Motional capacitance

Equivalent (motional) capacitance of bare quartz crystal... 

Motional capacitance Electrical (parallel) capacitance Dissipation, D = Q" ... 

Fig. 14 Simplified Mason circuit (a) close to Fig. 13d. Since tan(fcqfiq) is large close to the resonance and, further, since this element is in parallel to the small load AZl, it may be neglected, b Close to resonance we have cot(fcqfiq 0) and the element - 2L4Zq cot(fcqfiq) can be approximated by a spring, a mass, and a dashpot. c Using the electromechanical analogy, the spring, the mass, and the dashpot may also be represented as a motional capacitance, Ci, a motional inductance, L, and a motional resistance, R ...

Figure 4.1.9. In the resonance state, large strain amplitudes and large capacitance changes (called motional capacitance) are induced, and the current can easily flow into the device. On the other hand, at antiresonance, the strain induced in the device compensates completely, resulting in no capacitance change, and the current cannot flow easily into the sample. Thus, for a high k material the first antiresonance frequency/a should be twice as large as the first resonance frequency/r.

Following the concepts of H. Helmholtz (1853), the EDL has a rigid structnre, and all excess charges on the solntion side are packed against the interface. Thus, the EDL is likened to a capacitor with plates separated by a distance 5, which is that of the closest approach of an ion s center to the surface. The EDL capacitance depends on 5 and on the value of the dielectric constant s for the medium between the plates. Adopting a value of 5 of 10 to 20 nm and a value of s = 4.5 (the water molecules in the layer between the plates are oriented, and the value of e is much lower than that in the bulk solution), we obtain C = 20 to 40 jjE/cm, which corresponds to the values observed. However, this model has a defect, in that the values of capacitance calculated depend neither on concentration nor on potential, which is at variance with experience (the model disregards thermal motion of the ions). 

A common dimensionless number used to characterize the bubble formation from orifices through a gas chamber is the capacitance number defined as Nc — 4VcgpilnDoPs. For the bubble-formation system with inlet gas provided by nozzle tubes connected to an air compressor, the volume of the gas chamber is negligible, and thus, the dimensionless capacitance number is close to zero. The gas-flow rate through the nozzle would be near constant. For bubble formation under the constant flow rate condition, an increasing flow rate significantly increases the frequency of bubble formation. The initial bubble size also increases with an increase in the flow rate. Experimental results are shown in Fig. 6. Three different nozzle-inlet velocities are used in the air-water experiments. It is clearly seen that at all velocities used for nozzle air injection, bubbles rise in a zigzag path and a spiral motion of the bubbles prevails in air-water experiments. The simulation results on bubble formation and rise behavior conducted in this study closely resemble the experimental results. 

Sodium contamination and drift effects have traditionally been measured using static bias-temperature stress on metal-oxide-silicon (MOS) capacitors (7). This technique depends upon the perfection of the oxidized silicon interface to permit its use as a sensitive detector of charges induced in the silicon surface as a result of the density and distribution of mobile ions in the oxide above it. To measure the sodium ion barrier properties of another insulator by an analogous procedure, oxidized silicon samples would be coated with the film in question, a measured amount of sodium contamination would be placed on the surface, and a top electrode would be affixed to attempt to drift the sodium through the film with an applied dc bias voltage. Resulting inward motion of the sodium would be sensed by shifts in the MOS capacitance-voltage characteristic. 

There is an equivalence between the differential equations describing a mechanical system which oscillates with damped simple harmonic motion and driven by a sinusoidal force, and the series L, C, R arm of the circuit driven by a sinusoidal e.m.f. The inductance Li is equivalent to the mass (inertia) of the mechanical system, the capacitance C to the mechanical stiffness and the resistance Ri accounts for the energy losses Cc is the electrical capacitance of the specimen. Fig. 6.3(b) is the equivalent series circuit representing the impedance of the parallel circuit. 

Fig. 11.1. The transmission line circuit used to model these data. The left hand end of the transmission line is at the electrode/film interface. The right hand end is at film/electrolyte interface. The extended resistances, RP and Rx, correspond to the resistance to motion of electrons between trimer centres and ions through the pores respectively, (a) The potential in the central line of the diagram is the potential within the film, and the connecting capacitors modify this potential to produce the driving potentials to drive current through the resistors. The CR kinetic circuit elements for the interfacial process can be seen at each end of the transmission line, (b) The modified circuit when the capacitance, C in equation (9) is not negligible. The potential at the trimer and in the pores is given by E and E ...

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 simple series RLC electrical circuit of Fig. 9.2 consists of a direct-current (DC) power source (here a 3-V battery), a relay, and three loads in series a resistor of resistance R, a capacitor of capacitance C, and an inductor of inductance L. Assume first a DC potential E = E0, in series with R, C, and L the capacitance stores charge, the inductance stores current, and the resistance dissipates some of the current into Joule13 heating. The arrow shows the direction of the current (which, thanks to Franklin s unfortunate assignment, is the direction of motion of positive holes—that is, the opposite of the flow of negative electrons) the relay across L avoids conceptual difficulties about an initial current through the inductor. The current is usually denoted by I (from the French word "intensite"). These three components (R, C, and L) will be explored in sequence. 

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