Transfer function capacitance

A capacitance transfer function based on the total double-layer capacitance can be... 

As the thermal capacitance and resistance of the hotplate provide a thermal low-pass transfer function (with the dominant pole corresponding to a characteristic time of 10-20 ms, depending on the fabrication process), the ZA modulator driving the hotplate constitutes a linear noise-shaping DAC with an output in the thermal domain. 

In this model, it is assumed that the energy of the oscillator is stored in a capacitor C, its losses are represented by resistor R and the mass loading by inductor L. The static capacitance of the crystal with the electrodes is shown as capacitor C0. The transfer function of this oscillator is governed by two equations.The first is... 

The important point to note here is that the capacitive terms determine the overall shape of the transfer function, whereas the frequency f and the sheet resistance Rg occur only within the parameter a, and only as a product. Therefore, an increase in frequency at fixed sheet resistance can cause the same change in transfer function as an increase in sheet resistance at fixed frequency. This property can be exploited in visualizing the transfer function by plotting the amplitude of Equation 1 against its phase shift, using... 

Figure 4. Calculated transfer function for CFT device illustrating effect of sheet capacitance on shape of calibration curve. Reproduced with permission from reference 18. Copyright 1982 Institute of Electrical and Electronics Engineers.

Another passive method is the transference function method (TFM) introduced by Muramatsu [6]. The method consists of an oscillator that drives a crystal through a known measuring impedance and a radiofrequency voltmeter which measures the transference modulus of the system. Muramatsu [6] neglected the effect of the parasitic capacitance and his expression for the quartz impedance resulted in a nonlinear relationship between the measured resistance R with the ac voltage divider and the value of R measured by an impedance analyser. Calvo and Etchenique [74] improved the method and introduced an analytical expression to fit the entire transfer function around resonance in order to obtain the same values of R, L and C as measured by a frequency response analyser. 

The transfer function i/V and the transfer function Cn/V can be neglected at this frequency. Thus the capacitance Cji may be considered to have its steady-state value. If Re can be neglected with respect to 1/ (/cuhfQi)/ the capacitance is obtained from the expression... 

The inverse of impedance is called admittance, Y s) = HZ s). These are transfer functions that transform one signal (e.g., applied voltage), into another (e.g., current). Both are called immittances. Some other transfer functions are discussed in Refs. 18, 35, and 36. It should be noted that the impedance of a series coimection of a resistance and capacitance, Eq. (6), is the sum of the contributions of these two elements resistance, R, and capacitance, HsC. 

IMPS uses modulation of the light intensity to produce an ac photocurrent that is analysed to obtain kinetic information. An alternative approach is to modulate the electrode potential while keeping the illumination intensity constant. This method has been referred to as photoelectrochemical impedance spectroscopy (PEIS), and it has been widely used to study photoelectrochemical reactions at semiconductors [30-35]. In most cases, the impedance response has been fitted using equivalent circuits since this is the usual approach used in electrochemical impedance spectroscopy. The relationship between PEIS and IMPS has been discussed by a number of authors [35, 60, 64]. Vanmaekelbergh et al. [64] have calculated both the IMPS transfer function and the photoelectrochemical impedance from first principles and shown that these methods give the same information about the mechanism and kinetics of recombination. Recombination at CdS and ZnO electrodes has been studied by both methods [62, 77]. Ponomarev and Peter [35] have shown how the equivalent circuit components used to fit impedance data are related to the physical properties of the electrode (e.g. the space charge capacitance) and to the rate constants for photoelectrochemical processes. 

What is a first-order system, and how do you derive the transfer functions of a first-order lag or of a purely capacitive process ... 

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. 

The actuation model obtained in Section 5.3.1 is an infinite-dimensional transfer function. All parameters in the model are already fundamental material parameters and actuator dimensions except the double-layer capacitance C and the resistance R. Scaling laws for C and R can be further derived to obtain a fully scalable model. In particular, G is proportional to the area A of polymer/electrolyte interface. The resistance R can be obtained as a function of material resistivity and dimensions using a transmission line model [Fang et al. (2008d)]. Fig. 5.5 shows the experimental verification of the scaling laws for C and R, respectively. 

The nanostructured electrodes can store a large number of electrons, which impUes that the photocnrrent is driven into a capacitive element. This introduces an additional time constant, the RC time, in the photocurrent response. To deal with it, Eqs (51) and (52) mnst be multiplied by the transfer function of the measuring system, presented in Figure 4.3.31. The transfer function of the measuring system and the measured signals in the frequency domain are given by Eqs (32) and (33), respectively. 

Here, Cp is a parasitic capacitance stemming from the combination of fhe coaxial cable. Bayonet Neill-Concelman (BNC) connector, printed circuit board (PCB) trace, lead frame, bonding pad, electrostatic discharge (BSD) cell, and input-pair transistors of the core amplifier. A(s) is the amplifier s transfer function, Aq/(1 + s/cOq), where Aq and cOq are its open-loop gain and bandwidth, respectively. 

The order of a filter is equal to the number of poles in the filter network transfer function. For a lossless L-C (inductance- capacitance) filter with resistive (nonreactive) termination, the number of reactive elements... 

The stray capacitances between turns of the coils are represented by a capacitor placed across the terminals of each coil. This capacitance is larger for coUs with more turns. Although the capacitance is actually distributed, it is lumped for the equivalent circuit, in order to simplify the analysis. The capacitance from one coil to the other is represented by another capacitor placed in parallel with the leakage inductance and resistance. The transfer function of the complete equivalent circuit is given in Eq. (10.16),... 

With the above simplifications and introduction of CPE instead of capacitances, the transfer function (Eq. (9-15)) can be described by the following equations... 

Using the model parameters, one can now calculate the transfer functions for the models, Eqs. 35 and 36 respectively, to derive the voltage across the actuator capacitance (Eq. 37). 

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