## Understanding Copper Losses in a Permanent Magnet Motor Design

Assuming a motor has been designed with a certain number of poles and slots. Further, let us assume that we wish to change the number of turns to see if the design can be optimized with respect to copper losses.

As you begin to change the number of turns, the following relationships hold true:

Torque α N*I                                      [1]

Resistance α N2                                   [2]

BackEMF α N                                     [3]

BaseSpeed α (Vbus/2 – I*R)/Ke        [4]

In the above equations, we have assumed Ldi/dt = 0 at peak values. Ke is the BackEMF constant in V/rad/s. Vbus is the bus voltage of the inverter. All the above values can be assumed to be peak.

Now, the question is as follows:

Q: Can one reduce copper losses in the motor at a specified torque level by changing the number of turns? One is not allowed to change the number of poles or slots or motor diameter or lamination stack height.

Now if one were to compute copper losses at a specified copper level, we arrive at the following :

Torque = Constant (as we wish to calculate at a specified torque level);

Hence, NI = Constant

Therefore, as N goes up, I goes down. As N goes up, R goes up. Let us look at the following equations:

N1I1 = N2I2                              [5]

N12R1 = N22R2                         [6]

Calculating, I22R2, substituting for I2 and R2 below

I22R2= (N1I1/ N2)2*( N2/ N1)2* R1 = I12R1       [7]

As can be seen in [7] above, the copper losses can be increased or decreased for a specified torque level.

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