# What's the Q of a pole at the origin of the s-plane?

I'm writing a simple function in Python to return the Q of a circuit, given its poles.

The Q of a system with pole p, can be found using $Q=-\frac{\left|p\right|}{2*Re(p)}$. Obviously, in a natural system, such a pole must come as part of a pair, and both would yield the same result.

For coincident poles on the real axis, this gives a Q of 1/2 (critically damped). For poles on the imaginary axis, that gives a Q of $\infty$ (undamped).

Writing the function left me with the question, as a matter of completeness, of what to do if the pole is at the origin. Is there a customary value to assign to poles at the origin?

Background material:

For anyone interested in where the equation relating pole location to Q comes from, It derives from the 2nd order transfer function:

Graphically described here:

A decent discussion of Q and damping in 2nd order systems can also be found here:

which gives an excellent review of how the pole placement corresponds to various damping characteristics.

• have a look to the properties here: en.wikipedia.org/wiki/Pole_and_polar Mar 21, 2016 at 8:14
• @BehindTheSciences: Would you be able to answer the question after having read the article you linked to? Mar 21, 2016 at 8:45

Note that the formula given in your question is valid for a system with a complex-conjugate pole pair $p$ and $p^*$ with $\text{Re}(p)\neq 0$. As you've correctly pointed out, if for $|p|>0$ the real part $\text{Re}(p)$ approaches zero, the $Q$ factor approaches infinity. This is the case for a pole pair on the $j\omega$-axis with $|p|>0$, i.e. not at $s=0$.

Also note that you can't use that formula for two distinct real-valued poles. That situation corresponds to an overdamped system with $Q<\frac12$.

The problem with a double pole at $s=0$ is that there is no reasonable way to evaluate the formula given in your question. The corresponding limit does not exist.

Having said that, we can think of a reasonable definition of $Q$ for the case of a double pole at $s=0$. For $p=p^*=0$ the transfer function of the corresponding system is (assuming no finite zeros)

$$H(s)=\frac{A}{s^2}\tag{1}$$

with some constant $A$. The impulse response is

$$h(t)=\mathcal{L}^{-1}\{H(s)\}=A\,t\,u(t)\tag{2}$$

We have to ask ourselves what we mean by $Q$ in this case. Since the damping is zero, you could say that $Q=\infty$. I haven't come across a definition of $Q$ in the case of a double real pole at $s=0$, and I think that for this degenerate case there cannot be any useful definition. If I had to decide on a value, I'd say $Q=\infty$, in analogy with all other pole pairs on the $j\omega$-axis.

• @MattL.: Thanks for taking the time to show the inverse Laplace transform that supports your opinion-- I should have thought to try that. I think it's pretty clearly not critically damped when you look at it that way. Mar 22, 2016 at 18:37