## Abstract

The high frequency electrochemical impedance measurements with positive imaginary components in the impedance complex plot of a polymer electrolyte fuel cell (PEFC) are attributable to the inductance of the electrical cables of the measurement system. This study demonstrates that the inductive effect of the electrical cables deforms the high frequency region of the cathode impedance spectrum and as such leads to an erroneous interpretation of the electrochemical mechanisms in the cathode catalyst layer (CCL). This study is divided into a theoretical analysis and an experimental analysis. In the theoretical analysis a validated model that accounts for the impedance spectrum of the CCL as reported in the authors’ previous study is applied with experimental impedance data reported in the literature. The results show that the ionic resistance of the CCL electrolyte which skews the oxygen reduction reaction (ORR) current distribution toward the membrane interface is masked in the cathode impedance spectrum by the inductive component. In the experimental analysis cathode experimental impedance spectra were obtained through a three-electrode configuration in the measurement system and with two different electrical cables connected between the electronic load and the PEFC. The results agree with the theoretical analysis and also show that the property of causality in the Kramers-Kronig mathematical relations for electrochemical impedance spectroscopy (EIS) measurements is violated by the external inductance of the measurement cables. Therefore the experimental data presenting inductance at high frequencies do not represent the physics and chemistry of the PEFC. The study demonstrates that a realistic understanding of factors governing EIS measurements can only be gained by applying fundamental modeling which accounts for underlying electrochemical phenomena and experimental observations in a complementary manner.

## Introduction

The polymer electrolyte fuel cell (PEFC) converts the chemical energy of a fuel such as hydrogen gas into electrical energy. The main components are two electrodes (anode and cathode) which are separated by a polymer electrolyte membrane (PEM). Each electrode consists of a gas diffusion layer (GDL) and a catalyst layer (CL) as a minimum. Hydrogen is supplied to the anode electrode and oxygen to the cathode usually as a component of air. The power output, hence performance of the PEFC, depends on the phenomenological processes occurring inside its individual layers.

Electrochemical impedance spectroscopy (EIS) is an in situ experimental technique that can unveil the electrochemical and diffusion processes that occur at different rates within the PEFC. Impedance measurements are commonly carried out in a PEFC, in either galvanostatic or potentiostatic mode. The former superimposes a small ac perturbation onto the dc bias electrical current drawn from the PEFC. The latter does so onto the potential that controls the driving force for electrochemical reactions taking place on the working electrode. The resulting impedance is shown in a complex plane and represents the electrochemical and diffusion mechanisms of the PEFC in the frequency domain. Impedance spectra have to be consistent with the Kramers-Kronig (K-K) mathematical transformations which relate the real part to the imaginary part of the frequency response and vice versa. EIS is commonly applied as a single channel or in a one-dimensional cell through a two-electrode configuration. A two-electrode configuration makes it difficult to identify, to separate, and to reveal the processes related to each electrode (anode, cathode) in the impedance spectrum of a cell. A three-electrode configuration allows the separation of the anode and cathode contributions in the PEFC impedance spectrum.

The EIS measurements with positive imaginary components at the high frequency end of the PEFC spectrum have been attributed to the inductance of the electrical cables of the measurement system. According to Ampere’s law, inductance is caused by the magnetic field generated by electric currents. Inductance is commonly related to inductors formed by a wire wound in a coil; however a small straight piece of wire can present some self-inductance. The effect of inductance is often seen at the highest frequencies. The impedance of an inductor increases with increasing frequency and its effect can be noticed in low impedance systems <1 Ω such as the PEFC. The inductive effect of the electrical cables in the EIS measurement system deforms the high frequency region and leads to an erroneous structural interpretation on the impedance spectrum [1,2]. Special care with the electrical connections must be taken during impedance measurements, as this could lead to irrelevant information normally represented as inductance and can lead to an incorrect interpretation at high frequencies of the impedance complex plane. Ciureanu and Roberge [3] reported in their work that PEFC experimental impedance spectra in the high frequency range showed an inductive behavior characteristic to the experimental setup. To avoid complications resulting from such inductive characteristics, they limited the high frequency range to 1 kHz. Fouquet et al. [4] reported that inductive behavior from the wires in EIS measurements was predominant above 1 kHz. Makharia et al. [5] concluded that the practice of using a specific high frequency value to estimate the ohmic resistance of the PEFC is subject to large errors (10%–50%). The size of these errors depends on the frequency used and the size of the inductance. Merida et al. [6] reported in their work that it is possible to minimize the inductive effect by maintaining equal lengths in the cables, twisting the large current leads to the electronic load, and polishing and clamping the metal contacts. Other studies have reported that it is necessary to optimize the cables of the measurement system to reduce interference signals. In the study of Asghari et al. [7] an inductive characteristic at high frequencies in the EIS spectrum resulted from a nonuniform contact (assembly) pressure in a PEFC. The inductive characteristic was diminished gradually as the level of uniformity of the assembly pressure increased (uniform tightening of the supportive tie rods in the PEFC).

The objective of this work is to reveal the inductive effect on the high frequency region of the complex plot for the cathodic side of a PEFC. A validated model that accounts for the impedance spectrum of the cathode catalyst layer (CCL) as reported in the authors’ previous study is applied with experimental impedance data reported in the literature. A further validation of the inductive effect is carried out by comparing the mathematical model with cathode experimental impedance data obtained with a three-electrode configuration and the use of conventional electrical cables (power flexible cable) and special low inductive cables (low inductive cable with fusion lug technology, TDI POWER).

## Inductive Effect on the Cathode Impedance Spectrum

In this study the analysis of the inductive effect on the cathode impedance spectrum is divided into a theoretical analysis and experimental analysis. The theoretical analysis studies the effect of inductance on EIS measurements reported in the literature using the impedance model reported in the authors’ previous study that accounts for the cathodic side of the PEFC. In the experimental analysis, an analysis of the Kramers-Kronig consistency for linear, steady, and causal electrochemical systems was evaluated to analyze the inductive effect on the experimental impedance results.

### Theoretical Analysis.

In the authors’ previous study [8,9], a model was developed based on fundamental electrochemical and diffusion theory to simulate the impedance spectrum of the CCL of a PEFC operated in any zone of the polarization curve, and also the current and oxygen distributions through the CCL in the spatiotemporal domain. The theoretical treatment considered the following main assumptions.

- 1.
Only effects in the CCL are considered.

- 2.
All processes proceed under isobaric and isothermal conditions through the thickness of the CCL.

- 3.
Spatial gradients are only considered through the thickness of the CCL.

- 4.
The resistance to ion transfer in the electrolyte of CCL is much greater than the resistance to the electron transfer in the carbon of the catalyst layer. Therefore electronic ohmic loss in CCL can be regarded as being negligible [5,10,11].

- 5.
The rate of oxygen distribution is considered as a nonsteady process.

$RP$ is the resistance to the flow of ions in the electrolytic phase of the CCL. *R _{C}* represents the charge transfer resistance presented in the oxygen reduction reaction (ORR) defined as $RC=b/j0exp(\eta S/b)$, where $b$ is the Tafel slope, $\eta S$ represents a value of voltage in activation overpotential, and $j0$ is the exchange current. $ZW$ is defined as the Warburg impedance and describes diffusion across a finite dimension in the frequency domain [4] $ZW=RWtanh(i\omega TW)0.5/(i\omega TW)0.5$, with $RW=RT\delta /(z2F2co*D)$ defined as resistance for the diffusion process and $TW=\delta 2/D$ defined as the time constant to diffuse oxygen through the CCL. $Y(i\omega )P$$Y$ represents a parameter related to constant phase element (CPE) and superscript $P$ represents a parameter to correct the inhomogeneity in the distribution of charge. $\omega $ is the angular frequency. $i$ is the imaginary component in impedance. $L$ represents the inductance in the electrical cables of the measurement system. $Re$ represents the total ohmic resistance to flow electrons and ions in the bipolar plate, GDL, and PEM.

Makharia et al. [5] estimated the ionic resistance in the CCL for a 5 cm^{2} PEFC (H_{2}-O_{2}) using MEAs with 0.8 and 0.4 Nafion/carbon (N/C) ratios through impedance measurements. The impedance measurements were tested from 100 kHz to 0.01 Hz through a Zahner IM6e potentiostat. The authors considered that no losses across the electrolyte in the anode catalyst layer were expected because the hydrogen oxidation reaction (HOR) in the anode is so facile that any ionic electrolyte resistance would have the impact of shifting the anode current distribution close to the membrane, without requiring significant anode overpotential to do so. Therefore the PEFC impedance spectrum only accounts for the cathode electrode. In these published data the lack of a reference electrode could add an uncertainty to the interpretation of the data. In this theoretical analysis the published data are considered to represent the cathode impedance spectrum obtained through a reference electrode. The experimental analysis in the next section will consider a reference electrode to separate the cathode impedance spectrum from the PEFC impedance spectrum. In the work of Makharia et al., the ionic resistance of the CCL was estimated by fitting the experimental PEFC spectrum to the equivalent electrical circuit type transmission line using Zview software (Transmission Line-Open Circuit Terminus, DX-type 6, Scribner Associates, Inc., version 2.3). The parameters of the equivalent circuit extracted from the experimental data of a MEA with 0.8 N/C ratio operated at 0.03 A/cm^{2} are shown in Table 1.

L (H) | $Re$ (Ω cm^{2}) | $RP$ (Ω cm^{2}) | $RC$ (Ω cm^{2}) | ||
---|---|---|---|---|---|

j (A/cm^{2}) | inductance | ohmic resistance | ionic resistance | Capacitance (mF cm^{−2}) | charge transfer resistance |

0.03 | 1.32 × 10^{−6} | 0.0687 | 0.122 | 19.67 | 1.215 |

L (H) | $Re$ (Ω cm^{2}) | $RP$ (Ω cm^{2}) | $RC$ (Ω cm^{2}) | ||
---|---|---|---|---|---|

j (A/cm^{2}) | inductance | ohmic resistance | ionic resistance | Capacitance (mF cm^{−2}) | charge transfer resistance |

0.03 | 1.32 × 10^{−6} | 0.0687 | 0.122 | 19.67 | 1.215 |

The parameters in Table 1 were substituted into Eq. (1) to simulate the experimental impedance spectrum with no mass transport effect $ZW=0$ as reported by Makharia and as shown in Fig. 1(a). The resistive parameters from Table 1 should be reflected in the real component *Z*′ (Ω cm^{2}) at *Z*″ (Ω cm^{2}) = 0 of the simulated impedance spectrum obtained from the mathematical model.

In the literature, the real component *Z*′ (Ω cm^{2}) at high frequency where the imaginary component *Z*″ (Ω cm^{2}) of the impedance spectrum is equal to zero has been reported as the ohmic resistance for the PEM, GDL, and plate [3,13,14], which in this study corresponds to $Re$ from Table 1. A 45 ° region representing the ionic resistance in the CCL at high frequency when the PEFC is tested for H_{2}/N_{2} operation has been also reported in the literature [15,16] and has been estimated by projecting the 45 ° region onto the real part *Z*′ and calculated as $RP/3$. The simulated impedance spectrum at high frequency represented in Fig. 1(b) reflects a value of *Z*′ = 0.084 Ω cm^{2} at *Z*″ = 0 Ω cm^{2}, which is different from $Re$ in Table 1, and the 45 ° region accounting for $RP/3$ is difficult to estimate from Fig. 1(b). This difference between $Re$ in Table 1 and the real component *Z*′ = 0.084 Ω cm^{2} at *Z*″ = 0 Ω cm^{2} high frequency of the simulated impedance spectrum is attributed to the inductive effect $L$ presented in the impedance measurements.

The inductance $L$ given in Table 1 was increased by one order of magnitude and decreased by one, two, and three orders of magnitude in Eq. (1) to demonstrate its effect on the high frequency region of the complex plot, as shown in Figs. 2(a) and 2(b),_{.} Figure 2(b) shows that the real part *Z*′ of the simulated data at high frequency where the imaginary part *Z*″ is zero in the complex plot changes for different inductance values. Inductance has been attributed to the external measurement system. Figure 2(b) demonstrates that the spectra with different inductance values lead to an incorrect interpretation of the cathode mechanisms at high frequencies. Inductance of the measurement system does not have an effect on low frequencies where the mass transport effect is manifested, however this inductance can misrepresent the high frequency mechanisms such as the ionic (and electro) conductivity in the real part of the complex plot *Z*′ at *Z*″ = 0. Kurz et al. [17] predicted the voltage drop for dehydration in different PEFC stack impedance spectra through a specific high frequency value (1 kHz). This methodology reported by Kurz et al. to compare impedance data at high frequency through one high frequency value taken as a reference would seem a reasonable method, as for different inductance presented in the system, the real component *Z*′ will be kept constant and just the imaginary component *Z*″ will be changed. Nevertheless this procedure would not correctly interpret the PEFC mechanisms presented for such frequency.

#### Electrochemical Mechanisms in the Cathode Spectrum at High Frequencies.

This study demonstrates that if the inductive component $L$ represented in Table 1 is reduced by three orders of magnitude in Eq. (1), the 45 deg line which has been reported in the literature [15,16] as the ionic resistance in the CCL is revealed in the cathode spectrum, as shown in Figs. 3(a) and 3(b). Figure 3(*b*) shows that at the high frequency end of the spectrum (100 kHz), the real impedance data *Z*′ approximately reflects the value $Re$ reported in Table 1, and the ionic resistance is estimated by projecting the 45 deg region onto the real part *Z*′ (Ω cm^{2}). This projection between 100 kHz and 490 Hz is approximately $RP/3$, and the difference between $Re+RP/3$ and the real impedance data *Z*′ at a frequency of 0.01 Hz reflects the charge transfer resistance $RC$ of Table 1. The resistive parameters *Z*′ at *Z*″ = 0 Ω cm^{2} presented in the simulated impedance spectrum yield a 2% error compared with the parameters reported in Table 1. This error is attributed to the capacitor *C* in Table 1. Instead a constant phase element (CPE) $Y(i\omega )P$ should have been used in Eq. (1), which is not reported in the published data and appears in the transmission line equivalent circuit in Zview software to correct the inhomogeneity of the charge distribution between the electrode-electrolyte interfaces in the CCL [18]. This study demonstrates that the cathodic mathematical model represented by Eq. (1) is reliable in correctly estimating the inductance effect and the electrochemical mechanisms of the CCL.

where $RP$ represents the resistance to the flow of ions in the electrolytic phase of the CCL, $RC$ is the charge transfer resistance presented in the ORR, and $C$ is the charge capacitance between the electrode-electrolyte interface. The first term on the right-hand side of Eq. (3) defines the current distribution by the ratio of the ionic resistance to the charge transfer resistance. Neyerling et al. [20] defined a similar relationship, where the higher the nondimensional ratio is, the more the current distribution will be skewed toward the membrane interface, while low values of this ratio predict a homogeneous current distribution. If the ratio between ionic resistance and charge transfer resistance increases, as expressed in Eq. (3), the ions may react closer to the membrane. However, in the case where the ratio between ionic resistance and charge transfer resistance is reduced, the ions will react through the entire CCL thickness leading to a uniform current distribution.

The parameters from Table 1 that represent the electrochemical mechanisms of a MEA with 0.8 N/C ratio and for a MEA with 0.4 N/C ratio reported in Makharia’s study operated at 0.03 A/cm^{2} were substituted into Eq. (3) to simulate the current distribution through the CCL thickness. Figure 4 shows the current distribution through the CCL thickness. The current distribution increases from the *x* = 0 thickness fraction (GDL-CCL interface) to the *x* = 1 thickness fraction (CCL-PEM interface). This reflects the transfer of ions from the PEM and the anode catalyst layer, which results in maximum current at the interface between the CCL and the PEM and a minimum current at the interface between the CCL and GDL as ions are consumed due to the ORR through the thickness of the CCL. A high ionic resistance (0.4 N/C) within the CCL electrolyte tends to skew the current distribution towards the membrane interface. Therefore, as reported by Thompson et al. [19], the same average current has to be provided by fewer catalyst sites near the membrane. The increase in ionic resistance results in greater kinetic losses. The ionic resistance in the CCL related to the magnitude of the 45 deg region ($RP/3$) in the cathode impedance spectrum at high frequency can bias the ORR current density distribution toward the membrane interface, as shown in Fig. 4. Overall this study demonstrates that it is necessary to take extra care in the hardware of the measurement system in order to quantify the cathode mechanisms revealed at high frequencies.

### Experimental Analysis.

The contributions of the anode and cathode in the PEFC impedance spectrum using a two-electrode configuration in the EIS measurements are difficult to interpret, as this information is masked in the PEFC impedance spectrum. A three-electrode configuration for EIS allows the measurement of half-cell impedance (anode or cathode) to be carried out. Some studies [21,22] have applied the three-electrode configuration in EIS measurements. Under such conditions, the signals are measured between the working electrode (WE) and the reference electrode (RE), and the current induced is collected by the counter electrode (CE).

#### EIS Measurements.

A 25 cm^{2} commercially available fuel cell and test rig acquired from Baltic Fuel Cells were used for the experimental tests. The MEA was a membrane DuPont Nafion-115 with a platinum loading of 0.4 mg/cm^{2} on both electrodes. The operational temperature was 50 °C and the back gas pressure was held to 0.9 bar(g) for both anode and cathode. Flow rates were constant during all the experiments, hydrogen in the anode with a stoichiometry of 2.5 and air in the cathode with a stoichiometry of 3. The PEFC was operated with 100% hydrogen relative humidity (RH) in the anode, the air supplied in the cathode was dry. In terms of optimal performance of operational PEFCs, the effect of RH in the reactant gases is not definite. For instance, Yan et al. [23] reported that optimal performance in a PEFC can occur at low air relative humidity and high hydrogen relative humidity.

EIS measurements were carried out through a multichannel frequency response analyzer FRA (Z#106 WonATech Co). The multichannel FRA consists of five channels and simultaneously measures five impedance spectra through one induced dc current value. The multichannel system is connected with a RBL488 Dynaload load bank. The multichannel FRA superimposes a small ac sinusoidal perturbation onto the bias current induced by the RBL488 Dynaload unit and measures the ac voltage signals resulting from the PEFC. The impedance measurements were carried out in a galvanostatic mode with 5% ac amplitude of the dc current [24–26] to obtain a linear answer from the system and at frequencies from 10 kHz to 0.2 Hz, the high frequency in the EIS measurements was limited due to the bandwidth of the electronic load. The low frequency is not relevant in the inductive effect of the measurement cables. To separate the impedance spectrum of the cathode from the impedance spectrum of the cell, a reference electrode made of a platinum wire of fixed potential was inserted such that it was in direct contact with the polymer electrolyte membrane of the cathode side. Two channels from the Z#106 FRA were used to simultaneously measure the impedance of the PEFC and the impedance solely accounting for the cathode electrode.

Based on the results from the theoretical analysis in Sec. 2.1, EIS measurements were carried out at 0.2 A/cm^{2} using two different electrical cables, a conventional electrical cable (power flexible cable) and a special electrical cable (low inductive cable with fusion lug technology, TDI POWER) connected between the PEFC and the RBL488 load, as shown in Fig. 5. A reference electrode and the use of the multichannel FRA allowed the separation of the cathode impedance spectrum from the impedance spectrum of the entire cell. For this study high inductance spectra are dominant when the conventional electrical cable was used, and low inductance spectra when the special low inductive cable referred to above was used in the EIS measurements.

It has been proposed that the PEFC impedance spectrum largely represents the electrochemical processes in the cathode, and as discussed because the electrochemical mechanisms in the anode are fast and facile are not expected to have a contribution [3,4]. Figure 6(a) shows a difference between the spectrum of the cell and the spectrum of the cathode. Figure 6(b) shows that the real part *Z*′ that accounts for the cathode at high frequency when *Z*″ = 0 is smaller than that for the cell. The semicircular part of cathode spectrum presents roughly the same features as the PEFC spectrum and a shift over the real part *Z*′. This difference is expected as the cathode is hydrated by the water produced by the ORR and due to electro-osmotic drag from the anode. This difference in the real part *Z*′ between the cell and cathode could be attributed to the electro-osmotic effect in the anode. The transport of water from the anode to the cathode by electro-osmotic drag can play a role in dehydrating the membrane and inducing performance degradation [27]. The magnitude of the inductive effect (EIS measurements with positive imaginary components *Z*″) on the spectrum of the cell was higher than that of the cathode using the conventional cable (high inductance spectra). The sensing cables from the FRA to measure the cell impedance spectrum were directly connected to the electrical cables. The reference electrode allowed the reduction of the inductive effect on the EIS measurements for the cathode by placing the sensing cables from the FRA as far apart as possible from the inductive source.

Figure 6(b) shows the high frequency region of the experimental impedance spectra. As expected from the theoretical analysis in Sec. 2.1 low inductance in the measurement system reveals the 45 deg region at a frequency of 336 Hz in the cathode spectrum which accounts for the CCL ionic resistance and plays an important role in the ORR current density distribution through the CCL thickness, as discussed in the previous section. The PEFC spectrum with low inductance in Fig. 6(b) also presents a 45 deg region at 545 Hz which is slightly different from that of the cathode. This spectrum of the entire cell not only accounts for the ionic resistance in the CCL electrolyte but also the mechanisms presented in the anode.

#### Kramers-Kronig Analysis on EIS Measurements.

One key advantage of the EIS technique is that it is noninvasive and can be applied in situ. Another advantage is that the frequency response tests are simple to carry out and can be easily tuned for greater accuracy by using readily available sinusoidal generators and precise measuring equipment. Impedance experiments involve the conversion of time-domain input and output signals into a complex quantity that is a function of frequency. The resulting experimental impedance has to be consistent with Kramers-Kronig (K-K) relations. These relations are mathematical properties that relate the real and imaginary part of the frequency response resulting from the electrochemical system studied. The derivation of the K-K mathematical relations begins with an application of Cauchy’s integral theorem which defines the integral around a closed contour and evaluates the real part or imaginary part of the impedance at a particular frequency with poles created at –ω and ω in the real axis of the frequency domain. K-K relations are applied to systems which are linear, causal, and stable and have been applied in electrochemical impedance data over the last 20 years [28–30]. If the experimental imaginary component of the PEFC impedance does not coincide with the transformed imaginary component (real to imaginary transformation), nor the experimental real component with the transformed real component, then the experimental data are not obtained under either a causal system (the measured impedance response is due only to the ac perturbation applied), linear system (for small ac perturbations), or stable system (the system returns to its original state after the perturbation is removed and does not change with time) [29]. The K-K relations cannot be used to determine if the impedance measurements were corrupted by instruments artifacts. In the work reported by Macdonald et al. [30] different amplitudes of the ac sinusoidal perturbation with a potentiostatic mode were changed from 5 to 150 mV to test the linearity condition of data for a rapidly corroding aluminum alloy through the K-K relations. The results showed a successful K-K transformation of the impedance data, in spite of a clear violation of linearity when the amplitude was increased. This success in the K-K transformation was an artifact of the experimental equipment used for the impedance measurements. The authors concluded that some commercial frequency response analyzers are narrow band devices that reject inputs at frequency other than that designated; in other words some frequency response analyzers can reject harmonic components of the current response that manifest nonlinearity.

An analysis of the K-K consistency for linear, steady, and causal systems was evaluated to analyze the inductive effect on the experimental impedance results of this study. The K-K consistency was evaluated through ZMAN software [31] in which the K-K transformed data are calculated by Maclaurin’s series method. The difference or residual in phase angle between the K-K transformed data and the measured impedance data $\Delta Phase=\varphi ZKK-\varphi ZEXP$ provides excellent sensitivity to discrepancies in the experimental values; this also can indicate a need to remove experimental data that are not consistent with the K-K relations [32].

The residual phase in Fig. 7 is presented because one or more of the conditions to satisfy the K-K relations such as causality, linearity, and stability are not fulfilled. Since the PEFC polarization curve is a nonlinear response, the use of a low ac amplitude in the impedance measurements allows the evaluation of the PEFC polarization curve as a pseudolinear response. A high ac amplitude would not be consistent with the K-K condition for linear systems [30]. The PEFC under study was steady, thereby the EIS measurements were repetitive. The causality condition is not satisfied in the EIS measurements with inductance. Hence the measured impedance response is not only due to the ac perturbation but also to the external inductance of the measurement cables. The condition of stationarity is implicit in the requirement of causality and vice versa [32]. The statement mentioned before can be demonstrated in the EIS data presenting high inductance where the difference in phase is presented at high frequencies (not causal) and low frequencies (not steady), as shown in Fig. 7. The K-K relations, based on the property of causality, demonstrate a useful method for data validation [33].

The residual phase in Fig. 7 is reduced for the data with low inductance. The cathode presents a minor difference in the residual phase compared to the PEFC because the reference electrode allows the reduction of the inductive effect on the EIS measurements because the sensing cables from the FRA are located away from the inductive source. The 45 deg high frequency region in the cathode low inductance impedance spectrum was revealed at a frequency of 336 Hz and for the PEFC was revealed at a frequency of 545 Hz, as shown in Fig. 6(b). In Fig. 7 the residual phase of the EIS data presenting low inductance is less at frequencies of 336 and 545 Hz for the cathode and the PEFC, respectively, than for EIS data with high inductance. The measured EIS data at frequencies where the residual phase is different from zero do not reflect the physics and chemistry of the electrochemical system studied. If the external inductance of the experimental system could be eliminated the remaining inductance would be due to any intrinsic inductance presented in the porous media of the electrode; therefore the K-K transformation would be satisfactory achieved as this inductance would be a physical property of the system studied. K-K transforms represent a convenient mean of evaluating the validity of experimental impedance data.

## Validation of Inductance Effect on the Cathode Impedance Spectrum at High Frequencies

In these experimental results the use of a reference electrode ensures that the data accounting for the processes in the CCL are captured for analysis and interpretation. To validate the experimental results, the mathematical model devised in Eq. (1) was compared with the cathode experimental spectra obtained using a graphical user interface (GUI) developed in Matlab^{®}. The GUI allows the fitting of the parameters from Eq. (1) to achieve a good agreement between the experimental and simulated data. The least-squares fitting method was used in order to find the best fit between the model and the measured data. A good quality fit is obtained when the sum of the deviations squared (least-square error) between the simulated and measured impedance data has a minimum value, for instance <0.1. Figures 8(a) and 8(b) show the comparison between the simulated data and the measured data.

Scattered data at high frequencies were presented for both experimental cathode spectra, as shown in Fig. 8(b), and were more noticeable for the low inductance spectrum. This behavior could be attributed to external noise at high frequency during the EIS tests. The noise disappeared when the magnitude of the inductance was increased, for instance in the high inductance PEFC spectrum the shape of inductive effect presented a straight line, as shown in Fig. 6(a). To quantify the inductive effect on the experimental data, the biggest value of the data presented in the positive imaginary component *Z*″ of the spectrum was taken as a reference. Equation (1) was first fitted to the experimental cathode spectrum for low inductance and the results are shown in Table 2. Mass transport effects at low frequencies were apparent during the EIS measurements, as shown in the parameters from Table 2 defining $ZW$ in Eq. (1). This effect is not relevant in the inductive effect at high frequencies.

j (A/cm^{2}) | L (H) | $Re$(Ω cm^{2}) | $RP$ (Ω cm^{2}) | Y CPE × 10^{−3}S/(Ω cm^{P}^{2}) | P | $RC$ (Ω cm^{2}) | $RW$ (Ω cm^{2}) | $TW$ (s) |
---|---|---|---|---|---|---|---|---|

0.2 | 6.77 × 10^{−7} | 0.195 | 0.12 | 38.2 | 0.8282 | 0.41 | 0.26 | 0.003 |

j (A/cm^{2}) | L (H) | $Re$(Ω cm^{2}) | $RP$ (Ω cm^{2}) | Y CPE × 10^{−3}S/(Ω cm^{P}^{2}) | P | $RC$ (Ω cm^{2}) | $RW$ (Ω cm^{2}) | $TW$ (s) |
---|---|---|---|---|---|---|---|---|

0.2 | 6.77 × 10^{−7} | 0.195 | 0.12 | 38.2 | 0.8282 | 0.41 | 0.26 | 0.003 |

Once defined the inductive component which simulates the low inductance cathode spectrum was increased from $L=6.77\xd710-7H$ to $L=2.07\xd710-6H$ and the remaining parameters from Table 2 were kept constant. This allowed the simulation of the cathode spectrum with high inductance as shown in Fig. 8(b). It is noticed that the calculated inductance of the curve with high inductance presents the same order of magnitude as the inductance from the data taken from the literature [5] shown in Table 1 of the theoretical analysis in Sec. 2.1 of this study. Therefore taking these values as a base, an inductance of the order of $10-9H$ and using a sinusoidal generator and precise measuring equipment allowing the increase of the frequency up to 100 kHz would clearly show the 45 deg region that extends from the real component *Z*′ and whose projection onto the real part *Z*′ represents the ionic resistance of the electrolyte in the CCL. The inductance that remains present in the data for the low inductive cathode spectrum can be attributed to the hardware of the PEFC or any intrinsic inductance presented in the porous media of the CCL. Hampson et al. [34] concluded on their work that the presence of porosity on electrodes may lead to the impedance becoming inductive at high frequencies.

This experimental study complements the theoretical analysis presented in Sec. 2.1 by separating the cathode impedance spectrum from the PEFC and comparing the same system under the influence of the inductance on the measurement impedance. The inductance of the electrical cables of the measurement system and hardware of the PEFC or for a special case an intrinsic inductance due to the nature of the porous media in the electrode can mask the ionic conductivity, which plays an important role in the ORR current distribution through the CCL in the high frequency region of the cathode impedance spectrum.

## Conclusions

This study has identified and demonstrated a key technical problem of PEFC impedance measurements. The inductance in the electrical cables deforms the high frequency region and leads to an incorrect interpretation of the PEFC impedance spectrum in the high frequency range. This study is divided into two sections. The first section simulates the effect of inductance on cathode impedance measurements reported in the literature and reveals the mechanisms in the CCL masked by the inductive effect through the mathematical model reported in the authors’ previous study. Second, an experimental analysis is presented to validate the theory established in the theoretical analysis. This study demonstrates that the cathode mathematical model represented in Eq. (1) and reported in the authors’ previous study can be applied reliably in order to correctly estimate the inductance effect and the electrochemical mechanisms of the CCL at high frequencies of the complex plot. This study demonstrates that the practice of using the real part *Z*′ of the complex plot where the imaginary part *Z*″ is equal to zero or a through a single frequency as a reference to quantify the ohmic resistance in PEFC can be subject to an erroneous interpretation due to the contribution of the size of the inductance. The Kramers-Kronig mathematical relations were applied to the experimental data presenting inductance and demonstrated that the measured data at high frequencies do not represent the physics and chemistry of the PEFC and cathode. The experimental analysis through the multichannel FRA has demonstrated that there is a difference between impedance spectra of the PEFC and cathode electrode which contrasts the theory in which there is no anodic contribution in the PEFC measured impedance. Future work is expected to fill the gap that exists between impedance in a PEFC and the cathode to accurately estimate the factors that influence the nature of polarization curves such as kinetic losses, ohmic losses, and mass transport losses from experimental EIS measurements.

## Acknowledgment

The authors thank the Mexican National Council for Science and Technology (CONACYT) for the sponsorship of the Ph.D. research study of S. Cruz-Manzo (Grant No. 183195).