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Torque calculation and definition of flux unit (V/Hz)

Jonathan Lock
Intellectual 775 points

Hey there,

Could anyone clarify how the rotor flux (V/Hz) is defined? In particular, I'm trying to determine the rotor torque based on the rotor's flux, and I haven't quite been able to determine things to my liking. In this thread; http://e2e.ti.com/support/microcontrollers/c2000/f/902/t/277190.aspx Dave Wilson notes at one point

"...the motor torque can be calculated by simply multiplying the rotor flux linkage times the q-axis stator current times the number of rotor poles/2.  This is how InstaSPIN-FOC calculates torque..."

Which doesn't add up for me -- I've tested a motor with a flux linkage of 0.017 V/Hz with 7 pole pairs, when driven with a quadrature current of 10A (verified with an oscilloscope on the current feedback) I measured an output torque of 0.25Nm, giving a Kt or 0.025Nm/A. This closely matches the value reported by FAST; approximately 2.2lb-in, or 0.249 Nm (0.0249Nm/A). Dave's example would give a torque of 0.017*10*14/2 = 1.19 (I'm guessing Nm).

I've also looked around briefly and tried to use the textbook relation Kt = Ke (= 1/Kv). As Ke is typically expressed as Vs/rad this would give a relation of Kt = 1/(2*pi) * V/Hz. This resolves to 1/(2*pi) * 0.017 = 0.0027 Nm/A, which is off by a factor of 10 or so. More importantly, this definition has no dependence on the number of pole pairs whatsoever, unlike the previous snippet.

Does anyone happen to know how to accurately determine the torque constant based on the flux linkage estimated by FAST? Note that I can't use the estimation given by FAST because I'm using a previously stored value for the flux in order to limit the output torque during motor startup (so given the torque constant it would be trivial to determine a maximum quadrature current).

Thanks!

  • 0
    TI__Guru** 119500 points

    Johnathon,

    I've sent this to Dave - I'm not sure why his equation is giving a different value, I suspect some sort of scaling difference - but he's on vacation this week.

  • 0
    TI__Expert 4760 points

    Hi Jonathan,

    The equation for calculating torque for a non-salient machine in SI units (including all appropriate scale factors) is as follows:

    Torque (Newton-Meters) = Rotor Flux (Webers) * Iq (Amps) * Rotor Magnet Poles * (3/4)

    There are several important points I need to mention:

    1.  For this equation to work, rotor flux must be in units of Webers, NOT V/Hz.  In SI units, "Webers" is exactly equal to volts-second/radians.  So to convert from V/Hz to V-sec/rad, you need to divide by 2p.  InstaSPIN-FOC reports motor flux in units of V/Hz.  However, internal to the FAST observer, it converts it to Webers.  This unfortunately creates some confusion when trying to explain how FAST calculates torque.

    2.  The value for Iq assumes you are using the "Amplitude Invariant" form of the Clarke transform.  In most cases, this isn't an issue since InstaSPIN-FOC does indeed use this form of the Clarke transform.  However, if you use a different form (such as the POWER Invariant form), you would need to use a different scale factor.

    3.  Just to clarify, the above equation uses the number of rotor poles, NOT pole pairs.  So you would need to plug 14 into the above expression.

    4.  This is the expression for "electromagnetic torque", NOT shaft torque.  To get shaft torque from electromagnetic torque, you must subtract the losses in your motor, including core losses, windage losses, and bearing friction.

    Solving for your situation, we get:

    Tem (N-M) =   0.017 (V/Hz) * (1/2p) * 10 (Amps) * 14 (rotor poles) * (3/4) = 0.28 NM.  This would suggest that all of your losses add up to 0.03 NM.

    I apologize for the confusion from my previous statement, as I did not include the scale factor assumptions.  I hope this clears up some of the confusion.

    Best Regards,

    Dave

     

  • 0 in reply to Dave (Wisconsin) Wilson
    Intellectual 775 points

    Hi Dave,

    Thanks for your extensive reply! Could you elaborate a little on the cause of the (unitless, unless I'm mistaken) 3/4 scale factor? Is it due to something intrinsic to/internal in FAST? I'm just curious as I haven't seen the need for a scale factor when modeling ideal motors in textbook literature.

    Thanks again for your reply,
    //Jonathan

  • 0 in reply to Jonathan Lock
    TI__Expert 4760 points

    Hi Jonathan,

    No, this scaling is not endemic to just FAST.  Unfortunately, part of the confusion stems from the many different forms of the Clarke transform, and the lack of standardization on any particular one.  I would be curious to know what expression you have been using for torque for nonsalient PMSM machines.  From a mathematical point of view, you can use ANY scaling you wish to translate between 3-phase to 2-phase, as long as you use 1/X of that scaling to transfer back from 2-phase to 3-phase.  The most popular scaling is "amplitude-invariant", where the amplitude of the alpha and beta waveforms is equal to the amplitude of the 3-phase sine waves.  This is very convenient because with this scaling, if you set Id = 0 and Iq = "x" amps, this means that the 3-phase current  sinewaves will have a peak amplitude of "x" amps.

    However, there is a lesser known form which is vector "magnitude" invariant for lack of a better term.  If you take the 3-phase currents and add them together as space vectors to obtain the true net current vector, you will see that its amplitude is 3/2 larger than the amplitude obtained by using the amplitude invariant form.  This 3/2 scale factor difference must be accounted for when calculating torque.  So if we use the amplitude invariant form of the Clarke Transform, the Iq value will be 2/3 too small, and we must correct for this by multiplying Iq by 3/2:

                                    Tem = (P/2)lr(3/2)(Iq)

    Combining coefficients, we get:

                                     Tem = (3/4)PlrIq

    Hopefully this clears up the mystery behind the 3/4 scaling coefficient.

    Best Regards,

    Dave

     

  • 0 in reply to Dave (Wisconsin) Wilson
    Intellectual 775 points

    Hi Dave,

    Thanks for the clarification! I just realized that I had been looking at brushed DC motor constants, which I suppose would be reasonably well modeled as a trivial ampliture-invariant "scaling" (simply 1:1) of Iq = Iwinding. As the return current in the motor is equal in magnitude to the input current this seems to match well with the expression Tem = PlrIq.


    Thanks for your clarification!

    //Jonathan