I have some practical problems with your Time Constants calculation. First of all, your definitions for both the Mechancical and the Electrical Time Constants are for the Motor only and assumes that a "Stepped" input Voltage is being applied to the motor. In practice neither assumption is correct since the drive electronics (including a battery) adds a finite electrical resistance in series with the motors. Including the Drive significantly changes both the Electrical and Mechanical Time constant for the Motor-Drive System. I have seen several clients and students confused by this when they attempt to measure a Motor-Drives' mechanical time constant in the laboratory and find that it is significantly "longer" in time than the value obtained using your equations. The problem is one must include a value for the Drives' "output" resistance in series with the motor to calculate correct values for both the Electrical and Mechanical time contants of the Motor-Drive system.
Second, modern PWM type drives do not generally apply a "stepped" input Voltage to a motor. Rather, a PWM type drive looks more like a "current" source than a "voltage" source to the motor. Hence a "stepped" input velocity command to the drive typically translates into a current wavew form that is closer to a stepped "current" input to the motor not a stepped "voltage". The reason being that the "buss" voltage in the drive is usually 30-40% higher than the final "steady state" voltage applied to the motor. Therefore, the motor's dynamic velocity response is often times closer to a "linear" rise with time as opposed to an exponential rise of velocity with time predicted by your Time Constants Calculation.
Third, I disagree with your Mechanical Time Constant equation for a "Brushless" dc motor. There are actually different equations for the different Motor-Drive combinations that are commercially available. For a "Trapezoidal" back EMF motor with a Six-step "Trapezoidal" drive the Mechanical time Constant is given by:
Tm (Trap-Trap) = RJ/KE*KT
This is identical to a brush type dc motor with a dc drive. However, for a "Sinusoidal" back EMF motor with a "Sinusoidal" drive, the correct expression for the Mechanical Time Constant is:
where KE = 0-peak value of the phase-phase back EMF (Volt/rad/sec).
Next, you also have the possibility of operating a "Sinusoidal" back EMF motor with a six-step "Trapezoidal" drive and the correct Mechanical Time Constant expression is given by:
Hence, to correctly calculate the Mechanical Time Constant for a "Brushless" dc motor you must specify both the Motor's back EMF function and the Drive used to electrically power the motor. Each combination produces a different Mechanical Time Constant value.
Finally, I totally agree with you that these time constants change with temperature due to both an increase in electrical resistance with increasing winding temperature and a decrease in KE, KT, with increasing magnet temperature due to thermal demagnetization of the motor magnets. I have had several students and clients fail to take both of these afects into account when attempting to correlate measurements with calculations.
This paper is a condensed version of a 6 page paper that I will e-mail to you. We are not sure why you reached the conclusions you did, but 1)the voltage input is not a step, it is a function (e), 2)the inertia is the motor plus the load, and 3) the input resistance from the amplifier wiring is included.
If anyone else wants the full paper e-mailed, just let us know at [email protected]. The paper is entitled "Electric Servo Motor Transfer Functions and Time Constants".
Tom and George.
Thank You very much for the paper. I plan on responding to you directly by E-mail as to how I reached the conclusions I did. However, after reading your paper I have a couple of comments that I think all of our viewers should be aware of.
Page 1, paragraph 3 of your paper states:
"The brushless dc motor is a three phase ac motor (usually a synchronous motor) having a position transducer inside the motor to transmit shaft position to the drive amplifier ...".
First of all, brushless dc motors are not limited to 3-phase. Motors with only 2-phases are commercially available and I have worked with them. Brushless dc motors containing more than three phases are also available and several papers have been written describing the the way in which they reduce harmonic distortation.
Second, I take exception with your comment "(usually a synchronous motor)". A brushless dc "is" a synchronous motor and will not operate correctly if its' permanent magnet rotor is not synchronous with the rotating magnetic field created by its' stator At least that's the way NEMA describes this type of motor in their definitions (see NEMA MG7 definitions).
Third, you don't need a position transducer inside a brushless dc motor for proper operation. "Sensorless" operation of brushless dc motors is currently being done in certain applications such as robots, blower, and radiator fans. Also, there are commercially available IC chips from companies such a Motorola, Allegro,International Rectifier, etc. that provide complete sensorless operation of brushless dc motors.
Finally, I have a question regarding the last sentence on page 6 of your paper. You state:
"In general, to raise the mechanical time constant to a 40°C rating, multiply the cold rating (25°C) by 1.8." How did you arrive at this 1.8 factor since I get a number of lesser value?? As you derive; tm = RJ/Ke*Kt. So let's assume the motor contains "Ferrite" or "Ceramic" magnets which would be worse case. The thermal coefficient of electrical resistance for copper wire is +0.0039/°C. Hence R(40°C)=1.0585xR(25°C).
Now, the thermal demagnetization coefficient for Ferrite magnets is -0.002/°C. Therefore, the reductions in both Ke and Kt amounts to:
Ke(40°C)= 0.97xKe(25°C). Combining the increase in resistance with the decrease in Ke, Kt; I find that the mechanical time constant at 40°C for a brushless dc motor containing Ferrite mangets amounts to:
tm(40°C)= 1.0585/0.97^2 tm(25°C)=1.125 x tm(25°C).
Since Ferrite has the highest coefficient of thermal demag for the three different magnet materials (NdFeB, SmCo, Ferrite) currently used in brushless dc motors, could you please explain how you came up with your 1.8 factor and for what magnet material it applies to ??