# Starting Torque of Synchronous Motors

Dr.D
TL;DR Summary
I'm interested in calculating the transient starting torque of salient pole synchronous motors. Can someone point me to a paper or other reference that develops the theory for this calculation, please?
It is pretty well known among electrical machines folk that large, salient pole electrical machines start typically with the field winding shorted and starting torque developed by current induced in the damper windings, acting much like a squirrel cage induction machine. The net machine torque pulsates rather strongly at twice slip frequency, a pulsation that often causes problems in the connected mechanical load. I read (on the internet) that motor manufacturers can typically provide curves for the "steady (accelerating) torque" and for the "pulsating torque," each as functions of slip. This raises several questions for me.

1) I'd like to know how these torque amplitudes are calculated for each slip value?
2) Is there any way to know the phasing of these torque pulsations?
3) Is there a way to estimate these torque amplitudes (as functions of slip) without detailed design information, say using just name plate data for the machine.

Direction to any sources (books, papers, web sites) would be much appreciated.

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• Abdullah Almosalami

Abdullah Almosalami
Alright. Though I am not able to answer your question directly, as I have not reached the level of knowledge needed, I'll share something I recently tried to answer myself that kind of is in the same ballpark as your question, albeit this is a bit long-winded:

1) The electromagnetic theory will give you the equations and relationships that describe quantities like torque. A book like "Electric Machines and Power Systems" by Del Toro seems to me to develop a decent amount of theory on developing the equations describing torque in electric machines; specifically towards the end of Chapter 3 and then a little bit more concerning torque in an induction-motor situation in Chapter 4-5. I'll admit though I haven't gotten to read the book yet as I am still going through my more introductory one.

After you've got your equations, the second half of the study of electric machines comes in, and this is something it seems to me introductory texts totally ignore: THE DYNAMICS! Once you are able to define a quantity like torque, then given some set of initial conditions and some idea of the main forces that come into play, you should be able to solve for the rotational kinematic quantities with respect to time; of biggest interest to EM folk would be the rotational speed. The only reasonable way it seems to me to solve for such quantities are numerical techniques since you will most definitely encounter some crazy nonlinear equations. If you are interested, I have come up with some neat MATLAB code that can solve and also somewhat simulate these quantities, and once you have that, you can see exactly what steady-state operation looks like, and how transients appear, etc.

So that'd be where I'd start. It seems like the general equation guiding what the torque in the machine will be is independent of the operating condition, so if you know that equation, you can just simulate it out, play around, and increase your insight tenfold.

2) An example of what I mean with point #1:
Recently I tried to understand why the rotors of synchronous machines are locked into synchronous frequency to begin with. To me it seemed like the only way the rotor stays at a constant rotational speed is if the net torque on it was zero, which means induced torque exactly cancels out friction/windage; but why would friction/windage cancel out induced torque EXACTLY at synchronous speed? Induced torque is purely dependent on flux density vectors, and friction/windage would be proportional to the speed (or the square of the speed or whatever) of the rotor. It is totally not obvious why they would cancel out at that point. And it seems overall the texts are trying to imply that if the rotor was spinning faster than synchronous, the induced torque and friction would bring it back down to synchronous speed; and again that is not apparent to me because induced torque is a function of the magnetic fields, not speed. Also, this would imply that even without friction, the rotor will be locked into synchronous speed; which again. Nope. I am totally not convinced by that.

To better explain my point, introductory derivation of torque will produce something like ##\vec \tau_{ind} = k \vec B_{rotor} \times \vec B_{stator}##. Now, imagine this: say your rotor was at synchronous speed; this means ##\vec B_{rotor}## and ##\vec B_{stator}## are rotating together with zero slip. This is the assumed steady state operation most people and books tell you. However, when I imagined that, I was like wait. Hold up. Unless the vectors were in-line with each other, there is some induced torque, and that induced torque, depending on the relative positioning of the vectors, will either try and cause the rotor to slow down or speed up, and there is no way you will convince me that friction will oppose this enough to cancel things out here because synchronous speed could be any speed but the induced torque, once you set the currents to produce the field, will have a finite specific range and that's it (i.e., from 0 to some max).

So I was like alright. I'm done with these lame explanations. Let's see if I can simulate this and get some better insight. Setup:
1) ##\vec \tau_{ind} = k \vec B_{rotor} \times \vec B_{stator}## will be the guiding equation for defining induced torque, and in this VECTOR form exactly so that you can actually do the cross product. You can play around with this parameter k.

2) I'll set up some friction that applies counter-torque that is proportional to the rotational speed.

3) I'll play around with what the initial state is along with the system's parameters. So for example, I'll have ##\vec B_{rotor}## initially be 45 degrees ahead of ##\vec B_{stator}##, or whatever, and the initial speed ##\omega_{rotor}## will be something. You can really play around with everything here. That's the beauty of simulation.

3) The general computational technique is as follows: You will split up the total time-period you are trying to simulate into however many small intervals of size T. Then going from step to step, you will assume torque and acceleration are constant if your T is small enough. Accordingly (with subscript i meaning step i):
a) Firstly, the stator's magnetic field and its kinematics is set. There is no angular acceleration. It will rotate at synchronous speed because it is the stator! The only tricky part is the rotor.
b) Compute the current induced torque and frictional torque at time ##t_i##. The net torque is just the sum of these.
c) Compute the resulting rotational acceleration based on ##\Sigma \tau_i = I \alpha_i##, and again you can play around with what the moment of inertia of the rotor is. The more it is, the less the angular acceleration will be for a given torque, and vice versa.
d) Compute the next step's rotational speed: ##\omega_{i+1} = \omega_i + \alpha_i*T##
e) Now compute the next step's angular position: ##\theta_{i+1} = \theta_i + \omega_i*T + 0.5*\alpha_i^2##
f) Now compute the next step's magnetic field vectors: ##B_{rotor, i+1} = B_{r,max}*[\cos\theta_{i+1}, \sin\theta_{i+1}, 0]## (I'm assuming they lie on the xy-plane). Likewise for the stator.
g) Iterate away.

Again. I can share with you the MATLAB code I wrote to implement this. And actually, I also use the drawnow function of MATLAB so that I can also see the vectors in 3d space along with the data play out after it's been computed which helps make sure the result make some sense and is also just pretty cool to see.

So, as one possible scenario, I set it up so that the initial rotational speed of my rotor was AT synchronous speed (I set that as 60Hz on a two-pole machine), which was what I was initially curious about to begin with, and had the initial angle be 90 degrees. Now there's A LOT of other parameters going on, and it would be insane to try and show you everything. Here's a setup with some friction, the rotor having some decent moment of inertia, and decent magnetic fields, over 1000s: The graphs on the left are for the rotational speed of the rotor ##\omega_{rotor}## and the graphs on the right are for the angular acceleration of the rotor ##\alpha_{rotor}##. And what I meant by the visualization (I uploaded to YouTube a 20 second thing to show you here more clearly: quick example).

Firstly, it's clear there is definitely some oscillation about synchronous frequency (the grey line for the graphs on the left), and the angular acceleration (which you can of course also see takes the same shape as the torque) is constantly oscillating all throughout. Furthermore, in this setup, the motor's steady state speed was actually HIGHER than synchronous speed.

Now I'm just showing this example as where you might yourself explore this so you can get more insight. I am certainly not saying my simulation is fully complete, because it is clearly evident experimentally that synchronous speed is the speed the rotor spins at without much oscillation. So it's a matter of improving the model, but that improvement will definitely not come about from my current textbook, because purely based on ##\vec \tau_{ind} = k \vec B_{rotor} \times \vec B_{stator}##, I am not getting the actual expected behavior! However, once that is done, you can see that I will be able to have a more accurate picture of transient and steady state behavior, and also gain the insight that comes with playing around with all these parameters! Looking forward to that as I increase my knowledge on the field, and all the while, this invaluable tool of simulation will surely come in handy TONS.

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Dr.D
@Abdullah Almosalami Thanks for your response; as you can see, you are the only one to take a shot at the question thus far.

There are a number of excellent electrical machines books out there. The one I like best is by P.C. Krause of Purdue University. I've read and studied Krause rather carefully, but even this source does not adequately address the transient torque question.

When operating at synchronous speed, the field current is in operation driving the magnetic field of the rotor while the stator current drives the stator field. If these two are perfectly aligned, there is no torque produced. It is only when a slight misalignment is introduced, called the torque angle, that torque is developed. A synchronous machine running at synchronous speed under external load will operate with a small torque angle. If the load increases, the torque angle increase. If the increase is excessive, the rotor can be pulled out of synchronism. All of this is well known and well documented in many places. But it only addresses operation at synchronous speed.

My question was specifically addressed to the torques developed during the starting transient period, that is, zero speed up to synchronous speed. During the run-up, the field is usually not energized. The field winding is either open circuited or shorted through a resistor. The machines of interest a salient pole machines (not round rotor machines), so that the air gap flux density varies widely around the circumference. The starting torque is developed by currents induced in the damper bars that are embedded in the pole faces. This is geometrically messy, but that is part of the source of the torque oscillations I'm interested in learning about. I did find one helpful IEEE paper by Godwin and Merrill.
→τind=k→Brotor×→Bstator\vec \tau_{ind} = k \vec B_{rotor} \times \vec B_{stator}
I do not see the connection between this expression and actual synchronous motor transient torque, but even if it is correct, it does not allow for all the geometry unless you work that into the B field calculations. Something more is needed. I'm afraid you are going on a wild goose chase with this approach.

Abdullah Almosalami
There are a number of excellent electrical machines books out there. The one I like best is by P.C. Krause of Purdue University. I've read and studied Krause rather carefully, but even this source does not adequately address the transient torque question.

That is indeed one of the books I have lined up, though a little more down the line! I am just currently on "Electric Machinery Fundamentals" by Chapman. The book I mentioned in my answer will be one I go into next as it has some more mathematical rigor in it.

When operating at synchronous speed, the field current is in operation driving the magnetic field of the rotor while the stator current drives the stator field. If these two are perfectly aligned, there is no torque produced. It is only when a slight misalignment is introduced, called the torque angle, that torque is developed. A synchronous machine running at synchronous speed under external load will operate with a small torque angle. If the load increases, the torque angle increase. If the increase is excessive, the rotor can be pulled out of synchronism. All of this is well known and well documented in many places. But it only addresses operation at synchronous speed.

I am familiar with what you mention here, and I know this is not only well documented but easily demonstrated. My issue is this explanation is going off of the equation ##\tau_{ind} = k \vec B_r \times \vec B_s##, at least from what I understand, and these explanations are implying the relative position of the magnetic field vectors (that torque angle) remains constant given some loading, which to me is just weird. That seems to me to require some very special set of circumstances where the net torque RIGHT at that orientation of field vectors is 0 because RIGHT at that moment, the friction counter-torque cancels out the induced torque, or somehow this state is achieved asymptotically. In any other case, the induced torque will cause change in the rotor's speed, which in turn will change the orientation of the field vectors, which in turn will change the torque, and so on. And clearly, when I try and simulate this out, you see this. I'm not sure if you saw the video to show you my visualization but along with the plots I posted, it does demonstrate what I am saying. So I feel as though there has to be more to the explanation here beyond just that "this is what happens".

Anyways. My apologies. I know am not answering your question so this is not helpful to you, so I don't mean to take up too much of your time.

Staff Emeritus
First regarding starting. There is no single answer because there are numerous methods. A good starting point is here
https://en.wikipedia.org/wiki/Synchronous_motor#Starting_methods

Next regarding synchronous operation. Your thought analysis with so many fluxes makes my head hurt. If you step back and look at it from the basis of energy, it is much simpler. Voltages, currents, fields, and fluxes need not be mentioned.
1. The rate of change of generator speed (and thus frequency) is proportional to the net mismatch between mechanical power, electrical power, and various friction and resistance losses.
2. The rate of change of angle is proportional to the mismatch between frequency and grid frequency. In the USA think of grid frequency fixed at 60 hertz.
3. Electrical power is proportional to angle. Feed this back to step 1)
The thing that makes synchronous machines different than say induction machines is step 3. A synchrnous machine has an angle relative to the grid which is constant in a steady state. For that angle to stay constant in time, the motor speed and grid speed must be exactly equal. An induction machine has a constant difference in speed relative to grid frequency. To make torque, that speed difference must be non-zero.

Now, consider the added complexity of magnetic attraction, I think the synchronous machine is the simplest of all. The trick is to visualize is from the frame of reference of the rotor. Think of a cowboy stradling the rotor like a horse.

The rotor's magnetic field appears constant to the cowboy. It has a North pole and a South pole, just like a permanent magnet.

The stator's magnetic field we like to say is rotating. But from the cowboy's view, it is not rotating, not changing in time. But it does change with position. If the looks to his left or right he will see a static magnetic field gradient, increasing or decreasing. So the force of attraction/repusion of those two magnetic fields depends on the position of the rotor field relative to the peaks/valleys of the stator field. To an outside observer (not the cowboy), that position is described as an angle.

essenmein
Summary:: I'm interested in calculating the transient starting torque of salient pole synchronous motors. Can someone point me to a paper or other reference that develops the theory for this calculation, please?

It is pretty well known among electrical machines folk that large, salient pole electrical machines start typically with the field winding shorted and starting torque developed by current induced in the damper windings, acting much like a squirrel cage induction machine. The net machine torque pulsates rather strongly at twice slip frequency, a pulsation that often causes problems in the connected mechanical load. I read (on the internet) that motor manufacturers can typically provide curves for the "steady (accelerating) torque" and for the "pulsating torque," each as functions of slip. This raises several questions for me.

1) I'd like to know how these torque amplitudes are calculated for each slip value?
2) Is there any way to know the phasing of these torque pulsations?
3) Is there a way to estimate these torque amplitudes (as functions of slip) without detailed design information, say using just name plate data for the machine.

Direction to any sources (books, papers, web sites) would be much appreciated.

My synchronous machine experience is predominately VFD driven, so this starting issue is basically non existent. Well starting does have issues, but they are completely different.