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# Z-SpecNNet: A Real-Time Embedded NN-Based Parameters Estimation for WPT Systems

Thomas Boulanger, Vincenzo Cirimele, Mattia Ricco, Eric Monmasson

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Thomas Boulanger, Vincenzo Cirimele, Mattia Ricco, Eric Monmasson. Z-SpecNNet: A Real-Time Embedded NN-Based Parameters Estimation for WPT Systems. IEEE Transactions on Industrial Electronics, In press, 10.1109/TIE.2024.3515264. [hal-04862736](https://hal.science/hal-04862736v1)

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# Z-SpecNNet: A Real-Time Embedded NN-Based Parameters Estimation for WPT Systems

Thomas Boulanger <sup>D</sup>, Vincenzo Cirimele <sup>D</sup>, Senior Member, IEEE,
Mattia Ricco <sup>D</sup>, Senior Member, IEEE, and Eric Monmasson <sup>D</sup>, Senior Member, IEEE

**Abstract**—This article proposes a fully embedded real-time mutual inductance and load estimation technique (Z-SpecNNet) applicable to wireless power transfer (WPT) systems. The technique consists of a three-step process, starting with online noise injection from the supplying converter to excite the system over a wide bandwidth. During the noise injection, the voltage and current at the converter output are recorded, allowing the system impedance to be calculated by fast Fourier transform. Finally, a neural network computes an estimate of the desired parameters. In this work, the Z-SpecNNet is applied to a series-series compensated system as it is one of the most popular compensation topologies in the literature and because it is the topology for which the information of load and mutual coupling result most correlated and therefore more difficult to estimate. The proposed Z-SpecNNet offers significant advantages because impedance spectroscopy is a straightforward and model-free method for characterizing system behavior. Furthermore, the neural network can be rapidly trained on a known transfer function. The technique has been demonstrated to be effective on a low-cost microcontroller that integrates the control of the converter. Experimental results indicate a mean relative estimation error of 7.81% with a total estimation time of 85 ms.

**Index Terms**—Embedded control, embedded neural network, inductive power transmission (IPT), real-time parameters estimation, wireless power transmission (WPT).

Received 24 June 2024; revised 22 September 2024 and 29 October 2024; accepted 30 November 2024. (Corresponding author: Vincenzo Cirimele.)

Thomas Boulanger is with N. Tesla Department, École Nor- male Supérieure Paris-Saclay, 91190 Gif-sur-Yvette, France (e-mail: thomas.boulanger@ens-paris-saclay.fr).

Vincenzo Cirimele and Mattia Ricco are with the Department of Elec- trical, Electronic, and Information Engineering, University of Bologna, 40126 Bologna, Italy (email: vincenzo.cirimele@unibo.it; mattia. ricco@unibo.it).

Eric Monmasson is with the Systèmes et Applications des Technolo- gies de l'Information et de l'Energie laboratory, 91190 Gif-sur-Yvette, France (e-mail: eric.monmasson@u-cergy.fr).

Digital Object Identifier 10.1109/TIE.2024.3515264

## I. INTRODUCTION

W HETHER it is for charging electric vehicles (EVs) [1], unmanned aerial vehicles (UAVs) such as drones [2] or medical implants [3], wireless power transfer (WPT) sys- tems have grown in popularity during the past decade due to various advantages such as improved user experience, safety, and flexibility. Currently, WPT systems need a bi-directional communication system to properly manage and optimize the charging process [4]: the receiver side periodically reports var- ious parameters such as the load voltage, and the received power to the transmitter side which then controls the charging process accordingly. This communication system increases the number of components, cost, and complexity of both hardware and software. Therefore many research works focus on methods and techniques aimed at eliminating the communication system.

The equivalent load and the mutual inductance are the two pa- rameters that need to be known to properly control the charging process as they can vary due to misalignment of the coils and the state of charge of the battery. Indeed several control techniques such as [5], [6], [7], and [8] require an estimation of such param- eters to manage the charging process. Other parameters such as the coils and compensation circuit are typically supposed to be known as the charging system should follow a standard such as the Qi or the SAE J2954 [9], [10]. On these premises, various methods have been proposed to estimate mutual inductance and equivalent load.

Some of these methods rely on additional hardware. For example, Su et al. [11] use an additional capacitor and two MOSFETs to change the tuning of the transmitter compensation circuit, allowing both the mutual inductance and the equivalent load to be monitored. Li et al. [12] instead propose the use of a third coil to measure the parameters and control the charging process. All these techniques provide fast and accurate results, but at the cost of increased cost and complexity and reduced reliability of the system.

Offline identification techniques such as energy injection [13] are based on observing the system self-oscillation after injection of pulses to estimate the parameters. The method proposed in [14], takes advantage of the soft-start process to recover the parameters. These techniques do not require additional hard- ware, making them more suitable for parameter identification. However, they only work when the system is switched OFF, so
they cannot follow the variation of the parameters during the
charging process.

Topology-specific identification techniques solve model-
specific equations to recover the parameters. In [15], the res-
onance frequency tracking controller and system equations are
used to compute the parameters of a series–series compensated
system. Su et al. [16] solves similar equations but for the series–
parallel compensation topology and Guo et al. [17] for the
LCC–LCC compensation topology. Whilst these methods are
fast and accurate, they have a major drawback in that they
cannot work with any other topology. Hence, they cannot be
used independently of the system architecture. In relation to
this aspect, it is worth noting that the current standards do not
advocate for a specific compensation topology.

Curve fitting-based identification techniques overcome this
limitation by measuring the system impedance and then com-
puting the system parameters by solving an inverse problem via
gradient descent. In [18], frequency sweeping is used to recover
the system impedance and then gradient descent is used to com-
pute the parameters. However, frequency sweeping forces the
system to operate at frequencies other than resonance. This can
affect the performance of the system and introduce unwanted
stresses on the reactive components. In [19], pulse density
modulation is used to recover the impedance while keeping the
inverter frequency at the resonance frequency. However, it is
claimed that gradient descent takes between 2 and 5 s, making
these techniques poorly suited to real-time applications. This
article proposes an enhanced curve fitting-based identification
technique that employs noise injection in lieu of frequency
sweeping and a neural network in lieu of gradient descent. The
noise injection does not necessitate a change in the inverter fre-
quency and is capable of recovering the system impedance with
greater rapidity and across a broader bandwidth than frequency
sweeping or pulse density modulation. The neural network is
also faster than gradient descent and requires only a few mil-
liseconds to compute the parameters in a deterministic manner,
as opposed to iterative methods, rendering it well-suited for
sensitive real-time applications.

The results obtained show a total estimation time of about
85 ms. This estimation time could make this identification
technique suitable for quasi-static charging of EVs, such as at
traffic lights or bus stops, where the vehicle is almost stationary
but slow variations of the parameters can occur. The param-
eters can therefore be considered constant during the signal
measurement (which lasts a few milliseconds), as the vehicle
and battery dynamics [20] are an order of magnitude slower
than the measurement time. This technique is based on a three-
step process that starts with a noise injection that stimulates
the system and then computes the system impedance from the
voltage and current signals. Finally, the values of mutual in-
ductance and equivalent load are estimated from the impedance
using a neural network. As shown in the Experimental Results
section (Section V), this technique can be embedded into the
dc/ac converter controller requiring no additional hardware.
This technique has been named Z-SpecNNet precisely because
it is based on impedance spectroscopy (Z-Spec) and the use of
a neural network (NNet).

<figure><img src="image_3.png" /><figcaption>Fig. 1. Circuit diagram of an SS compensated WPT system.</figcaption></figure>

The remainder of the article is organized as follows. Sec-
tion II briefly introduces the WPT system and its descrip-
tive equations. Section III details the different steps of a
Z-SpecNNet. The verification of the technique is summarized
in Sections IV and V where the experimental setup and the
derived results are presented and discussed. Last, a conclusion
with proposed improvements is drawn in Section VI.

II. SYSTEM ANALYSIS

The heart of a WPT system lies in the two coupled coils,
named the transmitter and the receiver. The main components
of the system are depicted in Fig. 1 in which the source is an
H-bridge having a dc bus voltage Vdc, all transmitter-side com-
ponents are denoted by subscript 1, and receiver-side compo-
nents are denoted by subscript 2. The coils are represented
through their self-inductances L1 and L2 and the mutual in-
ductance M. The coils are connected to the capacitors C1 and
C2. The two resistances R1 and R2 represent the equivalent
series resistances (ESRs) of coils and capacitors. The load at
the receiver side is represented through the equivalent resistance
RL. Due mainly to the large air gap between transmitter and
receiver coils, their mutual coupling is weak, i.e., the coupling
factor k is usually lower than 0.5. This forces WPT systems to
operate at frequencies that go well beyond the tens of kilohertz.
Consequently, the use of the capacitors is necessary to com-
pensate for the coils impedance and allow maximizing power
transfer. By choosing the value of capacitors to compensate for
the inductive components on the two sides of the circuit at a
single frequency, i.e., at a system resonance frequency, it is
possible to minimize the VA rating of the H-bridge. At the same
time, the converter can work under soft-switching conditions to
the benefit of efficiency [21].

Compensation can be carried out by arranging different
topologies. The simplest ones only have a capacitance on the
transmitter side and one on the receiver side either in series
or in parallel with the coil leading to the four classic compen-
sation topologies [22]: series–series (SS), series–parallel (SP),
parallel–series (PS), and parallel–parallel (PP). These basic
topologies can be combined into more complex topologies,
such as the LCC-S or the LCC-LCC [23], [24] each of which
has distinctive features. In general, for different topologies, the
resonance frequency of the system is strongly dependent on
both load and mutual inductance variations. The only exception
is the series–series compensation. While this feature makes
SS compensation among the most widely used [25], it makes
parameter identification much more complex. This is why the
<figure><img src="image_4.png" /><figcaption>Fig. 2. Impact of variation of *M* on the system equivalent impedance.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. Impact of variation of *R*<sub>L</sub> on the system equivalent impedance.</figcaption></figure>

analysis developed in this article is based precisely on the SS compensation depicted in Fig. 1.

The impedance that the system shows toward the dc/ac converter can be expressed as [22]

$$ \bar{Z}_{\text{eq}} = R_1 + j \left( \omega L_1 - \frac{1}{\omega C_1} \right) + \frac{\omega^2 M^2}{R_2 + j \left( \omega L_2 - \frac{1}{\omega C_2} \right) + R_L} \quad (1) $$

At the resonance frequency, the components in round brackets compensate by canceling each other out. The impedance then becomes purely real and its value is simultaneously and exclusively dependent on the values of *M* and *R*<sub>L</sub>. The effect that varying these parameters has on the system is shown in Figs. 2 and 3 representing amplitude and phase of *Z*<sub>eq</sub> as frequency changes. These graphs are made using the values of the system used for experimental validation and indicated in Table I of Section IV. The nominal conditions of the system refer to a mutual inductance *M* = 4.8 µH and an equivalent load *R*<sub>L</sub> = 3 Ω. The system's resonance frequency is chosen as 85 kHz.

It can be noted that small variations of load and mutual inductance have a noticeable impact on the impedance. Most of

<table><caption>TABLE I
REFERENCE VALUES OF THE TEST SETUP</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Bus voltage V<sub>dc</sub></td><td>12 V</td></tr><tr><td>Resonance frequency f<sub>0</sub></td><td>85 kHz</td></tr><tr><td>Transmitter self-inductance L<sub>1</sub></td><td>24 μH</td></tr><tr><td>Receiver self-inductance L<sub>2</sub></td><td>24 μH</td></tr><tr><td>Transmitter compensation cap C<sub>1</sub></td><td>146 nF</td></tr><tr><td>Receiver compensation cap C<sub>2</sub></td><td>146 nF</td></tr><tr><td>Transmitter resistance R<sub>1</sub></td><td>75 mΩ</td></tr><tr><td>Receiver resistance R<sub>2</sub></td><td>75 mΩ</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Flowchart of a Z-SpecNNet estimator.</figcaption></figure>

the variations occur at the resonance frequency *f*<sub>0</sub> and remain pronounced in the frequency range [*f*<sub>0</sub>/1.7; *f*<sub>0</sub> · 1.7].

As commonly done in the domain of WPT [26], the H-bridge output voltage is controlled via phase shift modulation. This modulation is here performed employing a normalized phase-shift angle α ∈ [0; 1] where 0 would create a constant 0 V output and 1 would correspond to a full square wave.

### III. IDENTIFICATION TECHNIQUE

The Z-SpecNNet is based on a three-step process described through the flowchart shown in Fig. 4. The identification starts by injecting white noise using a pseudo-random binary sequence (PRBS) in order to stimulate the system. Then the system impedance is computed from the recording of the current and voltage signals. Last, a neural network (NN) estimator is used to compute an estimation of the wanted parameters. The different steps are separately discussed in the following sections.

#### A. Noise Injection

As the WPT system is a linear time-invariant (LTI) system, it is possible to stimulate it on a wide bandwidth by injecting white noise in its input. A common way to do so is to use a PRBS in a maximum length sequence (MLS) configuration [27], [28], [29], [30], [31] as it is an excellent approximation of a discrete white noise with a variance σ<sup>2</sup> = 1. From (2) and (3) that represent the autocorrelation function of a discrete white
<figure><img src="image_7.png" /><figcaption>Fig. 5. Diagram of a PRBS-10 implemented with an LFSR.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Diagram of the noise injection.</figcaption></figure>

noise and the autocorrelation function of a PRBS respectively, it can be seen that the two become more similar as the length L of the PRBS increases. n denotes the n-th value of the sequence

$$R_{\text{WN}}[n] = \begin{cases} \sigma^2 & \text{if } n = 0 \\ 0 & \text{if } n \neq 0 \end{cases} \quad (2)$$

$$R_{\text{PRBS}}[n] = \begin{cases} 1 & \text{if } n = 0 \\ -1/L & \text{if } 0 < n < L. \end{cases} \quad (3)$$

A PRBS is usually implemented with a linear-feedback shift register (LFSR) (see Fig. 5) enabling a simple, fast, and efficient implementation. At each step, only the least significant bit (LSB) is considered to output a binary value. This value is then offset in order to have either +e or -e where e is a chosen amplitude (Fig. 6).

Unlike frequency sweeping which stimulates the system one harmonic at a time, the injected noise signal directly stimulates multiple frequencies enabling a faster recovery of the system impedance. As shown in [32] and highlighted with the power spectral density plot in Fig. 7, no spectral power is injected at the frequency at which the PRBS is injected and at the multiples of this frequency. Therefore, to stimulate the system in the bandwidth of interest, it is mandatory to inject the PRBS at a higher frequency than the system working frequency. To do so, the PRBS is injected on each half wave of the voltage signal $v_1(t)$ which enables an injection at a frequency equal to twice the resonance frequency of the system. In the present case, with a resonance frequency of 85 kHz, this means injecting noise at 170 kHz. This allows the stimulation of the whole bandwidth of interest.

As recommended by [33], the length of the PRBS is chosen so as to obtain an injection period longer than the system time response. As the sequence length of a PRBS of N bits is L = 2^N - 1, a PRBS-10 is chosen with a sequence length of 1023 values which equals an injection period of 1023/170 000 = 6.0 ms. The action on the voltage modulation during the noise injection can be appreciated through Fig. 8 where PRBS[n] is the n-th value of the PRBS. During the noise injection, two sequences are injected on top of the main modulation for a total duration of 12 ms. For measurement stability, the current $i_1(t)$ and voltage $v_1(t)$ start to be recorded only during the second sequence, this let the time for the system to set in a steady state.

<figure><img src="image_9.png" /><figcaption>Fig. 7. Power spectral density of a PRBS-10 injected at 85 and 170 kHz.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. H-bridge output voltage modulation during the noise injection.</figcaption></figure>

The amplitude of the noise needs to be chosen by considering several aspects. It needs to be adjusted high enough so that the analog-to-digital converters (ADCs) can effectively read the signals to record the system impedance. On the other hand, it needs to be low enough to avoid saturation of the sensors, saturation of the command or non-linearities of the system behavior. Finally, it is necessary to make sure that the current increase caused by noise injection remains well below the maximum rating of the switches.

## B. Impedance Computation

To obtain the system impedance from voltage and current measurement, the FFT of both recorded signals is first computed. The impedance is then recovered using equation (4) in which $V(j\omega)$ is the FFT of the voltage signal, $I(j\omega)$ the FFT of the current signal, and $Z(j\omega)$ is the impedance

$$Z(j\omega) = \frac{V(j\omega)}{I(j\omega)}. \quad (4)$$

In practice, the real fast Fourier transform (RFFT) [34] is computed instead of the FFT enabling a reduction in both computation time and RAM consumption.

Clearly, the raw impedance $Z(j\omega)$ obtained from this first step results in noise (see Fig. 9), hence a moving average filter is used to smooth the impedance samples and have a
<figure><img src="image_11.png" /><figcaption>Fig. 9. Effect of the smoothing filter with a smoothing factor $S = 1.06$. This result refers to the experimental system described in Section IV.</figcaption></figure>

more reliable identification. The filter is based on a smoothing factor $S$, and the impedance at each angular frequency $\omega_i$ is then smoothed with a fixed-size window $[\omega_i/S; S \cdot \omega_i]$ on the logarithmic scale. The smoothed impedance is then computed as

$$Z_{\text{smooth}}(j\omega_i) = \frac{1}{|A|} \sum_{\omega_k \in A} Z(j\omega_k) \quad (5)$$

where $A = \{\omega_k | \omega_k \in [\omega_i/S; S \cdot \omega_i]\}$. The smoothing factor needs to be tuned in order to remove noise while maintaining the details of the impedance. An example of the resulting smoothed impedance is shown in Fig. 9.

### C. Neural Network Estimator

The final estimation of $R_L$ and $M$ from the impedance $Z_{\text{smooth}}(j\omega)$ is achieved by training a neural network estimator. The input tensor of the NN is composed of 15 pairs of values (amplitude and phase) from 15 points of the impedance equally spread across the bandwidth of interest. The desired parameters are made available in the output tensor. A synthesis of this process is provided through Fig. 10.

In this work, as in those already presented in the literature and mentioned in the Introduction, the working frequency value (i.e., the resonance frequency) as well as the self-inductance values of the coils are assumed to be known. This assumption reflects the current state of standardization of these systems. In fact, different standards such as the mentioned SAE J2954 and Qi, already provide precise references for coil geometries and materials as well as expected self-inductance values [35]. From these, it is possible to derive the relative values of compensation capacitors and an estimation of the ESRs.

Hence, a dataset of 100 000 data is created using (1) and the known parameters ($L_1, L_2, C_1, C_2, R_1$, and $R_2$). The wanted parameters are randomly chosen in realistic ranges, $R_L \in [0.1 \, \Omega; 10 \, \Omega]$ and $M \in [1.2 \, \mu\text{H}; 24 \, \mu\text{H}]$, corresponding to a coupling factor between 0.05 and 1. A small Gaussian noise was added to each feature of the input tensor to improve robustness [36]. Data are iteratively created and added to the

dataset until the desired dataset size is reached (see Fig. 11). As the neural network is intended to be embedded in a small microcontroller, it is proposed to keep a relatively simple model composed of only four fully connected layers composed of 128 neurons each with the rectified linear unit (ReLU) as the activation function. Different models have been tested and the four layers 128 neurons was found to achieve a good speed-to-accuracy ratio. As the inference time was already very low, no quantization was done on the model, weights and biases are therefore encoded with the single-precision floating-point format (float32).

Unlike the least-square approximation or the Newton-Raphson method, an NN estimator is not based on an iterative process and can estimate the parameters in a deterministic time. The operations needed are also very simple [i.e., addition, multiplication, and ReLU($x$) = $\max(0, x)$] enabling a fast computation of the parameters. The main difference is that once trained offline, the neural network has learned to minimize the error while the least-square approximation does an online iterative curve fitting.

It is worth noting that the NN can also be re-trained to estimate other parameters such as the receiver resonance frequency which could change due to capacitor aging. This feature could be interesting according to the indications of the mentioned standards which advocate for both sides to agree on an operating frequency before starting the power transfer. Last, as the equivalent impedance equation is known for all other compensation topologies [22], [23], [24] a dataset could easily be created to fit another compensation topology [37] making the Z-SpecNNet estimator extremely versatile and widely applicable.

### IV. EXPERIMENTAL SETUP AND IMPLEMENTATION

For experimental validation of the Z-SpecNNet, the experimental setup shown in Fig. 12 was implemented. The coils used, are commercially available coils compliant with the Qi standard. They are both circular and laid on a plate of ferromagnetic material. As can be seen from the picture in Fig. 13, these coils have a different size and a different number of turns. The smaller coil, used as the receiver coil, has the windings distributed over two layers, while the larger coil, used as the transmitter coil, has the windings distributed over a single layer. Despite these differences, both coils are made so that they present an identical self-inductance value of 24 $\mu$H with an uncertainty of $\pm 10\%$. Each coil self-inductance is compensated with a capacitor bank whose value is chosen to achieve a system resonance frequency of 85 kHz. The air gap between the coils is regulated by using 5 mm thick plexiglass spacers. The H-bridge used to power the system consists of two half bridges Infineon EVAL_1EDF_G2_HB_GAN based on CoolGaN transistors. The control of the system as well as the Z-SpecNNet are implemented on the embedded platform Nucleo-G474RE built around an STM32-G474RE microcontroller running at 170 MHz and equipped with a custom shield. The measurement of the H-bridge output current is performed through the LEM LA 55-P current transducer integrated into the custom shield
<figure><img src="image_12.png" /><figcaption>Fig. 10. Working principle of the neural network estimator.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11. Flowchart of the creation of the dataset for the NN training.</figcaption></figure>

while the output voltage is indirectly computed from the states of the PWM commands. The PRBS is implemented with a software LFSR, a subroutine is then called after each switching to add the appropriated noise value to the phase shift modulation. The initialization of the identification procedure is triggered via a push button still integrated into the shield (simply indicated as acquisition board in Fig. 12). To make it easier to read, the parameters obtained at the end of the identification are shown in a small Oled display mounted on the custom shield. The microcontroller then manages the PWM signals generation for the H-bridge, the injection of the noise, the recording of the signal, the computation of the parameters $R_L$ and $M$, and their displaying.

The main values of the experimental WPT system setup are collected in Table I. The custom shield PCB source files and

<figure><img src="image_14.png" /><figcaption>Fig. 12. Experimental setup.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13. Detail of the transmitter (on the right) and receiver (on the left) coils. Both coils are Qi-compliant and present the same value of self-inductance.</figcaption></figure>

the source code for the embedded platform and neural network training are made publicly available<fnref n="1" />.

<footnote n="1">¹Shield PCB source files, source code for the embedded platform and neural network training is publicly available at https://github.com/LeRatonLaveurSolitaire/Wireless-power-transfer-system_nucleo</footnote>
<figure><img src="image_16.png" /><figcaption>Fig. 14. H-bridge output current signal during the noise injection acquired via an oscilloscope.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 15. Detail of the H-bridge output voltage and current signals during the noise injection acquired via an oscilloscope.</figcaption></figure>

In this implementation, the H-bridge control is handled in open loop, and for each different relative position of the coils, the phase-shift angle $α$ is set to have an output current of about 2 amperes peak before the starting of the identification procedure. The amplitude of the PRBS was fixed at 0.15, as this was found to be the lowest amplitude that could accurately recover the impedance. This resulted in an increase of 18.5% in the current amplitude, which corresponds to a peak current of 2.37 A during the noise injection.

The variation of the current signal during the noise injection can be observed in Fig. 14 while a detail of both the voltage and current is given in Fig. 15 which highlights the impact of the noise injection on the modulation of the H-bridge output voltage.

The signals are recorded with 12-bit ADC at the sampling frequency of 1 MHz for 4 ms during the injection of the second sequence of the PRBS. The FFT is computed using the open-source CMSIS DSP library [38] which is optimized for ARM microprocessors. Single Instruction Multiple Data (SIMD) is used to perform parallel processing and vector operations enabling a faster computation of the parameters.

Raw impedance values obtained in the range of interest $[f_0/1.7; f_0 \cdot 1.7] = [50 \text{ kHz}; 144.5 \text{ kHz}]$ are filtered by setting a smoothing factor $S$ of 1.06. To further reduce computation

<table><caption>TABLE II
TRAINING PARAMETERS OF THE NN MODEL</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Learning rate</td><td>0.0001</td></tr><tr><td>Number of epoch</td><td>100</td></tr><tr><td>Batch size</td><td>128</td></tr><tr><td>Training dataset size</td><td>80 000 data (80%)</td></tr><tr><td>Test dataset size</td><td>20 000 data (20%)</td></tr><tr><td>Loss function</td><td>Mean squared error (MSE)</td></tr><tr><td>Optimizer</td><td>Adam [39]</td></tr></tbody></table>

<figure><img src="image_18.png" /><figcaption>Fig. 16. Computation timeline of an identification (total time = 84.3 ms).</figcaption></figure>

time, the smoothing filter is not applied to all raw samples, but only to the ones used to construct the NN input tensor.

The neural network is trained on a computer before being embedded in the microcontroller. The training parameters are shown in Table II. During the training, the loss over the training dataset is monitored to ensure a good learning of the model. The accuracy over the test dataset is also monitored to prevent over-fitting.

The whole computation time required to estimate the parameters is detailed in Fig. 16. As it can be seen, the neural network only required 8 ms which is a huge improvement compared to the 2 to 5 s required by the least-square approximation [19] on a computer. The entire identification process takes 84.3 ms.

## V. EXPERIMENTAL RESULTS

To test the accuracy of the estimation, 12 configurations have been tested with three different values of mutual inductance obtained by varying the distance between the coils. The resulting couples of the air gap and values of mutual inductance are (15 mm, 8.22 μH), (20 mm, 6.01 μH) and (25 mm, 4.68 μH). For each different coil position, four different load resistance values were tested whose nominal values are 1 Ω, 3 Ω, 5 Ω, and 7.5 Ω. The reported mutual inductance values were measured through an LCR meter. Similarly, the actual resistive load values were measured with the same instrument at the working frequency of the WPT system.

The measured couples of values and the corresponding estimation for each configuration can be found in Table III. The relative error of the estimation is reported in Table IV.

A comparison between the impedance obtained by the microcontroller and the one expected by the model [i.e., (1)] can be seen in Fig. 17. The geometric mean of the relative error between the measured impedance and the model impedance results in about 12.4% over the bandwidth of interest.
<table><caption>TABLE III
PARAMETERS ESTIMATION FOR EACH
COILS CONFIGURATION</caption><thead><tr><th colspan="2">Measured Value Pair</th><th colspan="2">Estimated Value Pair</th></tr><tr><th>R<sub>L</sub> (Ω)</th><th>M (μH)</th><th>R<sub>L</sub> (Ω)</th><th>M (μH)</th></tr></thead><tbody><tr><td>1.04</td><td>4.68</td><td>1.31</td><td>4.66</td></tr><tr><td>1.04</td><td>6.01</td><td>1.25</td><td>5.52</td></tr><tr><td>1.04</td><td>8.22</td><td>1.13</td><td>7.15</td></tr><tr><td>3.00</td><td>4.68</td><td>3.39</td><td>4.94</td></tr><tr><td>3.00</td><td>6.01</td><td>3.18</td><td>5.94</td></tr><tr><td>3.00</td><td>8.22</td><td>2.80</td><td>6.88</td></tr><tr><td>4.99</td><td>4.68</td><td>5.19</td><td>5.64</td></tr><tr><td>4.99</td><td>6.01</td><td>5.65</td><td>6.88</td></tr><tr><td>4.99</td><td>8.22</td><td>5.14</td><td>7.74</td></tr><tr><td>7.46</td><td>4.68</td><td>7.17</td><td>5.53</td></tr><tr><td>7.46</td><td>6.01</td><td>8.50</td><td>7.19</td></tr><tr><td>7.46</td><td>8.22</td><td>6.77</td><td>7.83</td></tr></tbody></table>

<table><caption>TABLE IV
RELATIVE ESTIMATION ERROR FOR EACH CONFIGURATION</caption><thead><tr><th>Measured Value Pair<br>(R<sub>L</sub>, M)</th><th>Relative Error<br>on R<sub>L</sub></th><th>Relative Error<br>on M</th><th>Global Relative Error<br>(Geometric Mean)</th></tr></thead><tbody><tr><td>(1.04 Ω, 4.68 μH)</td><td>25.96%</td><td>0.43%</td><td>3.33%</td></tr><tr><td>(1.04 Ω, 6.01 μH)</td><td>20.19%</td><td>8.15%</td><td>12.83%</td></tr><tr><td>(1.04 Ω, 8.22 μH)</td><td>8.65%</td><td>13.02%</td><td>10.61%</td></tr><tr><td>(3.00 Ω, 4.68 μH)</td><td>13.00%</td><td>5.56%</td><td>8.50%</td></tr><tr><td>(3.00 Ω, 6.01 μH)</td><td>6.00%</td><td>1.16%</td><td>2.64%</td></tr><tr><td>(3.00 Ω, 8.22 μH)</td><td>6.67%</td><td>16.30%</td><td>10.42%</td></tr><tr><td>(4.99 Ω, 4.68 μH)</td><td>4.01%</td><td>20.51%</td><td>9.07%</td></tr><tr><td>(4.99 Ω, 6.01 μH)</td><td>13.23%</td><td>14.48%</td><td>13.84%</td></tr><tr><td>(4.99 Ω, 8.22 μH)</td><td>3.01%</td><td>5.84%</td><td>4.19%</td></tr><tr><td>(7.46 Ω, 4.68 μH)</td><td>3.89%</td><td>18.16%</td><td>8.40%</td></tr><tr><td>(7.46 Ω, 6.01 μH)</td><td>13.94%</td><td>19.63%</td><td>16.54%</td></tr><tr><td>(7.46 Ω, 8.22 μH)</td><td>9.25%</td><td>4.74%</td><td>6.62%</td></tr></tbody></table>

<figure><img src="image_19.png" /><figcaption>Fig. 17. Impedance measured by the microcontroller ($R_L = 3 \ \Omega$, $M = 6.01 \ \mu H$).</figcaption></figure>

The results show an average error in the estimation of 7.81%. The maximum error of 25.96% was found for the lowest load ($R_L = 1.04 \ \Omega$ measured) and lowest mutual ($M = 4.68 \ \mu H$ measured). It is interesting to note that, the same conditions led to the lowest error on the estimation of the mutual inductance.

The achieved average error is higher than that reported by other curve-fitting estimations. E.g., [18] claims an estimation error lower than ±2% while [19] reports an error of less than ±5%. Nevertheless, the estimation error is within the tolerance

<table><caption>TABLE V
COMPARISON OF THE MAIN FEATURES OF THE IDENTIFICATION
TECHNIQUES PRESENTED IN THE LITERATURE</caption><thead><tr><th rowspan="2">Feature</th><th colspan="4">Identification Technique</th></tr><tr><th>Energy injection [13]</th><th>Frequency sweep [18]</th><th>Pulse density modulation [19]</th><th>Z-SpecNNet</th></tr></thead><tbody><tr><td>Online estimation</td><td>✗</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Compatible with any compensation topology</td><td>✗</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>No frequency variation</td><td>✗</td><td>✗</td><td>✓</td><td>✓</td></tr><tr><td>Computation in deterministic time</td><td>✓</td><td>✗</td><td>✗</td><td>✓</td></tr><tr><td>Measured impedance points and range</td><td>-</td><td>31 points around f<sub>0</sub></td><td>14 points around f<sub>0</sub></td><td>f<sub>0</sub>/1.7-1.7f<sub>0</sub> continuous</td></tr><tr><td>Estimation time</td><td>Not specified</td><td>Not specified</td><td>2 to 5 s</td><td>84.3 ms</td></tr><tr><td>Estimation average error</td><td>2%</td><td>2%</td><td>5%</td><td>7.81%</td></tr></tbody></table>

range of other parameters such as the compensation capacitors (±10%). Consequently, the control system can operate with a set of parameters exhibiting the same tolerance.

In case better accuracy is required, a Z-SpecNNet estimator can be used to quickly set the initial values for a least-square approximation. As the estimated values will be closer to the final values than fixed initial values, the Z-SpecNNet estimator should reduce the number of iterations until convergence and therefore computation time.

In case a faster estimation is required, the PRBS could be changed for one with a lower number of bits such as a PRBS-8. This will reduce the noise injection time and the length of the recorded signals thus reducing the computation time of the FFT, impedance, and smoothing. However, this might impact the accuracy of the recovered impedance and therefore the accuracy of estimated parameters.

In case both the accuracy and estimation time require enhancement, the hardware can be modified to accommodate a faster microcontroller.

## A. Comparisons of Z-SpecNNet and Other Existing Methods

At the conclusion of the work presented, Table V provides a comparative summary of the advantages of the Z-SpecNNet in relation to the other discussed identification techniques, outlined in the Introduction section.

Compared with identification techniques that rely on additional hardware [11], [12], the Z-SpecNNet doesn't require any additional hardware and is able to run on low-end microcontrollers eliminating the need for an additional computing unit. Compared to offline identification techniques [13], [14], the Z-SpecNNet is able to work online and therefore follow the variation of the parameters during the charging process. While topology-specific identification techniques [15], [16], [17] are faster than the Z-SpecNNet, they cannot work with topologies other than the one for which they were designed, whereas a Z-SpecNNet could be trained to estimate the parameters for any topology. Compared with other curve fitting techniques [18], [19], Z-SpecNNet is able to recover the system impedance over a wider bandwidth and, due to the neural network, is able to compute the parameters faster and in a deterministic manner.
## VI. CONCLUSION

This article has presented a novel technique, called Z-SpecNNet, for estimating the load and mutual inductance of a WPT system based on impedance spectroscopy and a neural network estimator. The Z-SpecNNet was demonstrated to be fast, with an estimation time of about 85 ms, and accurate enough, with an average estimation error of 7.81%, to be used effectively in controlling the charging process. In addition, the proposed technique can run directly on a microcontroller obviating the need for additional hardware, and is applicable to any compensation topology.

Several improvements could be made to enhance the accuracy of the identification. One potential approach is the utilization of a least-squares approximation downstream of the neural network. Furthermore, the estimation can be extended to other parameters, such as the effective resonance frequency of the receiving side, through a straightforward retraining of the NN. It is also worth noting that a Z-SpecNNet estimator could also be employed to estimate the parameters of numerous LTI systems controlled by a PWM-like modulation and a known transfer function. As impedance spectroscopy is a model-free method of measuring system behaviour, only the neural network would need to be retrained to learn about different system behaviour.

## ACKNOWLEDGMENT

The authors wish to thank Würth Elektronik to have provided the coil samples used for the experimental verification.

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<figure><img src="image_20.png" /></figure>

**Thomas Boulanger** received the double B.Sc. degrees in engineering sciences and mathematics and physics in 2022 from École Normale Supérieure Paris-Saclay, Gif-sur-Yvette, France, where he is currently working toward the M.Sc. degree in electrical and computer engineering with an expected graduation in 2025.

His research interests include embedded systems, neural networks, signal processing, and power electronics.

<figure><img src="image_21.png" /></figure>

**Vincenzo Cirimele** (Senior Member, IEEE) received the M.Sc. (summa cum laude) and Ph.D. degrees in electrical engineering from the Politecnico di Torino, Turin, Italy, in 2013 and 2017, respectively, and the Ph.D. (Hons.) degree in electronics engineering from the Université Paris-Saclay, Gif-sur-Yvette, France, in 2017.

Currently, he is an Associate Professor with the Department of Electrical, Electronic, and Information Engineering, Alma Mater Studiorum University of Bologna, Bologna, Italy. From 2020

to 2021, he was a Technical Responsible for the R&D and Innovation Group of the company Movyon s.p.a., Florence, Italy of Autostrade per l'Italia Group where he supervised projects related to energy sustainability and development of highway electric mobility. From 2017 to 2020, he held the position of an Assistant Professor with the Department of Energy, Politecnico di Torino. His research interests include technologies for electric mobility, inductive power transmission, electromagnetic modeling and simulation, and power electronics.

Prof. Cirimele acts as a Deputy Editor for *IET Power Electronics* and an Associate Editor for the journals *IEEE TRANSACTIONS ON TRANSPORTATION ELECTRIFICATION* and *Wireless Power Transfer*.

<figure><img src="image_22.png" /></figure>

**Mattia Ricco** (Senior Member, IEEE) received the master's degree (cum laude) in electronic engineering from the University of Salerno, Fisciano, Italy, in 2011, and the Ph.D. double degrees in electrical and electronic engineering and in information engineering from the University of Cergy-Pontoise, Cergy-Pontoise, France, and University of Salerno, in 2015.

Since 2021, he has been an Associate Professor with the Department of Electrical, Electronic, and Information Engineering, University of Bologna, Bologna, Italy, and from 2018 to 2021, he was a Senior Assistant Professor (Tenure Track). From 2015 to 2018, he was a Post-doctoral Research Fellow with the Department of Energy Technology, Aalborg University, Aalborg, Denmark. His research interests include modular multilevel converters, battery management system, electric vehicle chargers, field-programmable gate array-based controllers, and photovoltaic systems.

Prof. Ricco is an Associate Editor for the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS and an Editor for *IET Power Electronics*.

<figure><img src="image_23.png" /></figure>

**Eric Monmasson** (Senior Member, IEEE) received the Ing. and Ph.D. degrees in electrical engineering from École Nationale Supérieure d'Ingénieurs d'Electronique d'Electronique d'Informatique et d'Hydraulique de Toulouse, Toulouse, France, in 1989 and 1993, respectively.

Since 2003, he is a Full Professor with the University of Cergy-Pontoise, Cergy-Pontoise, France. He is also with the Systèmes et Applications des Technologies de l'Information et de l'Energie laboratory (SATIE, UMR CNRS 8029), Cergy-Pontoise. He is the Author or Co-Author of three books and more than 200 scientific papers. His research interests include the control of power electronics, electrical motors and generators, and FPGA-based and SoC-based industrial control systems.

Dr. Monmasson was the Chair of the technical committee on Electronic Systems-on-Chip of the IEEE Industrial Electronics Society from 2008 to 2011. He is also a member of the steering committee of the European Power Electronics Association and he was the Chair of the ELECTRIMACS technical committee of the International Association for Mathematics and Computers in Simulation (2011-2017). He is an Associate Editor of IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS and an Editor-in-Chief of *IEEE Industrial Electronics Magazine*.