Characteristic Equation

[simonbramble]

In control theory, the characteristic equation (and hence the response of the circuit) is determined by putting the denominator of the transfer function equal to 0.

Why is this? Why do we ignore the numerator?

 

[birdman0_o]

What you are determining is the stability of the system by finding the roots (poles) of the characteristic equation.

 

[simonbramble]

Hi Birdman

Thanks for this. This would make sense. I'll look deeper into this....

 

[MrAl]

In control theory, the characteristic equation (and hence the response of the circuit) is determined by putting the denominator of the transfer function equal to 0.

Why is this? Why do we ignore the numerator?

Hi there,

We dont really ignore the numerator, we just dont need it for certain types of analysis. When we look for certain things sometimes the numerator doesnt matter. When we look for other things then it does matter.

If i said to you, "I have a circle drawn on my paper here". You would know what i was talking about, even though you cant see the circle. You would know it is a line drawn around some central point and it is connected, and there is nothing in the center.
If i said to you, "I also have a bigger circle". You would be able to visualize this too, as another object like the first but it would look to be larger than the first.
In this way, i've told you things that matter in many circumstances yet i never gave you any numerical data such as the diameter or circumference.
In other words, all i have given you was the shape and you knew what i was talking about right away.

The characteristic equation shows us something similar to this idea. We dont have to know the amplitude of the transfer function in many cases, we just have to know the shape of the transfer function. We can then compare the shape alone to other transfer functions we've seen in the past and determine if we like this new one or we need to do something to the circuit to change it.

There's only a couple different ways that the shape can change too so it makes some analysis a little easier. The shape can either be gradually increasing or decreasing and also may contain various bumps along the way. Most importantly, it may happen to increase without end or it levels off at some point like an airplane that takes off and then the height levels off. If it never levels off but increases indefinitely, we say that it is unstable. If it does level off, we say it is stable.
If it has bumps we say it has oscillatory terms and is underdamped, but if not we say it only has exponential terms and is overdamped.

Looking at the characteristic equation we get all this information without having to know what amplitude it actually went up to. We never had to talk about whether or not it went up to 10 volts, or 100 volts, etc. All we had to know was those things above. In this way, we can tell in both systems some very important information about the systems without having to go into the actual numerical detail of their actual plots in time.

What we do usually do though is we solve for the roots and that tells us a lot about the shape of the response. Once we know this, we know if it will fit our intended purpose in the real life application.

There's a little more to it than that though. Because it simplifies part of the analysis of a system in general, it makes certain things simpler when we go to design a control system. That means that because the characteristic equation simplified some things it makes it simpler to do more advanced things too.