Electrochemical Impedance Spectroscopy 301 Zstack is the sum of the cell impedances plus the impedance of the wire connected to the unipolar plate of the first cell, Z: IIIIIIIV( )( )( )( )( )( )stackZsZsZ sZsZsZs=+(6.5) Then, the stack impedance can be written as IIII( )( )( )( )( )load stackloadZs ZsZsZsZs=(6.6) Figure 6.35 shows the stack impedance (Figure 6.35C) calculated from ZII (Figure 6.35A) and Zload (Figure 6.35B). Similarly, through measurements of Ha, Hb, Hc, and Hd, the values of ZI, ZII, ZIII, and ZIV can be calculated. The calculation procedure is shown below. (1) According to the measurement of Ha, IIIIIIIVZ /(ZZZZ )+ can be calculated: )()()()()()(1 11sIsIsZsIsEsZstacke =(6.7) )()()()()()(1 sIsIsZsIsEsHa a=(6.8) )()()()(sIsEsEsKae a=II( )( )aZsHs= I IIIIIIIV( )( )( )( )( )( )I sZ sZsZsZsI s=+(6.9) Then IIIIIIIVZ /(ZZZZ )+ is obtained by the division of Equation 6.9 by Equation 6.8: IIIIIIIVII( )( ) ( )( )( )( )( )( )aaHsZs Z sZsZsZsZsHs=+(6.10) (2) Based on the measurements of Hb, Hc, Hd, (Z + ZI) / (ZII + ZIII + ZIV), and (Z + ZI + ZII) / (ZIII + ZIV), (Z + ZI + ZII + ZIII) / ZIV can be calculated. The calculation process is similar to that ofIIIIIIIV/ ()ZZZZZ+: 302 X-Z. Yuan, C. Song, H. Wang and J. Zhang IIIIIIIVII( )( )( ) ( )( )( )( )( )bbHsZsZ s ZsZsZsZsHs+=+(6.11) IIIIIIIVII( )( )( )( ) ( )( )( )( )ccHsZsZ sZs ZsZsZsHs+=+(6.12) IIIIIIIVII( )( )( )( )( ) ( )( )( )ddHsZsZ sZsZs ZsZsHs+=(6.13) (3) Using the four equations containing IIIIIIIV/ ()ZZZZZ+ (6.10), IIIIIIIV()/()ZZZZZ+ (6.11), IIIIIIIV()/()ZZZZZ+ (6.12), and IIIIIIIV()/ZZZZZ+(6.13), combined with Equation 6.5, the five unknown parameters, Z, ZI, ZII, ZIII, and ZIV are therefore obtained separately. Consequently, the impedance of the individual cell of a PEMFC stack and the total impedance of the stack have been determined separately. The results are shown in Figure 6.36. Figure 6.36. Three-dimensional representation of impedance diagrams calculated for each cell of the fuel cell stack. The solid line links the low-frequency limits (RLF) of the diagrams. The large dots indicate the 100 Hz frequency for each diagram 37. (Reproduced by permission of ECSThe Electrochemical Society, and the authors, from Diard JP, Glandut N, Le-Gorrec B, Montella C. Impedance measurement of each cell of a 10 W PEMFC stack under load.) It is observable that the impedance increases from the cell at the gas inlet to the cell at the gas outlet (the gases having entered the stack on the cell IV side). This indicates that gas starvation occurs in the stack, cell by cell, from the inlet to the outlet. In addition, it is possible to foresee that a failure in a stack can be detected from one or any of the individual cells. Electrochemical Impedance Spectroscopy 303 6.1.4.2 Direct Determination As described in the previous section, the impedance of the stack and individual cells can be determined through a complicated calculation. Since integral measurements of the entire stack do not provide information about single cells, direct measurement of the individual cell impedance is of more interest. Impedance of individual cells under heavy load has been introduced in Section 5.3.3, but the spectra obtained were measured consecutively one by one. By using a multi- channel frequency response analyzer system, the impedance spectra of all single cells in a fuel cell stack can be measured simultaneously, which is suitable for evaluating the operating state of each cell concurrently and thereby explaining overall stack performance. For the purpose of simultaneously obtaining EIS measurements of single cells, Hakenjos et al. 38 utilized a Solartron 1254 frequency response analyzer (FRA) with two 1251 multichannel extensions, which is capable of measuring the impedances of up to 19 single cells, to investigate their self-designed short stack consisting of four cells with an active area of 53 cm2. A Kepco BOP 20-20M bipolar power supply was used for the load. The load was operated in current control mode with the AC perturbation above a DC bias analog programmed by the generator output of the 1254 FRA. The chosen amplitude of 250 mA for the current excitation was small enough to measure impedances up to 0.04 . The stack was operated at a current of 4 A, and impedance spectra were taken approximately every 7 min. The duration required for the measurement of one spectrum was about 4.5 min. The first impedance spectra were taken 12 min after the start of the experiment and are shown in Figure 6.37(M0). All four spectra are similar in shape, with the emergence of a second arc in the frequency range below 1 Hz suggesting minor mass transfer losses. There are no significant deviations in the high-frequency resistance. Cells 1 and 3 show larger charge transfer arcs causing slightly lower cell voltages, which is consistent with the voltage measurement resul