I have 3 separate areas of discussion below
1 – Deriving Linear assumption torque vs current
2 – Optimal approach
3 - Questions
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1 - Deriving Linear assumption torque vs current
I was not familiar with the method described by Bill or that link. It seemed a little odd to me that we can draw a direct relationship between current and torque accross a wide range of speeds from starting to full load, so I studied it a little to come up with the explanation based on the equivalent circuit:
In the IM Eq Ckt, we can replace the leakage reactances and magnetizing reactance with a Thevinin equivalent impedance.
V1---Zth ---- R2tot'/s—ground
Where the prime (') in R2tot' implies that R2tot is expressed on a primary (stator) basis... We know the ratio between primary and secondary voltages is slip times a turns ratio. The turns ratio is equal to the voltage ratio at slip=1. Therefore it is V2/V1 where V2 is the open-circuit locked-rotor rotor voltage V2, given on the motor nameplate.
So we can express everything on the secondary (rotor) side as follows:
V2---Zth'' ---- R2tot/s—ground
where now Zth'' indicates that it is referred to the secondary side... and again V2 is open-circuit locked-rotor rotor voltage from nameplate.
Now we make a big assumption (*) that R2tot/s >>Zth'' over the range of interest from starting to full load. Under that assumption, we can rewrite the circuit as:
V2---- R2tot/s—ground
From which we can write:
I2 = V2/[R2tot/s] [eq 1]
Additionally, as per the post dated 25 Mar 11 14:11 above, we know
Torque = Pairgap/wsync
Under our assumption, Zth is negligible, so the circuit is resistive and
Pag = 3*V2*I2
Putting together the 2 equations above:
Torque = 3*V2*I2/wsync.
Since V2 and wsync are constant, we can simply say
Torque proportional to I2 (where I2 computed from eq1 above)
* The assumption R2/s>>Zth would not apply for SCIM during start since in that case Zth >> R2. But I believe it does apply pretty well for SCIM from start through full load.
By the way, conclusions based on equivalent circuit accross a range of conditions from start to run are much more reliable for WRIM than SCIM because WRIM does not have the deep bar effect.
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2 - optimal approach
New subject - I suspect I was wrong to suggest that the optimal solution involves an initial starting resistance that moves breakdown torque to s=0. (By the way I didn't invent that idea, I have read it several places). One reason that I think I'm wrong is that many industry sources suggest to use Bill's approach above shooting for initial torque of 1.5*rated using linear approach (1.5 is used by the link above and also by EASA). This puts will put the slip of breakdown torque far above 1.0 (for example 1.7 slip) for two reasons:
1 - actual breakdown torque is likely around 2.5, much higher than the 1.5 target
2 – the actual slip at breakdown torque is higher than we predict by this particular linear extrapolation method used. i.e. the linear extrapolation carries and error and direction of the error adds to error 1.
My initial thought was that we strive to maximize Torque/Current ratio, especially at low speeds. (The reason for emphasizing low speeds is that for constant acceleration rate, we have constant airgap power... kinetic energy varies with speed squared so that airgap power goes mostly into increasing kinetic energy of rotor at high speeds and mostly into rotor I^2*R at low speeds.) If that thought was correct, we should strive for breakdown torque at start, because breakdown torque is the highest torque to current ratio.
But Torque/Current ratio isn't the whole story. Let's assume unloaded start and we want to compute rotor I^2*R energy created during acceleration through a given speed range. It is I^2*R2internal*time. But time is inversely proportional to Torque. So it is I^2*R/Torque. Maximizing the ratio (Torque/I) doesn't minimize the heating amount. We want to maxmize (Torque/I^2).
Under the linear assumption Torque ~ I, maximinizing (Torque/I^2) means we want to
minimize current... as long as we meet our other constraints. Another new constraint we should probably impose is that motor torque remains above 100% throughout the start so that the motor will not stall even if 100% load torque is applied. So this leads to the logic of choosing some target only sightly above 100% (like 150%), and then switching to the next lower resistance step before torque would be expected to decrease to 100%.
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3 - Questions
1 – Does anyone have any comments on my discussion of justification for "optimum" strategy for selecting rotor resistors
2 – Does anyone have alternate discussion of optimum strategy?
3 - I see Mark E's site says:
"2. Determine the short circuit current of the rotor from the motor nameplate."
I thought the nameplate shows the locked-rotor open-circuit rotor voltage and the full load rotor current (not short circuit current). Are there some motors that show short circuit current?
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(2B)+(2B)' ?
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