How to calculate motor capacitance? 2

[lyen12]

I have done a quick search on this forum, but unable to find a thread on this.

Is this the right way to calculate "motor capacitance"?
Take C = Epsilon * A/d

with C = capacitance
Epsilon = Epsilon0 * relative_permitivity_of_air
A = Area of stator lamination bore
d = airgap

Why is it useful to know the motor capacitance?

 

[electricpete]

Yes, good clarification. It is rare that anyone is interested in capacitance by itself, so the only time I have ever seen that measurement is as part of an insulation power factor test (that is the sense that it is “typical”). But if all you want is capacitance, then it can be measured with less expensive and less specialized equipment.

But I have to pause to mention there would be a correction factor applied to any off-line measurement to account for the for the difference in voltage distribution within the windings between test and operation: Voltage accross the insulation during off-line test is uniform along the entire entire winding, whereas voltage accross the insulation during operation varies smoothly from VLG at the line terminal down to 0 at the neutral. This difference in voltage distribution affects the equivalent capacitance as can be shown with the energy approach described above. **

We can derive a correction factor to account for the difference in voltage distribution in test and during operation:
Let’s say we are given Capacitance Per Length of conductor: CPL.
If we apply uniform voltage, the capacitance is easy C = CPL * L
But if we apply graded voltage V0 on one end and zero on the other (V = V0-V0*x/L) where x is position along the line), we have energy We given as:
We = Int 0.5 * CPL * (V0 – V0*x/L)^2 dx for x=0..L
We = Int 0.5 * CPL * (V0^2 – 2*V0^2*x/L + V0^2*x^2/L^2) dx for x=0..L
We = 0.5 * CPL * [V0^2*x – V0^2*x^2/L + (1/3)*V0^2*x^3/L^2] evaluated at x=L minus x=0
Since the quantity at x=0 is 0, we have:
We = 0.5 * CPL * [V0^2*L – V0^2*L^2/L + (1/3)*V0^2*L^3/L^2]
We = 0.5 * CPL * [V0^2*L – V0^2*L + (1/3)*V0^2*L]
We = 0.5 * CPL * [(1/3)*V0^2*L]
Ceq = 2*We/V0^2 = 2*0.5 * CPL * [(1/3)*V0^2*L] /V0^2 = (1/3)*CPL*L
Ceq = (1/3)*CPL*L = 1/3 of what we would measure if uniform voltage was applied to the whole winding.

So there is a factor of 3 to account for the voltage distribution going to zero at the neutral. This is different than the factor of 3 for converting 3-phase test to per-phase basis. i.e. take your 3-phase-to-ground offline measurement and divide it by 9 to account for both effects.

The approach of the F.E. that I linked was appropriate for the problem I was tackling (finding the relative contribution of local turn insulation changes within overall phase-to-phase impedance measurements), but would definitely be overkill for determining ground capacitance considering all the limitations/inaccuracy of the input data / model to begin with. Your initial mentioned approach of capacitance per length of conductor might not be too far off, as long as:
1 – account for the conductors along the periphery of the slot (ignore the others in the interior)
2 – use a capacitance of a series of individual conductor adjacent to conducting plane which accounts for that configuration. You might be able to estimate it analytically. I would be inclined to do a simple F.E. of a single insulated conductor a specified distance above a bottom grounded boundary surface, with left/right boundary potentials matching each other as if we have mirror image of another conductor on left/right, and top boundary either same potential as conductor or symmetric boundary.
3 – Determine equivalent capacitance by “combining” all of the individual capacitances. i.e. by summing the energies 0.5*Ci*Vi^2 where Vi varies by conductor (probably good enough to assume all conductors of a given phase in a given slot are the same voltage). **

But I do think measurement (with correction factors) is the better approach. Or else try to find someone that has already taken measurements of similar motors.

** It may seem non-intuitive: after all... capacitance shouldn’t depend on voltage... we can model this as a bunch of capacitances in parallel between the conductor and ground, so why can’t we just add the parallel capacitances Ceq = C1+C2+C3....? The answer is that capacitance of each sub-element is a function of geometry and not voltlage, but if we want to combine them into an equivalent capacitance seen at a terminal with a different voltage, that equivalent capacitance does depend upon the internal voltage distribution. See appendix of my linked paper for further justification of the energy approach to determine equivalent capacitance.

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(2B)+(2B)' ?

 

[electricpete]

Now that I think about it, it is not straightforward to figure out which capacitance is more relevant for system grounding calcs – the capacitance based on on-line voltage distribution or off-line distribution. Presumably the analysis involves swinging of all three phases together with respect to ground. Now voltage to ground to ground along a given phase looks like a constant plus that linear gradient from line to neutral. Don’t know where that leads.

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(2B)+(2B)' ?

 

[electricpete]

The reference below gives means for calculating minimum neutral grounding resistance based on analysis of system capacitance. Section 4.9 is devoted to capacitance of various system elements. The data for induction motors is excerpted below:

“Industrial Power Systems” by Shoaib Khan, Sheeba Khan, Ghariani Ahmed.
ISBN 978?0?8247?2443?6.....

4 System Neutral Grounding...

4.9 System Capacitance Data...

4.9.9 Induction Motors

Zero-sequence capacitance (C0) in uf/phase is given in table 4.8.

Table 4.8
Zero-Sequence Capacitance Data for Induction Motors (uf/phase)
Rating, kW kV 1800 rpm 1200 rpm 900 rpm 600 rpm
225 4.0 0.009 0.012 0.014 0.016
300 4.0 0.01 0.013 0.015 0.017
350 4.0 0.012 0.014 0.016 0.018
400 4.0 0.014 0.016 0.017 0.021
Source: Westinghouse, “System Neutral Grounding and Ground Fault Protection,” publication PRSC- 4B-1979, Westinghouse, 1979.

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(2B)+(2B)' ?

 

[electricpete]

Since the relevant quantity is a “zero-sequence capacitance”, I tend to think the factor of 3 correction factor to apply to measured values that I derived above would not be relevant. (because the voltage gradient between line and neutral does not exist if zero sequence voltage is applied ... same voltage at all three phases and neutral).

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(2B)+(2B)' ?

 

[electricpete]

Can you give a complete description of your motor?

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(2B)+(2B)' ?

 

[lyen12]

The response overwhelmed me with lots of information. Basically I am required to give a budgetry fora motor cartridge for subsea compression. The motor which I am supposed to quote, excludes thrust bearing and pump which will be provided by other vendor.

The motor must be sized so that it could fit into a large housing. Therefore, the stator lamination outer diameter should be about 650mm, with its bore 300mm, and pack length of about 400mm. The motor is VF driven, rated at 600kW 3300V 2pole 50Hz. 3Hz starting, and max speed 53Hz.

I am able to provide all other information to the customer, except I am a little bit unsure on how to calculate the windings capacitance per phase, and therefore would like to have some ideas on how to go about it.

There are many ideas here, and most of it looks complicated to calculate. Since the motor is for a RFQ, I would thought an estimate of capacitance would suffice....?