Here we want to look at the assumption that the system is a first order linear system.  Here's Al's data along with some data we generated using a theoretical solution that was based on the assumption that the system was a first order linear system.  We started the theoretical solution at 32, and used 68 as the final value, and the time constant was 30 seconds.  It's a reasonable match isn't it?  Click on your response below.  If you have any question about the starting point, you should click here.  The linear model here assumes that we are dealing with temperature rise above ambient.

        The expression for the blue curve is given below.  Makes you eager to get to second order systems, right?

Here's the same expression plotted on a different time scale.

Notice how the second order response - with 2 real poles - starts out more slowly.  In fact, the initial slope is zero.  As the "quick" time constant dies out, the response approaches the kind of approach found in first order linear systems.

        The question that you need to answer is this.

• How do you know that you don't have this kind of behavior?  If you have this type of behavior, then the system is second order.  If the slope changes instantaneously, you probably have a first order system.

• One thing you can - and should - do is to be sure that you take enough data that you can see the first part of the transient.
• Secondly, you need to know what kind of model will suffice.  If you need to make very detailed predictions, then you need to take "dense" data near the start of the transient.  Otherwise, the kind of data plotted above will suffice.  Be clear about that.  You need to make sure that your model is adequate for the job intended.  Don't do more than is necessarsy - and engineers always want to do more than necessary.  However, be sure that you do enough.  You must make the judgement and you need to call it correctly, so you need to understand how the model will be used.