What are poles and zeros in Laplace Transform?

[MrOmnos]

SO, I understand the mathematical meaning of poles and zeros. You put Numerator= 0 of the transfer function and solve for s, the answer is the zero. You put denominator = 0 and you get the poles. But what do they actually mean? When someone says Butter worth is an all pole filter and the zeros only exists at infinity what do they mean? What is the physical meaning of pole? Please help me understand. How can I intuitively understand these things?

[T3sl4co1l]

There is no physical embodiment, it's a description of the frequency response.  Abstract.

An all-pole filter simply means its transfer function is of the form: H(s) = 1 / [ (s/w1 - 1) * (s/w2 - 1) * ... * (s/wn - 1) ], i.e., the numerator is constant.

A filter with zeroes will have, well, zeroes (notches) at certain frequencies.  The elliptical filter is a pole-zero filter, meaning it has notches in the stop band.

Examples of all-pole systems are simple transmission line* structures, and most filter designs (any that do not have zeroes (notches) in their response).

Examples of pole-zero systems include phase or amplitude equalization networks (i.e., there is a modest boost or cut over a limited frequency range), overlapping transmission line structures (you can think of a flat solenoid inductor as a parallel-wire transmission line, looped in on itself, so that each turn acts as a transmission line against the turn near it, but the turns are also connected by the delay of that transmission line), etc.

*A transmission line is a continuous element, which cannot be expressed as a finite, rational network (i.e., lumped L and C).  It can be approximated by an LC network, but only up to a cutoff frequency.  An ideal transmission line has infinite bandwidth, so requires an infinitely long LC network with infinite cutoff frequency.   Given that drawback, we can at least say: if we aren't interested in stupendously high frequencies, then we can use a finite limit, rather than infinity.

A pole or zero at infinity is usually an analytical convenience; some analysis methods require equal numbers of poles and zeroes to factorize the expression.  A finite number of poles or zeroes at infinity has no effect on the expression (IIRC; I haven't thought about this in a while).

Tim

[MrOmnos]

I was stuck thinking that they actually meant something in physical world. Now it's clear.

[orolo]

In brief, the closer you are to a pole, the more the system responds to the excitation. The closer you are to a zero, the more insensible the system is to the excitation.

If a system has a zero that means that the excitation at that frequency is completely ignored by the system. For example, a voltage divider made with a capacitor C followed by a resistor R has a transfer function sRC/(1+sRC), with a zero at s=0. This means that the dividier blocks any DC signal: any DC excitation will be ignored by the divider.

Having a pole means that the system responds to the excitation without bounds: the response to a sinusoidal excitation of the right frequency is unlimited. Since the Laplace transform is time-independent, that suggests that an infinite lasting sinusoidal excitation causes an infinite absortion of energy. Imagine a series LC divider, the transfer function is \$\frac{1}{1 + s^2LC}\$. You have poles at \$ \omega = \pm 1/\sqrt{LC}\$. How does the divider behave in the time domain at the pole frequency?



It is an increasing excitation, infinite (or indeterminate) in the time-independent picture.

To further complicate matters, the above examples have zeros and poles that are in the s-axis. Much more typical is to have them outside that axis. In that case, the general answer is that being near a pole turns the system on, and being near a zero turns it off. To complicate matters even more, different poles and zeros interact with each other in a complicated manner. That's the idea used in filters: make the poles and zeros interact to give the required transfer function.

An intuitive vision of your Butterworth low pass filter: it has a lot of poles in semicircular position around the origin (DC frequency), and the zeros are in the infinite. Since it is a LPF, it is logical that the zeros are in the infinite: high frequencies are a turn off for a LPF. The poles around the origin are cleverly positioned so, at low frequencies, they cooperate to generate a smooth passband: you are always more or less equally turned on by the whole bunch, never too close to a single pole (ripple). But once you go away from the semicirle, their influence fades away quite fast, as you transition into the stopband.

I'm not sure if that is physical enough as an intuitive explanation. A more mechanical explanation: zero -> the system reflects the excitation. Pole -> the system absorbs all the excitation. In between, you get all the intermediate behavior.

[T3sl4co1l]

Yeah, so whenever poles are talked about in a physical sense, the meaning is in regards to the position of points located on a 2D graph: the complex plane (\$\omega\$ vs. \$j\omega\$ in the Fourier domain, or \$-js\$ vs. \$s\$ in the Laplace domain; same thing, more or less).  This allows a geometrical interpretation, so that semicircles, and angles, and distances, become meaningful.

So it's at least two levels of abstraction below the physical: first you measure the system (layer 1), then you decompose or curve-fit or approximate the measurements as a pole-zero system (layer 2).

Electronics is a bit overwhelming if you aren't good at abstraction..

Tim

[MrOmnos]

I don't really want to blame my professors but teaching here is sub par. I use MITOpencourse ware and text books to learn. When both don't work I come here. Out mathematics professor who taught us Laplace , Fourier and Z Transform didn't really give us any insight. He just taught us how to solve the equations. He left the insight part to Circuit Theory and Filter Design professors who assumed we knew all about it and expected us to just grasp everything.