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Stepper Motor Inductance / Torque

Mariss Freimanis says:

If you put two 1mH chokes (inductors) in series, the inductance would be 2 mH. Series inductance in a step motor is 4 times higher because the two inductors share a common magnetic flux path; the current in one inductor affects the current in the other one. This coupling effect is called mutual inductance and it doubles the inductance. Two inductors in series have double the inductance, mutual inductance between them doubles the inductance again for a 4-fold increase.

Batteries have a voltage polarity and inductors have a magnetic polarity. Magnetic polarity is indicated by a round dot at one end of the coil just like the "+" sign indicates battery polarity. Two separate inductors in series have no polarity effect provided their magnetic fields don't interact (no mutual inductance). You can reverse the wires on one of the mentioned 1mH inductors and still have 2mH in series.

The coils (inductors) in a step motor have mutual inductance; wiring a motor in series with the dotted ends connected is the same as wiring two batteries in series with "+" of one going to "+" of the other one. You get zero volts from the battery connection and you get zero inductance if the windings are connected that way.

Major Rule 1: Inductance increases as a square of the number of turns of wire. Compare two windings of 10-turns of wire and 30-turns of wire. The 30-turn inductor will have 9 times the inductance of the 10-turn coil (10 squared is 100, 30 squared is 900).

Major Rule 2: The relationship between motor power output, power supply voltage (V) and motor inductance (L) is:

Power = Volts / Square Root of L

This can be understood intuitively. Compare a 1A motor and a 2A motor of the same physical size, manufacturer and holding torque.

Holding torque equals Ampere-turns. This means multiply current times the number of turns of wire the current passes through. The 1A motor has twice the number of turns of wire. Half as much current passes through the winding. It is the same as 2A passing through half as many turns of wire. Call them equal for holding torque.

The 1A motor has 4 times the inductance of the 2A motor. Let's see how they compare when they are turning at higher speeds. Inductance has a property called inductive reactance just like a resistor has a property called resistance. Both properties are measured in Ohms. Unlike a resistor, inductive reactance is proportional to frequency AND inductance; double the speed of a motor and its inductive reactance in Ohms doubles as well.

Inductive reactance obeys Ohm's Law, (Amps equals Volts divided by Ohms). The 1A motor has 4 times the inductance of the 2A motor so at a given speed, only 1/4 the current flows through its winding compared to the 2A motor. However, the current passes through twice as many turns of wire so it's torque (Ampere-turns) is 1/2 of the 2A motor. This doesn't look so good.

Let's double the power supply voltage to make it look better. Now the current (I=V/R) is half that of the 2A motor but it still is passing through twice as many turns. The Ampere-turns is equal to the 2A motor now. The 1A motor delivers exactly the same torque at any speed as the 2A motor does. Now they are equal except the 1A motor needs twice as much voltage to achieve equality with the 2A motor.

Let's stand back and look at this. The 1A motor needs twice the voltage at half the current to be identical to the 2A motor. Twice the voltage at half the current is the same Watts (voltage times current). Both motors use the same power to deliver the same power. What is the difference?

The difference is impedance. The 1A motor is a high impedance compared to the 2A motor. The 1A motor at 48V is identical to the 2A motor at 24V. Both deliver identical power and consume identical Watts from their respective 48V and 24V supplies.

In practical terms. A NEMA-23 (58mm) motor can deliver about 125W mechanical (1/6 HP) to a load just short of overheating. It can be a 7A 1mH rated motor that can take a maximum supply voltage of 32VDC or a 3.5A 4mH motor that can take a maximum of 64VDC. The results will be identical.

Where does the 32V or 64V come from?

Max supply voltage is 32 times the square-root of inductance in mH. It's an equation we derived for NEMA-23 motors without a heatsink where the motor temperature rise will be 85C above ambient. That's a nasty-hot motor.


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