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Kv (rpm/V) = max speed (rpm) / bus voltage (V)
Coil | Eco/PWM Graph | Kv (rpm/V of back-emf) |
|---|---|---|
HI (short coil - high speed) | PWM | 5.6412 |
LO (long coil - low speed) | PWM | 4.8702 |
HI (short coil - high speed) | ECO | 6.0153 |
LO (long coil - low speed) | ECO | 5.2061 |
I believe this to be reasonable accurate - the Kv estimation is a little wonky since we have not idea of the test setup or procedures that Nomura was using, but I believe this to be within 10% of the actual value.
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|---|---|---|
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BLDCs - From Class Notes in MTE322
Max speed we can achieve is when the back-emf is equal to the supply voltage. This condition means we can no longer have current flowing through the coils, and thus no torque is produced.
Variable | Symbol | Units | Value | Reasoning |
|---|---|---|---|---|
Supply Voltage | V | Volts | 90 to 150V | This is the voltage range of our 36S battery pack. |
Back-Emf (or the voltage produced by the coils rotating in a magnetic field) | E | Volts | ||
Motor Current | I | Amps | ||
Motor Resistance, LOW coil | R_low | Ohms | 0.997 | According to motor graphs given in datasheet. See spreadsheet above. |
Motor Resistance, HIGH coil | R_high | Ohms | 1.33 | According to motor graphs given in datasheet. See spreadsheet above. |
Motor Speed | n | rpm | 1rpm = 2*pi/60 rad/s | |
Motor Speed | w | rad/s | ||
Back-Emf Constant, LOW COIL | Kv_low_rpm | Volts / rpm | 1/4.87 = 0.20533 | According to motor graphs given in datasheet. See spreadsheet above. |
Back-Emf Constant, HIGH COIL | Kv_high_rpm | Volts / rpm | 1/5.64 = 0.177267 | According to motor graphs given in datasheet. See spreadsheet above. |
Back-Emf Constant, LOW COIL | Kv_low_rad | Volts / (rad/s) | 1.96075 | Calculated from Kv_low_rpm 1/4.87 (Volts/rpm) * (1rpm/(2*pi/60) rad/s) |
Back-Emf Constant, HIGH COIL | Kv_high_rad | Volts / (rad/s) | 1.69277 | Calculated from Kv_low_rpm 1/5.64 (Volts/rpm) * (1rpm/(2*pi/60) rad/s) |
Torque Constant, LOW COIL | Kt_low | Nm / Amp | 1.96075 | Same value as Kv_low_rad (https://micromouseonline.com/micromouse-book/motors/motor-equations/ ) |
Torque Constant, HIGH COIL | Kt_high | Nm / Amp | 1.69277 | Same value as Kv_high_rad (https://micromouseonline.com/micromouse-book/motors/motor-equations/ ) |
1: V = E+IR
2: E = Kv_rpm * n = Kv_rad * w
3: I = T/Kt
Subbing 2 and 3 into 1
4: V = Kv_rad * w + (T / Kt) * R
If we rearrange the equation, we can get a maximum torque as a function of speed.
5: T = (Kt / R) * (V - Kv_rad * w)
We can plot this in python for both the high and the low cables with the parameters from the table above. These are just simple linear equations, but it is easier to visualize in a graph. NOTE - the minimum battery voltage of 90V was used to generate these curves. Higher battery voltage pushes the curves upwards vertically.
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We also need to pay attention to the maximum battery current, as that will limit the maximum torque that we can produce. Higher battery voltage extends these curves along the same lines to achieve higher possible torque.
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Now, we need to have a discussion about the maximum motor phase currents.
There is a start of an investigation into that here: Determining Motor Current Limits