Quartz Crystal Microbalance - Equivalent Circuits

Equivalent Circuits

Modeling of acoustic resonators often occurs with equivalent electrical circuits. Equivalent circuits are algebraically equivalent to the continuum mechanics description and to a description in terms of acoustic reflectivities. They provide for a graphical representation of the resonator’s properties and their shifts upon loading. These representations are not just cartoons. They are tools to predict the shift of the resonance parameters in response to the addition of the load.

Equivalent circuits build on the electromechanical analogy. In the same way as the current through a network of resistors can predicted from their arrangement and the applied voltage, the displacement of a network of mechanical elements can predicted from the topology of the network and the applied force. The electro-mechanical analogy maps forces onto voltages and speeds onto currents. The ratio of force and speed is termed “mechanical impedance”. Note: Here, speed means the time derivative of a displacement, not the speed of sound. There also is an electro-acoustic analogy, within which stresses (rather than forces) are mapped onto voltages. In acoustics, forces are normalized to area. The ratio of stress and speed should not be called "acoustic impedance" (in analogy to the mechanical impedance) because this term is already in use for the material property Z_ac = ρc with ρ the density and c the speed of sound). The ratio of stress and speed at the crystal surface is called load impedance, Z_L. Synonymous terms are "surface impedance" and "acoustic load." The load impedance is in general not equal to the material constant Z_ac = ρc = (Gρ)1/2. Only for propagating plane waves are the values of Z_L and Z_ac the same.

The electro-mechanical analogy provides for mechanical equivalents of a resistor, an inductance, and a capacitance, which are the dashpot (quantified by the drag coefficient, ξ_p), the point mass (quantified by the mass, m_p), and the spring (quantified by the spring constant, κ_p). For a dashpot, the impedance by definition is Z_m=F / (du/dt)=ξ_m with F the force and (du/dt) the speed). For a point mass undergoing oscillatory motion u(t) = u₀ exp(iωt) we have Z_m = iωm_p. The spring obeys Z_m =κ_p/(iω). Piezoelectric coupling is depicted as a transformer. It is characterized by a parameter φ. While φ is dimensionless for usual transformers (the ratio of the number of loops on both sides), it has the dimension charge/length in the case electromechanical coupling. The transformer acts as an impedance converter in the sense that a mechanical impedance, Z_m, appears as an electrical impedance, Z_el, across the electrical ports. Z_el is given by Z_el = φ2 Z_m. For planar piezoelectric crystals, φ takes the value φ = Ae/d_q, where A is the effective area, e is the piezoelectric stress coefficient (e = 9.65·102 C/m2 for AT-cut quartz) and d_q is the thickness of the plate. The transformer often is not explicitly depicted. Rather, the mechanical elements are directly depicted as electrical elements (capacitor replaces a spring, etc.).

There is a pitfall with the application of the electro-mechanical analogy, which has to do with how networks are drawn. When a spring pulls onto a dashpot, one would usually draw the two elements in series. However, when applying the electro-mechanical analogy, the two elements have to be placed in parallel. For two parallel electrical elements the currents are additive. Since the speeds (= currents) add when placing a spring behind a dashpot, this assembly has to be represented by a parallel network.

The figure on the right shows the Butterworth-van Dyke (BvD) equivalent circuit. The acoustic properties of the crystal are represented by the motional inductance, L₁, the motional capacitance, C₁, and the motional resistance R₁. Z_L is the load impedance. Note that the load, Z_L, cannot be determined from a single measurement. It is inferred from the comparison of the loaded and the unloaded state. Some authors use the BvD circuit without the load Z_L. This circuit is also called “four element network”. The values of L₁, C₁, and R₁ then change their value in the presence of the load (they do not if the element Z_L is explicitly included).