Synchronous machine confusion

[phantom_photon]

Hello, I'm a little confused about synchronous machines. Specifically, I don't understand how the armature coils are wound. In discussing how the windings are configured in a three-phase synchronous machine, my textbook provides several diagrams which suggest that the windings of a coil go between opposite slots.

Are all synchronous machines constructed in this way or is it possible to have other configurations of armature windings? For example, could the coils be wound between adjacent slots?

Also, I don't understand the derivation my textbook gives for EMF induced in the coil. It obtains the air gap flux linked per coil by integrating air gap flux density around the interior of the stator from x=0 (x is an arc length co-ordinate fixed with respect to the stator) to x=2*Pi*R/p (R is the stator interior radius, p is the number of rotor poles) and multiplying the result by the axial length of the rotor. The induced EMF is then obtained by application of Faraday's law.

My problem with this derivation is that I don't understand how the air gap flux is linked to the coil. It's possible I have misunderstood the meaning of "flux linkage". I understand it to be equal to:

$$N\Phi = N\int_A{{\bf B}\cdot{\bf dS}$$

where N is the number of turns in the coil, $$\Phi$$ is the flux linked to one turn of the coil, $${\bf dS}$$ is a unit vector normal to the plane of the coil and A is the planar surface bounded by the coil. This would mean that air gap flux is only linked (and therefore EMF-inducing) when it flows through the loop of a coil, not when it simply intercepts the edges of a coil as seems to be the case with coils wound between opposite slots. Could somebody please explain this to me?

 

[berkeman]

Your interpretation of flux linkage is correct -- it's only the flux that goes through the coil area that induces the coil voltage. Can you post a link to a picture of the synchronous machine coil arrangement?

 

[phantom_photon]

Can you post a link to a picture of the synchronous machine coil arrangement?

Please find attached a diagram of a two-pole, three-phase synchronous machine. In this diagram a', b' and c' are connected to neutral while a, b and c actually represent independent sets of coils distributed around the stator interior. They are connected in series so that the EMF induced in each coil is slightly out of phase with it's neighbour and the voltage for a particular phase is found by summing the EMF phasors of each coil in that phase.

My understanding is that, at the moment captured in the diagram, the flux through coil a-a' wil be zero. However, the rate of change of flux will be greatest at this point hence the induced EMF greatest. If the rotor were to turn a further 90 mechanical degrees the flux linked through coil a-a' would be greatest but it's time derivative would be zero, therefore the induced voltage would be zero. This is what I think my textbook is trying to tell me. Have I understood this correctly?

What confuses me about the flux linkage concept is that I've always imagined the linked flux as being uniformly distributed over the area of a coil (as in a tranformer or inductor) but that's not the case here. The equation for flux linkage indicates that this makes no difference to the flux linked to the coil (and hence the induced EMF) but it seems to me intuitively that it should. If a sinusoidal magnetic flux threads through a coil several meters in diameter would the induced voltage be the same as if the coil were tightly wrapped around an iron core carrying the same flux? Is there some limitation inherent in the derivation of the aforementioned equation that I am unaware of?

 

[berkeman]

It's the dB/dt that generates the output voltage of the coil. Whatever configuration you use to get the same total dB/dt over the coil area, you will get the same V(t) out. Remember that B=mu*H, so that's why motors and electromagnets use ferrous cores. Sounds to me like you have a pretty good understanding of all this!

 

[phantom_photon]

Whatever configuration you use to get the same total dB/dt over the coil area, you will get the same V(t) out.QUOTE]

Okay, thanks, I guess that makes sense. I'm glad I decided to take an electromagnetics class this year - hopefully I'll have a better knowledge of these things by the end of the course.