# 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: (source)

Graphically described here: (source)

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

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 – Behind The Sciences Mar 21 '16 at 8:14
• @BehindTheSciences: Would you be able to answer the question after having read the article you linked to? – Matt L. Mar 21 '16 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. – Omegaman Mar 22 '16 at 18:37

My research is involving some aspect underdampness lower dampness and almost nodampness the root locus diagram of a planet by its spin forming Q reonance as the pole moves along jw axis a circle above a plus code.Even this oscillate between plus and minus underdamped lower damped with middle zero damped in understanding diffeten Q factors of panetary emissions promoting the Q factor of planets of our solar system playing the role of genetic hologram espeially that of Jupiter takes up the second quadrant and strangely Saturn takes up the fourth Quadrant perhaps forming a helical resonance also is attached with it. Sankaravelayudhan Nandakumar.