Load impedance

As one can see, there is the familiar choke input filter (T-C) on the output, which is characteristic of the buck and all forward-mode converters. The configuration shown in Figure 4—10 is called a parallel resonant topology because the load impedance (the T-C filter acting as a damping impedance) is placed in parallel to the resonant capacitor. The input to the T-C filter stage... 

Consider first the series junction of N waveguides containing transverse force and velocity waves. At a series junction, there is a common velocity while the forces sum. For definiteness, we may think of A ideal strings intersecting at a single point, and the intersection point can be attached to a lumped load impedance Rj (s), as depicted in Fig. 10.11 for TV= 4. The presence of the lumped load means we need to look at the wave variables in the frequency domain, i.e., V(s) = C v for velocity waves and F(s) = C / for force waves, where jC denotes the Laplace transform. In the discrete-time case, we use the z transform instead, but otherwise the story is identical. 

For example, Ft(s) could be pressure in an acoustic tube and 1 (s) the corresponding volume velocity. In the parallel case, the junction reduces to the unloaded case when the load impedance Rj(s) goes to infinity. 

Z1 internal grid impedance Z2 load impedance RF fault resistance T1, T2 protective devices Q1, Q2 isolating switches K1, K2 short-circuiting devices R1, R2 resistances ofK1, K2 11 fault current generated by the grid I2 fault current generated by a reactive load. 

Short-circuit current (/sc). This is the current output when the load impedance is much smaller than the device impedance. 

To know what the measured voltages Vl and Vn are, we now look at Figure 9-3. The voltage due to the common mode component is 25 2 multiplied by the current flowing in the earth connection (i.e. 50 Q times the current in each leg). The voltage due to the differential mode component is 100 2 times the differential mode current. Therefore the LISN provides the following load impedances to the noise generators (in the absence of any input filter)... 

Electrical admittance Electrical impedance Acoustic load (impedance)... 

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 ... 

Figure 6a shows the transmission hne representing a viscoelastic layer [64]. Every layer is represented by a T . The apphcation of the Kirchhoff laws to the Ts reproduces the wave equation and the continuity of stress and strain. The detailed proof is provided in [4]. To the left and to the right of the circuit are open interfaces (ports). These can be exposed to external shear waves. They can also be connected to the ports of neighboring layers (Fig. 6b). Alternatively, they may just be short-circuited, in case there is no stress acting on this surface (left-hand side in Fig. 6c). Finally, if the stress-speed ratio Zl (the load impedance, see below) of the sample is known, the port can be short-circuited across an element of the form AZl, where A is the active area (right-hand side in Fig. 6c). Figure 6c shows a viscoelastic layer which is also piezoelectric. This equivalent circuit was first derived by Mason [4,55]. We term it the Mason circuit. The capacitance, Co, is the electric capacitance between the electrodes. The port to the right-hand side of the transformer is the electrical port. The series resonance frequency is given by the condition that the impedance of the acoustic part (the stress-speed ratio, aju) be zero, where the acoustic part comprises all elements connected to the left-hand side of the transformer.

Apart from these practicalities, there is an important new concept contained in the equivalent circuit representation, which is the load impedance, Zi. The load impedance in this context is the ratio of the stress, a, and the speed, ii, at the crystal surface. The load impedance is normaUzed to area (unlike the mechanical impedance). 

The load impedance, Zl, in general is not equal to the material constant Z = pc = (Gp) /. Only for propagating plane waves in an infinite medium are the values of Zl and Z the same. The ratio of stress and speed in this case is given as... 

A few other comments on the Mason circuit and the BvD circuit are provided in the Appendix. Here, we move on and discuss the role of the load impedance in data analysis. 

Relation Between the Frequency Shift and the Load Impedance... 

The load impedance is the ratio of stress and speed at the crystal surface. From the BvD circuit, one can read how the resonance frequency responds to the load. Below we derive a relation between the frequency shift and the stress-speed ratio. We use a complex spring constant, ic = ic + itwl p, and a complex eigenfrequency of the bare crystal, u5o = ( p/mp) /, for computational convenience. From Fig. 14b one reads the resonance condition as ... 

Equation 51 is the most important equation of the physics of the QCM. As long as the frequency shift is small compared to the frequency, the complex frequency shift is proportional to the load impedance at the crystal surface. We term Eq. 51 the small-load approximation. At this point, we have not made any statement on the nature of the sample. We have only stated how the frequency shift depends of the stress-speed ratio at the crystal surface. 

Assume that the sample does not consist of planar layers, but instead of a sand pile, a froth, an AFM tip, an assembly of spheres or vesicles, a cell culture, a droplet, or any other kind of heterogeneous material. There are many interesting samples of this kind. The frequency shift induced by such objects can be estimated from the average ratio of stress and speed at the crystal-sample interface. The latter is the load impedance of the sample. The concept of the load impedance tremendously broadens the range of applicability of the QCM. The load impedance is the conceptual link between the QCM and complex samples. If we want to predict the frequency shift induced by a complex sample, we must ask for the average stress-speed ratio. If this ratio can be estimated in one way or another, a quantitative analysis of the experimental QCM data is in reach. Otherwise, the analysis must remain qualitative. 

Equation 94 holds regardless of the overtone order and is a simple result. However, Eq. 94 is not reproduced when applying the small-load approximation (Eq. 51) and using the load impedance of a viscoelastic film as expressed in Eq. 72 ... 

In the following, we derive the Butterworth-van Dyke (BvD) equivalent circuit (Fig. 7) from the Mason circuit (Fig. 6c). The Mason circuit itself is derived in detail in [4]. The BvD circuit approximates the Mason circuit close to the resonances. The BvD circuit accounts for piezoelectric stiffening and can also be extended in a simple way to include an acoustic load on one side of the crystal. In the derivation of the BvD circuits, one assiunes small frequency shifts as well as small loads and apphes Taylor expansions in the frequency shift (or the load) whenever these variables occur. The condition of A/// material constant). Zq sets the scale of the impedances contained in the Mason circuit. Generally speaking, the QCM only works properly if ZL Zq.ii... 

The extension of the previous models to a sphere coupled to the plate via a spring and a dashpot is straightforward. The coupling can be achieved either via a Voigt-type circuit (viscoelastic solid, Fig. 2e) or via a Maxwell-type circuit (viscoelastic liquid, Fig. 2f). Below, we assume that the object is so heavy that it does not take part in the motion. When the mass is infinite, the inertial term drops out of the load impedance. An infinite mass is graphically depicted as a wall. For Voigt-type couphng we find ... 

Quantitative analysis of the impedance data recorded with and without cells provides the change in the complex load impedance AZl that is due to the presence of cells on the resonator surface relative to medium loading. Table 2 compares the magnitude of the load impedance A Zl for seven different cell types that had been grown to confluence prior to the QCM experiment. The change in load impedance varies considerably for the var-... 

Table 2 Change in the load impedance of quartz resonators covered with confluent layers of different cell types relative to resonators in contact with culture meditun only. The complex load impedance AZl is expressed by its magnitude A Zl as weU as in its real and imaginary components...

---