Calculate AC PM synchronous motor
The calculation method will be explained by the hand of an example case. The user input is given and a suitable motor plus transmission is randomly picked. For this motor configuration the results will be calculated as is done by specAmotor. The motor datasheet is given in table 2. The transmission parameters are given in table 3. The user has given the next workpoint:
The Motor Datasheet
An AC synchronous motor is a 3-phase motor. The given resistance R2ph and self-inductance L2ph are those as measured between two phases. Cv is a velocity dependant friction and is zero and will therefor not return in the calculations. Rth,total is given in table 2 what indicated that the simple thermal model is applied. The thermal capacity Cth also is part of the simple thermal model, but this information is not provided by the manufacturer. Therefor it is not possible to determine after which time the end temperature will be reached. This incompleteness is noticed in the results under tab 8 Quality Index.
In figure 1 the thermal model is shown, and from this figure the formula is determined for the motor temperature (formula 1).
Figure 1 Simple thermal model of a motor
When there is thermal equilibrium the motor temperature can be determined as described in formula 1. This formula applies for a simple thermal model as gegeven in figure 1.
For an AC PM synchronous motor the next formulas apply for the resistance R2ph and motor constant k.
Formula 3 Motor constant k for AC PM synchronous motor
The transmission parameters
The transmission parameters are provided table 3.
Workpoint table
In table 4 the Workpoint tabel is shown as it will be provided in the search results. The quantities that are given in table 4 will be calculated hereunder. The user gave the workpoint as given in tabel 1, notice that these values return in table 4.
Calculation motor load Tmotor and current I
A transmission is being used (see table 3) so the workpoint must be translated in a load on the motor axis.
The calculation is as follows:
The parameters i and η (the efficiency) are parameters from the transmission (see table 3). kend is the motor constant at the end temperature of the motor housing. This value is given in the dissipation table (table 5) and calculated later on.
Calculation voltage and P.F.
The AC synchronous motor is a 3-phase motor feb by alternating voltage. The amplitude of the alternating voltage between two phases U2ph can be determined. The voltage is a factor √3 larger than the voltage over one single phase. The current is in phase with the electro-motoric voltage, this voltage is indidacte as Uq. Due to self-inductance of the stator windings the applied voltage will lag behind the current. There is a voltage component Ud to be pointed out that is at a right angle with Uq and is causing the lagging behind. A quantity for the lagging is the power factor or P.F. By definition it is the cosinus of the angle between the phase-voltage and the current. The following can now be declared:
Because no acceleration is occuring IRMS does not change in time and the quantities become:
The values R2ph,end and kend are given in table 5 and will be calculated later on.
Calculation maximum current
Ilimit is the maximum current for the motor for a given speed ω and duty cycle DC. At this current the maximum allowed winding temperature is reached, for the given motor this maximum temperature is Θmax=155 oC=428 K (see table 2).
Ilimit = 1.128 A (table 4) and filling in this value in the formula for dissipatoin and formula 1 should yield the given maximum temperature Θmax= 428 K again. See this calculation:
Pfric is given in table 5 (shown as Pmech, motor). The calculated maximum temperature is equal to the given value so this proves that Ilimit in table 3 is correct.
Dissipation table
The dissipation table is given in table 5.
All sources that contribute in the dissipation are shown. Also the temperature of the motor is given and the resistance R and motor constant k at the end temperature (the temperature at thermal equilibrium).
Calculation of R2ph and k
The resistance measured between two phases R2ph and the motor constant k can be determined according to formula 2 and 3 when the temperature is known. This is calculated hereunder, the values match those of table 5.
Calculation of the sources of dissipation and input and output power
The sources of dissipation are shown in figure 2.
Figure 2 Power distribution in an electric motor
The required values are calculated hereunder, the values for U and I are provided by table 4:
These values matche those of table 5. As a check the sum of the sources plus the output power can be determined, that should be equal to the input power of Pin=98.2 W. This also proves to be correct:
Calculation of the motor temperature
The motor temperature is not validated yet, but now that all quantities are known this is just a small step. Formula 1 filled in yields:
This value is the same as in table 5 so from this can be concluded that all values are correct.
Load on motorside
In figure 3 a torque-speed characteristic is given from the motor as described in table 2. Also the workpoint is given as experienced by the motor, the reduced workpoint (ω;T)=(502.8 rad/s; 0.540 Nm) as given in table 4.
Figure 3 Load on motorside
The torque-speed characteristic indicated the working area which is enclosed by a line indicating the maximum torque and a line indicating nominal voltage. The workpoint is right to the slanting line that indicated the nominal voltage. That means that the voltage of the workpoint is larger than the nominal voltage. The voltage is 341 V (U2ph in table 4) and the nominal voltage is 322 V (Unom in table 2). This is allowed for a synchrnous motor (in fact the voltage for every workpoint is limited by Umax, an optional motor parameter that is not provided by the manufacturer in this case).
The maximum torque in figure 3 is determined by Imax= 2.52 A (or 1.79 Nm given the k at surround temperature). It be noticed that figure 3 is determined for a motor temperature which is equal to the surround temperature.
The black line in figure 3 is the thermal limit and indicates the maximum continously allowed current at the maximum motor temperature (in this case 155 oC, see table 2). This current will decline at larger speeds because the dissipated power caused by friction increases with speed, thus will be causing the motor to heat up more.
Load on loadside
In figure 4 the torque-speed characteristic is given of the motor as experienced at the output side of the transmission. Also the workpoint is shown as experienced by the user in table 1 (ω;T)=(125.7 rad/s; 2 Nm).
Using a transmission on a motor will alter the working area of the motor. The maximum allowed torque will alter reversely proportional to i (see table 3) and thus become larger. The speed will alter proportional to i and thus become smaller. The steepness S (see table 2) is the line that indicates the orientation of the voltage and is quadratically proportional to i and will thus become much steeper.
Figure 4 Load on loadside
In figure 4 two lines are shown (light red and dark red) that differ somewhat in position. The difference is explained by the friction of the motor and the transmission. The light red line gives the 'netto' woring area on the loadside; this means that the friction of the motor and th transmission are subtracted from the theoretically available torque. Friction on the motor axis or in the transmission after all will decrease the netto torque available for driving a load.