System Dynamics - Time Constants
Where Are Time Constants Found?
Measuring Time Constants

Time Constants - Where are they found?

        Time Constants are ubiquitous.  They are found in many different kinds of systems, including the following.


How Do Time Constants Come About?

        Time constants are parameters of systems that obey first order, linear differential equations.  That would be a differential equation like this one.

t(dx(t)/dt) + x(t) = Gu(t)

        In this situation, the variables are:

        The constants in the differential equation are:         The responses you need to know about for this sytem include:



Measuring Time Constants

        In many experimental situations you need to measure the time constant of a system.  In this section we will examine properties of time constant response that permit you to get a measurement of a time constant.

        We start by examining some typical time-constant behavior.  Here is a response of a first order system that exhibits time response behavior.

We need to think about the features of this response that will permit us to get measurements that look like this and determine the time constant of the system from these kinds of measurements.

        If this is the impulse response of a system, it would have this for.

x(t) = (Gdc/t)e-t/t

The general form is given by:

x(t) = x(0)e-t/t

We can look for important points in this response.  The most obvious is when t = t.  That's when the elapsed time is one time constant.  When one time constant has elapsed the response is:

x(t) = x(0)e-1 = x(0)(0.36788)


The 37% Method

This gives us the first method for finding a time constant.  Here is the method.

That is a simple method that can be applied to numerous situations.  You need to note the following.         The first note above has some implications.  Here is a response that exhibits time constant behavior.

This response does not decay to zero.  However, it does apparently decay to a value of 17, so if you subtracted 17 from the response you would get a response function that did decay to zero.  That might make some sense physically in many situations.  For example, the plot above might be the temperature above ambient (in Celsius degrees) and the temperature rise above ambient would fall to zero as the temperature fell to ambient in a thermal system.  Look for situations like that.

        The same kind of reasoning would also apply to finding time constants from step responses.  Here is a step response.

Here the steady state is 50, and you could take the difference between the steady state (50) and the response.  That difference decays to zero as the response approaches the steady state, and that gives you the kind of function that you want.

Here is the method.

You need to note the following.
The Initial Slope Method

       There are other methods that you can use.  One other method usese the initial slope of the response.  Consider the equation for the response curve.

x(t) = x(0)e-t/t

The slope is given by:

dx(t)/dt = [-x(0)/t]e-t/t

Interestingly, if the response continued to decay at this rate - and we know that it doesn't - the response would decay to zero in t seconds.  Since the rate of change is -x(0)/t, if it continued t for seconds the total change would be -x(0), which would bring the response to zero.

        The implications of the argument above are that you can draw a line from the initial point at the initial slope and check where that line intersects the final value, and the time when that happens is the time constant.  That is better when you have a visual representation.  Here is a plot.

In this plot, the initial slope has been extended (blue line) and intersects the final value (0) at about 25 seconds, so the time constant would be 25 seconds.

The Logarithmic Method

       There are other methods that you can use.  Both of the previous methods really depend upon values at a single time.  There is another method that uses all of the data.  Consider the equation for the response curve.

x(t) = x(0)e-t/t

Take the logarithm of the response curve and we have the following.

ln[x(t)] = ln[x(0)] - t/t

This is the equation of a straight line when plotted against t.  That leads us to the following strategy.

Note the following about this method.         What happens if you don't have a decaying exponential?  In that case, you need to isolate the decaying exponential that is buried in the data.  Here is an example.

In this response, the data does not decay to zero.  However, if you take the following steps, you can get a data set that you can use with this method.
Problems


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