is depicted as a transformer. It is characterized by a parameter φ. While φ is dimensionless for usual transformers (the turns ratio), it has the dimension charge/length in the case of electromechanical coupling. The transformer acts as an impedance converter in the sense that a mechanical impedance, Zm, appears as an electrical impedance, Zel, across the electrical ports. Zel is given by Zel = φ2 Zm. For planar piezoelectric crystals, φ takes the value φ = Ae/dq, where A is the effective area, e is the piezoelectric stress coefficient (e = 9.65·10−2 C/m2 for AT-cut quartz) and dq 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. ZL is the load impedance. Note that the load, ZL, 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 ZL. 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 ZL is explicitly included). Small-load approximation The BvD circuit predicts the resonance parameters. One can show that the following simple relation holds as long as the frequency shift is much smaller than the frequency itself: ff is the frequency of the fundamental. Zq is the acoustic impedance of material. For AT-cut quartz, its value is Zq = 8.8·106 kg m−2 s−1. The small-load approximation is central to the interpretation of QCM-data. It holds for arbitrary samples and can be applied in an average sense. Assume that