A Time Constant Is A Time Constant, Right?

        There is a temptation to say that a certain device has a time constant, and it is important to remember that when you talk about time constants you need to be specific about the situation and the function you are calculating.  It is NOT CORRRECT to say that a certain device has a time constant and not say more about it.  We'll look at a specific example.

Example

        Imagine that you have a motor that is slowing down because you turned off the power.  We can look at the rotational velocity, w(t).  If the motor is slowing down, it starts out at some initial rotational velocity (so many rpm, etc.) and eventually stops moving altogether.  It's easy to believe that the rotational velocity is given by:

w(t) = wo(0)e-t/t

That's the expression for an exponential decay, with some time constant, t.  It has the form of an exponential decay, and in that form, the constant that is divided into the time variable, t, in the exponential is always the time constant.  Let's just imagine that we have a time constant of 10 seconds, i.e. t = 10 sec.  We would be tempted to say that the time constant of this system is 20 seconds.

        Is that true?  Let's consider another aspect of the situation.  In this system, there is probably some rotational inertia, call it J, for the shaft and any load attached.  It's a well-established fact from rotational mechanics that the rotational kinetic energy (at any instant of time) is given by:

Rotational kinetic energy = Jw(t)2/2

Then, since we know the rotational velocity as a function of time, we can just plut in the formula for the rotational velocity.  Then, we have:

Rotational kinetic energy = wo2(0)e-2t/t/2

Now, if we look at the form for the rotational kinetic energy we would conclude that the time constant is now  t/2.  That's half of what it was before, and would be 5 seconds in our example.

        So, the question we are tempted to ask is this.  "What is the time constant of this system?"  The point of this note is that the question is not a valid question.  You just can't ask that question.  You have to ask another question, and there are choices.  Here are two valid questions.

        In other words, you have to be more specific about what it is that you want the time constant of.  When you ask for the time constant of the velocity you get one answer, and when you ask for the time constant of the kinetic energy you get another answer.

        The problem here is that when you have a functional transformation of the time varying variable - for example, when you square the rotational velocity - the resulting function of the time varying variable will not exhibit the same time constant behavior.  And, if you think about it further, if the functional transformation is not a power (squaring, cubing, taking the square root) the result is probably not even an exponential and is not going to exhibit a form where it is valid to extract a time constant.

        Where could this come into play?  One place is in a thermistor circuit, either a voltage divider or a bridge circuit.  If you have a thermistor in a voltage divider there are some things to note.

What can you do in this situation?  There's really only one answer.  You should only deal with the temperature.  It would be interesting to look at the output voltage of a voltage divider or bridge circuit, plot the data and get the time constant, and then compare the results there with what you get when you calculate the temperature and get the time constant of the temperature behavior (i.e. the temperature time constant).