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Transversal capacitive transducer

Movable electrode is displaced off-plane, perpendicularly to the electrode surface
Original distance between the electrodes d varies by the displacement u_x when the transducer is in function
Instantaneous transducer capacity is:

where e is the permittivity of the dielectric between the electrodes

A is the electrode surface area

When the voltage U (or the charge Q) is applied between the electrodes, the energy of the system can be expressed as:

Differentiation of the preceding Equation relating to the displacement u_x and to the voltage U gives:
Principle of virtual work

In a practice, capacitive transducers operate between these two modes
Quadratic relation between the force and the voltage or the charge can be linearized by the application of the polarization voltage or by the polarization charge
Alternating quantities are superposed on the polarization
If we consider small signals, we can express the mechanical and the electrical quantities of the transducer in a frequency domain. We thus have for a

Transversal capacitive transducer with a constant voltage:
Supposing small signals

where is a transduction factor

Q₀ is the static charge

is the distance between the electrodes at a rest

is the negative compliance

C₀ is the capacity at a rest

I is the current passing through the condenser

V is the velocity of the movable electrode

Analogue circuit representation of the preceding equations:

Impedance Z_m represents the effect of mechanical components attached to the transducer

Presence of the negative compliance is related to the transducer stability - without collapsing the movable electrode towards the fixed electrode

In order to provide the stable operation of the transducer, the external compliance c must be added to the system.

This compliance that is a part of the impedance Z_m must satisfy the condition:

This condition can be expressed by the following relation in term of the value of the polarization voltage:

The limiting value of the polarization voltage U₀ is called as pull-in voltage.

Transversal capacitive transducer with a constant charge:

Similar analogue circuit representation as in the case of constant voltage
Difference: negative compliance doesn?t appear

Transducer polarization

Linearize the transducer performance
Serves as a source of energy in the case of the sensor
Can be realized by
- external source
- dielectric material, called electret
  conserves permanently the electric polarisation
  Application of an electret can be realized by a deposition of an electret layer on of two electrodes