formula for airgap flux based on voltage, dimensions, winding configur 7

[electricpete]

1 – Can anyone tell me a formula to approximate the airgap flux Bg in an induction motor, given the stator voltage, winding configuration and geomtery.

2 – I copied directly from an EPRI text below.
I agree with equation 2.1 (derived from v=dPhi/dt 4.44=2*Pi/sqrt(2), sqrt(2) converts peak flux to rms voltage)).
I agree with equation 2.2 (this is simply N/2).
I agree with equation 2.3 (solves 2.1 for Phi).

I am confused by equation 2.4.
I would start with B = Phi/ (A/P) where Phi is flux per pole, A/P is area per pole

(A/P) = (Pi * ID * LEN) / P

B = Phi /(A/P) = Phi / {(Pi * ID * LEN) / P}
= Phi * P / ( Pi * ID * LEN)

But this does not agree with their 2.4. I vaguely remember also a factor of 3/2 for 3-phase machines associated with the overlap of three phases. I'm not sure if that belongs here but it still wouldn't make it work exactly. Any ideas how they came up with equation 2.4?

========= BEGIN QUOTE =================
The strength of the magnetic field is given by the following relationship:
E = 4.44 * f * N * Phi * Kd * Kp * 1E-8 (Equation 2.1)

Where:
E = internal voltage (applied volts, V, minus the IX voltage drop
of the stator winding)
f = frequency in Hertz
N = series turns per phase
Phi = lines of magnetic flux, flux per pole
Kd = winding distribution factor (value 0.9 to 1.0)
Kp = winding pitch factor (value 0.9 to 1.0)

The series conductors per phase, Z, is determined by Equation 2-2. Then, the total flux per pole in lines of flux, Phi, is determined,
equation 2-3. With the total flux per pole, the magnetic-flux density in lines per square inch can be determined for the air gap, Bg; the stator teeth, Bt; and the stator core, Bc.

Z = 2 * Q * T / (Paths * m) Equation 2-2

Phi = 10^5 * Volts / (2.22 * f* Z * Kd* Kp) Equation 2-3

Bg = Phi * P / ( 2 ID * LEN) Equation 2-4

Bt = Phi * P / (0.605*WT*Q*LEN) Equation 2-5

Bc = Phi / (1.9 * HI * LEN)

Where:
Z = series conductor per phase
Q = total number of slots
T = number of turns per coil
PATHS = number of parallel paths in the winding
m = number of phases
Volts = volts per phase
f = frequency
Phi = kilolines per pole
HI = core height in back of teeth
ID = bore diameter
WT = tooth width
N = 1/2 (Z) = series turns per phase
Kd = winding distribution factor
Kp= winding pitch factor
[LEN = Effective core length]
[ P = Number of Poles]
Constants, 0.605 and 1.9, include an insulation between lamination factor of 0.95.
Note: All dimensions are in inches.
============= END QUOTE ============

 

[rhatcher]

electricpete, I will warn you that I do not have any references with me to make sure that I am thinking straight. Looking at what you came up with, I would think that the equation should be:

Bg = (Phi * P/2)/A where A = Pi*ID*LEN
= (Phi*P)/(2*Pi*ID*LEN)

My line of thinking (remember no references) is that the flux in the air gap is contributed by pole pairs...ie. the lines of flux from the north pole go to the corresponding south pole. As example, for a two pole motor if phi=10, then the air gap flux would be 10 lines, not 20. Likewise, the stator core flux would be 5 lines (parallel paths) and the tooth flux (lines per tooth) would be 10/(# of teeth per pole).

With respect to the quoted material, I have stared at that for quite some time and I have a couple of problems with it. Maybe I am missing something.

1 - what is Z, series conductors per phase (eq 2-2)? Equation 2-2 would appear to calculate series turns per phase, (although I cannot resolve where the 2 comes from). And, if this is turns per phase, then is it equal to the term N (also turns per phase) from equation 2-1? If so, I think it is in poor taste to change variables (N-Z) in midstream.

2 - How is eq 2-3 derived from 2-1? If we substitute Z for N and volts for E (another variable change in midstream?), there still is no equivalency between the 4.44 and 1E-8 in 2-1 and the 2.22 and 10^5 in 2-3.

3 - where is the pi term in 2-4?

4 - for eq 2-5, how is the 0.605 term derived?

5 - for eq 2-6 (Bc), where is the term P for # of poles? Also, the parallel paths are disregarded.

I don't know pete, maybe it is just over my head (for now).......

 

[electricpete]

Ray – I always appreciate your insight and comments. And thanks for taking the time to read that long post and quote.

It seems like you look at equation 2.4 similar to me except that you add a factor of 2. I don't see the need for that factor of 2 yet. My simple way of thinking is to imagine only one full-span coil per pole-phase group. If it's a 2-pole motor than the angle spanned by that coil is 2*Pi/p = 2*Pi/2 = Pi. It seems like that area (Pi*LEN*ID / 2) and the flux Phi passing through it are all we need to know to apply to E = N*dPhi/dt. Return path doesn't seem important. The fact that flux from another pole phase group may be contributing to Phi doesn't seem important. Only total flux and related area. On that same basis I think the factor of 3/2 I mentioned is irrelevant. Then again maybe I am missing something.

Your question 1: As you point out N is the quanitity of real interest (the one in E = N*dPhi/dt). Z and N are defined such that Z=2*N (see definition of N). So Equation 2 calculates Z=2*N. Equation 2 could be re-written equivalently as N = Q * T / (Paths * m). I agree Z has no apparent physical significance and only adds an unnecessary variable. But I think Z has been used in a mathematically consistent manner.

Your question 2:
Start with equation 2.1:
E = 4.44 * f * N * Phi * Kd * Kp * 1E-8
(Phi was defined in lines.)

Solve for Phi
Phi = 1E8 * E / (4.44 * f * N * Phi * Kd * Kp)

Redefine Phi as kilolines per pole (change in definitions).
New Phi = 1E-3 * Old Phi
New Phi = 1E5 * E / (4.44 * f * N * Phi * Kd * Kp)

Substitute N=Z/2
Phi = 1E5 * E / (4.44 * f * [Z/2] * Phi * Kd * Kp)
Phi = 1E5 * E / (4.44/2 * f * Z * Phi * Kd * Kp)
Phi = 1E5 * E / (2.22 * f * Z * Phi * Kd * Kp)

I agree they have been shifty in their definitions. That is why I cut and pasted verbatim.

Your question 3 – that is my question as well.

Your questions 4 and 5: I haven't got there yet.

Although they did not specify detail on E, I think it would have to be line-to-ground voltage (for wye connected winding).

Any comments?

 

[rhatcher]

Pete,

I didn't read the post too carefully or I would have noted that the second variable list shows that N=1/2(Z) and that Phi was redefined to kilolines. Those account for the factor of 2000 that I was missing from eq 2-3 and they answer my second question.

I am still unclear what Z actually represents though. The term "series conductors per phase" doesn't mean anything to me, even knowing that it is twice the "series turns per phase", a term that I understand. Somehow I do not think it is just an arbitrary (made-up) variable with an arbitrary name and the factor of 2 added in.

To answer your question, this looks like a "per phase" thing, so I am assuming that E is voltage per phase, L-N for wye and L-L for delta. But, like I said, I am still trying to work all of this out myself so I am not certain of anything right now. I'll let you know when I come up with something. The good news is that I return to work tomorrow and I believe that I have a reference on this that could clear up the confusion without us having to rely on mental brute force to work it out.

By the way, you said "I cut and pasted verbatim." Is this something that you could email to me?

 

[jbartos]

Suggestion: Reference:
Gordon R. Slemon "Magnetoelectric Devices, Transducers, Transformers, and Machines," John Wiley and Sons, Inc., 1966.
Chapter 5 Polyphase Machines
Eq. 5.30: Magnetic Flux Density Bt in the air gap g':
Bt=mu,o x Ht = |B| x cos(ws x t + alpha,m - theta)
where
|B| = mu,o x FIm / 2g'
Eq.5.33: Flux linkage Lamda,ma = integral from -pi/2 to +pi/2 of [(Ns/2) x cos(theta) x Lambda,turns dtheta]=
= (pi/2) x Ns x |B| x l x r x sin(ws x t + alpha,m), in Webers
where
g' airgap in meters
Ns number of stator turns of winding
|B| magnitude of the airgap flux density
l is axial length in meters
r is radius of the rotor
Eq. 5.71: Induced voltage Ema in phase a:
Ema=dlambda,ma/dt
=ws x Lms x Ims x cos(ws x t + lambda,m)
or in phasor form:
Ems =j ws x Lms x Ims, in Volts
Now, using Eq. 5.32 in 5.71 yields Eq. 5.74:
Ema = ws x (pi/2) x Nx x |B| x l x r x cos(ws x t + lambda,m)

Similarly, the induced voltage EmA in winding A of the rotor is:
Eq. 5.76:
EmA=ws x (pi/2) x Nr x |B| x l x r x cos(ws x t + lambda,m - beta)
Now, by comparison Eqs 5.74 and 5.76, the induced voltage in the rotor and stator are related by the phasor expression:
Eq. 5.77:
Emr = (Nr/Ns) x /_-beta x Ems, in Volts,
where
Nr are rotor winding turns
ws is stator supply angular frequency in rad/sec.

Textbooks tend to have a little better explanations and are easier to use.

 

[UKpete]

Pete, in equation 2.4 Bg is the PEAK gap flux density, assuming a sinusoidal space-variation around the airgap. Your equation B = Phi * P / ( Pi * ID * LEN)
is actually the AVERAGE flux density around one pole pitch. To convert from peak to average for a sinusoidal quantity, multiply by 2/pi; hence if you multiply eqn 2.4 by 2/pi, you get back to your equation.

rhatcher, it is true that both terms N and Z get used in different texts, it's better to stick to one or the other. As Pete says Z = 2N (always) because each coil has 2 sides, hence it is "slot conductors ...". The 2 factor always has to come in somewhere, either in the voltage equation if using Z, or the slot fill calculations if using N.

That's my two-penny worth.

 

[electricpete]

UKPete, the factor of Pi/2 to convert my answer from average to peak would seem to make the answer work out properly. I have to stop and think for a second about the reason.

 

[electricpete]

OK, I thought for a second and it makes sense. B = Phi/A is an average value of field varying sinusoidally over space (even though Phi was calculated based on E converted to peak in time).

Thx UKPete, you are right again!

 

[electricpete]

Now with UKPete's info, it seems easier to determine the origin of the 0.605 and 1.90.

0.605 = 0.95 * (2/Pi). The 0.95 corrects axial length for insulation between lamination. The 2/Pi is again the factor to convert betweeen spatial average around the air-gap to peak.

1.90 = 0.95 * 2. The 0.95 is same as above. No factor of 2/Pi is required since we are no longer looking at sinusoidally flux flowing radially. (flux is constant accross the cross section where flux flows tangentially in back iron). I think the factor 2 will account for the fact that the flux which enters a pole radially can exit tangentially in two different directions.

For completeness, I should mention there is another axial dimension correction factor required if core vent ducts are present.

 

[RalphChristie]

Now my head is totaly spinning....


O.k. ... Maybe after New Year ..... rather take a few more beers ...




A Happy New Year to all, thanx for all the valueble questions and answers, I've learned a lot.

RCC

 

[rhatcher]

electricpete, it looks like UKpete gave you the info you needed to figure this out. (a star for his post)

 

[jbartos]

Suggestion: Please notice that:
Average value = 2/pi x Maximum value of pure sinusoid, i.e. Amplitude=Amax
or
Aav=(2/pi) x Amax = 0.6366 x Amax
This has been discussed in several threads in this Forum.
As it can be seen in numerous References, it is required in the rotating machine principle of operation or functioning.

Clarification to my previous posting:
|B| = magnitude of the air gap density is to be interpreted as Bmax (=Amax above, in general).
This is compatible with electricpete (Electrical) Dec 30, 2003 statement:
"0.605 = 0.95 * (2/Pi). The 0.95 corrects axial length for insulation between lamination. The 2/Pi is again the factor to convert between spatial average around the air-gap to peak."