Butterworth-Van Dyke Circuit

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This Demonstration shows plots of the admittance for the Butterworth–Van Dyke electrical circuit, commonly used to model the electro-acoustic admittance of a quartz resonator used in the quartz crystal microbalance (QCM).

Contributed by: Nicolas Murer and Jean-Paul Diard (June 2020)
(Bio-Logic SAS)
Open content licensed under CC BY-NC-SA



This Demonstration plots various representations of the admittance of the Butterworth–Van Dyke electrical circuit, commonly utilized to model the electro-acoustic behavior of a quartz resonator used in quartz crystal microbalance (QCM). The top branch of the electrical circuit is called the acoustic branch while the bottom branch is called the electrical branch.

The admittance can be written:


where is the frequency and .

The reduced admittance is then:

with , the reduced frequency,



The expressions of the real and the imaginary parts of the admittance are:



The resonance of the quartz crystal is reached when the reduced frequency




The green dots represent the reduced resonance frequency.


[1] A. Arnau, T. Sogorb and Y. Jiménez, "A Continuous Motional Series Resonant Frequency Monitoring Circuit and a New Method of Determining Butterworth-Van Dyke Parameters of a Quartz Crystal Microbalance in Liquid Media," Review of Scientific Instruments, 71(6), 2000, pp. 2563–2571. doi:10.1063/1.1150649.

[2] S. Butterworth, "On Electrically-maintained Vibrations," Proceedings of the Physical Society of London, 27(1), 1914 pp. 410–424. doi:10.1088/1478-7814/27/1/330.

[3] K. S. Van Dyke, Phys. Rev., 25, 1925 p. 895.

[4] K. S. Van Dyke, in Proceedings of the 1928 IEEE International Frequency Control Symposium, New York: IEEE, 1928 p. 742.

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