MME 3374a Study Notes on DC Motors

Course Title: MME 3374a - Introduction to Electrical Engineering for Mechanical Engineers
Focus: DC Motors (direct current motors)
Term: Fall 2025

Objective: To understand how DC brush-type motors work, both in theory and in practice.
We will cover two main ideas:
- How voltage (EMF) is created.
- How turning force (torque) is created.

Electrical Circuit: The sum of all voltages in a closed loop is zero.
$\Sigma V = 0$
Mechanical Circuit: The sum of all forces acting on an object is zero (if it's not accelerating).
$\Sigma F = 0$
Energy Conversion: These motors change electrical energy into useful mechanical energy (movement).
Magnetic Circuit: The sum of all magnetic forces in a closed loop is zero.
$\Sigma MMF = 0$

How Voltage is Created: Imagine a wire moving through a magnetic field. This movement creates an electrical voltage (EMF).
Equation for created voltage:
$e_{gen} = vB l$
Where:
- $e_{gen}$: The generated voltage (measured in Volts, V)
- $v$: How fast the wire is moving (measured in meters per second, m/s)
- $B$: The strength of the magnetic field in the air gap (measured in Teslas, T)
- $l$: The active length of the wire that is inside the magnetic field (measured in meters, m).
If a wire moves through a gap filled with a magnetic field at a speed $v$ , an electric voltage is created.
Conditions for Created Voltage: This process follows Faraday’s Law of Induction. Simply put, if a conductor (like a wire) moves through a changing magnetic field, it creates an electric current.
Direction of Created Voltage: We use Fleming’s Right Hand Rule to figure out the direction:
- Your Thumb: Points in the direction the wire is moving ( $v$ )
- Your Index Finger: Points in the direction of the magnetic field ( $B$ )
- Your Middle Finger: Points in the direction of the created voltage ( $e$ ).

Lorentz Force: This is the pushing force on a wire when electric current flows through it while it's in a magnetic field:
- $F = i (l \times B)$
- Where:
- $F$: The Lorentz force (measured in Newtons, N)
- $i$: The amount of current flowing through the wire (measured in Amperes, A)
- $l$: The length of the wire inside the magnetic field (measured in meters, m)
- $B$: The strength of the magnetic field in the air gap (measured in Teslas, T).
Direction of Force: We use Fleming’s Left Hand Rule to find the direction of this force:
- Your Thumb: Points in the direction of the force ( $F$ )
- Your Index Finger: Points in the direction of the magnetic field ( $B$ )
- Your Middle Finger: Points in the direction of the electric current ( $I$ ).

If a wire loop spins in a magnetic field, the voltage created is given by:
$e_{gen} = (v \times B) \times l$
As the loop rotates, the voltage it generates changes in a wave-like (sinusoidal) pattern.

For a current $I$ :
- The turning force (torque) on a wire loop carrying current ( $i$ ) is:
  $\tau = 2r i B l \sin(\theta) \text{ (Nm)}$
- Where $\tau$ is the torque (turning force), $B$ is the magnetic field strength, and $\theta$ is the angle.
We'll learn tips on how to use these equations in real-world situations.

What a Commutator Does: This is a special part of the motor that flips the direction of the current in the coils. This makes sure the motor keeps spinning in one direction constantly, instead of just wiggling back and forth.
It keeps a steady opposing voltage (back EMF) against the battery (power source).
It helps arrange the coils so the motor always produces a turning force in the same direction.
How it Works: We'll look at models showing how wire loops move and how forces are created at different points during their spin.

We can draw a simple electrical diagram for permanent magnet DC motors. It includes the motor, a battery, the generated voltage (EMF), and the motor's internal resistance.
The equation for the generated voltage ( $V_{gen}$ ) is:
$V_{gen} = K_e \omega$
Where $K_e$ is an electrical constant (a fixed number for that motor) and $\omega$ is the motor's rotational speed.

Generated Torque & Mechanical Power: The motor's turning force (torque) is linked to the current by a constant ( $K_T$ ):
$\tau_{Motor} = K_T I_{Motor}$
The power generated by the motor is calculated as: $\P_{Gen} = \tau_{Gen} \omega_{Motor}$
We'll also talk about what makes motors lose energy, such as electrical losses (like heat from resistance) and mechanical losses (like friction).

How we figure out energy losses in motors:
- Iron Losses: These are losses that happen in the motor's core, like wasted energy from changing magnetic fields (hysteresis) and small circulating currents (eddy currents).
- Mechanical Losses: These include friction in the bearings and other rubbing parts that slow down the motor.
We'll see how these losses affect how efficient the motor is.

Design Example: How to choose the right motor for things like moving a 10 kg cart up a hill at a certain speed.
We will calculate the necessary turning force (torque), rotational speed, and gear sizes to pick the best motor.
General Specifications: This includes details like the motor's size, shaft diameter, required input voltage, how fast it spins with no load, how much torque it produces when stalled, etc.

Torque-Speed Relationships: We'll look at graphs showing how the motor's turning force (torque) changes with its speed, both when it's just running freely and when it's under a heavy load.
We'll also learn about PWM (Pulse Width Modulation) control, which is a way to change the motor's speed.

We'll work through problems to find things like the motor's developed torque, power, and the voltage applied to it for a given DC motor with specific values.

Final thoughts on how to use this information in real life, and suggestions for more reading to understand DC motors better, including references and practice problems.

A list of books and online materials recommended for learning more about electrical engineering principles that apply to mechanical systems.