DC Machine Magnetics: Torque, Back EMF & Flux Explained

[cnh1995]

I have a question regarding this part from a book on electrical machinery (the author is highly reputed in India). While teaching the motor action at an elementary level, the torque developed in the motor is explained using the equation F=Bil. But this paragraph says that the torque developed is due to the interaction of the two magnetic fields and actual flux cutting the conductors is very small. The rotor is pulled by the poles at some angle and the tangential component of this pull is responsible for the torque. (wikipedia says something similar: https://www.google.co.in/url?sa=t&s...SgCMAA&usg=AFQjCNHxYQcr7f_2j_DYuUpo5OpNoLi5tw). I understand this part completely. But since the flux through the conductors is very small, shouldn't this affect the back emf? We write torque T∝ I*Φ and E_back∝Φ*N, but how can we take the same Φ in both the equations? For torque, the flux in the tooth section is responsible and for back emf, air gap flux linking with the conductors is responsible, which is practically much smaller than the teeth flux. What am I missing? Thanks in advance.

 

[jim hardy]

That is counter to what i was taught.

EDIT And it looks like have carried an incomplete understanding for fifty years ! see post #5 below

Professor Charles A. Gross* taught us boys in my DC machinery class, ca 1965, that the force is exerted on the conductors by Lorentz F = QV cross B .
It is responsible for both rotor torque and counter-emf.
Were that not so it would be unnecessary to wedge the stator conductors in huge central station generators.
Try a search on "generator stator bar wedge" to see how significant a matter it is.

A wire loop carrying current in a B field will experience torque in the complete absence of an iron rotor, see
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html

I'd write the book's publisher and suggest he re-assign that editor .

EDIT for the rest of the story, see post #5

*see http://www.eng.auburn.edu/~gross/

old jim

 

[cnh1995]

A wire loop carrying current in a B field will experience torque in the complete absence of an iron rotor

Yes, I agree. I guess that's why the basic motor model is explained using Lorentz force. If this is true for practical motors, the above equations would make sense to me.

But in case of iron rotor, flux in the tooth section should be greater than that linking with the conductors, because of the difference in reluctances.

 

[jim hardy]

This delightful old book is back in print, in India i think
"Dynamo Electric Machinery" by Sylvanus P Thompson
https://babel.hathitrust.org/cgi/pt?id=umn.31951000936579f;view=1up;seq=7
page 3:

my copy is an original 1901 US edition ( library discard )
i love his baroque style..

old jim

 

[jim hardy]

But in case of iron rotor, flux in the tooth section should be greater than that linking with the conductors, because of the difference in reluctances.

Well - it looks like i owe you and that author an apology
and i have to wonder if i owe Dr Gross a note

look what Sylvanus Thompson said about it on page 99 of that book i linked
(wow am i embarrassed - i have never been to that page, and i own a hardcopy !)

Theory adjusted for reality, "back to the future".
Does it suggest that there will be forces on the conductors at slot frequency ?
Here's actual flux in the air gap adjacent rotor surface , courtesy Generatortech
http://generatortech.com/B-Page2-Theory-Overview.html

If i learn something every day will i ever become learned ?

“In times of change, learners inherit the earth, while the learned find themselves beautifully equipped to deal with a world that no longer exists.” eric hoffer

old jim

 

[jim hardy]

flux in the tooth section should be greater than that linking with the conductors, because of the difference in reluctances.

That brings up a really interesting fine point, use of terms "flux linking" and "flux cutting" by different authors.
To your image in post 3

adjust it in your mind to the instant when the loop is perfectly horizontal
that is its plane is aligned with magnetic lines of flux
the conductors are being cut by flux at highest rate so voltage induced is maximum
yet the conductors enclose zero flux so the flux linking them at that instant is zero;
and we know sine function has greatest slope at zero crossing so d(flux)/dt is maximum and so is voltage.

Meaning -
whether you're in the " flux cutters" camp or in the "flux linkers" camp you arrive at the same result. There's no paradox.

But - observe how clever is the location of their commutator segment: brushes short out that particular turn at the very instant it has zero volts induced because flux linkage is maximum and d/dt of flux is zero.. when plane of coil is perpendicular to flux.

old jim

 

[cnh1995]

Well - it looks like i owe you and that author an apology

That's not at all true. I have said this before, you've made all this machinery stuff very interesting for me, so much so that I am planning to choose it as a career option.

Well - it looks like i owe you and that author an apology
and i have to wonder if i owe Dr Gross a note

look what Sylvanus Thompson said about it on page 99 of that book i linked
(wow am i embarrassed - i have never been to that page, and i own a hardcopy !)

View attachment 104671
Theory adjusted for reality, "back to the future".
Does it suggest that there will be forces on the conductors at slot frequency ?
Here's actual flux in the air gap adjacent rotor surface , courtesy Generatortech
http://generatortech.com/B-Page2-Theory-Overview.html

If i learn something every day will i ever become learned ?old jim

That is some interesting information! I'll have to think on this for a while. I'll post again if anything is unclear. Thanks!

 

[cnh1995]

Does it suggest that there will be forces on the conductors at slot frequency ?

Could you please elaborate? What does Sylvanus Thompson say further in the book about this paradoxial phenomenon? How do the conductors cut all the magnetic lines? Or do they really cut all the magnetic lines? I mean do we just assume the ideal case while writing the equations that the flux is uniform everywhere and the conductors are not protected?If I understand this graph correctly, the red waveform is of the field flux and the non-uniformity on its top is because of the iron rotor teeth, right? What are the blue waveform and the green arrows?

 

[jim hardy]

I mean do we just assume the ideal case while writing the equations that the flux is uniform everywhere and the conductors are not protected?

--I think that's the answer.
I run into this often - undergraduate degree gives one enough basics to get started in a variety of areas, work experience takes over after that and we learn the details from mentors in whatever specialty we are working .

So, i inquired of Dr Gross who replied

"
In my view F = (i x B)l and E = (u x B)l, where F, i, and B are mutually perpendicular vectors (as are E, u, and B) and can be used to explaine motor and generator operation on a single conductor of length l and zero diameter moving with velocity u, and by extension, in a dc machine armature.

In this idealized situation, “B” is the magnetic field (i.e. flux density) at the conductor location (with the current set to zero).
When the situation is more complicated (such as the addition of mixed media (iron, air, copper, etc) and geometry some approximations are necessary.
"

which upon reflection makes sense . To calculate exact flux anyplace would be a three dimension undertaking and surfing the internet turns up some highfalutin simulations.
this one's from a finite element program at a place called "imoose", http://imoose.sourceforge.net/

so we have a logical conflict, how do lines cut the conductor if they pretty much stay in the teeth ?

I muddled for days how to resolve this in my own alleged brain and the best words i can come up with are pretty much like Thompson's

and that is why i make myself remain bilingual when it comes to flux-cutting versus flux-linking.
Flux cutting works when i think of flux as discrete hard lines and am using QVcrossB
Flux linking works when i am thinking of flux as a soft fluid field that goes everywhere and am using NdΦ/dt .
And i know the two must give the same result.

I accept that since i don't know how to calculate flux in the slot where cutting takes place i have to resort to linking.
When flux is perpendicular to the plane of the coil it's all linked, irrespective of the teeth. And i trust that high math would resolve the apparent disparity.

So, to your question - i use the simplification of ideal. I remain aware that specialists do go further and calculate local flux density. When you hear terms "fringing" and "tooth heating" you know you're about to exit the realm of undergrad studies.old jim

 

[cnh1995]

and that is why i make myself remain bilingual when it comes to flux-cutting versus flux-linking.
Flux cutting works when i think of flux as discrete hard lines and am using QVcrossB
Flux linking works when i am thinking of flux as a soft fluid field that goes everywhere and am using NdΦ/dt .

Yes. But here's what I think.. When a conductor is moving and there is change in flux linking with the conductor loop, although E=dΦ/dt gives the magnitude of the induced emf, the physical reason behind the induced emf is "flux cutting" and E=BlvsinΘ. So, if the flux linking with a "moving" conductor is changing, it "must be cutting" the flux lines and motional emf BlvsinΘ= dΦ/dt. So, for a "moving" conductor in a "fixed" field (like in a dc generator), E=dΦ/dt would be true only if the conductor is physically "cutting" the flux such that E=BlvsinΘ. If the conductor loop is stationary, then dΦ/dt will be the "physical reason" behind induced emf, but in case of a moving conductor, the physical reason should be flux "cutting" and motional emf turns out to be equal to dΦ/dt. Does this sound correct?

 

[jim hardy]

If I understand this graph correctly, the red waveform is of the field flux and the non-uniformity on its top is because of the iron rotor teeth, right? What are the blue waveform and the green arrows?

Okay, a note about that waveform and how it works might be in order.
Firstly it's flux in the AC generator of a power plant. But flux is flux and that's why i put it up.
The objective of this measurement is to detect shorted turns in a power plant's generator rotor.
Please excuse my plodding, it was 33 years ago i had the good fortune to assist Don Albright in making that measurement on a 200 mw generator. Back then we used analog 'scope and Polaroid camera to capture the traces. Since then he and his sons improved their technique.

First some glossary:
The flux probe is just a small coil inserted in the air gap very close to the rotor surface.
Red trace is the voltage induced in that coil. So it's actually dΦ/dt, derivative of flux.

Observe the little red circles labelled "slot leakage flux" - a slot that has shorted turns will have less mmf so its leakage flux will be low compared to other slots.
Those squiggles riding atop the red trace are the individual rotor slots as they pass under the flux probe.

IF we could somehow get rid of the aggregate rotor flux and look just at the squiggles we'd get a picture of just the leakage flux, something like this:

Knowing how many turns are in each rotor slot you can see whether one is lacking mmf. Here slots 4 and 6 are deficient.

When we did my machine in 1983 we shorted the generator terminals at the switchyard side of stepup transformer. Forcing terminal volts to zero let's armature reaction force aggregate flux to zero , leaving just the squiggles.
But as you can imagine, conservative utilities find it disconcerting to short circuit a machine and it takes co-ordination with system folks to get a line crew out there to install the short circuit. (the wires are as big as your arm) .
So the Albrights figured out how to do it easier. That's where the blue and green lines come in.

Look at that cross section of a rotor above. Aggregate flux travels vertically through the rotor, at the instant shown none of it links the flux probe. So at that instant we'll have a good picture of the slot 6 squiggle.
That's what the vertical green line represents, the instant at which aggregate flux is zero at the flux probe location.
The red line represents aggregate rotor flux and he made it by integrating the flux probe voltage. Since e = dΦ/dt, Φ = ∫edt . As you see, it's zero at green line.
This picture is obviously from a different generator with 8 slots instead of 6,
but it's what they had in their explanation at http://generatortech.com/B-Page2-Theory-Sample.html
Observe zero flux occurs right in the middle of the slots, as in their fig 1 above .

So - if we could somehow shift the flux in the generator to be zero when the other slots pass tghe flux probe we could get a good look at heir squiggles too.
Well- armature reaction shifts flux, and the machine's rotor pulls forward as we add load,
So by applying load to the machine they shift the instant of zero flux to occur directly over the slot of interest and capture data with their computer.
And that's what is shown in that moving diagram in earlier post.
Quite an improvement in technique, i'd say. Especially not having to short the machine.

Spend some time on their explanation pages and apply your basics. I think you'll have fun.

Mr Albright senior is now deceased and his children run generator testing the business. They graciously gave me permission to use their images here for discussion purposes, i hope you find the topic as interesting as i do.

And i hope i addressed your questions. Sorry i don't have an better answer for what goes on inside the conductor hidden inside a slot, i suspect it's something to do with magnetic vector potential which i have yet to master.

You might enjoy this one
https://www.researchgate.net/publication/280734258_Localised_Flux_Density_Distribution_in_the_Stator_Core_of_05HP_Three_Phase_AC_Induction_Motor

old jim

 

[cnh1995]

Okay, a note about that waveform and how it works might be in order.
Firstly it's flux in the AC generator of a power plant. But flux is flux and that's why i put it up.
The objective of this measurement is to detect shorted turns in a power plant's generator rotor.
Please excuse my plodding, it was 33 years ago i had the good fortune to assist Don Albright in making that measurement on a 200 mw generator. Back then we used analog 'scope and Polaroid camera to capture the traces. Since then he and his sons improved their technique.

First some glossary:
The flux probe is just a small coil inserted in the air gap very close to the rotor surface.
Red trace is the voltage induced in that coil. So it's actually dΦ/dt, derivative of flux.
View attachment 104834
Observe the little red circles labelled "slot leakage flux" - a slot that has shorted turns will have less mmf so its leakage flux will be low compared to other slots.
Those squiggles riding atop the red trace are the individual rotor slots as they pass under the flux probe.

IF we could somehow get rid of the aggregate rotor flux and look just at the squiggles we'd get a picture of just the leakage flux, something like this:
View attachment 104835
Knowing how many turns are in each rotor slot you can see whether one is lacking mmf. Here slots 4 and 6 are deficient.

When we did my machine in 1983 we shorted the generator terminals at the switchyard side of stepup transformer. Forcing terminal volts to zero let's armature reaction force aggregate flux to zero , leaving just the squiggles.
But as you can imagine, conservative utilities find it disconcerting to short circuit a machine and it takes co-ordination with system folks to get a line crew out there to install the short circuit. (the wires are as big as your arm) .
So the Albrights figured out how to do it easier. That's where the blue and green lines come in.

Look at that cross section of a rotor above. Aggregate flux travels vertically through the rotor, at the instant shown none of it links the flux probe. So at that instant we'll have a good picture of the slot 6 squiggle.
That's what the vertical green line represents, the instant at which aggregate flux is zero at the flux probe location.
The red line represents aggregate rotor flux and he made it by integrating the flux probe voltage. Since e = dΦ/dt, Φ = ∫edt . As you see, it's zero at green line.
This picture is obviously from a different generator with 8 slots instead of 6,
but it's what they had in their explanation at http://generatortech.com/B-Page2-Theory-Sample.html
Observe zero flux occurs right in the middle of the slots, as in their fig 1 above .
View attachment 104836

So - if we could somehow shift the flux in the generator to be zero when the other slots pass tghe flux probe we could get a good look at heir squiggles too.
Well- armature reaction shifts flux, and the machine's rotor pulls forward as we add load,
So by applying load to the machine they shift the instant of zero flux to occur directly over the slot of interest and capture data with their computer.
And that's what is shown in that moving diagram in earlier post.
Quite an improvement in technique, i'd say. Especially not having to short the machine.

Spend some time on their explanation pages and apply your basics. I think you'll have fun.

Mr Albright senior is now deceased and his children run generator testing the business. They graciously gave me permission to use their images here for discussion purposes, i hope you find the topic as interesting as i do.

And i hope i addressed your questions. Sorry i don't have an better answer for what goes on inside the conductor hidden inside a slot, i suspect it's something to do with magnetic vector potential which i have yet to master.

You might enjoy this one
https://www.researchgate.net/publication/280734258_Localised_Flux_Density_Distribution_in_the_Stator_Core_of_05HP_Three_Phase_AC_Induction_Motor

old jim

That's some really interesting information! Thanks a lot! I learn more from you than from the books. I'll be thinking on it as soon as I finish my assignment.

 

[jim hardy]

Yes. But here's what I think.. When a conductor is moving and there is change in flux linking with the conductor loop, although E=dΦ/dt gives the magnitude of the induced emf, the physical reason behind the induced emf is "flux cutting" and E=BlvsinΘ. So, if the flux linking with a "moving" conductor is changing, it "must be cutting" the flux lines and motional emf BlvsinΘ= dΦ/dt. So, for a "moving" conductor in a "fixed" field (like in a dc generator), E=dΦ/dt would be true only if the conductor is physically "cutting" the flux such that E=BlvsinΘ. If the conductor loop is stationary, then dΦ/dt will be the "physical reason" behind induced emf, but in case of a moving conductor, the physical reason should be flux "cutting" and motional emf turns out to be equal to dΦ/dt. Does this sound correct?

I distinctly remember sitting in physics class about 1966 trying to resolve the same conundrum.
I took it back to force on an individual charge moving in a field F = QVcrossB which when integrated along the length of your wire gives BLvsinθ.

So, for a "moving" conductor in a "fixed" field (like in a dc generator), E=dΦ/dt would be true only if the conductor is physically "cutting" the flux such that E=BlvsinΘ. If the conductor loop is stationary, then dΦ/dt will be the "physical reason" behind induced emf, but in case of a moving conductor, the physical reason should be flux "cutting" and motional emf turns out to be equal to dΦ/dt. Does this sound correct?

I'm careful to keep in mind that flux isn't "lines" it's a continuum, we use lines to indicate its intensity.
I'm still adjusting to Thompson's assertion that slots and teeth shield the charges inside the wires from Lorentz's mechanical force QVcrossB because it seems at odds with counter-emf(as in your original question)..

You and i both want a mental model that treats charges like discrete particles, and Hall effect to explain the force on the conductors, and that model serves us well for 99% of what we're likely to encounter in a career.

I've always used the simple ideal formulas, approximations though they be.
I expect the explanation to our conundrum lies in dipoles and magnetic vector potentials...
Perhaps a better mathematician than i will chime in ?

http://www.eng.fsu.edu/~dommelen/quantum/style_a/elechamil.html

I'm just not fluent in Vector Calculus.

http://cds.cern.ch/record/630753/files/0307133.pdf
So far everything we have done has been entirely deductive, making use only of
Coulomb's law, conservation of charge under Lorentz transformation and Lorentz in-
variance for our physical laws. We have now come to the end of this deductive path. At
this point when the laws were being written, God had to make a decision. In general
there are 16 components of a second-rank tensor in four dimensions. However, in anal-
ogy to three dimensions we can make a major simpli cation by choosing the completely
antisymmetric tensor to represent our eld quantities. Then we would have only 6 inde-
pendent components instead of the possible 16. Under Lorentz transformation the tensor
would remain antisymmetric and we would never have need for more than six independent
components. Appreciating this, and having a deep aversion to useless complication, God
naturally chose the antsymmetric tensor as His medium of expression.

which leaves me in the dust...

old jim

 

[jim hardy]

Yes. But here's what I think.. When a conductor is moving and there is change in flux linking with the conductor loop, although E=dΦ/dt gives the magnitude of the induced emf, the physical reason behind the induced emf is "flux cutting" and E=BlvsinΘ. So, if the flux linking with a "moving" conductor is changing, it "must be cutting" the flux lines and motional emf BlvsinΘ= dΦ/dt. So, for a "moving" conductor in a "fixed" field (like in a dc generator), E=dΦ/dt would be true only if the conductor is physically "cutting" the flux such that E=BlvsinΘ. If the conductor loop is stationary, then dΦ/dt will be the "physical reason" behind induced emf, but in case of a moving conductor, the physical reason should be flux "cutting" and motional emf turns out to be equal to dΦ/dt. Does this sound correct?

I distinctly remember sitting in physics class about 1966 trying to resolve the same conundrum.
I took it back to force on an individual charge moving in a field F = QVcrossB which when integrated along the length of your wire gives BLvsinθ.

So, for a "moving" conductor in a "fixed" field (like in a dc generator), E=dΦ/dt would be true only if the conductor is physically "cutting" the flux such that E=BlvsinΘ. If the conductor loop is stationary, then dΦ/dt will be the "physical reason" behind induced emf, but in case of a moving conductor, the physical reason should be flux "cutting" and motional emf turns out to be equal to dΦ/dt. Does this sound correct?

I'm careful to keep in mind that flux isn't "lines" it's a continuum, we use lines to indicate its intensity.
I'm still adjusting to Thompson's assertion that slots and teeth shield the charges inside the wires from Lorentz's mechanical force QVcrossB because it seems at odds with counter-emf(as in your original question)..

You and i both want a mental model that treats charges like discrete particles, and Hall effect to explain the force on the conductors, and that model serves us well for 99% of what we're likely to encounter in a career.

I've always used the simple ideal formulas, approximations though they be.
I expect the explanation to our conundrum lies in dipoles and magnetic vector potentials...
Perhaps a better mathematician than i will chime in ?

http://www.eng.fsu.edu/~dommelen/quantum/style_a/elechamil.html

View attachment 104839

http://cds.cern.ch/record/630753/files/0307133.pdf
So far everything we have done has been entirely deductive, making use only of
Coulomb's law, conservation of charge under Lorentz transformation and Lorentz in-
variance for our physical laws. We have now come to the end of this deductive path. At
this point when the laws were being written, God had to make a decision. In general
there are 16 components of a second-rank tensor in four dimensions. However, in anal-
ogy to three dimensions we can make a major simpli cation by choosing the completely
antisymmetric tensor to represent our eld quantities. Then we would have only 6 inde-
pendent components instead of the possible 16. Under Lorentz transformation the tensor
would remain antisymmetric and we would never have need for more than six independent
components. Appreciating this, and having a deep aversion to useless complication, God
naturally chose the antsymmetric tensor as His medium of expression.

 

[jim hardy]

Yes. But here's what I think.. When a conductor is moving and there is change in flux linking with the conductor loop, although E=dΦ/dt gives the magnitude of the induced emf, the physical reason behind the induced emf is "flux cutting" and E=BlvsinΘ. So, if the flux linking with a "moving" conductor is changing, it "must be cutting" the flux lines and motional emf BlvsinΘ= dΦ/dt. So, for a "moving" conductor in a "fixed" field (like in a dc generator), E=dΦ/dt would be true only if the conductor is physically "cutting" the flux such that E=BlvsinΘ. If the conductor loop is stationary, then dΦ/dt will be the "physical reason" behind induced emf, but in case of a moving conductor, the physical reason should be flux "cutting" and motional emf turns out to be equal to dΦ/dt. Does this sound correct?

I distinctly remember sitting in physics class about 1966 trying to resolve the same conundrum.
I took it back to force on an individual charge moving in a field F = QVcrossB which when integrated along the length of your wire gives BLvsinθ.

So, for a "moving" conductor in a "fixed" field (like in a dc generator), E=dΦ/dt would be true only if the conductor is physically "cutting" the flux such that E=BlvsinΘ. If the conductor loop is stationary, then dΦ/dt will be the "physical reason" behind induced emf, but in case of a moving conductor, the physical reason should be flux "cutting" and motional emf turns out to be equal to dΦ/dt. Does this sound correct?

I'm careful to keep in mind that flux isn't "lines" it's a continuum, we use lines to indicate its intensity.
I'm still adjusting to Thompson's assertion that slots and teeth shield the charges inside the wires from Lorentz's mechanical force QVcrossB because it seems at odds with counter-emf(as in your original question)..

You and i both want a mental model that treats charges like discrete particles, and Hall effect to explain the force on the conductors, and that model serves us well for 99% of what we're likely to encounter in a career.

I've always used the simple ideal formulas, approximations though they be.
I expect the explanation to our conundrum lies in dipoles and magnetic vector potentials...
Perhaps a better mathematician than i will chime in ?

http://www.eng.fsu.edu/~dommelen/quantum/style_a/elechamil.html

View attachment 104839

http://cds.cern.ch/record/630753/files/0307133.pdf
So far everything we have done has been entirely deductive, making use only of
Coulomb's law, conservation of charge under Lorentz transformation and Lorentz in-
variance for our physical laws. We have now come to the end of this deductive path. At
this point when the laws were being written, God had to make a decision. In general
there are 16 components of a second-rank tensor in four dimensions. However, in anal-
ogy to three dimensions we can make a major simpli cation by choosing the completely
antisymmetric tensor to represent our eld quantities. Then we would have only 6 inde-
pendent components instead of the possible 16. Under Lorentz transformation the tensor
would remain antisymmetric and we would never have need for more than six independent
components. Appreciating this, and having a deep aversion to useless complication, God
naturally chose the antsymmetric tensor as His medium of expression.

i'm left in the dust...