Control Systems
System Dynamics - Time Constants

System Dynamics - Time Constants

Table of Contents

What are time constants and where are they found?

Time constants are everywhere. Since (almost) nothing happens instantaneously, but rather with a delay, processes are said to have time constants associated with them.

In electrical systems, purely resistive circuits are static (not dynamic) as there are no energy storing elements. Resistors do not store energy but rather dissipate it. However, consider these circuits:

In mechanical systems:

In thermal systems:


Measuring Time Constants

When would one want to measure the time constant of a system?

It should be noted that systems typically have more than one time constant. However, it is very typical for a system to have its time constants far apart from each other, resulting in a separation between fast transient responses and the dominant response (which is the slowest one).

E.g. a power electronics switch (say a silicon-carbide MOSFET) contains a junction within a package. The package is mounted on a heatsink. The switch junction temperature rises almost immediately with increased current since its time constant is very short; the package (middle picture) might take a few seconds. The heatsink will have a time constant on the order of minutes.

Picture is owned by GE

We will examine properties of time-domain response that will allow you to obtain an estimate of a time constant.

This write-up focuses only on time-domain. See the Further Reading section for links to relevant articles in frequency domain analysis.

A very standard first-order system response is as below:

Such shape looks like an impulse response of a system.

How does impulse response work?

Impulse response is a parameter of a system that dictates how the system reacts to its inputs. For example, when we take out a frozen fish from the refrigerator, the ambient temperature changes from 30F to 70F. The equations below show how the output of a system is related to the input and the impulse response (h(t) = impulse response, x(t) = input):

The figure above shows both the impulse response and the output signal when the input is a step function, which is quite often the case. The shape and duraration of the response is described with the time constant parameter "tau".

When time "t" is zero, no decay has happened yet and the signal is equal to y(0).

At time "t" equal to "tau", exactly one time constant has elapsed. The signal is now equal to:

Now that we have established the notion of time constant and the time-domain waveform, let's see how to obtain time constant from experimental data.

There are three simple methods to estimate the time constant from time-domain data:

  1. The 37% method
  2. The initial slope method
  3. The logarithmic method

The 37% Method

The 37% method is a widely used method for finding a time constant. It is quite simple:

  1. Determine the initial value of the signal (y(0) as above).
  2. Determine the time when the signal has decayed to 37% of the initial value.
  3. The elapsed time is the time constant.

E.g., for this system, the initial value is 5. 37% of 5 is 1.85, which happens at approximately 1 second. Hence, the time constant tau is 1 second.

However, there are some restrictions to this method:

The first restriction is explained in more detail here:

Thought Nuggets

Q: Is the slowest response always dominant?

A: Not necessarily. If the fast transient has higher magnitude then the slow transient, the slow transient might not be perceived by the control system.

E.g., climate change causes the ocean level to slowly rise. However, tides due to the Moon are much more perceptible.

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