An “origin pole”

bode plotfeedbacklaplace transformswitch-mode-power-supplytransfer function

I'm studying Christophe Basso's book Designing Control Loops for Linear and
Switching Power Supplies.

In the book, he often uses the term "origin pole". This is what I think
I understand about it so far:

  • When a transfer function contains an "integrating" element, that element
    represents an origin pole. An integrating element is
    a denominator element with an \$s\tau\$ factor on its own, one that is not
    part of a \$(1 + s\tau)\$ factor. This is consistent with the idea that
    the Laplace transform for an integral is \$1/s\$. \$\tau\$ is commonly an
    RC time constant. This would be an example of an integrating element:

$$\frac{1}{sR_2C_2}\text{ from, say, }\frac{(1+sR_1C_1)}{sR_2C_2(1+sR_3C_3)}$$

  • Mathematically, an origin pole has infinite gain at DC (\$s = 0\$), from which
    "point" the gain declines at 20dB/decade. In practice, this rise to infinity
    is halted at some point, such as when the available gain of the op amp is
    reached.

  • (Not completely sure about this bit): The gain curve of the origin pole, if
    unaffected by other poles or zeros, crosses 0dB at \$\omega_o\$, the
    frequency of the pole, \$\frac{1}{2\pi\tau}\$, which is perhaps typically
    \$\frac{1}{2\pi RC}\$. This is markedly different than a "regular" pole, whose \$\omega_p\$ is the point of a downward inflection in the gain, a so-called breakpoint.

Before starting the book, I thought that all poles were located at a 3dB breakpoint and looked like this, but maybe I slept through the day origin poles were mentioned in class 🙂 :

enter image description here

so this idea kind of threw me for a loop (no pun intended 🙂 while I've been working to make sense of the book.

So here's my question:

  • Am I understanding this correctly so far?
  • Do other folks use the term origin pole or is it something Christophe has
    introduced? The term doesn't seem to pull up too much on search.
  • Is there anything else interesting about origin poles that I and other
    curious readers yearning for knowledge might like to know, particularly in the
    realm of control loop transfer functions? 🙂

Best Answer

The "origin pole" is indeed the \$1/s\$ term in the transfer function \$H(s)\$. In the bode plot it results in a first order transfer that does NOT flatten out for low frequencies.

Your Bode plot is that of a low pass filter $$H(s) = \frac{1}{1 + s}$$ Note how this \$H(s)\$ would result in \$H(0) = 1 = 0\text{ dB}\$ like in your Bode plot.

\$H(s) = 1/s\$ is different, \$H(0) = \infty\$! In theory at least. So the -20 dB /decade line in the Bode plot keeps going on forever to both sides. Note that a Bode plot has a logarithmic X-axis, where would that place the 0 Hz point ? At minus infinity!

I call this \$1/s\$ an integrator or pole at zero, they are useful in feedback loops to eliminate static errors. Almost every PLL has an integrator consisting of a charge-pump (switched current source) feeding current into a capacitor. What happens to the capacitor's voltage when you feed a current into it? Yes, it keeps rising forever. That's integrator behavior.

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