# Analytical Equivalent Circuit Model

Mechanical systems can be converted into electrical circuits by using the analogy between the mechanical and the electrical domains. One way to implement this analogy is to replace the forces in the mechanical domain by voltage sources and velocities by electrical currents. Then, an equivalent circuit of the system is constructed. This method becomes an even more powerful tool for the analysis of electromechanical systems where some parts of the system are already in the electrical domain. For example, equivalent circuit analysis is successfully used for piezoelectric transducers for their design and optimization [1].

Recently, the equivalent circuit model was employed for the characterization of CMUTs [2, 3]. Fig. 1 shows the equivalent circuit for a CMUT transducer. In the electrical part, C0 is the clamped capacitance of the device at the bias voltage. Spring softening capacitance and the mechanical membrane impedance constitute the mechanical part. The two parts are coupled together through an electromechanical transformer.

Compared to the FEM, the equivalent circuit provides quick solutions for many analyses. For example, collapse voltage is one of the critical parameters of a CMUT device. It may very well take about several hours to calculate the collapse voltage of a specific design. Therefore, there is a need for a fast, yet reliable, design tool. We have developed software, specifically for the CMUT, that outputs most of the important parameters, within an acceptable error margin for most of the practical cases. For example, compared to FEM, this software is capable of calculating the collapse voltage in couple of seconds with less than five percent error. The heart of this software is the ability to find the static displacement profile of the CMUT membrane under a uniform pressure (i.e. atmospheric pressure) and a specific DC voltage. It relies on plate theory to find the exact displacement profile [4]. The calculated profile using this software matches very well the result of FEM simulation. For dynamic simulations, this software uses the well-known equivalent circuit models, depicted in Fig. 1. The value of parameters associated with CMUT in these circuit models (i.e. C0, n and Zmem) are calculated using the displacement profile of the membrane and plate theory. Using these circuits, it is easy to find the input impedance, output pressure, output receive signal, signal-to-noise ratio, etc.

**FIGURE 1.** (a) Equivalent circuit model of CMUT in transmit. (b) Equivalent circuit
model of CMUT in receive. R
_{s}: Source Impedance, L
_{TUN}: Tuning Inductance, C
_{P,S}: Parasitic Series Capacitance, R
_{P,S}: Parasitic Series Resistance, R
_{P,P}: Parasitic Parallel Resistance, C
_{P,P}: Parasitic Parallel Capacitance, C
_{0}: Device Capacitance, n: Electromechanical Conversion Factor, Z
_{mem}: Membrane Mechanical Impedance, R
_{loss}: Mechanical Loss, Z
_{med}: Medium Impedance, C
_{amp}: Amplifier Input Capacitance, R
_{amp}: Amplifier Input Resistance.

## References:

**[1]** Mason WP, “Electromechanical Transducers and Wave Filters”. New York, NY: Van
Nostrand, 1948.

**[2]** Ladabaum I, Jin X, Soh HT, Atalar A, and Khuri-Yakub BT, "Surface
Micromachined Capacitive Ultrasonic Transducers,"
*IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control*, vol. 45, pp.
678-690, May 1998.

**[3]** Khuri-Yakub BT, Cheng CH, Degertekin FL, Ergun S, Hansen S, Jin XC, and
Oralkan Ö, “Silicon Micromachined Ultrasonic Transducers,”
*Japanese Journal of Applied Physics*, vol. 39, pp. 2883-2887, 2000.

**[4]** A. Nikoozadeh, B. Bayram, G. G. Yaralioglu, and B. T. Khuri-Yakub, “Analytical
calculation of collapse voltage of CMUT membrane”, Proc.
*IEEE Ultrasonics Symposium*, Vol. 1, Aug. 2004, pp. 256-259.