## Fundamentals

This section reviews some fundamental concepts that are central to the operation of electrical machines. The field of electrical machines requires some knowledge of theory from a range of topics. These include basic circuits, rotational mechanics, electromagnetism, and three phase systems.

We often think of electric motors or generators. As you become more familiar with electric machines, you will see that machines can operate as either motors or generators, depending on the direction of the torque that they develop. If an electric machine prodices a torque that supports the mechanical rotation, it is a motor. If an electric machine opposes the mechanical rotation, it is acting as a generator.

### Rotational Mechanics

In linear mechanics, the relationship between force and acceleration is given by Newton's familiar equation:

$\Sigma F = m a$

under constant velocity, there is no acceleration and the sum of the forces is zero. In rotational dynamics, the equivlent equation is given by:

$\Sigma \tau = J \dot{\omega_m}$

where $$\tau$$ is torque, $$J$$ is mass moment of inertia $$\dot{\omega_m}$$ is angular acceleration and $$\omega_m$$ is mechanical angular velocity.

This indicates that if a system is rotating at constant speed, the sum of torques will be zero. Following convention, we define positive rotation as being in the counter clockwise direction, therefore positive torque is counter-clockwise and negative torque is clockwise. Considering the case with an electric motor, we often say e.g. the motor drives a load of 10Nm. At steady state this means that there is a 10Nm mechanical torque that is opposing the rotation (i.e. clockwise, or negative) and there must be an equal and opposite positve (counter clockwise) torque produced by the motor. Similarly in a generator, an external mechanical system is driving the rotation, therefore the mechanical torque from the driving system will be in the direction of motion (positive, counter-clockwise) and in steady state the electrical machine will produce an equal and opposite torque (negative, clockwise).

Now consider the case when a machine is not operating in steady state. If a motor creates a torque with a magnitude that is greater than the load

$|\tau_{motor}|-|\tau_{load}|=J \dot{\omega}$

indicates that the motor will accelerate. If the magnitude of the load torque is greater than the torque thath motor is capable of providing, the motor will deccelerate.

In many cases, as electrical engineers, we carry out the math for a problem without considering the sign or direction of mechanical torques, but simply think about the operation:

• If we are dealing with a motor, the electric machine produces torque that supports the motion
• If we are dealing with a generator, the electric machine produces torque that opposes the motion

### Mechanical Power

Under constant speed operation:

$P_{mech} = \tau \omega_m$

### Mechanical Speeds

The SI unit of speed is angular velocity in radians per second. However we typically think about speeds in revoutions per minute. $$omega_m$$ is mechanical speed in rad/s and $$n_m$$ is mechanical speed in rpm.

\begin{aligned} \omega_m & = \frac{2\pi}{60} n_m \\ n_m & = \frac{60}{2\pi} \omega_m \end{aligned}

### Summary

• The idea of equal and opposite torques in steady state
• Motors support rotation
• Generators oppose rotation