2.2 Different models
2.2.1 Equivalent circuit method
Quartz is an oxide (SiO2) of triclinic crystal symmetry. It has a high melting point, with a phase transition at around 570 oC; this places a limit on the viable temperature range for both the application and process. Moreover, quartz is an anisotropic material, i.e. its material properties vary within the crystal. Therefore it stands to reason that by cutting the quartz plate from different angles it will exhibit varying resonant behaviour. In general, AT-cut quartz is used for QCM's due to its low temperature coefficient around room temperature, with only minimal frequency changes at higher temperatures compared to other cuts of quartz crystals. AT quartz is cut from a specific orientation (35o15') with respect to the optical z-axis of the parent crystal. If an alternating electric current is applied perpendicular to the surface of quartz, it will result in shear deformation, referred to as 'thickness-shear mode' as illustrated in Figure 2.2.1. 
Elements of the parallel RLC resonator are explicitly linked to the properties of ink, and can be interpreted in terms of elastic energy and viscous damping. This model leads to a simple relationship between of the coupling of the quartz and ink dot resonances and facilitates understanding of the resulting responses. These responses are compared with predictions from the transmission-line and Sauerbrey models. 
A mechanical structure can be represented as a simple mass-spring-damper (MSD) system and an equivalent circuit can be used to model this electrically. QCM's are no different and can be modelled using a modified Butterworth Van Dyke (BVD) circuit.  The mass M, spring constant K, damping coefficient D and electrodes of quartz crystal are replaced by inductance L1, dynamic capacitance C1, resistance R1 and static capacitance C0, respectively. Since an inductor resists current changes by self-induction, the current will continue to flow until it is has been spent in order to charge the capacitor C1, but with opposite polarity. In terms of the mechanical model, when the capacitor (C1) is fully charged the potential energy and the shear displacement is at a maximum, while the kinetic energy is zero. The potential energy is completely converted to kinetic energy when the shear displacement of the crystals surface is zero. The displacement will again approach its maximum value as all the kinetic energy reverts to potential energy. The relationship between the electrical and the mechanical models is illustrated in the equations below. 
The BVD circuit shown in Figure 2.2.2 was used to simulate each element in the series structure. The BVD model is one of many electronic representations of a resonator and has been used successfully in a variety of applications [6, 7]. The BVD model shown in figure 2.2.3 represents a "lossy" resonator because it makes use of a resistance R1, which provides means for energy absorption in the device as a result of acoustic attenuation.
The mechanical behaviour of the quartz crystal can be expressed by the electrical equivalent circuit shown in figure 2.2.2 . The inductance, L1, represents the vibrating quartz mass, the resistance, R1, relates to the mechanical energy losses, the capacitance, C1, is related to the mechanical elasticity of quartz and the capacitance, C0, is related to the "disk capacitance" formed by the electrodes and the dielectric substrate. Z, is the electrical impedance which is used to model mass loading conditions.
Considering the relationship between the electrical parameters and the mechanical properties of the resonator, the values for R, L and C can be obtained as follows: 
The area of the QCM electrode affects the capacitance value Cc which will be substituted later to find the other electrical constants. For that reason, the electrode geometry must be specified to derive the equivalent circuit parameters for the each crystal design.
In the mass loading tests, ink with a density of 1090 kg/m2 was placed on the surface of the electrode. By relating this to the resistance R1, it can be seen that the impedance Z is roughly 4 times larger. The ink dot markedly alters the resonant frequency of the QCM.
After this property is calculated, the QCM can be employed as a mass sensor. With the ink dot placed at the centre of the electrode, it causes a reduction in the oscillating frequency of the crystal. The mass of the ink dot was assumed to be around 0.021µg ± 0.002.  Equation 2.2.4 leads to the extended Sauerbrey equation, which is free from any assumptions of invariability of certain circuit elements, and represents the mass calculation of the ink dot using the values of the equivalent circuit previously specified.  Given this, we derived the equation below which represents the mass of ink droplet respect to the electrical components of equivalent circuit of loaded QCM.
The quality factor (Q) can be used to describe the resonance of a damped oscillator. Q is defined as the ratio between the stored and dissipated energy of an oscillating system during one cycle, multiplied by a factor of 2p. The quality factor of the QCM can be derived using the equivalent circuit parameters previously calculated.
A high quality factor implies the system is efficient.
2.4.2 Computer analyser method
By using a vector network Analyser and a 5MHz crystal, it was possible to produce a number of charts which demonstrate the operation of such devices. Figure 2.2.4 shows the apparatus used to collect this data.
In the equation below, Z denotes the total impedance of the circuit, with XL and XC representing the inductive and capacitive reactances respectively.
Figures 2.4.5 shows the admittance phase angle (blue) and the admittance (Y=1/Z) measured for a 5MHz AT cut QCM. From this it can be seen that the phase angle is equal to zero at two points. The first occurs at the series resonant frequency, also coinciding with the maximum for the vibration amplitude. The second marks the parallel resonant frequency, which corresponds to the anti resonance. At low frequencies the capacitive reactance dominates and the circuit behaves as a pure capacitor. The gain phase angle (blue) is close to 90o indicating that the voltage is leading the current. As the frequency approaches the resonant region, the inductive reactance becomes more evident.  Exactly at resonance the contributions from the capacitive and the inductive reactance cancel one another, so that the phase angle becomes zero and the impedance reaches a minimum. The gain (red), on the other hand, is at its maximum. At this point the circuit behaves as a pure resistor. As the frequency increases, the gain phase angle decreases until the circuit becomes totally inductive (gain phase angle -90o).
Figure 2.2.5: Magnitude of gain (red) and phase angle of gain (blue) for a 5MHz AT-cut quartz crystal measured with an impedance analyzer
Increasing the driving frequency causes the crystal to resonate once again when the parallel inductive and capacitive reactance's cancel. At parallel resonance the gain phase angle passes through zero again and for higher frequencies goes back to 90o. In addition to this trace, the admittance plot was taken which can be broken into its imaginary (susceptance) and real (conductance) components as shown in figure 2.2.6. The relationship between the susceptance (Bo) and the conductance (Go) can be expressed as follows: 
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