Motional impedance

The equivalent circuits (Figure 3.5) can be used to describe the electrical response of the perturbed device. The lumped-element model. Figure 3.Sb, is most convenient to use. When the resonator has a surface perturbation, the motional impedance increases, as represented by the equivalent-circuit model of Figure 3.7. This model contains the elements C , Li, C, and Ri corresponding to the unperturbed resonator. In addition, the surface perturbation causes an increase in the motional impedance Z(n as described by the complex electrical element Ze in Figure 3.7a. This element is given by [12]... 

Letting = / 2 + allows the complex element Z to be represented by a real motional resistance R2 and inductance L2, as indicated in Figure 3.7b. From Equation 3.19, the motional impedance elements Lz and Rz can be related to the components of the surface mechanical impedance as [14]... 

Combining Equations 3.19 and 3.2S gives the motional impedance elements arising from the ideal mass layer [9,14] ... 

The motional impedance elements arising from liquid loading are found from Equations 3.21 [9,14,17] ... 

The equivalent circuit model of Figure 3.7 can be used to describe the near-resonant electrical characteristics of the quartz resonator coated by a viscoelastic film. The surface film causes an increase in the motional impedance, denoted by the complex element Zg. From Equation 3.19, this element is proportional to the ratio of the surface mechanical impedance Zj contributed by the film to the characteristic shear wave impedance Zq of the quartz. 

Equations 3.19 and 3.36 can be combined to find the change in (electrical) motional impedance that arises from a viscoelastic film on a thickness-shear mode resonator [40] ... 

Equation 2 can be rewritten in a way that Z x can be presented as a parallel arrangement of Co, the only genuine electrical parameter in Eq. 2 (formed by the two electrodes with quartz as dielectric), and a so-called motional impedance, Z Z = Co Z (Pig. 4a). Zm contains two elements in series. The first summand, Z q, includes only crystal parameters and describes the motional impedance of the quartz crystal as a fimction of frequency co = litf. The second summand expresses the transformation of the acoustic load, Zl, into the (electrical) motional load impedance, Z il- We therefore call the fraction in front of Zl transformation factor. Applying some assumptions reasonable in most sensor applications Z becomes ... 

Consequently, the second term in Eq. 3 is the important one for sensor applications. Obviously a small coupling factor and a small wave velocity increase the electrical representation of the acoustic load. On the other hand, the quartz motional impedance also alters upon changes in Vq and K. It is therefore helpful to rewrite Eq. 3 for our purposes ... 

As shown in the previous section, the mechanical properties of a quartz crystal close to resonance frequency can be expressed by means of a motional impedance. To complete the equivalent circuit of a quartz crystal, the capacitance, Co, must be added in parallel to the motional impedance. It results in the Butterworth-Van Dyke (BVD) equivalent circuit of a quartz crystal, as shown again in Fig. 8 for an unloaded quartz crystal [32]. In this notation common in electronic Hterature, Is is the dynamic inductance and is imder-stood here as a representation of the oscillating mass of the quartz crystal. Cs is the dynamic capacitance and reflects the elasticity of the oscillating body. Rs is the dynamic resistance and returns friction of the quartz slice as well as all kinds of acoustic damping. 

Fig. 14 Phase of impedance Z((p=XIR) and phase of motional impedance 2 ipm = Im(2m)/Re(2m)) and conductivity G of a 10 MHz blank quartz crystal with one surface in contact with water...

There are two electrical equivalent circuits in common usage, the transmission line model (TLM) and a lumped element model (LEM) commonly referred to as the Butterworth-van Dyke (BvD) model these are illustrated in Figs. 2(a and b), respectively. In the TLM, there are two acoustic ports that represent the two crystal faces one is exposed to air (i.e. is stress-free, indicated by the electrical short) and the other carries the mechanical loading (here, a film and the electrolyte solution, represented below by the mechanical loading Zs). These acoustic ports are coimected by a transmission line, which is in turn connected to the electrical circuitry by a transformer representing the piezoelectric coupling. For the TLM, one can show [18, 19] that the motional impedance (Zj ) associated with the surface loading can be related to the mechanical impedances of... 

The movement of the fast electrons leads to the fonnation of a space-charge field that impedes the motion of the electrons and increases the velocity of the ions (ambipolar diffusion). The ambipolar diffusion of positive ions and negative electrons is described by the ambipolar diffusion coefficient... 

Intermetallics also represent an ideal system for study of shock-induced solid state chemical synthesis processes. The materials are technologically important such that a large body of literature on their properties is available. Aluminides are a well known class of intermetallics, and nickel aluminides are of particular interest. Reactants of nickel and aluminum give a mixture with powders of significantly different shock impedances, which should lead to large differential particle velocities at constant pressure. Such localized motion should act to mix the reactants. The mixture also involves a low shock viscosity, deformable material, aluminum, with a harder, high shock viscosity material, nickel, which will not flow as well as the aluminum. 

The object to be propelled carries the material to be pushed against, and usually carries the energy source as well (fuel and oxidant). Some examples of this type are rockets, inflated balloons released to exhaust their air, and ion engines. The inflated balloon would be propelled by interaction with its released air even if it were released in space. It does not push against the air around itself. In fact, the ambient air only tends to impede its motion. 

Different lengths of chains play different roles in controlling polymer properties. For instance, shorter chains flow more readily.in the molten state and are more readily incorporated into crystallites because they have fewer entanglements to impede their motion. Conversely, longer chains tend to resist flow and impede crystallization. 

Other methods for impeding dislocation motion are the introduction of grain boundaries, and/or twin boundaries. While these impediments may increase the hardness, they are also likely to decrease the tensile strength. 

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