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Niall McMahon © 2013
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These notes contain some further thoughts about the electrical aspects of wind turbines. See Wind Energy Explained for expanded descriptions of some of the material.
Please review last year's lecture.
It's important always to remember how energy flows through a wind turbine, from rotor to grid. An appeal to first principles is always useful. For example, the rotor uses the air flow to generate lift forces across the blades. This force gives rise to a very large axial thrust force that seeks to push the turbine over and a tangential driving force that seeks to turn the rotor. Since the tower is strong and relatively stiff, it moves very little, and so no work can be done in this direction: work is done only when a force moves something. The tower acts as a dam to the flow of energy in the axial direction. The rotor is free to turn, and so all the captured energy is turned to doing work in this direction. If the rotor is free-wheeling, the driving force channels the portion of kinetic energy in the wind captured to accelerate the rotor. The rotor will continue to accelerate until aerodynamic losses (essential heat transfer) and heat transfer via the bearings dissipate all the energy flowing through the rotor. A portion of the energy will remain as kinetic energy in the rotor. At high rates of rotation, the efficiency of the rotor drops, and so the energy flow is lower than at moderate rates of rotation. If the rotor is connected to a well-matched, loaded generator, the rotor will slow to its most efficient rate of rotation as the generator channels all the energy captured by the rotor and converts it into electrical energy. This is transferred to the load where some work is done.
A schematic of a permanent-magnet synchronous generator, in cross-section. Starting at the centre, the layers are, in order: (i) rotor; (ii) magnet layer; (iii) air gap; (iv) stator windings.
In a typical modern small wind turbine permanent magnet generator, the magnets are attached to the surface of a cylindrical rotor. Between the magnet layer and the stator windings is a small air gap. As shown in Figure below, the windings are assumed to form a layer attached to the surface of the stator, immediately adjacent to the air gap, opposite the magnets.
As the generator rotor is driven, opposing forces are generated in the magnets and the windings. Taking an axial section through the generator, we can think of these forces acting on a tangent to the radius of the mid-point of the air-gap. A back-torque acts on the rotor about the axis of rotation. For a particular geometry, all else remaining the same, the generator has a peak continuous torque.
The peak tangential force acting on the rotor, per unit length of generator, Fl, can be calculated by summing the force contributions from infinitesimal circumferential elements, i.e.
Fl ∝ Δ Fl 2 π R
Fl = C Δ Fl R
Where C is a constant, Δ Fl is the force contribution per unit circumferential length, per unit length of generator, for an element, R is the radial distance out to the air gap and 2 π R is the circumference of the rotor. Fl depends in a simple way on R, i.e. Fl ∝ R. Remembering that
T = F R,
the total associated peak torque per unit length of generator can be written as,
Tl ∝ R2.
Multiplying Tl by L, the length of the generator, gives the total peak torque acting on the rotor.
We assume that the overall radius of the generator scales with the radial distance to the air gap and that the magnetic loading and electric loading are both held constant, i.e. that Δ Fl remains constant.
It follows that,
Ttotal ∝ R2 L
For a regular radial flux machine, R and L are well defined; L is the length of the magnet layer. This is often close to the total length of the generator.
It is evident that doubling the generator radius will quadruple the peak torque. Doubling the length of a radial generator will double the peak torque.
All else remaining the same, then,
T1 total / T2 total = (R12 L1) / (R22 L2)
This allows us to compare the performance of different sized generators.
(Axial flux generators are a little more complicated. For axial flux machines, L can be roughly approximated as twice the length of the magnets, measured radially, and R can be taken as the distance from the centre of rotation to the middle of the magnets. The effect on the peak torque of having a relatively large radius for a constant L explains the attraction of axial flux machines for electrodynamic braking applications: generator geometry matters for electrodynamic braking: axial flux machines have a significant torque advantage.)
See the draft paper on electrodynamic braking for more discussion and examples.
Moving power from generator to load more often than not involves the transfer of much more power across the network; this is a consequence of the phase shifts associated with capacitors and inductors. (See last year's lecture.) This can be perhaps imagined as power sloshing between reservoirs (inductors/capacitors) across the network. This excess power does no net work but it does increased the current flowing through the electrical network. The situation is described by the following equation (which results from a consideration of the behaviour of capacitors and inductors),
S2 = P2 + Q2
S: Apparent power [VA]
P: Real power [W]
Q: Reactive power [VAR]
The reactive power is the component of the total power transmitted that does no useful work. The total (apparent) power and the reactive power are both given the units of volt-amps to distinguish them from the power delivered to the load (the real power). An "R" for "reactive" is added to the VA units for the reactive component.
The power factor is defined as P/S.
If this is 1, then all the power is transferred to the load. This is the most efficient use of the power.
The power factor can be corrected back to 1 by careful placement of compensatory capacitors close to the source of the reactive power delivery or draw.
Please see here.
Most material © Niall McMahon. See legals and disambiguation for more detail. Don't forget that opinions expressed here are not necessarily shared by others, including my employers.