Impedance high-frequency limit

With this extension, the complex impedance response of the CCL could be calculated. The model of impedance amplifies diagnostic capabilities— for example, providing the proton conductance of the CCL from the linear branch of impedance spectra (in Cole-Cole representation) in the high-frequency limit. 

Often a non-blocking interface will behave like a resistance (/ ct) and capacitance (Q,) in parallel. This leads to a semicircle in the impedance plane which has a high frequency limit at the origin and a low frequency limit at Z = (Fig. 10.4). At the maximum of the semicircle if the angular frequency is then ctQin>max = fro which dl can be evaluated. 

A high-frequency limit for the applied potential is encountered above several kilohertz where the impedance of the conductance cell again begins to deviate from the resistance R. Since the solution medium itself is a dielectric situated between two parallel charged surfaces, it can assume the characteristics of a capacitor placed in parallel across the solution resistance as shown in Figure 8.9a. The magnitude of this capacitance is given by... 

Figure 3.6. Complex plane graph of the total impedance at low frequencies 3.3.2.2 High-Frequency Limit...

Remember 11.2 The Warburg impedance, equation (11.52), applies for diffusion in an infinite stagnant domain. This expression applies as a high-frequency limit for diffusion in a finite domain. 

The high-frequency limit for the imaginary part of the impedance given in equation (16.8) is identical to that given for the series arrangement as equation (16.3) for all frequencies. 

In the case where only oxygen diffusion transport limitations are considered and proton transport limitations are neglected, the linear impedance response prescribes a perfect semicircle in the complex plane without showing a linear branch in the high-frequency limit. The response given by Eq. (110) is, thus, an exclusive feature of proton transport limitations, which, thereby, provides a feasible tool for their characterization. 

Thus, the graphical representation of impedance spectra provides definite tools for the determination of catalyst layer parameters via the identification of the linear branch in the high-frequency limit and the branches of semicircular character. 

If the high-frequency limit of the real part of the impedance is known, the real part of the impedance can be obtained from the imaginary part of the impedance using... 

Ro and Roo, which represent the low and high-frequency limits. The impedance of this circuit can be expressed as... 

Limiting Regimes High-frequency limit, ft) S2c. In this regime, the complex impedance has a linear branch at an angle of 45° in the Cole-Cole representation, which is prescribed by... 

Impedance spectroscopy Features in impedance spectra, like the linear branch (in Cole-Cole representation) due to proton transport limitations in the high-frequency limit or the semicircular character of the response, observed at lower frequencies, give information about the actual morphology of the layer. 

The high-frequency limitation imposed on the operation of reactively substituted Wheatstone bridges by unavoidable stray capacitances prompted the development of the transformer ratio arm bridge (Calvert [1948]). By substituting a transformer for orthodox ratio arms, a bridge was produced for which the impedance ratio is proportional to the square of the number of turns and which was capable of accepting heavy capacitive loads with virtually no effect on the voltage ratio. 

Series leakage inductances in the transformers within the bridge result in an impedance measurement error that is proportional to frequency. This effect has been examined by Calvert [1948], but is seldom likely to impose high-frequency limitations in electrochemical apphcations. 

In the high-frequency limit the circuit behaves like the two resistances in parallel. The impedance is dominated by the smaller of the resistances as the current takes the most conducting pathway. The second term in Eqn. 23 describes a 45° Warburg line, but now it is displaced from the origin by Eqn. 24. It has the same type of dimensionless frequency term as the classical line. 

Figure 4.5 shows calculated impedance plots for the general model for different values of p. As resistances become more equal, the high-frequency limit on the x axis for Zl(Rx + Re) increases. For the extreme case when Rx = Re, a maximum value of 1/4 is found. There is then only a small difference between the real component at the high-frequency limit (1/4) and the real component at the low-frequency (1/3) limit. Under these conditions the Warburg region almost disappears, and the transmission line appears to be a capacitor. 

At the high-frequency limit of the spectrum the intercept on the resistive axis specifies the serial ohmic component in the measured system, since this element does not introduce a phase shift. Similarly the low-frequency limit approaches the steady-state condition corresponding to the d.c. characteristics of the cell. Each spectral feature detected between these limits represents a dissipation process with the specific time dependence indicated by the inverse of the frequency at which it occurs. It should be noted, therefore that two processes with similar time constants in the anodic system will not be distinguishable by impedance spectroscopy. 

The high-frequency limit is defined as the frequency where the impedance of the cell becomes lower than indicating a largely capacitive current it can be determined with the following equation, a derivation of Eq. 6-3 ... 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

Whichever physical interpretation is chosen, the difference between the high-frequency real axis intercept [Z (high) and the low-frequency limiting real impedance [Z (low)] is one-third of the film s ionic resistance (i.e., R[ = 3[Z (low) - Z (high)]). Ideally, the real component of the... 

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