mmme2051 notes on dc motors

uni/mmme/2051_electromechanical_devices/dc_motors.md

@@ -0,0 +1,57 @@
+---
+author: Akbar Rahman
+date: \today
+title: MMME2051 // DC Motors
+tags: [ dc_motors, motors ]
+uuid: b8313ef8-ef1e-486d-8031-52c39ac88751
+lecture_slides: [ ./lecture_slides/MMME2051EMD_Lecture9.pdf ]
+lecture_notes: []
+exercise_sheets: [ ./seminar_worksheets/Exercise Sheet 12 - DC Motors.pdf ]
+---
+
+Relevant lecture slides are 15-26.
+Two worked examples can be found in the lecture slides 24-26.
+
+In a simple DC motor, the stator is either a permanent magnet or a coil with a
+current flowing through it.
+Both create a constant magnetic field.
+
+![](./images/dc_motor.png)
+
+The rotor is just a coil with current flowing through it by a (different) DC power supply:
+
+![](./images/vimscrot-2023-03-30T11:18:31,204273426+01:00.png)
+
+Note the commutator and brushes.
+This flips the current direction inside the coil, flipping the direction of the induced magnetic field,
+and therefore flipping the direction of the resultant force.
+
+The split in the commutator is placed such that the current flips just as the force would.
+This is [visualised in this video (timestamped to 50s)](https://youtu.be/LAtPHANEfQo?t=50).
+
+# Torque, $T$
+
+$$T = rBIl = KI$$
+
+using $F = BIl$ and $T = rF$, where $B$ is magnetic field strength, $I$ is current flowing through coil, $l$ is length
+of coil in the magnetic field, 
+$K$ is a constant which varies with motor designs, and $I$ is the current through the motor.
+
+# Back EMF, $E_b$
+
+$$ E_b = Bvl = K\omega $$
+
+where $B$ is magnetic field strength, $v$ is linear velocity, $l$ is length of coil in field,
+$K$ is a constant defined by the motor, and $\omega$ is angular velocity.
+
+# Relating Voltage to Motor Performance
+
+There are two equations which govern the motor's performance:
+
+$$V_\text{in} = E_b + IR$$
+
+\begin{equation} \label{eqn_torquespeed}
+V_\text{in} = K\omega + \frac{T}{K}R
+\end{equation}
+
+![Equation \ref{eqn_torquespeed} can be plotted to show how speed varies with load. Plotting multiple voltage also show how speed varies with coltages.](./images/vimscrot-2023-03-30T11:33:14,679355133+01:00.png)