Mechanical Model for Electrical System - Odd Harmonics

^{"I'm looking for a reasonable way, in the form of a demonstration, to display the effects of odd harmonics in a network. Most of people who need to appreciate the explanation have no capacity to follow the mathematics required to understand this issue (and I'm not sure I want to disturb the brain cells required to crank through the analysis). It seems, however, that a lot of (non-believers) people could benefit by an effective demonstration. I'm sure it has been done before, but I'd like to know how to reproduce the demonstration."}

Hmmm... Think of system of water troughs flowing in from a lake to a main large trough and multiple smaller troughs flowing out. The outflow troughs feed to paddle wheels on machinery and back into the main trough (for the sake of this demonstration, ignore gravity for a moment here). If the outflow is moving smoothly through the paddle wheels, the flow back into the main trough is smooth as well. But if one of the paddle wheels is working a ratcheting machine and jerking from pawl to pawl, what happens? The stopping and starting paddles cause waves to slosh around in the trough and back to the main. The waves eventually start to compound on one another, forming new waves at higher peaks than the originals. Too may waves and it starts to cause water to spill out of the troughs and wash away the supporting soils around them (my analog to heating?) leading to failure of the troughs. It also affects the ability of the other smoothly running paddle wheels to function smoothly as well, so the machinery they operate starts acting erratically.

Good luck on your second wish. Nobody, not even UL, tests appliances (machinery, controls etc.) for issues related to ability to withstand power line disturbances. Indirectly, if all devices were rated on their EFFECTS ON the power lines, the overall reliability of everything would be more deterministic because you could mitigate power line disturbances by selection. CE listing supposedly does this, as does IEEE 519. But devices. i.e. appliances, are not yet REQUIRED to meet any specific requirements along those lines, it's all still essentially voluntary and self-certified.

I use the water analog for some of your third "wish", with the obvious caveat of the gravity issue (electricity can "flow up hill"). Generation is like a dam delivering water, voltage is like the pressure, current is the flow, volume is the power, pipes and hoses are wires, spigots and valves are switches, diodes are check valves, restriction is resistance, turbulence is impedance, etc. etc. There is no good analog to heating of conductors unfortunately (erosion is maybe as close as I could get in the above example), nor is there one for inductance or capacitance that I can think of. But most lay people don't really need to get to that level of detail in these types of discussions, they are usually only interested in how things affect THEM.

So what I have said to people wanting to add more stuff into a system than it can handle is to describe something to them they can relate to: a toilet and a shower. If you have a small water line that feeds just enough water to take a shower, and then you add a toilet on that same line, what happens if you are taking a shower and someone flushes the toilet?

inertia of mass may be the counterpart of inductor and capacitor in mechanical system as they both introduce time constant(tao) in the system. Inductor may be the analog of inertia of flow and capacitor may be the analog of rotatinal inertia of mass.

Hello JRaef,

Thank you for the analogies. I have admired your contributions many times and I really like how different your approach is to the many questions presented on CR4 forums. And while I am at it, thanks to CR4 for providing such an interesting site where ideas can be discussed so freely.

As to the shower/toilet issue, I recently corrected that very problem with some new plumbing. So that one was so close to home it was actually there.

The analytic equivalent for capacitors and inductors are the models that seem to be the most difficult. It is the ability to store energy that makes them so difficult to model. Some people suggest a spring as a capacitor, but I think that is most a better model for an inductor. A good model for a capacitor seems to be an air cylinder. I like to think of resistance as friction between some mass and a flat surface.

I think what I was hoping to describe may best be described in a Bode plot of some system where stability was considered to be fairly normal but under certain conditions (i.e. an impulse) certain poles and zeros would become apparent. That is to say that at line frequency everything looks great. But when pulses are introduced, as in your paddle wheel example, then normal containment is no longer sufficient. The LaPlace transform of a pulse illustrates the source of the harmonics and effective magnitudes at those frequencies. And a given combination of capacitive and inductive components with some amount of stored energy exchange provides a natural resonance condition that can overload a containment system.

It seems like a system of springs and (plugged) air cylinders connected in such a way as to act like a filter would provide the circuit, while the mass (net energy) would then determine the proximity to the natural frequency to oscillation or instability.

I know that some mechanical systems have accidentally been created in the past like the famous Tacoma Narrows bridge failure. What I would like to set up is a demonstration of a system that appears to be stable in the presence of most disturbances until one of just the right frequency causes the instability. Your trough example may very well be able to do this if it is constructed correctly. I'll give that some more thought.

For last question:

Review the differential equations describing an RLC circuit and a mechanical spring-mass-damper system. You will see from the equations that electrical components have equivalent mechanical analogs.

resistance R is analogous to damping factor "delta" (arbitrary label for this discussion)
inductance L is analogous to mass M
capacitance C is analogous to the inverse spring constant 1/k

Links below are good basic reviews of the 2nd order differential equations for a spring-mass-damper mechanical system and an RLC electrical circuit.

http://en.wikipedia.org/wiki/Damping

http://en.wikipedia.org/wiki/RLC_circuit

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