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.

• Electrical Systems.
• Resistor-Capacitor circuits have time constants.
• Resistor-Inductor circuits have time constants.
• Mechanical Systems
• Many motions exhibit time-constant behavior.  For example when a motor speeds up, the measurements of speed would reveal time-constant behavior.
• Thermal Systems
• Heating up and cooling down in simple thermal systems shows time-constant behavior.  Lakes cooling off after the summer slow down the approach of winter near the lake, and later lakes slowly heating up in the spring retard the approach of summer.  That behavior can be explained using time constants.
• Physiological Systems
• The inner ear has dynamics that can be explained using two time constants
• Psychological Systems
• Psychologists have measured the time constant that determines how much learned material you retain as a function of time after learning.

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.  In the lesson on first order system behavior, we encountered this response.

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.

• Determine the initial value of the variable, x(0) in the above.
• Determine the time when the variable has decayed to 37% of the initial value.
• That time is the measured time constant.
That is a simple method that can be applied to numerous situations.  You need to note the following.
• You need a response that dies out to zero.  If you have a response that has the right shape, but which does not die out to zero, then you may have to redefine a component of the response in order to meet this requirement.
• The method depends upon the value of the response at only one point.  In essence, you throw away all of the data that you have except for the starting value and the 37% value.
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.

• Determine the initial slope of the variable, x(0) in the above.
• Determine the time when the variable would have decayed to zero if the initial rate of decay (slope) had continued.
• That time is the measured time constant.
You need to note the following.
• You do not need a response that dies out to zero.  If you have a response that decays to some non-zero final value, just extend the initial slope line until the extended line hits the final value.
• The method depends upon the slope of the response at only one point.  In essence, you throw away all of the data that you have except for the starting value and the initial slope.

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.

• Take a set of data at different times.  That data set would be x(tk) and would comprise all of the values and all of the times.
• Plot the values of the data, x(tk), against the times, tk.  That should give you a straight line.
• Determine the equation of the line.
• The slope of the line is the time constant.
• The intercept is the natural log of the initial value.
Note the following about this method.
• This method uses all of the data.
• If the data is noisy, you can use regression to get the straight line parameters (slope and intercept).
• You must (repeat MUST) have data that decays to zero.  If you do not have data that decays to zero, see the note above about creating a set of data that decays to zero by extracting the essential part of the data from the data set.

Problems

Links to Related Lessons

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