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. June 12,2006 6.11s June 2006 L1

Take this as a prototypical machine form June 12,2006 6.11s June 2006 L1

June 12,2006 6.11s June 2006 L1

June 12,2006 6.11s June 2006 L1

June 12,2006 6.11s June 2006 L1

June 12,2006 6.11s June 2006 L1

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 June 12,2006 6.11s June 2006 L1

Force on an object is the integral of force density: June 12,2006 6.11s June 2006 L1

Forces due to fields normal to a surface Assume these are highly permeable poles Force is UP on lower pole, DOWN on upper pole June 12,2006 6.11s June 2006 L1

June 12,2006 6.11s June 2006 L1

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 June 12,2006 6.11s June 2006 L1

June 12,2006 6.11s June 2006 L1

June 12,2006 6.11s June 2006 L1

Using the two induction equations, we find Of course J is flowing in something, and if that is stationary: June 12,2006 6.11s June 2006 L1

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) June 12,2006 6.11s June 2006 L1

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: June 12,2006 6.11s June 2006 L1

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 June 12,2006 6.11s June 2006 L1

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?) June 12,2006 6.11s June 2006 L1

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 June 12,2006 6.11s June 2006 L1

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 June 12,2006 6.11s June 2006 L1

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, June 12,2006 6.11s June 2006 L1

Hysteresis (nonlinear and hard to characterize) Iron Loss Consists of Eddy Current (linear) Hysteresis (nonlinear and hard to characterize) Semi-Empirical ‘Curve Fit’ June 12,2006 6.11s June 2006 L1