Calculate PM brushless 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 datasheet is given in table 3. The user has given the next workpoint:
The Motor Datasheet
A brushless motor is a 3-phase motor. The given resistance R2ph andself-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.
Rth1 and Rth2 are provided in table 2 which indicates that the elaborate thermal model is applied. This model is shown in figure 1. In formula 2 and 3 the formulas are derived for resp. the winding and the motor housing temperature.
Figure 1 Detailed thermal model for PM brushless motor
When there is a thermal balance (the temperature does not change anymore) the temperatures can be determined as described in formula 1 and 2. The formules are valid for a detailed thermal model as given in figure 1.
For a PM brushless motor the following formulas are valid for the resistance R2ph and motor constant k.
The transmission Datasheet
The transmission datasheet is given in 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 windings. This value is given in the dissipation table (table 5) and calculated later on.
Calculation voltage
Because the given workpoint does not have an acceleration the voltage U is calculated as follows:
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=125 oC=398 K (see table 2).
Ilimit in general is smaller than the maximum current because usually no thermal effects are taken into account when determining the maximum current. The choice for the workpoint around the maxium allowed temperature is not a smart thing to do, so Ilimit is a limit that must be avoided. The ratio between the calculated current I and Ilimit is a unit for the safety margin applied.
Ilimit = 5.269 A (table 4) and filling in this value in the formula for dissipatoin and formula 4 should yield the given maximum temperature Θmax= 398 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 windings and the motor housing is given and the resistance R2ph 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 3 and 4 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=17.28 W. This also proves to be correct:
Calculation of the temperatures
The temperatures are not validated yet, but now that all quantities are known this is just a small step. Formula 1 and 2 filled in yield:
These values are 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)=(1767.2 rad/s; 0.022 Nm) as given in table 4.
Figure 3 Load on motorside
The torque-speed characteristic indicates the working area, which is enclosed by a line indicating the maximum torque, a line indicating maximum voltage Umax and a line for maximum speed. The workpoint lies within this area.
The maximum torque in figure 3 is determined by the torque at standstill (stall) Tstall= 0.184 Nm, the maximum speed is ωmax=5236 rad/s. 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 which the motor temperature reaches its maximum temperature of 398K or 125 oC (from table 2).
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; 0.2 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.