From hardware to equations
A model keeps the mechanisms that matter for your question and leaves the rest out.
Write a balance law and use it to predict the direction and rate of change.
Feedback tells us what a controller does with error. A model tells us how the physical system responds to its command.
Review the previous lesson: Open loop and closed loop: why feedback?First, picture it
A motor does not turn voltage straight into speed. Voltage changes current. Current produces a turning force called torque. The torque speeds up the rotor while friction and the load oppose it. A model writes this chain of causes as equations.
Build the idea step by step
Start by choosing what you want to predict. To predict speed, write a torque balance: driving torque minus opposing torques equals inertia times acceleration. Inertia measures how hard it is to change rotational speed. The dot over speed means its rate of change. To predict current as well, add an electrical balance for voltage, resistance, and inductance. Keep only the effects needed for the question, and check the units on both sides.
Read the equations
Motor torque accelerates the rotor after friction and load torque are subtracted.
What the symbols mean
- rotor inertia, in kilogram metres squared
- shaft speed, in radians per second
- motor torque, in newton metres
- viscous friction coefficient, in newton metres per radian per second
- load torque, in newton metres
- angular acceleration: change in speed per second, rad/s²
This balance gives acceleration at an instant. Finding speed later requires adding up that acceleration over time. Real parameters must come from measurements or a stated example; a smooth simulation is not evidence that the model fits a real motor.
Find the initial acceleration
Use a rotor with inertia 0.02 kilogram metres squared. At this instant the motor makes 0.10 newton metres, while friction and load each use 0.02 newton metres.
The net torque is 0.06 newton metres.
Divide by inertia. Speed initially rises by 3 radians per second each second; this says nothing yet about the final speed.
More detail and model assumptions
A model is a question-specific simplification
The motor equations keep current and shaft speed because voltage changes current, current makes torque, and torque changes speed. They omit many real effects such as thermal drift and detailed commutation. That is reasonable for some learning questions but not evidence that every real motor behaves identically.
Before choosing a state, name the output you need and the input you can apply. The cart-pole needs more states than angle alone because cart velocity and position can signal danger while the pole is momentarily upright.
Trace units through the motor
In L di/dt=v−Ri−Kₑω, every term has units of volts. In J dω/dt=Kₜi−bω−τL, every term has units of torque. This quick units check catches many modeling mistakes before simulation. The current state cannot normally jump instantly because electrical inductance resists sudden current change.
A fitted model should predict a new test, not merely reproduce the trial used to choose its parameters. Later lessons change the actual motor while holding the nominal frequency model fixed; that is a controlled way to expose model mismatch.
Connect the example to the lab
A larger rotor inertia J means the same torque produces less acceleration. Compare how fast the motor can change speed, not just its final speed.
Inspect the lab modelMake a prediction before moving a control.
- PREDICTWhich term immediately opposes a new load torque?
- CHANGE ONE THINGInspect the two motor state equations and apply a short load pulse in the Lab.
- OBSERVETrack speed, error and applied voltage at one replay cursor.
- EXPLAINExplain which equation receives τL and how feedback later changes v.
A smooth animation does not prove that a model matches a particular real motor. Parameters and validation matter.
Check your understanding+
With the same net torque, what happens to initial acceleration if rotor inertia doubles?
It halves, because acceleration is net torque divided by inertia. This comparison holds at the same instant and load.Why the next lesson follows
A changing system needs enough current values to predict its next motion. We call those values its state.