Consequences of increasing the length of the iron core in electric motor 7

[EngRepair]

Hypothetical, theoretical question about electric motors.
Let's say we have a fully functional three-phase squirrel-cage LV motor.
Let's imagine we made another one with exactly the same geometry of stator and rotor lamination, exactly the same winding (turns/coil, wire size, pitch, etc...).
The only difference should be the length of the stator and rotor cores.
Let's say the length is increased by 10%.
Also, the motor load will remain the same as before.
What changes will this cause in terms of hp, torque, rpm, FLA, NLA, efficiency, and power factor?
It would be greatly appreciated to hear some expert opinions.

 

[electricpete]

This link shows Example 3.1 from the textbook "Electric Machinery" by Fitzgerald which solves torque associated with a single coil rotor (as shown in Figure 3.2) below:

For context, someone has uploaded the entire Chapter 3 of the same textbook here

The last line of the solution of Example 3.1 from Fitzgerald's Electric Machinery gives the following:

His torque equation is:
T = -2 I Bo R L sin (alpha)

Let's compare to my torque equation
T = B I cos(theta) R N L Kdp

What do they have in common:
[ul]
[li]T on the left side[/li]
[li]B, I, R, and L on the right side[/li]
[/ul]

What are the differences:
[ul]
[li]Fitzgerald uses angle alpha to represent instantaneous angle between the loop and the field. I use theta to represent the spatial angle between the fundamental current wave and the fundamental flux wave. Due to the defined labeling of the angle alpha, it represents 90-theta. sin(90-theta) = cos(theta). This factor plays the same role.[/li]
[li]The N in my equation takes the place of the 2 in Fitzgerald's equation. In my earlier linked attachment, I defined N as total number of conductors in the slot section. For Fitzgerald's loop, N=2.[/li]
[li]Fitzgerald's equation does not include Kdp. It is not needed since Kdp=1 in his geometry where there is only one coil with full span for a 2-pole configuration. [/li]
[li]Fitzgerald's sign has a minus sign to indicates a polarity of torque. My equation does not assert any torque polarity.[/li]
[/ul]
As you can see they are basically the same equation. Including the factor L for length.

 

[edison123]

Bill

"Surely the length enters into the torque equation."

No, the length, dia, turns, flux etc are already subsumed in EMF equation. Finally, the motor torque is the direct product of flux density and the rotor (and in turn stator) current.


Muthu

http://www.edison.co.in

 

[electricpete]

EDISON - It appears your approach is to just keep saying the same thing over and over without providing proof.

You tried to provide proof in the form of a proportionality T ∝ B I cosɸ which is a simplification of a longer equality which also includes L, as I outlined in detail on 11 Mar 23 15:49. I provided the longer equality, but you rejected the equality on the bizarre basis of labeling it an EMF equation. And yet it is equivalent to the equation shown in Fitzgerald's Electric Machinery (do you think he also gave an EMF equation and labeled it as torque?... i.e. you're claiming that Fitzgerald just happened to make the exact same mistake in his published textbook which has survived many editions?). Similar relations are given in many reputable textbooks... I also have it in an old Liwschitz Garik AC Machinery textook (ref 9 of my previous attachment) in a form which is not just talking about a single loop in a dc field (like Fitzerald) but in an ac motor. (Photo of the relevant textbook page will be coming soon if we don't resolve this)

If you truly believe what you're saying then surely you can answer some simple questions I have already asked and you seem to have ignored:

To recap I am waiting for one or both of the following:
1. What equation would you propose to use for torque instead of the equation that I provided?
2. If you had to quantitatively estimate torque developed by a given motor given B and I, tell me how you would do it, including what other parameters would you need. (Are you really sure you don't need L?)
3. EDIT - BONUS QUESTION. Let's say I have 2 loops carrying the same current in the same field, each one similar to this. The only difference between the two loops is that one is 1m long and the other is 10m long. Does the 10m loop experience the same torque as the 1m loop? (PS - if you are looking for R and L within the link I just posted earlier in this bullet, they are hidden in area A = 2 R L)

I believe tackling the questions above is the only way to resolve the disagreement. Technical disagreement cannot be resolved by making statements and ignoring the questions that arise in response to those statements. I think one or the other of us stands to learn something if you answer these questions. That would be a useful outcome imo.

 

[edison123]

pete

What part of "you are not a winding designer or a rewinder" and "I am done" do you not get? Read the room. Again, I am done.

Muthu

http://www.edison.co.in

 

[electricpete]

So I guess the implication is that you are an experienced winding designer and that the problem is simply that I don't understand what you're saying. If that is the case, then why don't you teach me something by answering my 3 simple questions above?

Eng-tips should be about learning. There is no one here who can't learn from someone else here.

 

[waross]

Motor torque as a function of a formula.
The formulae presented by both of my friends, as far as I understand, are based on static conditions.
The describe the directed force on a conductor that is at a particular angle in a magnetic field.
A static condition.
So much about a motor's parameters are not static.
While the theoretical torque may be calculated for a given tangent angle of force, that may not represent the forces in a running motor.
In a running motor, the torque is dependent on the load.
For a given load, the conductors assume an angle in relation to the flux that satisfies the equation.
The equations are a result, not a cause.
So, what happens in a working motor if the torque equations predict a reduction in torque?
That will affect the shape of the torque curve with the probable result of dropping the breakaway toque, and the peak torque.
So I would expect slightly less acceleration torque and slightly longer acceleration times.
Sort of like the millions of times a motor is started with 10% low voltage.
Aside from starting, a running motor typically runs at 40% or 50% of available torque.
The motor operating point may move up the torque curve with a slight drop in RPM until the equations are satisfied.
A couple of possible issues.
Less starting torque:
This is a pump motor.
The pump laws tell us that as the speed drops, the torque required drops as the inverse of the cube.
I don't see starting a pump as a problem.
Magnetizing current;
The longer air gap may require more magnetizing current.
However, if that is so, remember that the magnetizing current is reactive current acting at 90 degrees to the load current and will have negligible effect at full load.
The longer stator will have more heat rejecting area and as the limit of full load current is determined by the ability of the motor to reject heat.
I suggest that the added heat rejection will more than offset the slightly greater load current caused by an increase in magnetizing current.
Conclusion;
I strongly suggest that there will be little change in the motor and the pump's operating characteristics.
I hope that this presents a view that is acceptable to both my learned friends.
It is rare to see such a disagreement between two such learned experts.
Respectfully.

--------------------
Ohm's law
Not just a good idea;
It's the LAW!

 

[wcaseyharman]

Okay, I'll play, now that everyone is tired of this thread. And for the record, I profess no expertise in this area, it's just very interesting to me.
From Alfred Still's "Elements of Machine Design", 3rd addition, article 102:

Hp=5.81(N*B"g*D*la*d*Z*10^-8)*(q*pi*D/(n*Z))*cos(theta)*eff*n/746

and thus in order to determine diameter and length of the machine the following equation can be used:

D^2*la=4.07*hp*10^11/(B"g*N*d*eff*cos(theta))

where

D=air gap diameter
la=armature length (effective length)
Z=number of conductors per phase
q=specific loading (ampere conductor per in of armature periphery
d=winding factor
B"g=average flex density
N=speed (rpm)

My interpretation of these relationships in regard to the original question is the length of the armature and the diameter of the core can be a tradeoff with the desired flux density. If the core were lengthened it appears from this literature that, all things being equal, the flux density would drop in the iron but the overall machine ratings would stay the same, although the saturation of the core and teeth would clearly change due to the lower flux densities. If the machine was well designed to begin with it would appear to be additional iron with no real benefit to the user, making the motor more expensive for no reason.

Again, no expertise here, just to follow and learn like everyone else.

 

[electricpete]

wcaseyharman – good comments. Thanks for wandering into the discussion.
Bill those are good comments to enlarge the discussion.

The formulae presented by both of my friends, as far as I understand, are based on static conditions

Absolutely. And I think I have made it clear at various points along the way I was referring to steady state hp rating or roughly equivalent steady state torque rating. And to give additional context to the prior discussion, the statement by edison that I quoted and objected to which he vigorously defended was: “Same turns with more core means the current rating remains the same (since no change in copper area) and hence no change its part on torque while the lower flux density (due to same turns) will lower the torque.”

...that has been the subject of the debate in this thread from my perspective because we never got past it.

To my mind if we want to get to the question of horsepower rating (which I treat roughly equivalent to steady state torque rating), we can use a 2-step process where first step compares torque assuming current is constant and 2nd step compares whether steady state current might be changed based on thermal considerations.

First step (constant current)
T = B L N I Kdp cos(theta). The product B L is constant so for a given current T is the same (all other variables constant. cos(theta) is related to the power factor of the circuit and is very much a 2nd order effect (change of 10% load near full load will cause negligible change in this factor).

2nd step (can we change current)
The 2nd step, how would the currents compare at the steady state limit. I have previously opined it’s a wash (resulting in same steady state rating) and I think waross Gr8blu and wcasyharman all agree. Here are the factors I see on each side with respect to which core can sustain a higher current
Benefits for the longer core (in thermal performance at a given current)
[ul]
[li] More heat transfer area as you point out[/li]
[li] Less core loss per core length due to lower B[/li]
[li] Less losses due to non-productive harmonic current and fluxes (since the core is further out of saturation there are less non-linearities leading to fewer harmonics). [/li]
[/ul]
Benefits to the original length core (in thermal performance at a given current relative to the longer core)
[ul]
[li]More length to accumulate core loss[/li]
[li]More length to accumulate slot I^2*R loss for rotor and stator[/li]
[/ul]
All in all it's probably a wash for steady state rating.

Now what about starting performance. The equation T = B L N I Kdp cos(theta) also applies at any speed. It will provide a valid quasi-static analysis (in the same way the torque vs speed and current vs speed curves provide quasisteady state information). As in the case of running, I don't foresee a large change in the angle associated with the change. But whereas in steady state the factor controlling the magnitude of the current was the load, at start the factor controlling current is the equivalent circuit parameters and in particular the leakage reactance. The biggest factor affecting starting current magnitude is the rotor and stator leakage reactance. The stator leakage reactance further affects B. The end turn and end-ring components are unaffected but the slot components of leakage reactance would increase. With a decrease in I and a further decrease in B (beyond initial 10% drop) associated with voltage drop across stator leakage reactance during start it is reasonable to expect degradation in torque performance during acceleration. These would be important considerations for someone using the motor to look at if the application poses a challenging starting duty or causes momentary torque overloads.

 

[waross]

Or, if the core of a 50 HP motor is lengthened 10% we now have a 55 HP core.
This core may be wound for 50 HP with a slight change in peak torque.

Respectfully, Pete. The winding and the core determine the torque curve, not the operating torque.

I suggest to both my friends to forget about calculating the torque from the formulae.
Rather, from speed and HP determine the rated torque output.
Then work back through the formulae to determine the angle that the conductor will be to the magnetic field.

--------------------
Ohm's law
Not just a good idea;
It's the LAW!

 

[electricpete]

> The winding and the core determine the torque curve, not the operating torque.

I'm not sure exactly what you are trying to say or how it relates to what I said. We all know the operating torque follows the load. That doesn't invalidate the formula. If we know B and I and can deduce constant angle then we can predict torque. Part 1 compares performance for the same current I. Part 2 asks if we have a reason to consider different I (we do not).

There is a complicated vector relationship involved to analyse how we expect that angle to change which can be inferred from the equivalent circuit. I am certainly not saying the angle is constant under all conditions. What I do feel very comfortable to say is that for the small changes under consideration near full load conditions the cos of the angle is not going to change meaningfully. In a analogous way, the relationship between torque (load) and stator current is highly non-linear at low load but becomes approximately linear near full load and we are justified to use the simpler relationship which neglects changes in stator current power factor angle near full load (moreso on a 2-pole than a low-speed). No doubt there are a multitude of minor effects to consider and computer provides the best solution but the analysis I provided seems quite reasonable to me.

It is a dicier proposition during start where of course the angle changes over the course of the start, but I focused on the starting torque (locked rotor torque) and again I think we can make a similar assertion that for small changes under consideration there is not going to be a significant difference in cos of that angle at starting conditions among the scenarios considered. But what there will be is decrease in both the current I and flux density B factors for the longer core which tend to decrease torque which I think probably won’t be overcome by the increase in length L. So I do see reason to suspect some degradation in starting starting performance.

 

[waross]

As I understand it, the conductor angle relative to the field approaches zero degrees at no load, and is at 90 degrees to the field at maximum torque.
I expect the angle to be around 45 degrees at full load.
Hi Pete.
I am mostly in agreement with your position.
I see misunderstanding as to total flux versus flux density.
I see misunderstanding as to calculated peak torque and operating torque.
I am trying to suggest that working backwards from HP may be a way to resolve the misunderstandings.
I see the conductor angle used in the formulae.
I don't feel comfortable assuming a conductor angle under load.
The actual conductor angle at full load is dependent on design decisions.
1. What has the designer decided is a safe maximum current,
2. Related to safe current, how far up the torque curve relative to pull-out torque has the designer designated full load.

--------------------
Ohm's law
Not just a good idea;
It's the LAW!

 

[electricpete]

> I see misunderstanding as to total flux versus flux density.

Please point it out, preferably by quoting so I can understand better what you're talking about. B is flux density which is the variable I have worked with. I don't think I've mentioned total flux (flux density integrated over area).

> I see misunderstanding as to calculated peak torque and operating torque.

Please point it out, preferably by quoting so I can understand better what you're talking about. The only place I talked about peak torque is where I mentioned "these would be important considerations for someone using the motor to look at if the application poses a challenging starting duty or causes momentary torque overloads."

> I don't feel comfortable assuming a conductor angle under load.
> The actual conductor angle at full load is dependent on design decisions.

I didn't assert anything about the angle other than that the cos of the angle would remain roughly constant when considering small changes near full load. Strictly speaking the current in the torque formula would not include any magnetizing current (it would be the rotor current, or load component of stator current). I have not made the distinction of which current we're talking about, but within the context that I have used it (comparing small changes when near full load or at starting / zero speed), the distinction is again not particularly important.

 

[waross]

Please point it out, preferably by quoting so I can understand better what you're talking about. B is flux density which is the variable I have worked with. I don't think I've mentioned total flux (flux density integrated over area).

Others have.

I didn't assert anything about the angle other than that the cos of the angle would remain roughly constant when considering small changes near full load.

Let's compare to my torque equation
T = B I cos(theta) R N L Kdp

Is not 'cos(theta)' assumed to be 45 degrees?

My understanding is that the angle varies from 90 degrees, when the motor is over-driven at synchronous speed, to 0 degrees at peak torque.
If rated torque is 50% of peak torque, then rated torque will be developed when the conductor is at a 60 degree angle to the field rather than 45 degrees.
As I suggested, use HP and speed to calculate rated torque. Then work backwards to determine the angle required to produce that torque.

Load change from zero to 100% will cause an angle change of 30 degrees.
While not linear, a 3.3% change in load will cause an angle shift in the ballpark of 1 degree.
I am not arguing against you, Pete.
I am trying to present the problem in a way that may lead to agreement.

--------------------
Ohm's law
Not just a good idea;
It's the LAW!

 

[electricpete]

Is not 'cos(theta)' assumed to be 45 degrees?
My understanding is that the angle varies from 90 degrees, when the motor is over-driven at synchronous speed, to 0 degrees at peak torque.
If rated torque is 50% of peak torque, then rated torque will be developed when the conductor is at a 60 degree angle to the field rather than 45 degrees.

I would suggest you are looking at a different angle, let's call it delta. P ~ |V1| |V2| sin(delta) / X applies to real power transfered through a series inductance. Often used for syncrhonous machines (where X is synchronous reactance) or for power transfer through tranmission lines.

There are a variety of ways we can characterize induction machines which may include an angle associated with the fields and I'd be glad to try to explore those later...

But for the angle theta that I'm talking about in my torque equation, I will now proclaim it to be the "power factor angle" of the rotor impedance (I should've done that earlier). We can view that in either frame but in the rotor frame it is the power factor of the series impedance Z2''= j(wsync-wrotor)L2 + R2. As such theta would not change directly with load, but only indirectly through speed (wrotor). So we see a big change in theta from starting to running conditions but a relatively small change in angle as load changes cause wrotor changes near full load and an even smaller change in the cos of the angle as wrotor changes near full load (since the slope of the cos curve is relatively flat near the peak).

I can support this interpretation of angle theta as the power factor angle of the rotor through 2 different lines of reasoning (I'll call them proofs). The first proof is through the paper attached to my post 7 Mar 23 14:21 where the equation was "derived". The second proof is through a textbook reference.

First proof based on paper posted 7 Mar 23 14:21 section where the torque is "derived" from the Lorentz force on conductor equation as T = R N L I B p.f. or at least that's my suggestion for an intuitive derivation (the devil is in the details when we look closely at which B we're using and where the torque is applying, but nonetheless for reasons I'd be glad to discuss it is well accepted we can calculate the motor torque as if the conductors were exposed to the airgap flux).

The abbreviated version of the derivation:
[ul]
[li]For a single conductor: F= I B L[/li]
[li]For N conductors all carrying current I in field B: F= N I B L[/li]
[li]If we allow both I and B to vary sinusoidally as a function of angle around the circumference, then if N is large when we integrate the product of I and B over all those conductors we get approx N I B cos(theta).... (it is the same math we use when we integrate the product of sinusoidal voltage and sinusoidal current... we end up with V times I times cos of the angle between them): F= N I B L cos(theta)[/li]
[li]To convert force to torque we multiply by R: T= N I B L cos(theta) R[/li]
[li](Kdp is not derived since a sinusoidal approximation to the fields was used).[/li]
[/ul]
So from that derivation we saw theta was the angle between the max of the B wave and the max of the I wave. The B wave of interest is in the airgap so it represents the T point in the equivalent circuit where L1, L2, and Lm all attach together. And that represents the voltage applied to the rotor circuit (*). we know the relationship between the rotor current and voltage, it is given by the power factor angle of the rotor impedance.

(*) Ok, I made a jump from B to V, how did I do that? Let's say the maximum of the B wave is at some reference angle call it 0 degrees in the rotor reference frame. That B is rotating so as it changes in magnitude it induces voltage in rotor loops formed by adjacent bars. The shape of that induced voltage if we plotted it for all the loops around the rotor is related to the rate of change of B with respect to the coordinate angle, so it is the spatial derivative which introduces a 90 degree phase shift. In other words the loop voltages peak 90 degrees away from B. But for bar currents we don't care about loop voltage, we care about the difference between adjacent loop voltages among the two loops sharing the bar (which is what drives the current in the bar). This again corresponds to spatial differentiation which causes another 90 degree phase shift and puts the voltage driving bar currents back in phase with the B (0 or 180 degrees). So the airgap V (which drives bar currents) is in phase with B and the angle between either of those and the max current would be the same.

It seems like I'm preoccupied with the rotor, what about the stator? Well from the Tee equivalent circuit the load component of the stator current is the same as the rotor current (they have the same phase and same amp turns). So indeed regardless of whether we combine airgap flux with rotor current or stator load component of current, we would see the same angle and same load amp turns and calculate the same magnitude of torque (in opposite direction)... which is something that would make Newton and his third law happy.

ok, i'm done with the first proof. Somehow I'm not sure how many will be satisfied but feel free to comment or poke at it however you please.

Second proof. Attached to this message is page 179 of ref 9 of the previous attachment, which is "Electrical Machinery" by Liwshitz-Garik and Whipple, Copyright 1946, Van Nostrand Company. (let me know if that particular jpg format from my pixel phone is not readable, i can convert it if not readable).

Reference 9 (Volume 2, page 179) gives the following equation for total torque-producing force as equation 7.21a
F = 6.25E-8 L N Bgap Irms Kdp

The 6.25E-8 is associated with the units selected as I showed in my previous attachment section 12.4.2. (I love how he makes things "simpler" by assuming the units for us [/sarcasm])

In going from 7.21a for force to 7.22 for torque, he multiplies by radius (which is the obvious requirement). He makes some other changes in 7.22 which complicate things unnecessarily for my purposes, but I hope you can see that in in multiplying 7.21a by radius, it will match my earlier torque equation.

He hasn't addressed the angle yet but he does so in the last full paragraph at the bottom of the page:

This equation has been derived under the assumption that the rotor current is in phase with its emf. If this is not the case, the cosine of the angle between them (Ψ_2s) has to be introduced into the torque equation. The considerations are the same as for the determination of a power of a circuit where voltage and current are in phase or out of phase. Thus in general... ...

So he resolves this by adding the cos of the angle Ψ_2s between rotor emf and rotor current which I think is easy to recognize as rotor power factor angle.

 

[waross]

Cut through the confusion, Pete, and stop arguing with yourself.
The torque is related to he conductor's angular position in the magnetic field.
That angle will change through a range approaching 90 degrees as the load changes.
That angle must be considered when the rated working torque is calculated.
Forget the formula and go back to basics.
As a winding rotates in a magnetic field, the EMF developed follows a sine curve.
If a current is flowing in the conductor, the same sine curve may be used to demonstrate the torque developed.
I will accept that the angle theta is the power factor of the rotor current. (Is that the absolute PF or the PF relative to the PF of the stator current?)
I submit that a formula to calculate the maximum torque is invalid to calculate the working torque.
There may be an angular displacement of the winding in the order of 45 degrees, and the rated torque may be 1/2 or less than the calculated maximum torque.
The original question was to do with the HP of the modified motor.
HP implies rated working torque.
Please use the correct formula, or method.
I have outlined a possible method in previous posts.

--------------------
Ohm's law
Not just a good idea;
It's the LAW!

 

[electricpete]

ok, well that's an interesting comment. I'm not going to react to the personal nature of your assumption that I'm confused. I have to wonder if you actually tried to read either of my proofs (I realize they are long winded but I invited questions and there are none) and if you think Liwschitz Garik was also confused when he wrote his textbook.

> That angle will change through a range approaching 90 degrees as the load changes.

What you're describing is the relationship for real power transfer through an inductive reactance
P ~ |V1| |V2| sin(delta) / X
or divide thru by radian speed to get torque
T ~ |V1| |V2| sin(delta) / ( w*X).

That is widely used in synchronous machines where X is synchronous reactance, V2 is terminal voltage. V1 represents the stator output of an idealized generator (whose phase matches the rotor). The approaching 90 degrees concept is what limits the output of a synchronous machine (beyond that you have poles slips).

It is not often used in induction machines. If you intended to apply that relationship to an induction machine, which reactance will you choose to measure the angle delta across... the stator leakage reactance? We have other ways of calculating torque speed curve, and the max torque of the curve (breakdown torque) has nothing to do with reaching any critical angle.

There are at least two other angle relationship to predict the torque in an induction motor which involve angles (I will respectfully suggest maybe you've seen these discussed for induction motors and incorrectly assumed they were they same as the sync motor relationship above... they are not).

1: T = -0.5 p/2 Is * Ir * Lsr sin(alpha) * P/2 (Paul Krause textbook).
Where alph is angle between Is and Ir. This one proportional to Is, Ir and sin of the angle between them.

2: T = K MMFs MMFr sin(alpha)
Where alpha is again sin of the angle between them.

Let's call the above two the Krause equations (I don't recall where the 2nd came from but you can find many like this).

Both of the Krauss equations are of the form of a stator quantity (I or MMF) times a rotor quantity times the sin of the angle alpha between them.
It is a different form than the sync motor relationship above with sin(delta).
Delta is the voltage angle change across an inductive impedance (typically synchronous reactance), while alpha is angle difference between two different fields (or currents) in two different windings.

It is also a different form then what I have, which is a magnetizing B times rotor I (or stator load component current) times the cos of the angle theta between them.
I suspect my Torque equation reflects the same relationship as the one above from Kraus, just in different forms. I also suspect that Krause's delta and my theta are representing the same thing, except Krause's delta is 90 degrees larger than theta. I’m pretty sure I have converted these equation forms into each other in the distant past. I have to search my notes or spend some time on figuring it out again. Maybe next weekend.

By the way, maybe you think the rotor impedance angle is nowhere to be seen in those Krause relationships. It is indeed hidden in there if you break out stator as sum of magnetizing and load component. The magnetizing branch current can be viewed as establishing the voltage phase reference for the rotor and the rotor impedance determines the angle of the rotor current with respect to that. But the rotor impedance also determines the angle of the stator load component (they are the same angle with respect to magnetizing branch). So the rotor impedance angle plays a fundamental role in the angle between stator load branch and stator magnetizing branch and associated vectors.

 

[edison123]

pete

Ignoring all your theorizing (because I don't see any reason whatsoever to read through them all), as a very successful winding designer and rewinder of hundreds of motors and generators of various voltage, KW and speed ratings since 1985, I fully stand by my assertion that torque is proportional to the product of flux density B and the motor current regardless of motor core length and dia.



Muthu

http://www.edison.co.in

 

[electricpete]

as a very successful winding designer and rewinder of hundreds of motors and generators of various voltage, KW and speed ratings since 1985, I fully stand by my assertion that torque is proportional to the product of flux density B and the motor current regardless of motor core length and dia.

ok, it makes me wonder why people don't make really tiny motors since the length and diameter contribute nothing towards torque production.

But you're a successful winding designer, so it should be easy to respond to at least one or more of the following

To recap I am waiting for one or both of the following:
1. What equation would you propose to use for torque instead of the equation that I provided?
2. If you had to quantitatively estimate torque developed by a given motor given B and I, tell me how you would do it, including what other parameters would you need. (Are you really sure you don't need L?)
3. EDIT - BONUS QUESTION. Let's say I have 2 loops carrying the same current in the same field, each one similar to this. The only difference between the two loops is that one is 1m long and the other is 10m long. Does the 10m loop experience the same torque as the 1m loop? (PS - if you are looking for R and L within the link I just posted earlier in this bullet, they are hidden in area A = 2 R L)

 

[electricpete]

In my previous response, I didn't notice that you had a question, so responding here

I will accept that the angle theta is the power factor of the rotor current. (Is that the absolute PF or the PF relative to the PF of the stator current?)

Theta is the angle of the rotor impedance Z2''= j(wsync-wrotor)L2 + R2. i.e. arctan((wsync-wrotor)L2/R2). It represents the angle between the rotor voltage (which is also magnetizing branch voltage) and rotor current.

 

[electricpete]

I suspect my Torque equation reflects the same relationship as the one above from Kraus, just in different forms. I also suspect that Krause's delta and my theta are representing the same thing, except Krause's delta is 90 degrees larger than theta. I’m pretty sure I have converted these equation forms into each other in the distant past. I have to search my notes or spend some time on figuring it out again. Maybe next weekend.

ok, I've done that and it was easier than expected...

T = -0.5 p/2 Is * Ir * Lsr sin(alpha) * P/2... From "Analysis of Electric Machinery and Drive Systems" by Krause
where Is is stator current and Ir is rotor current.

It can be written as a proportionality to make things easier:
T ~ Is * Ir * sin(alpha).

Switch to vector quantities and use vector cross prodcut Is x Ir = |Is| |Ir| sin alpha
T ~ Is x Ir

From the equivalent circuit, apply KCL at the Tee node connecting braches with L1, L2, and Lm:
Is = Ir + Im where Im is the current that flows into the magnetizing branch.

Plug Is = Ir + Im into our previous torque equation
T ~ ( Ir + Im) x Ir

Since cross product is a linear operator we can apply it distributively:
T = Ir x Ir + Im x Ir

Bearing in mind the definition of vector cross product, Ir x Ir is 0.
T ~ Im x Ir

Switch back to scalar quantities:
T ~ Im Ir sin (alpha).

B is proportional to Im and shifted 90 degrees in space from Im.
If the angle between Im and Ir is alpha then the angle between B and Ir is gamma = alpha +/- pi/2.
I'm not keeping close track of polarities since I have no figure with polarities defined, but I'll choose gamma = alpha - pi/2 or equivalently alpha = gamma + pi/2
So sin(alpha) = sin(gamma + pi/2) = - cos(gamma)

Setting aside the negative sign, this leads to
T ~ B Ir cos (gamma)
It matches the form of my equation if we set gamma to theta (and why not, it's just a label... I made a new letter gamma to avoid presupposing it was the same as my theta but now that we see the matching form of the equation we can conclude they are the same).
T ~ B Ir cos (theta) [matches the form of my torque equation]

So my theta was conveying the same info as Krause's alpha, just with a 90 degree phase shift.

If there were some steps in there that you think need clarification, please let me know.

What does it all prove? It proves the form of my torque equation (derived from Lorentz force equation) matches exactly what we expect the form of a torque equation to look like based on Paul Krause's textbook. Maybe that goes a little way beyond the prior two proofs towards legitimizing my equation and resolving Bill's concerns, I'm not sure.

The proportionality in Krause's equation also includes an mutual inductance he calls Lsr. I'm quite certain it is proportional to length, which would further support the apparently-controversial proposition that the force on a current carrying conductor in a magnetic field depends on its length.