Reference

A compact reference using the same notation as the course. Convert inputs to SI before substituting them into equations.

Quantities and units

Symbol	Quantity	Unit
t	Time	s
u, e	Voltage and back EMF	V
i	Armature current	A
R	Resistance	Ω
L	Inductance	H
ω	Angular speed	rad/s
n	Rotational speed	rpm
J	Moment of inertia	kg·m²
Me, ML	Electromagnetic and load torque	N·m
b	Viscous friction coefficient	N·m·s/rad
kt	Torque constant	N·m/A
ke	Back-EMF constant	V·s/rad
W	Energy	J
P	Power	W

Core equations

u = Ri + Li′ + keω
Jω′ = kti − bω − ML
ω = 2πn/60
W = Li²/2 + Jω²/2

For constant inputs, set derivatives to zero to find equilibrium. Across switching, the final state of one interval becomes the initial state of the next. An input discontinuity does not by itself imply a state discontinuity.

Glossary

State variable: a quantity whose initial value is needed to continue the solution; here current and speed.

Input: an external model signal such as voltage, load torque or reference.

Equilibrium: a state with zero derivatives at constant inputs; it may be stable or unstable.

Time constant: a characteristic exponential time scale. Coupled dynamics cannot always be reduced to L/R or J/b.

Model stability: behaviour of small deviations under stated conditions. Numerical stability is a separate property of the algorithm.

Stiffness: fast decaying modes constrain explicit step sizes even when the response of interest is slow.

Saturation: an actuator command limit; the base speed-control example uses ±12 V.

Back EMF: rotational EMF coupling mechanical and electrical subsystems, not a heat loss in itself.

Overshoot: an excursion beyond the final value; any percentage needs an explicit normalization base.

Common mistakes

Using rpm in a rad/s equation.
Resetting current or speed at an input step.
Replacing a dynamic current equation with a static formula without justification.
Treating the final sample of a short run as equilibrium.
Comparing different initial conditions as one experiment.
Increasing plot samples instead of checking solver tolerances.
Calling braking regenerative without checking ui.
Tuning for a physically unreachable speed.

Further sources

The course is an independent introductory treatment. These primary educational and software sources support further study:

Course plots use teaching parameters and must not be taken as characteristics of a real product.