7.A Two port transducer (examination assignment November 2001)
Consider a two-port transducer with extensive variables x1 and x2 and intensive variables X1 and X2. The following constitutive relations are known (A, B and C are constants):
a)Show that the energy function U(x1, x2) of the transducer can be written as:
with UA a constant.
b)From the calculation of the energy it follows that the constants B and D cannot be chosen independent from each other. What relation between B and C has to be fulfilled?
c)Derive the characteristic equations of the transducer.
d)Give an expression for the coupling factor.
7.B Electromechanical transducer (examination assignment October 1998)
Consider the following electromechanical transducer.
Dielectric material with a relative dielectric constant εr can move between two capacitor plates. The chunk dielectric material is fixed by a spring (spring constant K, the spring is relaxed at position x=x0). When the dielectric material moves between the plates the capacitance of the capacitor will change. That is why this configuration can be used as an electromechanical transducer. The air gap between the dielectric material and the plates can be neglected. The plates have a length L and a width b. The distance between the plates is d.
a)Calculate the capacitance C between the plates as a function of the length x by which the dielectric material is pushed between the plates.
b)Give an expression for the energy in the transducer as a function of the charge q on the plates and the length x by which the dielectric material is pushed between the plates. Take the spring into account when you consider the transducer.
c)A constant voltage controls the transducer. Calculate the displacement x of the mechanically unloaded transducer as a function of the voltage. (Note: the spring is still a part of the transducer).
d)Is it possible that the transducer becomes unstable at a position x at a given constant control voltage u?