Resistance and capacitance in series

We exemplify the use of vectors in the complex plane with a resistance and capacitance in series (Fig. A2.2a). The total potential difference is the sum of the potential differences across the two elements. From Kirchhoff s law the currents have to be equal, that is 

The differences in potential are proportional to R and Xc respectively. Their representation as vectors in the complex plane is shown in Fig. A2.1. The vectorial sum of — iXc and of R gives the impedance Z. As a 

Often the in-phase component of the impedance is referred to as Z and the out-of-phase component, i.e. at jt/2, is called Z , that is Z = Z 4- iZ . Thus for this case 

This is a vertical line in the complex plane impedance plot, since Z is constant but Z varies with frequency, as shown in Fig. A2.2b. 

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For the impedance of the resistance and capacitance in series, the current will lead the potential phase angle by... 

FIGURE 12.12 Equivalent circuits with resistance and capacitance in series (a) and in parallel b). 

To find the relation between the values of R and measured experimentally in terms of the circuit of Fig. 12.11a and the parameter values in the circuit of Fig. 12.14a, we must first convert [with the aid of Eq. (12.23)] the parameters of the circuit with parallel elements Ry and Q into the parameters of a circuit with a resistance and capacitance in series, and to the value of resistance obtained we must add R.. As a result, we have... 

Variations of resistance with frequency can also be caused by electrode polarization. A conductance cell can be represented in a simplified way as resistance and capacitance in series, the latter being the double layer capacitance at the electrodes. Only if this capacitance is sufficiently large will the measured resistance be independent of frequency. To accomplish this, electrodes are often covered with platinum black 2>. This is generally unsuitable in nonaqueous solvent studies because of possible catalysis of chemical reactions and because of adsorption problems encountered with dilute solutions required for useful data. The equivalent circuit for a conductance cell is also complicated by impedances due to faradaic processes and the geometric capacity of the cell 2>3( . 

The experimental impedance is always obtained as if it were the result of a resistance and capacitance in series. We have already seen in (11.20) and (11.21) the relation between an RC series combination and the Rct + zw combination. It can be shown for the full Randles equivalent circuit for this simple charge transfer reaction, see Fig. 11.4, on separating the in-phase and out-of-phase components of the impedance, that... 

A2.4 Representation in the complex plane A2.5 Resistance and capacitance in series A2.6 Resistance and capacitance in parallel A2.7 Impedances in series and in parallel A2.8 Admittance... 

If the circuit can be regarded as consisting of resistance and capacitance in series, then ... 

Resistance and Capacitance in Series For the serial combination of a resistor and capacitor, the result is shown in Fig. 3(c). According to KirchhofFs law, the currents through both elements are the same. The total potential equals the sum of the potentials across the capacitor and resistor... 

In almost all SCP cases of interest, the time constant = RqoC will be appreciably shorter than other time constants present in the system. Furthermore, in most cases, C will also be much smaller than other capacitances present. Undlr these conditions, the parallel combination of C and R may be placed in series with the rest of the circuit to a high degree of approximation. For the special Cases A and B already discussed in connection with Fig. 3, the rest of the circuit may be well approximated by another resistance and capacitance in series, yielding an N = 2 Voigt-model circuit. Finally, whenever one... 

When the applied constant potential is superimposed by an alternating (sinusoidal) potential signal of small magnitude (Eq. (3.40)), the certain periodic variations in current density and surface concentrations arise (see Eqs. (3.41)-(3.45)). It is possible to derive expressions for impedance from these equations assuming that the electric properties of the electrode can be represented by the resistance and capacitance in series. Then, the real and imaginary parts of the faradaic impedance are respectively ... 

The next step, after all experimental parameters have been given their correct values, is usually a calibration. A dummy cell is used, consisting of electronic components that imitate the behaviour of the real cell as closely as possible. The simplest one, which also is in many cases a completely adequate one, is shown in Fig.5. It consists of a capacitance (double layer capacitance) in parallel with a resistance (charge transfer resistance), and then, in series with this circuit, another resistance (solution resistance). The admittance of the dummy cell is recorded in an ordinary experiment and the transfer function, T(u), of the instrument is set equal to the ratio of the calculated, 0( )5 to the measured, ym( )) admittance of the dummy cell i.e. 

Tan 5 is a measure of dielectric loss in a capacitor unit and is represented by the ratio of equivalent series resistance and capacitive reactance of a capacitor unit at the rated voltage and frequency (Figure 9.7) i.e. 

Capacitors are often combined in series or parallel, with the resulting circuit capacitance calculated as depicted in Figure 4. An important relationship is the time constant of a capacitor. The time constant is based on the product of the resistance and capacitance and is known as the RC time constant. A capacitor in a dc circuit will charge or discharge 63.2 percent in one RC time constant. The time dependence of a capacitor is shown in the equations. 

Galvanostatic evaluation is often conducted at current densities of about 10-50 mA/cm2 and capacitances in the range of 1 F/cm2 can be easily achieved. The discharge time is approximately 100-200s for a 2V range. A good series resistance is below 1 Q cm2 in organic electrolytes (a time constant of about 1 s). An important component of the series resistance may be the interfacial resistance between the current collector and the active mass. 

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

The admittance in this region is modeled using either a film capacitance in series with the solution resistance or a parallel resistance and capacitance, which is in series with the solution resistance. This is treated as a parallel R-C combination whose magnitude, expressed as an impedance, is given by... 

Z(a)i,ak) = Zre(a)i,ak) +jZim(cai,ak) is the model function, which can be altered using the adjustable parameters the model function can often be presented by an equivalent circuit, involving such elements as resistance, capacitance, and Warburg in series and/or in parallel ... 

Figure 11. Plots of log Z and / v.v. log / for a thiol-hexapeptide-coated mercury drop immersed in 5xlO 3M (a), I.3xlO 2M (b), 3.6xlO 2M (c), and 0.1M (d) KC1, as obtained at -1.000 V over the frequency range from 0.1 to 105 Hz. At frequencies <102 Hz all Bode plots coincide hence, only the experimental points for the lower KC1 concentration were reported. The solid curves are least-squares fits to the simple equivalent circuit of inset (1), which consists of the electrolyte resistance Ra, with in series a RSCS mesh representing the self-assembled monolayer and a further RjiCji mesh representing the diffuse layer. Rs = 0.14 Mfi cm2 C, = 11 pF cm-2 Ra = 4.53 (a), 4.17 (b), 1.27 (c) and 0.87 KO cm2 (d). CW 68 (a), 61 (b), 80 (c) and 84 pF cm 2 (d). Inset (2) shows the reciprocal, 1/Cji, of the experimental diffuse-layer capacitance vs. the l/C fajj = 0) value corresponding to the same KC1 concentration, as calculated on the basis of the Gouy-Chapman (GC) theory. The solid curves are 1 /Ca(OM) vs 1 /C,ii(ctm = 0) plots calculated from the GC theory for different charge densities afo on the metal, whose values are reported on each curve. (Reprinted from Ref.114 with permission from the Am. Chem. Soc.)...

Figure 3.30. Relation between the imaginary and real parts of the impedance (left) and admittance (right). The arrows indicate the direction of the increase of for V = R". Figure (b) is circular. Starting from the origin (where oj = 0) the frequency Increases. Solid lines for a resistance and ideal capacitance in series dashed for resistance and a constant phase element in series.

A schematic of the apparatus is shown in Figure 2. The series resistor senses current. When the overall circuit RC (R and C are circuit resistance and capacitance, respectively) time constant is short compared with the voltage developed on the sensing resistor is proportional to the instantaneous... 

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