↔2</td><td>2↔3</td><td></td></tr><tr><td>M<sub>L</sub> (pH)</td><td>-1485.9</td><td>628.86</td><td></td><td>-153.07</td><td>307.38</td><td></td><td>6154.2</td><td>224.64</td><td></td></tr><tr><td>β<sub>M</sub></td><td>-25.226</td><td>1.7019</td><td></td><td>-727.25</td><td>100.15</td><td></td><td>670.54</td><td>-0.03122</td><td></td></tr><tr><td>M<sub>C</sub> (pH)</td><td>0.9190</td><td>0.9064</td><td></td><td>1.0272</td><td>0.9077</td><td></td><td>0.8982</td><td>0.8378</td><td></td></tr><tr><td>α<sub>M</sub></td><td>1</td><td>1</td><td></td><td>1</td><td>1</td><td></td><td>1</td><td>1</td><td></td></tr><tr><td>C<sub>0</sub> (pF)</td><td>100.4</td><td>-819.27</td><td></td><td>260.73</td><td>-408.59</td><td></td><td>-481350</td><td>-2076.2</td><td></td></tr><tr><td>α<sub>M</sub></td><td>1005.8</td><td>-726.62</td><td></td><td>729.12</td><td>-320.94</td><td></td><td>-208.71</td><td>0.01169</td><td></td></tr><tr><td>α<sub>M</sub></td><td>0.9853</td><td>1.0625</td><td></td><td>0.9202</td><td>1.0663</td><td></td><td>0.7628</td><td>1.1117</td><td></td></tr><tr><td>α<sub>M</sub></td><td>1</td><td>1</td><td></td><td>1</td><td>1</td><td></td><td>1</td><td>1</td><td></td></tr><tr><td>α<sub>M</sub></td><td>22.048</td><td></td><td></td><td>32.039</td><td></td><td></td><td>268.94</td><td></td><td></td></tr><tr><td>α<sub>M</sub></td><td>7.6006</td><td></td><td></td><td>5.0671</td><td></td><td></td><td>1.5795</td><td></td><td></td></tr><tr><td>Error (dB)</td><td>-63.2</td><td></td><td></td><td>-71.3</td><td></td><td></td><td>-54.02</td><td></td><td></td></tr><tr><td>Error (dB)</td><td>-46.2</td><td></td><td></td><td>-50.2</td><td></td><td></td><td>-41.3</td><td></td><td></td></tr></tbody></table>

Nonetheless, first-order circuits exhibit quite commendable performance, with a fitting error that remains below −40 dB. However, the fractional-order models demonstrate a systematic improvement, achieving a fitting error that is 15 dB to 20 dB lower. It can be observed that in the first case (P = 2.7 mm, h = 1.5 mm), mutual coupling between the cells does not significantly improve the fitting error, which explains why mutual coupling was not initially identified as a necessity in the case of this particular antenna. The presence of mutual coupling results in an error of −63.2 dB, while the error without coupling is −62 dB (cf. Section II-C-III). This finding conceivably suggests that the different antenna modes may be weakly coupled. However, ascertaining this point from the elements values presented in Table 2 (no coupling) and Table 5 (the first three columns) is challenging, as the active parts contributed by mutual inductive and capacitive couplings may balance each other, or be dissipated within cells.

## B. LOOP ANTENNA

A small loop antenna [40] of 36.2 mm diameter and 1.12 mm wire diameter (Fig. 16, left) has been simulated (CST MWS) from 0.5 to 6 GHz. The extracted fractional equivalent circuit is based on four paralleled series cells, with partial mutual coupling (Fig. 16, right), and gives a fitting error of −45.2 dB, while the extracted first-order equivalent circuit could not achieve a fitting error any lower than −37.6 dB.

The compared impedances are shown in Fig. 17, and differences between the curves are almost imperceptible. The
<figure><img src="image_18.png" /><figcaption>**FIGURE 16.** The loop antenna (left, all dimensions in mm) and its 4-cell equivalent circuit with partial mutual coupling (right).</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**FIGURE 17.** Impedance of the fractional equivalent circuit (4 series cells, with partial mutual coupling) of the loop antenna, compared to full-wave result.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**FIGURE 18.** Reflection coefficient (Γ) of the loop antenna (Smith chart zoom).</figcaption></figure>

return losses ($S_{11}$ modulus) could be plotted, but an even more meaningful representation is the reflection coefficient on a Smith chart (Fig. 18), since it inherently includes the phase.

The concordance between the full-wave and model results is more than excellent, and Table 6 gives the extracted circuit parameters. It can be noticed that some of the coupling coefficients are negative, which reflects a phase reversal of the coupled voltages and currents into the cells.

<table><caption>**TABLE 6.** Extracted fractional and first-order circuit parameters for the Loop Antenna. (Gray background: Fractional-Order; white background: First-Order.)</caption><thead><tr><th>Element</th><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th><th>Cell 4</th></tr></thead><tbody><tr><td>C</td><td>19.366 nF</td><td>350.23 fF</td><td>2.7028 pF</td><td>204.94 fF</td></tr><tr><td></td><td>752.27 pF</td><td>366.1 fF</td><td>1.1262 pF</td><td>28.554 fF</td></tr><tr><td>α</td><td>1.0002</td><td>0.98783</td><td>0.8562</td><td>0.89297</td></tr><tr><td></td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>L</td><td>83.347 nH</td><td>351.34 nH</td><td>271.81 pF</td><td>24.959 nH</td></tr><tr><td></td><td>88.184 nH</td><td>226.48 nH</td><td>6.9627 nH</td><td>16.49 nH</td></tr><tr><td>β</td><td>1.0003</td><td>0.85449</td><td>1.1673</td><td>1.0177</td></tr><tr><td></td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>R</td><td>8.2698 mΩ</td><td>349.51 mΩ</td><td>6.6925 Ω</td><td>10.256 Ω</td></tr><tr><td></td><td>4.4032 Ω</td><td>62.484 mΩ</td><td>10.566 Ω</td><td>217.42 Ω</td></tr><tr><td>Coupling</td><td>1↔2</td><td>2↔3</td><td>3↔4</td><td></td></tr><tr><td>M<sub>L</sub></td><td>-3.8452 pH</td><td>-4.2111 pH</td><td>1.1958 pH</td><td></td></tr><tr><td></td><td>-127.39 nH</td><td>14.597 nH</td><td>25.821 pH</td><td></td></tr><tr><td>β<sub>M</sub></td><td>1.3214</td><td>1.002</td><td>0.9668</td><td></td></tr><tr><td></td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>M<sub>C</sub></td><td>993.87 fF</td><td>-215.02 fF</td><td>-576.01 fF</td><td></td></tr><tr><td></td><td>8.1769 pF</td><td>-529.58 fF</td><td>-24.437 fF</td><td></td></tr><tr><td>α<sub>M</sub></td><td>0.97919</td><td>0.96418</td><td>0.84508</td><td></td></tr><tr><td></td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>Error (dB)</td><td>-45.2</td><td></td><td></td><td></td></tr><tr><td></td><td>-37.6</td><td></td><td></td><td></td></tr></tbody></table>

It can also be noticed from Table 6 that the sums of fractional-order parameters for cells #1 and #3 fall in the case $α + β > 2$. This confirms the previously mentioned observation (Section III-B) that the $α + β ≤ 2$ passivity criterion is irrelevant in the case of mutually coupled – i.e., active – cells.

### C. STRIPLINE VIVALDI ANTENNA

Next, equivalent circuit modeling tests have been carried out on a wideband stripline Vivaldi antenna (Fig. 19). This 3-layer structure consists of two identical Vivaldi tapered slotlines (top and bottom), and a feeding stripline. Metallic walls all around the structure connect the upper and lower layers, except in the feeding port zone and in the antenna aperture. CST MWS simulations have been conducted from 4 to 7 GHz.

The extracted equivalent circuit is composed of six parallel cells in series, with partial mutual coupling between the cells. The in-band behavior of the impedance is embodied by four cells, while the remaining two cells respectively represent off-band low and high response (the antenna's first resonance lies below 4 GHz).

Fractional circuit parameters extraction yields a fitting error of approximately -40 dB, while the error is about -31 dB for first-order extraction, which is quite honorable. Therefore, the fitting is not as optimal as that observed for simpler antennas. However, given the complexity of both the antenna and circuit, the results are more than convincing: Fig. 20 compares circuit and full-wave impedances, and Fig. 21 shows the reflection coefficients on the Smith chart. The extracted circuit parameters are presented in Table 7. Once again, the proposed fractional equivalent circuit model
<figure><img src="image_21.png" /><figcaption>**FIGURE 19.** Design view of the stripline Vivaldi antenna (not to scale). Dimensions (in mm): L = 110, W = 60, La = 64, Wa = 52, d = 6.5, h = 1.524. Substrate is Rogers RO4350B (ε<sub>r</sub> = 3.66 and tgδ = 3.7e-3 at 10GHz).</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**FIGURE 20.** Impedance of the fractional equivalent circuit (6 parallel cells, with partial mutual coupling) of the Vivaldi antenna, compared to full-wave result.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**FIGURE 21.** Reflection coefficient (Γ) of the stripline Vivaldi antenna (Zoom on the Smith chart).</figcaption></figure>

proves its remarkable efficacy in accommodating the intricate frequency dispersion law of the impedance.

<table><caption>**TABLE 7.** Extracted circuit parameters for the Vivaldi Antenna (Gray background: Fractional-Order; white background: First-Order).</caption><thead><tr><th>Element</th><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th><th>Cell 4</th><th>Cell 5</th><th>Cell 6</th></tr></thead><tbody><tr><td>C (pF)</td><td>4.2264<br>10.487</td><td>37.261<br>4.0869</td><td>25.443<br>11.858</td><td>7.7709<br>3.0978</td><td>47.955<br>5.9323</td><td>0.3232<br>0.50117</td></tr><tr><td>α</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>L (pH)</td><td>699.33<br>74.265</td><td>78.247<br>165.75</td><td>151.04<br>67.248</td><td>618.65<br>781.78</td><td>20.474<br>205.02</td><td>1485.6<br>632.04</td></tr><tr><td>β</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>R (Ω)</td><td>115.23<br>4.8326</td><td>51.542<br>342.44</td><td>488.34<br>17.848</td><td>146.3<br>311.53</td><td>8.8148<br>37.115</td><td>147.15<br>97.581</td></tr><tr><td>Coupling</td><td>1↔2</td><td>2↔3</td><td>3↔4</td><td>4↔5</td><td>5↔6</td><td></td></tr><tr><td>M<sub>L</sub> (fH)</td><td>-6.8072<br>-1.6063</td><td>-132.33<br>-44.782</td><td>63.929<br>447.79</td><td>2026.4<br>7821.7</td><td>70.249<br>55.967</td><td></td></tr><tr><td>β<sub>M</sub></td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>M<sub>C</sub> (fF)</td><td>43.132<br>1327.6</td><td>208.95<br>869.62</td><td>7.7451<br>300.27</td><td>-462.02<br>-43.611</td><td>8.2562<br>67.548</td><td></td></tr><tr><td>α<sub>M</sub></td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Error<br>(dB)</td><td>-40.3<br>-31.1</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

## D. DIELECTRIC RESONATOR ANTENNA

Finally, the proposed modeling methodology has been applied to the Ku-band dielectric resonator antenna (DRA) described in [116]. This antenna (Fig. 22) consists of a slot-coupled rectangular dielectric resonator (ε<sub>r</sub> = 20). The feeding microstrip line is printed on an alumina substrate (ε<sub>r</sub> = 9.2). CST MWS simulations have been carried out from 10 GHz to 20 GHz, this frequency band encompassing four resonances of the antenna.

<figure><img src="image_24.png" /><figcaption>**FIGURE 22.** View of the DRA (not to scale) [116]. Dimensions (in mm): L = 17, W = 14, W<sub>f</sub> = 0.3, l<sub>f</sub> = 2.2, W<sub>S</sub> = 1.6, l<sub>S</sub> = 1.8, h = 0.5, a = 3.6, b = 3.6, d = 1.8.</figcaption></figure>

The equivalent circuit is composed of six paralleled series cells, with partial mutual coupling between the cells. As in the previous example, a low-frequency cell is added, since the first resonance occurs below 10 GHz, and a high-frequency cell represents the higher modes. The four remaining cells account for the in-band behavior of the impedance.

Circuit parameters extraction yields a fitting error of -58 dB, to be compared to -48 dB for first-order extraction. Fig. 23 compares circuit and full-wave impedances, and Fig. 24 shows the reflection coefficients on the Smith chart. Table 8 presents the extracted circuit parameters, showing positive and negative coupling coefficients, whereas passivity (α + β ≤ 2) is satisfied in each cell without specific
<figure><img src="image_25.png" /><figcaption>**FIGURE 23.** Impedance of the fractional equivalent circuit (6 paralleled series cells, with partial mutual coupling) of the DRA, compared to full-wave result.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**FIGURE 24.** Comparison of the reflection coefficient (Γ) of the DRA.</figcaption></figure>

enforcement. Once again, the fractional equivalent circuit model still demonstrates more than excellent performances and accuracy.

As a summary, all of these different example results demonstrate the power, accuracy and versatility of the proposed fractional circuit model. However, up to this point, the fractional-order circuit modeling has only been validated through comparisons with numerical EM simulations. For completeness, the subsequent section will apply the proposed approach to measurement data of an actual antenna.

## V. EXPERIMENTAL VALIDATION

The preceding results illustrate the excellent capacity of fractional-order equivalent circuits in modeling wideband antenna impedances. However, it should be noted from the aforementioned results that first-order equivalent circuits (with partial mutual coupling) also demonstrate quite commendable performance, with a fitting error that is on

<table><caption>**TABLE 8.** Extracted circuit parameters for the DRA (Gray background: Fractional-Order; white background: First-Order).</caption><thead><tr><th>Element</th><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th><th>Cell 4</th><th>Cell 5</th><th>Cell 6</th></tr></thead><tbody><tr><td>C (fF)</td><td>335.74<br>172.55</td><td>55.242<br>137.88</td><td>6.4965<br>0.98144</td><td>34.937<br>36.735</td><td>5.0724<br>57.296</td><td>87.086<br>36.913</td></tr><tr><td>α</td><td>1.0011<br>1</td><td>1.0014<br>1</td><td>0.98436<br>1</td><td>0.99854<br>1</td><td>0.98349<br>1</td><td>0.99429<br>1</td></tr><tr><td>L (nH)</td><td>1.7361<br>2.8781</td><td>3.5443<br>1.6211</td><td>27.223<br>123.94</td><td>2.9235<br>2.581</td><td>13.973<br>1.0481</td><td>0.51059<br>1.3572</td></tr><tr><td>β</td><td>0.99546<br>1</td><td>0.99624<br>1</td><td>1.0016<br>1</td><td>0.99719<br>1</td><td>1.0137<br>1</td><td>1.0053<br>1</td></tr><tr><td>R (Ω)</td><td>0.10108<br>0.79551</td><td>0.24747<br>0.53036</td><td>69.873<br>582.69</td><td>3.16<br>4.7694</td><td>11.708<br>2.9998</td><td>1.8609<br>0.57877</td></tr><tr><td>Coupling</td><td>1↔2</td><td>2↔3</td><td>3↔4</td><td>4↔5</td><td>5↔6</td><td></td></tr><tr><td>M<sub>L</sub></td><td>-36.136fH<br>4.7751aH</td><td>-16.674fH<br>10.077aH</td><td>10.499fH<br>-133.05aH</td><td>-43.588fH<br>-95.698aH</td><td>31.34fH<br>-437.38aH</td><td></td></tr><tr><td>β<sub>M</sub></td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>M<sub>C</sub></td><td>11.489fF<br>51.158fF</td><td>-26.808aF<br>-206.68aF</td><td>339.66aF<br>613.02aF</td><td>305.02aF<br>-199.85aF</td><td>-7.1619fF<br>-10.242fF</td><td></td></tr><tr><td>α<sub>M</sub></td><td>0.95833<br>1</td><td>0.96756<br>1</td><td>1.0351<br>1</td><td>1.0263<br>1</td><td>0.96001<br>1</td><td></td></tr><tr><td>Error<br>(dB)</td><td>-58.4<br>-48.2</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<figure><img src="image_27.png" /><figcaption>**FIGURE 25.** Realized U-Slot microstrip patch antenna on Rogers RO4003C substrate. All dimensions in mm (via diameter is 1 mm, substrate thickness h is 1.524 mm).</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>**FIGURE 26.** Impedance of the fractional and first-order equivalent circuit (5 paralleled series cells, with complete mutual coupling) of the U-Slot Microstrip Patch Antenna, compared to measured impedance.</figcaption></figure>

average 10 dB larger than that of the fractional-order circuits. This prompts the question of understanding the reasons why fractional-order circuits exhibit superior performance in comparison to their first-order counterparts.
<figure><img src="image_29.png" /><figcaption>FIGURE 27. Comparison of the reflection coefficient (Γ) of the U-Slot Patch Antenna for fractional and first-order circuits vs. measurements.</figcaption></figure>

A key benefit of fractional-order circuits is their ability to provide precise control over frequency dispersion through the fractional-order parameters $\alpha$ and $\beta$. However, all of the preceding results are derived from numerical EM simulations. And it is a well-established fact that EM solvers introduce numerical dispersion and, as a result, numerical phase and group velocities may deviate from the correct values [117]. Consequently, it is legitimate to wonder whether the primary contribution of fractional-order circuits over first-order ones is to compensate for the numerical dispersion in results. In order to dispel this concern, the fractional-order (and first-order) modeling approach has been applied to experimental data, which are genuinely free of numerical dispersion.

Therefore, a U-slot microstrip patch antenna has been designed and realized on a 1.524 mm-thick Rogers RO4003C substrate (Fig. 25). This type of antenna combines patch resonances and slot resonances, which are likely to be coupled [113]. As outlined in Section III-A, the presence of two distinct resonators (the patch and the slot) both implies the mutual coupling of the different modes of each resonator, as well as the mutual coupling between the resonances of the patch and the U-slot. As the nature of these resonance modes is inherently different, there is no a priori knowledge of the coupling mechanisms. Consequently, in lieu of the partial mutual coupling that has been employed heretofore, the complete coupling scheme (as illustrated in Fig. 12) has been utilized during the extraction process.

The impedance of this antenna has been measured from 2 GHz to 6 GHz, and fractional and first-order equivalent circuit extraction has been performed. The extracted equivalent circuit is composed of five paralleled series cells (three cells for the in-band behavior of the impedance, and the two remaining cells representing the off-band low and high response).

<table><caption>TABLE 9. Extracted circuit parameters for the U-Slot Patch Antenna. Self-inductances and capacitances of the cells are reported in the 'diagonal' elements of the coupling matrices (Gray: Fractional-Order; white: First-Order).</caption><thead><tr><th>Element</th><th>Cell 1</th><th>Cell 2</th><th>Cell 3</th><th>Cell 4</th><th>Cell 5</th></tr></thead><tbody><tr><td>R</td><td>522.8mΩ<br>19.001 Ω</td><td>21.315mΩ<br>499.94mΩ</td><td>673.24mΩ<br>3.1779 Ω</td><td>5.8204 Ω<br>60.446 Ω</td><td>2.9712 Ω<br>10.863 Ω</td></tr><tr><td>Cell 1</td><td>M<sub>C1j</sub><br>α<sub>M1j</sub><br>M<sub>C1j</sub></td><td>1.511nF<br>0.7831<br>3.0764pF</td><td>4.5341nF<br>0.6956<br>101ff</td><td>332.17ff<br>-98.121ff<br>331.17ff</td><td>144.41pF<br>0.2935<br>57.899ff<br>25.761ff</td></tr><tr><td>Cell 2</td><td>M<sub>C2j</sub><br>α<sub>M2j</sub><br>M<sub>C2j</sub></td><td>4.5341nF<br>0.6956<br>101ff</td><td>1.1907pF<br>1.0449<br>3.3933pF</td><td>-29.897ff<br>0.9763<br>48.604ff</td><td>484.45ff<br>0.4866<br>35.708ff<br>-56.743ff</td></tr><tr><td>Cell 3</td><td>M<sub>C3j</sub><br>α<sub>M1j</sub><br>M<sub>C3j</sub></td><td>332.17ff<br>0.8026<br>-98.121ff</td><td>-29.897ff<br>0.9763<br>48.604ff</td><td>74.86ff<br>0.9366<br>343.82ff</td><td>771.86aF<br>0.5689<br>15.217ff<br>-18.328ff</td></tr><tr><td>Cell 4</td><td>M<sub>C4j</sub><br>α<sub>M1j</sub><br>M<sub>C4j</sub></td><td>144.41pF<br>0.2935<br>57.899ff</td><td>484.45ff<br>0.4866<br>35.708ff</td><td>771.86aF<br>0.5689<br>15.217ff</td><td>2.7775ff<br>0.1743<br>10.509ff<br>-4.5165ff</td></tr><tr><td>Cell 5</td><td>M<sub>C5j</sub><br>α<sub>M1j</sub><br>M<sub>C5j</sub></td><td>-8.0087ff<br>0.7734<br>25.761ff</td><td>-102.55ff<br>0.9737<br>-56.743ff</td><td>-29.424ff<br>0.8745<br>-18.328ff</td><td>5.1054ff<br>0.5091<br>-4.5165ff<br>46.341ff</td></tr><tr><td>Cell 1</td><td>M<sub>L1j</sub><br>β<sub>M1j</sub><br>M<sub>L1j</sub></td><td>4.3503pH<br>1.2868<br>3.2311nH</td><td>-13.214μH<br>0.6139<br>54.423fH</td><td>38.387pH<br>1.1352<br>89.235fH</td><td>26.852fH<br>0.9812<br>46.56fH</td></tr><tr><td>Cell 2</td><td>M<sub>L2j</sub><br>β<sub>M2j</sub><br>M<sub>L2j</sub></td><td>-13.214μH<br>0.6139<br>54.423fH</td><td>31.765μH<br>0.5897<br>1.4548nH</td><td>-292.96fH<br>0.9552<br>-22.363fH</td><td>88.074fH<br>0.9917<br>65.214fH</td></tr><tr><td>Cell 3</td><td>M<sub>L3j</sub><br>β<sub>M3j</sub><br>M<sub>L3j</sub></td><td>38.387pH<br>1.1352<br>89.235fH</td><td>-292.96fH<br>0.9552<br>-22.363fH</td><td>20.895nH<br>1.0517<br>5.6293nH</td><td>-57.538fH<br>0.9932<br>-51.022fH</td></tr><tr><td>Cell 4</td><td>M<sub>L4j</sub><br>β<sub>M4j</sub><br>M<sub>L4j</sub></td><td>26.852fH<br>0.9812<br>46.56fH</td><td>88.074fH<br>0.9917<br>65.214fH</td><td>-57.538fH<br>0.9932<br>-51.022fH</td><td>106.42mH<br>1.3530<br>135.41nH</td></tr><tr><td>Cell 5</td><td>M<sub>L5j</sub><br>β<sub>M5j</sub><br>M<sub>L5j</sub></td><td>-248.24fH<br>1.0266<br>49.613fH</td><td>-214.89fH<br>1.0054<br>45.259fH</td><td>-193.5fH<br>0.9504<br>63.734fH</td><td>-26.509fH<br>0.9796<br>-70.312fH<br>16.664nH</td></tr><tr><td colspan="2">Error</td><td colspan="4">-38.3</td></tr><tr><td colspan="2">(dB)</td><td colspan="4">-29.5</td></tr></tbody></table>

Table 9 reports the extracted circuit parameters. Due to the use of complete mutual coupling, the table provides direct presentation of the symmetrical coupling matrices $[M_L]$ and $[M_C]$ for better readability. The self-inductances and capacitances of each cell are reported in the “diagonal” elements of the matrices.

The fitting error for fractional-order extraction is -38.3 dB, which is not as optimal as in the case of numerical results. However, this must be considered in the context of measurement uncertainties and noise. Nonetheless, the systematic improvement attributable to fractional-order formalism remains consistent, with a first-order extraction error of -29.5 dB, which is 9 dB larger than the error of the fractional-order extraction. Fig. 26 compares the fractional-order and first-order equivalent circuits impedances with the measured impedance of the antenna. While the agreement between curves may seem excellent, it can be noted that the impedance of the first-order extracted circuit faces difficulties to follow the measured impedance
in the upper or lower frequencies of the frequency band. This discrepancy is more apparent in Fig. 27, which compares the measured reflection coefficient with fractional and first-order circuits responses. The differences are clearly visible. Despite the increased number of degrees of freedom provided by the complete mutual coupling, the first-order equivalent circuit does not attain the same level of accuracy in reproducing the frequency variations of the experimental response. The fractional-order circuit, while not perfect, possesses a greater capacity to manage the complex frequency behavior of the measured reflection coefficient. And, most significantly, these results demonstrate that the systematic improvement provided by fractional-order equivalent circuits over first-order models is unrelated to a better handling of numerical dispersion.

## VI. CONCLUSION

This paper introduces a new equivalent circuit model of antennas impedance based on the utilization of fractional-order reactive circuit elements. To the best of the authors' knowledge, this is the first time these fractional circuit elements are used for modeling antennas impedance. These elements are associated to build parallel or series resonant cells, each cell being devoted to one resonance in the impedance data. When directly applied to a dipole and patch antenna, the resulting equivalent circuits achieve extreme modeling accuracy. However, some limitations have been encountered when trying to model more complex antennas. A thorough investigation led to the conclusion that the modeling challenges were attributable to the coupling of EM modes. The addition of new circuit elements, i.e., fractional mutual inductances and capacitances, allowed accounting for this mode coupling.

The complete modeling methodology has then been applied to various antenna impedances, either originating from EM computations or measurements. In all cases, extracted fractional-order equivalent circuits have been compared to their first-order counterparts. First-order equivalent circuits (including mutual couplings) consistently delivered reliable results with respectable accuracy. However, despite the increased complexity resulting from additional parameters, fractional-order equivalent circuits demonstrated systematic improvement over first-order models, and achieved unprecedented accuracy. In this regard, fractional-order equivalent circuits can be considered as a generalization of first-order circuits, offering more degrees of freedom to reproduce the complex frequency dispersion laws in antenna or, more generally, distributed EM structures.

The basic principle – used throughout this paper – of associating individual resonances to specific cells provides a physical foundation for equivalent circuits. Furthermore, this allows for the natural inclusion of mutual couplings and off-band resonances, which both appear to be essential for comprehensive analysis. In contrast, classical network-oriented approaches such as rational expansion or poles/residues decomposition (e.g., vector fitting) are known to yield supe-

rior fitting accuracy. However, the resulting synthesized circuits offer no insight into the underlying physics, and may even contain negative-valued elements. This issue can be mitigated through the implementation of physics-informed approaches, such as in [118], constraining the circuits synthesized from poles and residues to stick to the observed resonances. Nevertheless, these formulations do not implicitly incorporate mutual coupling. Incidentally, it is also interesting to notice that most of the resonant circuits synthesized through these network-oriented approaches involve resistor-damped inductances or capacitances [26], whereas the presence of a resistive part in reactive elements naturally stems from fractional-order inductances or capacitances (3). Most notably, this resistance is frequency-dependent, offering greater flexibility and degrees of freedom.

Nonetheless, a clear connection between the fractional-order parameters ($\alpha$ and $\beta$) and EM phenomena is still to be established. The primary hypothesis is that, given the consistent correlation between fractional order and distributed phenomena, it should facilitate the compact expression of the EM response of antennas. On this basis, intuitive assumptions can be posited: the EM fields distribution at the resonance modes of antennas or the nonlocality of fringing fields imply distributed losses in the substrate. Furthermore, the asserted transcendental nature of the radiation resistance of antennas, i.e., the radiation losses, is another indication of the limitations of classical circuit approaches. Consequently, the resulting frequency dispersion is more likely to be expressed using fractional-order formulations. However, it is impossible to relate the fractional orders to a particular phenomenon or a physical parameter. In fact, a fractional-order coupled circuit has to be considered as a whole, since looking at all these resonance and radiation distributed phenomena from the antennas' driving point inevitably leads to a loss of information on the distributed nature of the said phenomena.

Fractional-order circuit elements with fractional mutual coupling lead to elegant frequency-domain expressions, but on the downside, their time-domain equivalent is far more complex [104] and has not been fully investigated yet. Therefore, time-domain circuit simulations are not possible for the moment.

Nevertheless, fractional cells, along with fractional mutual coupling, constitute a powerful, high-accuracy and compact approach for wideband frequency-domain equivalent circuit modeling of the impedance of antennas. The use of fractional-order elements seemingly bridges the gap between lumped and distributed systems but, in the absence of a comprehensive physical explanation of the relationship between fractional systems and distributed EM phenomena in antennas, fractional circuit elements are nothing more than an elegant means of expressing the EM response of antennas.

Moreover, the physical aspect of the fractional-order equivalent circuit model is especially well-suited for parametric macromodeling of antennas. This feature will be
used in the companion paper (Part II) to develop such macromodels. The devised macromodels are characterized by an extreme accuracy in parameters interpolation and frequency extrapolation, and a limited parameters extrapolation ability.

# APPENDIX

## FRACTIONAL-ORDER EQUIVALENT CIRCUITS IMPLEMENTATION IN CIRCUIT SIMULATORS

All of the fractional-order equivalent circuit computations in this article have been performed using a MATLAB implementation of (20) or (21). However, such equivalent circuit calculations can easily be carried out using classical circuit simulators, such as Keysight's Advanced Design System (ADS) or SPICE.

ADS implementation is predicated on the utilization of equation-based linear blocks: $Z1P_Equ$, $Z2P_Eqn$, $Y1P_Eqn$, and $Y2P_Equ$. The $Z1P_Eqn$ and $Y1P_Eqn$ blocks are used within each cell to respectively represent the impedance of the fractional inductance and the admittance of the fractional capacitance through their equations (1) and (2). $Z2P_Eqn$ and $Y2P_Eqn$ blocks are, in fact, $2 \times 2$ [$Z$] and [$Y$] matrices, respectively. In this context, they are here used as partial mutual coupling matrices, i.e., sub-matrices of the global mutual coupling matrices [$M_L$] (18) and [$M_C$] (19). This allows implementing the partial mutual coupling between two adjacent cells. Each $Z2P_Eqn$ block (resp. $Y2P_Eqn$ block) contains the fractional mutual inductance (resp. capacitance) equations (17). As the self-inductances and self-capacitances of the cells already are implemented through $Z1P/Y1P$ blocks, the main diagonal terms of the $Z2P/Y2P$ blocks are set to zero.

Fig. 28 depicts the ADS implementation of the circuit topology involving parallel cells in series. Specifically, it presents the circuit of Fig. 14, which corresponds to the microstrip patch antenna of Fig. 6. The 3-cell topology is readily discernible, along with the inductive (in red) and capacitive (in green) partial mutual couplings. The values of the different parts correspond to the data in Table 5 for the specified parameter values of P = 2.7 mm and h = 1.524 mm.

<figure><img src="image_30.png" /><figcaption>**FIGURE 28.** ADS implementation of the equivalent circuit (3 parallel cells in series) of the microstrip patch antenna of Fig. 6 (P = 2.7, h = 1.524mm). The $Z2P_Eqn$ blocks (in red) represent the partial inductive mutual coupling, whereas the $Y2P_Eqn$ blocks (in green) implement the partial capacitive mutual coupling.</figcaption></figure>

Similarly, Fig. 29 presents the ADS implementation of the paralleled series cells topology, corresponding to the loop antenna described in Fig. 16, with the parts values of Table 6. As these implementations are the exact translation of the circuit equations implemented in MATLAB, ADS simulations of the two above circuits yield the exact same results as the MATLAB calculations. Therefore, impedance plots are not provided herein, as the corresponding results are already plotted in Fig. 15 and Fig. 17, respectively.

<figure><img src="image_31.png" /><figcaption>**FIGURE 29.** ADS implementation of the equivalent circuit (4 series cells in parallel) of the Loop Antenna of Fig. 16. The $Z2P_Eqn$ blocks (in red) represent the partial inductive mutual coupling, and the $Y2P_Eqn$ blocks (in green) implement the partial capacitive mutual coupling.</figcaption></figure>

Fractional-order circuits implementation in SPICE is contingent upon the behavioral modeling capabilities of the employed software, more particularly on the incorporation of Laplace controlled sources. Native SPICE programs do not allow for such modeling, but some SPICE-based software programs incorporate these extensions. To the best of the authors' knowledge, Cadence PSpice [119], the free ADI LTSpice [120], and the late MicroCap, which is still freely available [121], are the only SPICE-based programs offering these possibilities. PSpice (and LTSpice) implementation of fractional-order elements is described in [122], for instance. However, we have chosen to present a MicroCap implementation, which we found to be more flexible and intuitive.

In addition to Laplace function sources, MicroCap provides analog behavioral voltage and current sources that perfectly meet our needs. These sources enable a straightforward description of the fractional-order circuit elements' AC behavior by simply assigning them the Laplace formula associated with the element. Therefore, fractional inductances (resp. capacitances) are implemented using voltage (resp. current) sources based on (1) (resp. (2)) equations. In a similar manner, inductive (resp. capacitive) mutual couplings are described by voltage (resp. current) sources governed by (17). Analogous to Fig. 28, Fig. 30 illustrates the MicroCap implementation of the microstrip patch antenna of Fig. 6.

The schematic is quite similar to the symbolic representation of Fig. 14. The E1, E3 and E6 voltage sources represent the inductances (the branch currents are monitored through 0 Ω resistors Rs1-Rs3), and the G1, G3, G5 current sources embody the capacitances. The other voltage sources (E2, E4, E5, and E7, in red) represent the inductive mutual
<figure><img src="image_32.png" /><figcaption>**FIGURE 30.** MicroCap Spice implementation of the equivalent circuit (3 parallel cells in series) of the microstrip patch antenna of Fig.6 (P = 2.7, h = 1.524mm). The En elements in red represent the partial inductive mutual coupling, whereas the Gn blocks in green implement the partial capacitive mutual coupling.</figcaption></figure>

couplings, whereas the current sources (G2, G4, and G6, in green) are the capacitive mutual couplings. The equivalent circuit being supplied with a 1 A frequency-independent current source (I1), the total impedance of the circuit is simply the AC voltage across this source. As mentioned earlier, the AC Spice calculations yield identical results to the MATLAB computations, therefore no impedance plot is displayed (cf. Fig. 15).

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<figure><img src="image_33.png" /></figure>

antennas and system modeling.

**FRANÇOIS TORRÈS** received the Ph.D. degree in electronics from the University of Limoges, France, in 1991. He joined CNRS as a Researcher, in 1993. From 1986 to 2021, he was with the Electromagnetic Compatibility (EMC) Research Team, XLIM Research Institute, University of Limoges. Since 2021, he has been with the “Antennas and Signals” Research Team, XLIM Research Institute, where he works on numerical simulation, EM/circuit interaction modeling, and mainly on

<figure><img src="image_34.png" /></figure>

**MARC THÉVENOT** received the M.Sc. degree in microwaves and the Ph.D. degree in electronics from the University of Limoges, France, in 1995 and 1999, respectively. He joined CNRS, in 2001. He is currently a Researcher with the XLIM Research Laboratory, RF Systems Group, University of Limoges. His current research interests include antenna arrays, reconfigurable active antennas, parasitic element antennas, and reflectarrays.

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