Differential capacitance

P. G. Dargie, S. T. Hughes. A thick-film capacitive differential pressure transducer. Measurement Science and Technology, 5, 1994, pp. 1216-1220. 

Figure 11A.2 Variable capacitance differential pressure transducer, (a) Differential pressure sensing element, (b) Pressure signal transmission system.

The following two references can be consulted for further details on the variable capacitance differential pressure transducer and the pneumatic control valve ... 

Such sensors are used to measure the pressure of a process or the pressure difference which is employed to compute a liquid level or a flow rate (orifice plate, venturi tube). The variable capacitance differential pressure transducer has become very popular. Figure 11A.2 shows a schematic of such a device. Pressure differences cause small displacements of the sensing diaphragm. The position of the sensing diaphragm... 

Measuring device. This can be a variable capacitance differential pressure transducer (Section 13.3), measuring the pressure of a liquid column of height h. The dynamic response of the sensor is given by eq. (13.9). Let Ap = ah, where a is a constant. Then take... 

The variable capacitance differential pressure transducer is a very popular device which is used to sense and transmit pressure differences. Figure 11A.2 shows a schematic of such a device. A pressure signal is... 

IV.l Consider the flow control loop shown in Figure 13.2a. The following information is also available (1) An orifice plate is used to measure the flow (2) a variable capacitance differential pressure transducer is employed (see Appendix 11 A) to sense and transmit the pressure difference developed around the orifice plate (3) the controller is PI and (4) the control valve is of equal percentage, with the valve flow characteristic curve given by... 

Measurement of differential capacitance. Differential capacitances vary with potential, but they can be measured provided the amplitude of the applied a.c. potential E used for measurements does not exceed a few milivolts ([1], p. 29). Differential capacitances are independent of E provided E is small enough. The measurements of the differential capacitance can be erroneous because of contamination by traces of strongly adsorbed organic impurities in the electrolyte. Gra-hame introduced the systematic use of the dropping mercury electrode and was able, in this way, to considerably minimize electrode contamination. Adsorption of impurity traces is generally a slow process because of diffusion control, and frequent renewal of the mercury drop provides a clean surface [21-24]. 

Similar methods directly related to surface films are involved in double-layer capacitance, differential capacitance, and nuclear magnetic resonance techniques, described in the recent literature. As with ellipsometry, advantages are sensitivity in measurement, but equipment requirements limit these techniques to laboratory use, and therefore are mostly for highly theoretical, mechanistic studies. 

The variation of the integral capacity with E is illustrated in Fig. V-12, as determined both by surface tension and by direct capacitance measurements the agreement confrrms the general correctness of the thermodynamic relationships. The differential capacity C shows a general decrease as E is made more negative but may include maxima and minima the case of nonelectrolytes is mentioned in the next subsection. 

The Series 1151 differential pressure transmitter manufactured by Rosemount (MinneapoHs, Minnesota) uses a capacitance sensor in which capacitor plates are located on both sides of a stretched metal-sensing diaphragm. This diaphragm is displaced by an amount proportional to the differential process pressure, and the differential capacitance between the sensing diaphragm and the capacitor plates is converted electronically to a 4—20 m A d-c output. 

Fig. 11. Capacitive pressure sensor in the differential measurement configuration. Courtesy of Rosemount, Inc.

Liquid Level. The most widely used devices for measuring Hquid levels involve detecting the buoyant force on an object or the pressure differential created by the height of Hquid between two taps on the vessel. Consequently, care is required in locating the tap. Other less widely used techniques utilize concepts such as the attenuation of radiation changes in electrical properties, eg, capacitance and impedance and ultrasonic wave attenuation. 

Many of the variations developed to make pressure sensors and accelerometers for a wide variety of appHcations have been reviewed (5). These sensors can be made in very large batches using photoHthographic techniques that keep unit manufacturing costs low and ensure part-to-part uniformity. A pressure differential across these thin diaphragms causes mechanical deformation that can be monitored in several ways piezoresistors implanted on the diaphragm are one way changes in electrical capacitance are another. 

Heat flows from a heat source at temperature 6 t) through a wall having ideal thermal resistance Ri to a heat sink at temperature 62(1) having ideal thermal capacitance Ct as shown in Figure 2.14. Find the differential equation relating 6 t) and 02(0-... 

From the experimental results and theoretical approaches we learn that even the simplest interface investigated in electrochemistry is still a very complicated system. To describe the structure of this interface we have to tackle several difficulties. It is a many-component system. Between the components there are different kinds of interactions. Some of them have a long range while others are short ranged but very strong. In addition, if the solution side can be treated by using classical statistical mechanics the description of the metal side requires the use of quantum methods. The main feature of the experimental quantities, e.g., differential capacitance, is their nonlinear dependence on the polarization of the electrode. There are such sophisticated phenomena as ionic solvation and electrostriction invoked in the attempts of interpretation of this nonlinear behavior [2]. 

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

The above rule apphed to potential v yields for the differential capacitance... 

To our knowledge this is quite a new formula for the differential capacitance. It is vahd whenever charging is equivalent to a shift in space of the position of the wall. We can verify that it is fulfilled for the Gouy-Chapman theory. One physical content of this formula is to show that for a positive charge on the wall we must have g (o-) > (o-) in order to have a positive... 

The above formulas combined with Eqs. (74) and (75) taken at zero charge density yield Eq. (54) for the differential capacitance. Eq. (82) can be used recursively to generate the derivatives of the differential capacity at zero charge density to an arbitrary order, though the calculations become rather tedious already for the second derivative. Thus, in principle at least, we can develop capacitance in the Taylor series around the zero charge density. The calculations show that the capacitance exhibits an extremum at the point of zero charge only in the case of symmetrical ions, as expected. In contrast with the NLGC theory, this extremum can be a maximum for some values of the parameters. In the case of symmetrical ions the capacitance is maximum if + — a + a, < 1. We can understand this result... 

To calculate the profiles and the differential capacitance of the interface numerically we have to choose a differential equation solver. However, the usual packages require that the problem is posed on a finite interval rather than on a semi-infinite interval as in our problem. In principle, we can transform the semi-infinite interval into a finite one, but the price to pay is a loss of translational invariance of the equations and the point mapped from that at infinity is singular, which may pose a problem on the solver. Most of the solvers are designed for initial-value problems while in our case we deal with a boundary-value problem. To circumvent these inconveniences we follow a procedure strongly influenced by the Lie group description. 

FIG. 3 Differential capacitance C as a function of a for several values of parameters. The parameters are given for 1 m aqueous solution and are rescaled... 

The differential capacitance C has been calculated according to the formula... 

It is natural to consider the case when the surface affinity h to adsorb or desorb ions remains unchanged when charging the wall but other cases could be considered as well. In Fig. 13 the differential capacitance C is plotted as a function of a for several values of h. The curves display a maximum for non-positive values of h and a flat minimum for positive values of h. At the pzc the value of the Gouy-Chapman theory and that for h = 0 coincide and the same symmetry argument as in the previous section for the totally symmetric local interaction can be used to rationalize this result. 

A further inconsistency in the Helmholz model is revealed by the differential capacitance C which is given by... 

Fig. 20.7 Differential capacitance/mercury electrode potential relationships for potassium chloride at different concentrations showing (a) how minima are obtained only at low concentrations and (6) the constant capacitance at negative potentials (after Bockris and Drazic )...

The Stern model predicts that the total differential capacitance C will consist of two terms representing two capacitors in series... 

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