torque angle for an induction motor? 1

[electricpete]

Electric Machinery (Fitzgerald) describes torque in terms of a torque angle theta_m or delta_sr:

circuit viewpoint:
T = -P/2 * Lsr * Is*Ir*sin(theta_m*P/2) (equation 3-70)
where
T = torque
p = poles
Is = stator current
Ir = rotor current
Lsr = cross inductance between rotor and stator
theta_m = mechanical angle

fields viewpoint:
T = - k * mmfs*mmfr* sin(delta_sr)
where
mmfs = radial airgap mmf associated with stator current.
mmfr = radial airgap mmf associated with rotor current.
delta_sr = angle between mmfs and mmfr
k = P*mu0*Pi*D*Length/(4*gap)

We are familiar with applying this to sync machines, and we expect the angle to be near 0 and certainly much less than 90 degrees (pole slip occurs if machine reaches/exceeds 90 degrees during transient)

We also know that we can apply this concept to induction machines. First we recognize that even though the rotor rotates slower than the sync field by the slip speed, the rotor field moves at sync speed. The rotor physically lags the sync field by slip speed, the rotor frequency is slip frequency, the rotor field leads the rotor by slip speed, and rotor field therefore at sync speed. Alternatively we can simply imagine that the rotating stator field induces the rotor poles and the rotor poles must move with the rotating stator field. Finally we know that average non-zero torque can only occur when both fields (rotor and stator) travel at the same speed, so we shouldn't be surprised to know the rotor field moves at slip speed.

Now my question: What would you estimate the torque angle delta_sr to be for an inducton motor?

I believe that analysis of the equivalent circuit shows that the torque angle for an induction motor would be between 90 and 180 degrees. This seems supported by chapter 9 of Fitzgerald (although he uses a slightly different version of torque angle delta_r = difference between rotor angle and resultant field, the conclusion is the same. He shows delta_r > 90 degrees and since delta_sr>delta_r, then the torque angle detla_sr must also be > 90 degrees.

This doesn't create any stability problem for an induction motor as it would for sync motor, since the stability is ensured by the speed-torque characteric which raises torque if speed decreases in the operating range (slip << breakdown slip).

But it seems like a strange result to me (90 <torque angle <180 for induction motor). I'm wondering if it is the correct conclusion.

Any comments or thoughts?


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[cswilson]

The torque angle is between 0 and 90 degrees for an induction motor. Think of the rotor as a low pass filter with L/R time constant, and the frequency of interest as the slip frequency. The lag of the rotor's response to the stator excitation is the torque angle.

At the no-load synchronous speed, the lag, and hence the torque angle are zero, so no torque is generated. As you increase the load and get some slip, you both increase the magnitude of the transformer coupling and create some phase lag, so you have a (small to start) torque angle between rotor and stator to generate torque.

At the peak of the torque curve, the torque angle is 45 degrees. Beyond this, the low-pass filter nature of the response to the transformer coupling takes over, and the magnitude of the rotor response decreases faster than the sine of the torque angle increases.

Since the rotor is a first-order system, its phase lag never exceeds 90 degrees, no matter how large the slip frequency is.

Curt Wilson
Delta Tau Data Systems

 

[electricpete]

Thanks for responding.

I don't understand your logic.

I would say the torque speed curve can be explained with reference to slip and without reference to torque angle as you probably are already aware. At low slip, the slip-dependent transformation ratio dominates (doubling slip doubles rotor voltage) resulting in increasing torque with increasing slip. At higher slip, the leakage reactance terms dominate due to the higher rotor frequency (the low-pass filter effect that you mentioned).

I do not see a concrete way to tie torque angle into this. Where did you come up with torque angle of 45 degrees at peak of the curve?

I did develop a torque angle estimate myself using the equivalent circuit. I will try to post it tonight. In the mean time any other thoughts?


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[electricpete]

Here is my view of the phase angle relationships in an induction motor which gives rise to
90 < delta_sr<180

http://home.houston.rr.com/electricpete/eng-tips/InductionMotorTorqueAngle.ppt

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[cswilson]

pete:

I think you're getting hung up on the idea of the reversal of the rotor current/magnetic vector being opposite of the stator current/magnetic vector so that rotor magnetic north pole lines up with stator magnetic south pole and vice versa. When there is no load (quadrature) current in the stator and these two vectors are exactly opposed, this is a zero-degree torque angle, not a 180-degree torque angle. This case is exactly equivalent to that of a synchronous motor at no-load equilibrium, a zero-degree torque angle with the poles opposing. (And remember that the earth's North pole is a magnetic south pole, to attract the magnetic north pole of a compass needle.)

While the mechanisms for creating a magnetic field in synchronous and induction (asynchronous) motors are very different, once the field is generated the torque interaction between rotor and stator is identical in both cases.

Torque angle is vital to generating the torque/speed (or torque/slip) curve. From a matter of fundamental physics, the torque generated is proportional to the magnitude of the stator magnetic field times the magnitude of the rotor magnetic field times the sine of the torque angle. (Before magnetic saturation, the magnetic field is basically proporational to the current.)

When the rotor lags by 45 degrees, which corresponds to a 45-degree torque angle, the magnitude of the rotor's current response to voltage is still pretty large (0.707 or 3dB down), but the sine of the torque angle is also pretty large (0.707 again). Perturb the lag/torque angle in either direction, and the product of these two values falls.

 

[electricpete]

I do appreciate you responding to my question. But I heartily disagree with your response. It is my habit to respond to each point to try to achieve a better understanding between us. I hope this isn’t seen as overly argumentative.

"I think you're getting hung up on the idea of the reversal of the rotor current/magnetic vector being opposite of the stator current/magnetic vector so that rotor magnetic north pole lines up with stator magnetic south pole and vice versa."

The load component of the stator current generates equal and opposite mmf to the rotor current. Do you disagree?

"When there is no load (quadrature) current in the stator and these two vectors are exactly opposed, this is a zero-degree torque angle, not a 180-degree torque angle"

First, my vector diagram does not predict a 180 degree torque angle at no-load, it predicts a 90 degrees torque angle as load approaches 0. Second, it is not necessary for the load angle to approach zero at no load to achieve zero load… all that is needed is for the rotor current to approach 0 (which is what happens when slip goes to 0). What is your basis for stating that the torque angle of an induction motor must approach 0 as load approaches 0?

"This case is exactly equivalent to that of a synchronous motor at no-load equilibrium, a zero-degree torque angle with the poles opposing."

I agree with the way you describe a syncronous motor as can be read in any textbook (torque angle goes to 0 when load goes to 0). But why do you think that the torque angle of an induction motor approaching zero load must be identical to the torque angle of a sync motor approaching no load?

(And remember that the earth's North pole is a magnetic south pole, to attract the magnetic north pole of a compass needle.)

While the mechanisms for creating a magnetic field in synchronous and induction (asynchronous) motors are very different, once the field is generated the torque interaction between rotor and stator is identical in both cases.

Torque angle is vital to generating the torque/speed (or torque/slip) curve. From a matter of fundamental physics, the torque generated is proportional to the magnitude of the stator magnetic field times the magnitude of the rotor magnetic field times the sine of the torque angle. (Before magnetic saturation, the magnetic field is basically proporational to the current.)

I agree with all of this (with the comment that torque angles for induction motors are in a different quadrant than those for sync motors). How does any of the above disprove 90 < delta_sr < 180 for an induction motor?

When the rotor lags by 45 degrees, which corresponds to a 45-degree torque angle, the magnitude of the rotor's current response to voltage is still pretty large (0.707 or 3dB down)…..

I would be interested in how you came up with 0.707 “rotor current response” at 45 degree torque angle. I believe that what you are doing is borrowing from filter theory as you did in your earlier post, under the assumption that the curve torque vs speed represents a linear polynomial transfer function and the torque angle is the angle of that transfer function. I believe this is incorrect. The torque angle is not the angle of a complex torque quantity. The torque angle is the angle between two other vectors (I1 and I2), neither of which is torque (and in fact torque is not even a linear combination of these two vectors). The relationships that you are citing from filter theory do not apply. (T=K I1 I2 sin<delta_sr> does apply). If you can show some basis why we should consider the torque angle equivalent to the phase angle of a filter, please explain further.

Can you identify your specific area of disagreement with the vector diagram that I posted? Please point it out to me or propose a corrected diagram. I would be especially interested in how you could possibly construct a diagram that would be consistent with your statement that the induction motor torque angle goes to 0 as load goes to 0…. I cannot imagine what it would look like.

Respectfully,
electricpete

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[cswilson]

pete:

We come at this from very different backgrounds, so it is tough to find common ground in our analysis. I work in the realm of field-oriented control; you obviously work in the realm of running motors "open loop" or from a VFD. I am not used to looking at that kind of vector diagram, because FOC simplifies everything vastly.

For 20 years we have sold (very successfully) servo controllers with built in FOC algorithms that can easily be configured for either synchronous or asynchronous motors. In our case, the synchronous motors are generally permanent-magnet brushless servo motors, and the asynchronous motors are induction motors. In both cases, and encoder or resolver on the back of the motor typically serves as the feedback for both the FOC algorithm and the servo algorithm.

Our algorithm has a "magnetization current" parameter that the user sets to tell the controller how much stator current must be "in line" with the rotor magnetic field (this is called "direct" current in FOC analysis). For a PM servo motor, this parameter is generally set to zero, because the permanent magnets on the rotor provide the field. For an induction motor, this parameter must be positive, and up to the point of rotor magnetic saturation, the rotor magnetic field strength is proportional to this value.

The torque command current (usually the output of the position/velocity servo) is applied perpendicular to the rotor field (this is called "quadrature" current in FOC analysis). For PM servos, this is the only current, so the torque angle is 90 degrees, yielding the maximum torque per unit current. (With rotor angle feedback, we can run at 90 degrees all the time, something that can be hard for people from an "open-loop" background to wrap their heads around.)

For an induction motor, the total stator current is the vector sum of the torque (quadrature) current (torque angle of 90 degrees) and the magnetization (direct) current (torque angle of 0 degrees). So the resulting torque angle is between 0 and 90 degrees, with the value depending on the relative value of the two components. As the load decreases, the net torque angle approaches 0; as the load increases, it approaches 90 degrees.

A couple of sanity checks:

Yes, the torque angle of the magnetization current is 0 degrees, not 180 degrees. For open-loop operation of synchronous motors (stepper motors in our case), we set quadrature current to 0, and use magnetization current alone, but tied to commanded rotor angle, not actual. The more the physical rotor angle lags behind commanded, the larger the torque angle. At no load, the torque angle is 0. Zero degrees provides a stable equilibrium; 180 degrees would be unstable equilibrium.

Yes, torque-producing current must have a torque angle of 90 degrees, for any kind of motor. Creating torque in a moving motor requires actual work, and therefore actual power transfer from electrical to mechanical. Real power in an AC electrical system can only be delivered by current and voltage in phase with each other, which means perpendicular to the rotor field.

And no, magnetizing current cannot have a torque angle of 90 degrees, because this is reactive power, and so this current must be out of phase with the voltage.

Note that the above assertions are from very basic physical principles, and so true for any kind of motor, synchronous or asynchronous, with open-loop (direct from line) or closed-loop (FOC) control. So it just cannot be that the no-load torque angle in an induction motor is 90 degrees, approaching 180 degrees as load increases.

As I said above, I have not worked with your style of phasor diagram much. But it looks very much to me that you have a 90-degree "offset" somewhere in your analysis.

That's all I have time for now. I hope to get the time soon to bone up on phasor diagrams in my reference books, and to explain further on the action of the rotor as a simple L/R circuit reacting to slip frequency.

Curt Wilson
Delta Tau Data Systems

 

[electricpete]

You are correct that my question applies to simple squirrel cage induction motors driven from a balanced three phase sinusoidal power source. I don’t work with FOC. After reading your comments I know more about it than I did before, but still not a lot.

I suspect there may be terminology differences. What you call torque component is what I called load component. Note I have defined what I am calling torque angle = delta_sr in my first post.

Up until a week ago, I would have said an induction motor torque angle is < 90 (just like sync motor) based on what I have read in textbooks. Then when I read Fitzgerald’s Electric Machinery, it showed a diagram that seems to be a torque angle above 90, and I have just tonight saw an almost-identical diagram in Modern Power Electronics and AC Drives by Bose (2001) page 34 (more later) . I worked it out myself with the vector diagram I posted before and it seems to me that vector diagram supports my interpretation of what I think I see in Fitzgerald and Bose (that the torque angle is above 90).

The figures in Fitzgerald and Bose don’t show the total stator mmf... only the magnetizing component (also called resultant component since it is the total). I will call that mmf_m. (remember in my first post I also defined mmf_s and mmf_r). They don’t show the torque angle between total stator mmf and rotor mmf... instead they show an angle between the magnetizing mmf and the rotor mmf which I will call delta_mr .

Even though their diagram is missing the elements of my torque equation
[recall my torque equation was T = - k * mmfs*mmfr* sin(delta_sr) ],
their diagram still provides equivalent information to calculate the torque. Fitzgerald provides the trigonometry and figures to demonstrate that mmf_s sin(delta_sr) = mmf_m * sin(delta_mr). That means we can transform my torque equation above into

T = - k * mmfm*mmfr* sin(delta_mr)
which is similar to what you see below the figure in Bose.

Now I have redone my vector diagram to show delta_mr here

http://home.houston.rr.com/electricpete/eng-tips/InductionMotorTorqueAngleRev1.ppt

As you can easily see from my revised figue:
delta_mr = 90 + Phi2
(delta_mr>90)
and
delta_sr>delta_mr

Now, here is an excerpt from Bose

http://home.houston.rr.com/electricpete/eng-tips/ExcerptTorqueAngle.pdf

In figure A, the solid sinusoid is labeled airgap flux density wave. This is my mmf_m (when we add together all the mmf’s, the resultant is the magnetizing and it’s what is in the airgap).

In figures B and C, the same solid sinusoid is shown again. It’s not labeled again, but rest assured that it still represents mmf_m (a similar figurre appears in Fitzgerald and it is labeled the same in all three figures there.... Bose references Fitzgerald and probably copied this diagram from him).

Now look at the dashed lines in the three figues. In the first drawing it is rotor voltage which is in-phase with the airgap flux. In the second figure it is rotor current which lags rotor voltage (the first figure dashed line) by angle theta_r (my Phi2). In the third figure, it is rotor mmf, which lags rotor current (the second figure dashed line) by 90. In the third figure they also specifrically label the spacing between the solid line (airgap mmf which is my mmf_m) and the dashed line (mmf_r) as 90 + theta_r. This result from Bose is of course 100% consistent with my vector diagram that shows delta_mr = 90 + Phi2.

The consistency makes me think I am right. But I could have misinterpretted those references somehow. (Can anyone offer a different interpretation?)

It’s a strange result, so I posted it here to see if anyone can show me why my logic is wrong. You’re telling me I’m wrong based on setup of an FOC controller. I’m not very familiar with that, but it seems to me an extra level of complexity that takes the discussion farther away from my real question (maybe I wouldn’t feel that way if I worked with them). Even though I’m sure you are very competent, you know exactly how your product works and that it’s a great product, I am not convinced that there is not some hidden convention, simplification, or tricky terminology built into the FOC algorithm which obscures the physical reality. If I get a chance I’ll read up a little on it.

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[electricpete]

Also as a small clarification, I have shown all my angles in terms of currents. If I redrew my vector diagram with mmf's, all the vectors would shift by 90 degrees but the angle differences delta_mr and delta_sr would remain the same.

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[cswilson]

pete:

I've had some time to look things over and check references, and I think I can start closing the gap between us. I have two problems with the phasor diagram you show in the link above, one fairly minor, and the other going to the key point at hand.

The first objection is that I believe the angle Phi2 in your diagram must be zero. That is, the load (torque) current in the stator must have the same orientation as the stator voltage. As I explained above, load current does real physical work (power transfer), and only the component of current in phase with the voltage can do that. The total stator current is the vector sum of the perpendicular load and magnetizing currents, and this will always lag the voltages.

Second, I believe that the torque angle is the angle between I1m and I1total. The rotor flux is parallel to the stator magnetizing current, and the torque angle is the angle between the rotor flux and the total stator (not rotor) current. This angle will be zero at no load current and increase towards 90 degrees as load increases. In this sense it is just like a synchronous motor, although the mechanism to create this effect is very different.

Having a torque angle between 0 and 90 degrees is critcal for the fundamentally stable velocity operation that both types of motors exhibit. If the load torque increases, the torque angle and its sine increase to generate enough countervailing torque to balance it out (up to a point, of course). With a torque angle of over 90 degrees, you cannot get this type of stable operation.

You miss a key point when you say that increased slip creates more torque. What you are missing is that it does so by increasing the torque angle. This is the insight I was trying to get across by explaining the rotor dynamics as an L/R circuit. The bigger the slip frequency, the greater the lag of the current in the rotor relative to the stator current. This lag angle is in fact the torque angle, and for small values of slip, there is an almost linear relationship between slip frequency, rotor lag/torque angle, and generated torque.

I used field-oriented control principles to try to give you a different way of looking at things. It is not magic, nor does it use any other physical mechanisms than open-loop control. In the steady state (velocity and load), operation stabilizes exactly as it would for open-loop operation, so the phasor diagrams are the same. FOC just manages non-steady-state cases a lot better than open-loop control.

Curt Wilson
Delta Tau Data Systems

 

[electricpete]

The first objection is that I believe the angle Phi2 in your diagram must be zero. That is, the load (torque) current in the stator must have the same orientation as the stator voltage. As I explained above, load current does real physical work (power transfer), and only the component of current in phase with the voltage can do that.

I have defined Phi2 is the angle between Vm and I2L where I2L is rotor current. Vm is not the terminal voltage, it is what is marked on the drawing. The rotor has inductance. Therefore it's current (I2L) will lag it's voltage (Vm) by phi which is the power factor angle of the rotor circuit at the frequency of interest. You may be using a different definition of I2L, but I have defined it on the drawing. Mine is the rotor branch current (maybe I should not have called it load current). This is the appropriate current of interest for me because I am seeking the difference between rotor mmf and stator mmf. Each is 90 degrees from their associated current.

Second, I believe that the torque angle is the angle between I1m and I1total.

That is an alternate definition to be used with the alternate equation I provided in my post 2 Mar 07 23:48. I can work with either one, but my original question refers to delta_sr. But delta_mr is also above 90 as shown in my figure.

The rotor flux is parallel to the stator magnetizing current, and the torque angle is the angle between the rotor flux and the total stator (not rotor) current. This angle will be zero at no load current and increase towards 90 degrees as load increases. In this sense it is just like a synchronous motor, although the mechanism to create this effect is very different.

I hear what you're saying but I don't see any proof.

Having a torque angle between 0 and 90 degrees is critical for the fundamentally stable velocity operation that both types of motors exhibit. If the load torque increases, the torque angle and its sine increase to generate enough countervailing torque to balance it out (up to a point, of course). With a torque angle of over 90 degrees, you cannot get this type of stable operation.

I disagree. The induction motor operating slightly below sync speed has a stabilizing mechanism available to it that a sync motor does not. If load increases, slip increase, rotor current increases , and torque will decrease even if torque angle decreases.

You miss a key point when you say that increased slip creates more torque. What you are missing is that it does so by increasing the torque angle.

What happens to the torque angle per my diagram when load and slip increases is that sin(delta_sr decreases). BUT once again increasing slip increases rotor current, so torque can increase even though delta_sr decreases.

This is the insight I was trying to get across by explaining the rotor dynamics as an L/R circuit. The bigger the slip frequency, the greater the lag of the current in the rotor relative to the stator current. This lag angle is in fact the torque angle, and for small values of slip, there is an almost linear relationship between slip frequency, rotor lag/torque angle, and generated torque.

I heard your explanation but I saw no proof of your explanation and as I noted above, torque angle does not correspond to the phase angle of a linear transfer function (what would be the input and output quantities and what is the linear system? Torque magnitude = -P/2 * Lsr * Is*Ir*sin(theta) is not a linear combination of circuit currents, and there is no recognizeable circuit component since . It is the angle between two other vectors.

I would describe the torque-speed curve in two parts:
1 - Near sync speed, torque increases with slip because rotor current increases with slip.
2 - Near 0 speed, torque decreases with slip due to the increases in leakage impedance with slip frequency.
The peak of the curve (breakdown torque) is the transition between these two regions.
The first of these factors has nothing to do with any angle. Torque has a linear relationship with slip based on the fact that voltage induced into the rotor circuit increases linearly with slip resulting in rotor branch current rising linearly with slip (for very low values of slip at speeds between full load and sync speed). This is predicted by the equivalent circuit.


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[electricpete]

correction in bold:
"I disagree. The induction motor operating slightly below sync speed has a stabilizing mechanism available to it that a sync motor does not. If load increases, slip increase, rotor current increases , and torque will increase even if torque angle decreases."

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[electricpete]

Referring again to the attached excerpt form "Bose's Modern Power Electronics and AC Drives"

http://home.houston.rr.com/electricpete/eng-tips/ExcerptTorqueAngle.pdf

They define torque angle in the same way you do. Delta_mr = angle between magnetizing mmf (also called resultant mmf or total airgap mmf) and the rotor mmf.

The figure shows the "rotor mmf wave" lagging the "airgap flux density wave" by angle "delta = 90 +theta_r" (quoted items exactly as labeled on the figure. The text at the bottom of the page says

"Te = Pi * (p/2) * l * r * Bp * Fp * sin(delta)
where = number of poles; l=axial length; r=machine radius; Bp = peak value of airgap flux density [electricpete: mu0*mmf_m]; Fp = peak value of rotor mmf [electricpete: mmf_r], and delta= pi/2 + theta_r is defined as the torque angle"

This author has used your definition of torque angle and he says it is greater than pi/2 (ie. greater than 90 degrees). Is he wrong?

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[electricpete]

I do appreciate the comments. I hope I am not being too argumentative. Just trying to present the way I see it.

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[cswilson]

pete:

We are coming at this issue from two very different starting points: I from the basic physics, and you from a stator "equivalent circuit" analysis. Let's see if we can reconcile these.

First the basic physics:

An induction motor rotor is a transformer secondary, as any motor text will tell you. It "sees" the slip frequency. But the voltage output from a transformer secondary is not a function of frequency -- the AC magnitude of the secondary voltage output is simply the AC magnitude of the primary input (i.e. stator) voltage times the effective turns ratio. For a constant AC stator voltage, you get a constant AC rotor voltage. Again, any elementary text will tell you this.

What about the current response of the rotor? Back to the basic texts again. The rotor is a shorted transformer secondary, and so is really just a series inductance and resistance loop driven by an AC voltage signal at the slip frequency. The frequency response of this circuit can be described by

1 / (jwL + R)

where w is the frequency it sees (the slip frequency here). This is a simple first-order system, with a low-pass response. Note that since the frequency "w" term is in the denominator, the current magnitude actually decreases with increasing slip frequency. It certainly does not increase linearly with slip frequency.

As to angle, the current response lags the voltage by arctan(wL/R), and so of course this increases with frequency. Note that for small angles the tangent is basically equal to the angle in radians, so this angle is directly proportional to the slip frequency.

If you take the magnitude of this current response as a function of slip frequency and multiply it by the sine of the angle of the current response as a function of slip frequency, you get the classic induction motor torque curve. It's basically that simple.

This all makes sense if you consider the torque angle at zero slip to be 0 or 180 degrees (I don't care which you call it), approaching 90 degrees as slip frequency increases. The change in torque angle is the same as the change in current response lag, so as the lag increases, the sine of the torque angle increases (and for small angles, the sine, the tangent, and the angle are all basically the same. So torque increases with slip for small slip frequencies even though rotor current magnitude is decreasing.

So how do we square this with your equivalent-circuit analysis? First, remember what this is. It is an "equivalent" circuit of the stator (alone) to take into account "non-circuit" effects like the interaction with the rotor. The key thing we have to look at is the right hand branch with its "variable resistor" equal to R2/s, where R2 is a motor constant, and s is the slip frequency. Of course, this isn't really a variable resistor (potentiometer?) -- that's why it's called an "equivalent circuit" -- but it provides a reasonable description (at least in the steady state) of the circuit response to slip. So the current through this branch is proportional to slip frequency, and in phase with the voltage across this branch.

But -- and this is the important point -- this is stator current, not rotor current. Your assertion that the equivalent circuit shows that rotor current increases linearly with slip is a fundamental misinterpretation, and, I think, at the core of our disagreement. This component of stator current does increase linearly with slip frequency, but it is only one component of the stator current, and one that is perpendicular to the constant magnitude current flowing through the inductor in the left vertical branch. Because this variable component is orthogonal to the magnetizing current, it does not affect rotor current.

For small values of slip frequency, this small perpendicular component barely increases the magnitude of the overall stator current (and hence stator mmf), which is the vector sum of these two components, but it does change the angle almost linearly. (It changes the tangent linearly, and for small angles, the tangent, the angle, and the sine are virtually the same.) Since we know that the torque varies virtually linearly with the slip frrequency in this range, the only effect that we have that could cause this is the change in angle. Because it is the sine of the angle that affects torque, it means that we must be starting from an angle whose sine is 0 -- that is, from 0 or 180 degrees. Once again, I don't really care whether you call the no-load (synchronous) torque angle 0 or 180 degrees -- it's just a naming convention to me. (But I see no reason, other than perhaps convention, that this has to be treated differently from synchronous motors.) In the Bose reference you provide that shows the torque angle as 90 degrees plus ThetaR, I cannot tell how ThetaR varies with slip, so I cannot say whether I disagree with him.

This analysis agrees completely with my "basic physics" analysis above. Note that in that analysis, the reference was the stator angle, with the rotor lagging more at increasing slip; in the equivalent circuit analysis, the reference is the rotor angle, with the stator leading more at increasing slip.

But it just cannot be that the no-load torque angle is 90 degrees, as you have stated -- you have no mechanism for creating a roughly linear relationship between torque and slip for small values of slip frequency.

Sanity check: Does it make sense that the current through the "variable resistor" in the equivalent circuit is the stator component in phase with voltage and orthogonal to magnetizing current? Yes. At synchronous speed, the back EMF in phase with voltage is equal to the supply voltage, so there is no voltage "headroom" to create current of this orientation. The back EMF magnitude decreases linearly with slip frequency, so the "headroom" and hence resulting current in phase with voltage increase linearly with slip frequency. Also, current in phase with voltage does real work. At no-load synchronous speed, no work, no in-phase current. As slip increases, torque increases, in-phase current increases.

Curt Wilson
Delta Tau Data Systems

 

[electricpete]

Your assertion that the equivalent circuit shows that rotor current increases linearly with slip is a fundamental misinterpretation, and, I think, at the core of our disagreement

Right and wrong. It is at the core of our disagreement, but the fundamental misinterpretation is not mine.

Forget the equivalent circuit.

Consider the airgap flux relatively constant. The voltage induced into the rotor is proportional to rate of change of flux cutting a loop between adjacent rotor bars. This will naturally be proportional to slip speed. E2 = E1*s. At low slip, the rotor inductive reactance is negligible. I2 = E2 / R2 = E1*s/R2 is be proportional to slip. Period.

Now to derive the equivalent ciruit, add back in the inductive reactance j*wsync*s*L2. Voltage equation on the secondary circuit
E2 = s*E1 = I2 * (R2 + j*wsync*s*L2)
divide each side by s
E1 = I2 * (R2/s + j*wsync*L2)

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[electricpete]

As mentioned before, the above discussion of I2 proportional to slip applies for low slip where rotor inductive reactance is neglible.

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[electricpete]

Going back to your post 5 Mar 07 12:39

Second, I believe that the torque angle is the angle between I1m and I1total.

That is different than either of the torque angles that I have defined (the one I defined in my first post or the one used by Bose). The angle you describe is < 90 degrees as shown in my diagram.

If that's your definition of torque angle I have no disagreement with it. But again I have defined the torque angle differently in agreement with Fitzgerald (delta_sr as in original post) and Bose (delta_mr as in Bose).

I still have a big disagreement with your suggestion that rotor current does not go to 0 at no load.

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[cswilson]

Arrrrrgh! That's what I get for writing late at night (the only time I have available for this) and without consulting my notes.

Unlike a "regular" transformer, where the primary and the secondary "see" the same frequency, and so the secondary voltage is independent of this frequency, in an induction motor, the primary is working with the line frequency and the secondary is working with the slip frequency. So at a fixed line frequency, the rotor voltage is proportional to slip.

And at low slip frequencies, the rotor current is pretty much proportional to voltage and in phase with it. As slip increases, the relative current magnitude falls off and starts to lag, explaining why the torque stops rising linearly and even falls with increasing slip.

It's amazing that you can get basically the same end results with a 90 degree offset and substituting a constant for a ramp. I guess it's like those "fake but accurate" memos that were in the news a few years ago...

The first sources I consulted discussed synchronous and asynchronous motors in completely different terms and without direct reference to torque angle. I finally dug up a 30-year-old text that showed the two cases side by side (and with a DC motor with a fixed torque angle of 90 degrees) that completely agrees with your analysis.

There are a lot of places to fool oneself in the analysis because there are lots of 90 degree offsets (e.g. unloaded, stator voltage and current are basically 90 degrees out of phase) and issues like constant flux (key in FOC analysis) leading to linearly varying voltage and current.

Curt Wilson
Delta Tau Data Systems