Capstone: Motor Speed Control

Plug-in power tools hide a genuinely odd machine called the universal motor, and the name is honest: it runs on either AC or DC. It is a series DC motor, meaning the field winding and the armature carry the same current in series. Here is why that lets it shrug off AC. When the AC mains reverses, the current in the field and the current in the armature both flip at the same instant, and torque is the product of the two. A negative times a negative is still positive, so the shaft keeps pushing the same way no matter which half of the cycle the wall is on.

That one quirk has a wild consequence. An ordinary AC induction motor is chained to the line frequency: its top speed is set by the 50 or 60 hertz coming out of the wall. A universal motor is not, so it can spin far faster than that synchronous limit. High speed means a lot of power from very little metal, which is exactly why a drill, a router, a circular saw, and a vacuum cleaner all use one. They are loud and they wear their brushes, but they are light and they are cheap. The downside is the series wiring: a series motor must never run with no load. Unloaded, the current drops, the back-EMF goes weak, and the motor speeds up trying to balance the supply voltage, faster and faster, until it can tear itself apart. Bench grinders and drills lean on the friction of the work to keep that in check. Commercial universal motors are usually small, under about a kilowatt, but engineers once built huge ones to pull electric locomotives, fed by special low-frequency power lines to keep the commutator from arcing under the load.

A wound DC motor has two sets of windings, the field on the stationary part and the armature on the spinning part, and how you connect them sets the motor's whole character. There are three classic choices. Wire the field and armature in series and you get the muscle motor above: huge starting torque, speed that sags hard with load. Wire them in parallel, the shunt connection, and the field sees the full supply no matter what the armature is doing, so the speed barely budges as the load changes. That steady speed is why shunt motors ran machine tools and winding machines for a century. Splice the two together and you get a compound motor, which borrows the strong start of a series winding and the steady speed of a shunt one.

Here is the part most people miss. Armature voltage is not the only speed knob. Speed is set by the ratio of voltage to field strength, so you can also speed a motor up by weakening the field, dialing down the current in the field winding. It feels backwards, fewer amps in the magnet making the thing spin faster, but it falls right out of the back-EMF balance: a weaker field needs more RPM to generate the same opposing voltage. Big industrial drives, like the separately excited motors that once ran mine hoists on thyristor controllers, used voltage control down low and field weakening up high to cover a huge speed range. Those have mostly been swapped for AC motors on variable-frequency drives now, but the trick lives on in how EV traction motors stretch a single gear ratio across the whole speed band.

A DC motor's speed $\omega$ obeys a clean first-order torque balance. The driving torque from the input $u$ fights viscous friction $b\,\omega$ and any external load torque $\tau_\text{load}$, all divided by the rotor inertia $J$:

With no load and constant input this settles to a steady speed, and the input-to-speed relationship is the canonical first-order lag, with gain $K$ and time constant $\tau$:

We close the loop on the error $e = r - \omega$ (commanded speed minus actual) with a PI controller, proportional plus integral. The derivative term is dropped because speed measurements are noisy and a load step needs erasing, not anticipating:

Now the punchline. A constant load torque is a step disturbance. Plug it into the closed loop and ask for the steady-state error using the final-value theorem. With $K_i = 0$ the error settles to a nonzero value proportional to the load, the residual you saw in the demo. With any $K_i > 0$, that integrator contributes a pole at the origin (an $s$ in the denominator of the open loop), which forces the steady-state error from a step disturbance to exactly zero. That is the entire mathematical reason the integrator recovers the speed and the proportional term cannot. The demo integrates the plant and controller together by forward-Euler at $dt = 0.01\,\text{s}$, with the load torque switching on at $t = 5\,\text{s}$.