System Dynamics - 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.

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.

        In this situation, the variables are:

• x(t) = response

• u(t) = input or driving function

• Examples:

• u(t) = heat in, x(t) = temperature

• u(t) = voltage into a circuit, x(t) = output voltage

• u(t) = voltage driving a motor, x(t) = motor speed.

        The constants in the differential equation are:

• t = the Time Constant

• The time constant is a measure of how quickly the system responds.

• G = the System Gain

• At steady state, the response, x(t), can be calculated by multiplying the input by the gain, G.

             The responses you need to know about for this sytem include:

• Response when the input is zero (There is no input), but the output has some initial value.

• x(t) = x(0)e^-t/t, which looks like:

• Examples:

• Motor slowing down when the power is turned off.

• Temperature rise above ambient decaying to zero when the heat source is turned off.

• Response when the input is a constant (Commonly called the Step Response), which looks like:

• x(t) = G*u*(1 - e^-t/t)

• Examples:

• Motor speeding up when the power is turned on.

• Temperature rising when the heat is turned on.

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.

The general form is given by:

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: