1. Elements of Energy Flows in Electromechanics
6.11s Notes for Lecture 1
Elements of Energy Flows in Electromechanics
June 12, 2006
J.L. Kirtley Jr.
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2. Take this as a prototypical machine form
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7. Field Description of Forces: Maxwell Stress Tensor
If we include forces due to changing permeability,
We find force is the divergence of something we call a Tensor
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8. Force on an object is the integral of force density:
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9. Forces due to fields normal to a surface
Assume these are highly permeable poles
Force is UP on lower pole, DOWN on upper pole
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11. Poynting’s Energy Flow is:
Power flow INTO a volume is:
This means we must invoke our old friends, Farady’s Law and Ampere’s Law
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14. Using the two induction equations, we find
Of course J is flowing in something, and if that is stationary:
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15. If the material is moving,
If the magnetic material is linear and lossless, the magnetic term is clearly rate of change of energy stored. More on this later
If the material is moving,
And this last is clearly energy conversion
(velocity times force density)
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16. Since Faraday’s Law is:
Now look at a prototypical situation: like an air-gap. We are looking in the axial direction and there is no variation tha way. Excitation is z-directed and like a traveling wave
Since Faraday’s Law is:
The interesting part of this is:
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17. Now suppose the lower surface is moving
Now energy flows are related: In the stationary frame:
And in the moving frame:
The difference between these must be energy converted
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18. Energy into those terminals over a time interval is:
Now consider about any type of system with voltage and current defined at a pair of terminals:
Energy into those terminals over a time interval is:
And over a periodic cycle, energy input per cycle is:
Here we would have net energy IN to the system (possibly a motor?)
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19. There is more to this than fits on a page -- see the notes
If there is no motion so there are no mechanical terminals, we have an analysis of hysteretic material loss
There is more to this than fits on a page -- see the notes
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20. Here is a simple ‘cartoon’ model of a linear induction motor
Here is a simple ‘cartoon’ model of a linear induction motor. The analysis follows steps we have already outlined
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21. Surface Impedance is important: For energy flow,
For a layer of magnetically linear conductive stuff excited by a traveling wave:
In iron, if we can idealize the saturation curve,
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22. Hysteresis (nonlinear and hard to characterize)
Iron Loss Consists of
Eddy Current (linear)
Hysteresis (nonlinear and hard to characterize)
Semi-Empirical ‘Curve Fit’
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