Frequency-independent resistance

The resistor R and capacitor C shown in Table 16.1 can take on different meanings for different electrochemical systems. The resistance may, for example, be associated with the charge-transfer resistance of an electrochemical reaction, with the resistance of an oxide or porous layer, or with the electronic resistance of a semiconductor. The capacitor C may be associated with the double layer for an electrode in electrolyte, with surface capacitance of a film, or with the space-charge region of a semiconductor. The resistor Re may be associated with the Ohmic resistance of the electrolyte or with the frequency-independent resistance of a solid. 

An ideal capacitor with dimensions area 1 cm and thickness 1 mm is measured to have a frequency independent resistance 10 O and a capacitance of 15 x 10 F. Find the complex permittivity, conductivity, and resistivity of the dielectric. 

Figure 7.19 shows the same data for hydrogel and aluminum electrode metal. Series resistance flattens out at about 1500 O at higher frequencies left part of the diagram of Figure 7.19. This corresponds to a conductivity of the hydrogel equal to a = 6 mS/m (gel thickness 1 mm). Accordingly, the polarization impedance down to about 0.1 Hz of this hydrogel electrode is purely resistive and dominated by the frequency independent resistance of the gel.

All the circuits in this chapter are with ideal components, that is, frequency independent resistance, conductance, and capacitance. Derived parameters, however, are often... 

Accordingly, under these conditions, the impedance of the metal-fihn interface is likely to appear as a (small) frequency-independent resistance due to the transfer of electrons between the two phases. 

Semiconductor devices ate affected by three kinds of noise. Thermal or Johnson noise is a consequence of the equihbtium between a resistance and its surrounding radiation field. It results in a mean-square noise voltage which is proportional to resistance and temperature. Shot noise, which is the principal noise component in most semiconductor devices, is caused by the random passage of individual electrons through a semiconductor junction. Thermal and shot noise ate both called white noise since their noise power is frequency-independent at low and intermediate frequencies. This is unlike flicker or ///noise which is most troublesome at lower frequencies because its noise power is approximately proportional to /// In MOSFETs there is a strong correlation between ///noise and the charging and discharging of surface states or traps. Nevertheless, the universal nature of ///noise in various materials and at phase transitions is not well understood. 

The resistance R) may be defined as an impediment to the flow of electronic charge. Consider a pure resistor (that is, one having no capacitance whatsoever) its resistance when determined with a continuous current is R, and its impedance is frequency-independent. We can say that ... 

We first note that all the resistances are pure, so each impedance is frequency-independent. Secondly, we note that the impedances Z are the same as the resistances R. R = Z, R2 = Z2 and R2—Z2. 

We now check whether Eq. (1), with S /3 = e2/ and modified as above to account for finite propagation time, can explain our data. The unknown parameters are the resistance Rq and the effective environment noise temperature Tq. We checked that the impedance of the samples was frequency independent up to 1.2 GHz within 5%. Fig. 2 shows the best fits to the theory, Eq. (1), for all our data. The four curves lead to Ro = 42 12, in agreement with the fact that the electromagnetic environment (amplifier, bias tee, coaxial cable, sample holder) was identical for the two samples. We have measured the impedance Zenv seen by the sample. Due to impedance mismatch between the amplifier and the cable, there are standing waves along the cable. This causes Zenv to be complex with a phase that varies with frequency. We measured that the modulus Zenv varies between 30 12 and 70 12 within the detection bandwidth, in reasonable agreement with f o = 42 12 extracted from the fits. 

The geometrical capacitance Cg is of the order of 10 10 F cm 2 for the majority of films studied, whose thickness is 1 pm by order the corresponding impedance would be an order higher than the film resistance Rs (with exception of very low-doped films) and can be neglected. We then obtain a simpler three-element circuit (Fig. 10b) often called the Randles circuit [65], An essential assumption is that all elements of the circuits in Figs. 10a and b are frequency-independent. 

If Rsn(f) is frequency-independent in the range of/max to fan, then Rn will equal Rsn. If the spectral noise resistance is equal to the magnitude of the impedance, which is reasonable at low frequencies, then... 

Australia. Pyrethroid resistance in Heliothis spp. was first documented in Australia during the 1982-1983 production season, when field control failures occurred against Heliothis armigera (2). Daly and Murray (341, in a thorough evaluation of hypotheses proposed on the evolution of resistance in Australia, concluded that an increase in the frequency of resistance genes in the Emerald area was a result of interactions between high selection pressure, population density, crop phenology and weather, but the evolution of resistance elsewhere was independent of the situation in Emerald. 

The simplest circuit describing a relaxation process with a single relaxation time, i.e., the circuit whose R and C combination leads to the Debye equation, is obtained when the resistance R is associated in series with the capacitance Ci and the contribution of induced polarization due to atomic and electronic polarization, C , is associated in parallel with C Ri according to Figure 4 a) [16, 20] where all the components are frequency independent. 

Differential capacitance measurements by Niki et for cytochrome C3 from D. vulgaris, strain Miyazaki, were consistent with irreversible, diffusion-limited adsorption for 4-s drop times above a concentration of 10 fiM. The surface excess of cytochrome C3 was calculated to be 0.92 x 10 " mole/cm. Niki etal also investigated the a.c. polarographic behavior of cytochrome C3 at the reversible half-wave potential. The capacitive peak height was frequency independent while the resistive peak height decreased with increasing frequency to a value of zero above 2000 Hz. These results were fit to a Laitinen-Randles equivalent circuit yielding an n value of... 

Equation (IL5.36) shows that the Warburg impedance cannot be represented as a series combination of frequency-independent elements in an equivalent circuit. This is possible, however, by a semi-infinite resistive-capacitive transmission line with a series resistance R per unit length and a shunt capacity C per unit length (Fig. IL5.4). 

To discuss the results, the sensor is represented as a lossy capacitor, with both the capacitance C and the resistance R depending on frequency (Fig. 4 the frequency dependence of the equivalent-circuit elements is a consequence of the distributed nature of the processes in the sensor, which cannot be modeled appropriately by only two lumped elements with frequency-independent element values.). That simplifies the recognition of even small changes in the impedance, as changes at low frequencies become easily visible in the representation of the resistance R(f) and changes at higher frequencies become even more visible in the representation of the capacitance C(f). 

With a dripping mercury electrode the surface is ideal and the double layer is modeled as a pure, frequency independent capacitor, somewhat voltage-dependent. The capacitance values are very high because of the small double-layer thickness, Cdi is about 20 pF/cm. With solid electrode materials, the surface is of a more fractal nature, with a distribution of capacitive and resistive properties. The actual values are dependent on the type of metal, the surface conditions, the type of electrolyte, and the applied voltage. The capacitance increases with higher electrolyte concentration. The double-layer capacitor is inevitable it is there as long as the metal is wetted. Cdi may dominate the circuit if there are no sorption or electrode reaction processes, or if the frequency is high. 

Impedance is by definition a complex quantity and is only real when 0=0 and thus Z(m) = Z(a>), that is, for purely resistive behavior. In this case the impedance is completely frequency-independent. When Z is found to be a variable function of frequency, the Kronig-Kramers (Hilbert integral transform) relations (Macdonald and Brachman [1956]), which holistically connect real and imaginary parts with each other, ensure that Z" (and 9) cannot be zero over all frequencies but must vary with frequency as well. Thus it is only when Z(linear resistance, that Z(m) is purely real. 

Conductive-system dispersion (CSD) usually involves thermally activated conduction extending to zero frequency plus an always-present bulk dielectric constant, usually taken to be frequency-independent in the experimental range. Dielectric-system dispersion (DSD) often involves dielectric-level response with only weak temperature dependence, and it may or may not involve a non-negligible frequency-independent leakage resistivity, pc = Pdc = po= 1/ob- There may be cases where separate processes lead to the simultaneous presence within an experimental frequency range of both types of dispersion, but this is rare for most solid electrolytes. Further complications are present when conduction involves both mobile ionic and electronic charges, neither of whose effects are negligible (Jamnik [2003]). Here only ionic, dipolar, and vibronic effects will be further considered, with the main emphasis on conductive rather than on dielectric dispersion. 

The frequency dependence of a simple response of an electrochemical cell consists of three contributions (i) double-layer charging with a linear dependence on co (via the term coC) (ii) the frequency independent faradaic charge transfer (Ret) (hi) the diffusional contribution with dependence and, finally, (iv) the solution resistance acting in series with all contributions listed (i-iii). 

The elements of the equivalent circuit can be interpreted physically as follows R i represents the resistance of the electrolyte, which from the fitting has a value of 4040 Q Cei is the capacitance of the electrolyte, equal to 32 pF and Cgc represents the Gouy-Chapman capacitance that describes the diffuse layer of charge at the gold-electrolyte interface [12-13]. This is modelled by a universal capacitor, characterised by two frequency-independent parameters, Co and n. The complex capacitance Cgc (to) is given by... 

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