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| What's the minimum (physics first) to get an oscillator? |
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| RoGeorge:
As the question say, how to explain an oscillator, and what's the must have for an oscillator? By physics-first, I mean explaining based on energy, causality, time domain and alike easy to grasp notions. Going from physics to math would be OK, the other way around, not so much. |
| IanB:
From a mathematical and physics point of view, you need to have a second order time derivative in the differential equation that describes the system. This is the only way the solution will include a complex exponential which will involve functions of sin() and cos(). From an intuition point of view, the second order derivative will result from some concept of momentum in the system. Without momentum you cannot achieve oscillation. Momentum can be thought of as a form of energy storage due to velocity (call it "kinetic" energy). Oscillation is the result of energy cycling between storage as "kinetic" energy and storage as potential energy. If the losses in the system ("damping") are low enough, the transfer of energy back and forth between the "kinetic" and potential forms of storage can result in an underdamped or cyclic behavior. To make an oscillator you set up a system with cyclic (periodic) behavior, and then you feed in just enough energy to compensate for the losses, so that the oscillation continues indefinitely. |
| TimFox:
That is a reasonable description of an "harmonic oscillator". Note that any such device needs an external power source to maintain the oscillation. The prototype harmonic oscillator in physics discussions is a spring (force proportional to displacement) and a mass, where the position of the mass oscillates. Another oscillator type is the "relaxation oscillator", where only first derivatives appear in the analysis. A 555 or a neon bulb can provide the hysteresis that bounds the oscillation. |
| IanB:
I think even a relaxation oscillator would include a second order time derivative in its description (albeit hidden and very nonlinear). This is because a system containing more than one first order derivative can exhibit second order behavior (the converse is true also: a second order ODE can be decomposed into a paired system of first order ODEs). |
| CatalinaWOW:
--- Quote from: IanB on May 23, 2023, 01:58:36 pm ---I think even a relaxation oscillator would include a second order time derivative in its description (albeit hidden and very nonlinear). This is because a system containing more than one first order derivative can exhibit second order behavior (the converse is true also: a second order ODE can be decomposed into a paired system of first order ODEs). --- End quote --- The relaxation oscillator highlights the fuzzy line between ideal models and reality. The conceptual oscillator can't be fully described by a differential equation because the trigger circuit is non continuous and has no meaningful derivatives. But it can be approximated mathematically and the physical reality can be matched by those approximations. |
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