From the characteristic equations we can derive the so-called coupling factor. In the general case where the characteristic equation is given by:
|
(2.55) |
with x1, x2 and X1, X2 representing the extensive and intensive variables, respectively, the coupling factor 
is defined as:
|
(2.56) |
Note that you really need to start from the characteristic equations in the form of (2.55) with the extensive variables at the right hand side and the intensive variables at the left hand side. Otherwise you get the wrong result for the coupling factor.
We can also express the coupling factor in terms of the energy function:
|
(2.57) |
The coupling factor is a measure for the efficiency of the transducer. A coupling factor of zero means that there is no transduction at all, since the coefficients b12 and b21 in the characteristic equation are zero so that there is no coupling between the variables of the two domains. A larger coupling factor means a stronger coupling between the two domains, however the coupling factor should not be larger than 1. At a value of 1 the transducer will become unstable!
For the parallel plate capacitor with spring with the characteristic equations given by (2.53), we have a coupling factor given by:
|
(2.58) |
Note that 
increases with increasing q or decreasing plate distance x. There will be combinations of x and q where the value of 
exceeds 1 and the parallel plate capacitor can in principle become unstable, meaning that the capacitor plates will collapse onto each other. We have already discussed the phenomenon of instability in section 2.6. We then showed that at the point of instability the effective spring constant is equal to zero (equation (2.34)). For a parallel plate capacitor with spring it is relatively straightforward to show that (2.34) can be rewritten into (try this yourself!):
|
(2.59) |
Thus we immediately see that a coupling factor of 1 corresponds exactly to an effective spring constant of zero.