DC Motor Drives

The back EMF at rated conditions:

Ninety-four per cent of the terminal voltage is back EMF. The armature drop is a small correction, which is exactly why the speed–torque curve is nearly flat.

The machine constant:

One constant, and it does double duty: multiply by speed for back EMF, multiply by current for torque. That is a consequence of energy conservation — the units V·s/rad and N·m/A are the same thing.

Torque and power:

The 625 W of copper loss is the whole armature-circuit story. Field excitation, friction, windage and iron loss are extra — a real machine of this rating runs at about 88%.

Speed at half torque, same terminal voltage:

Halving the load raises the speed by 3% — and it rises because the armature drop halves, not because anything was commanded. The machine self-regulates.

Speed regulation:

Six per cent is entirely due to \(I_aR_a\), so it can be eliminated by a feedback loop that raises \(V_a\) as the current rises — or even by open-loop \(IR\) compensation, adding \(I_aR_a\) to the voltage command. That simple trick is why DC drives held zero-speed-error positions long before encoders were cheap.

The two control handles:

Two independent inputs giving two independent regions — and the second is available without any extra power hardware, because the field circuit carries only a few per cent of the machine's current. Set 34 will find that an induction machine achieves the same split with far more effort.

The duty ratio is set directly by the required armature voltage:

The chopper of Set 17 has become a speed controller with no change of any kind — the same volt-second balance, the same duty ratio, now interpreted as a speed command.

The maximum speed at rated torque:

Nearly twice base speed, but the machine's commutator and its 220 V insulation were not designed for it. In practice the armature voltage is capped at its rating and the extra speed comes from field weakening — the second control handle from Problem 1.

The current ripple. The chopper output is a square wave of \(V_{dc}\) against a smooth back EMF, so the inductance sees the difference:

Thirty per cent of rated current as ripple, at 1 kHz. Note the \(D(1-D)\) shape, which peaks at \(D = 0.5\) — so mid-speed is the worst case, and this operating point at 0.55 is close to it.

Raising the frequency:

Why the ripple matters more than it looks. It costs in four separate ways:

The third row is specific to DC machines and is what limits their life. A commutator segment carries the ripple as well as the mean, and the resulting sparking erodes both brush and copper — which is why DC drives specify a maximum ripple, typically 5–10%, rather than merely tolerating it.

If the frequency cannot be raised — for a large drive where switching loss binds — the alternative is added inductance:

So 13.3 mH of smoothing inductor added to the armature's 6.5. That inductor is a real cost and it slows the current loop — the time constant rises from 26 ms to 79 — so raising the frequency is preferable whenever the switching loss allows it.

What braking requires. To reverse the current while the speed and back EMF are unchanged, the applied voltage must fall below the back EMF:

A change of duty ratio from 0.55 to 0.4875 — six per cent — takes the machine from full motoring to full braking. The transition is that small because the armature drop is that small, which is also why the current loop must be fast and well controlled.

The power returned:

The same torque as motoring, in the opposite sense. That symmetry is not automatic in other machines and is one reason DC drives dominated hoisting and traction for so long.

The deceleration:

The energy actually recovered. Start with the stored kinetic energy and subtract what is lost:

Eighty-eight per cent recovered, and the loss is fixed at 625 W for the whole braking period because the current is held constant. Braking harder recovers more, because the same loss runs for less time.

The four quadrants, and what each demands of the converter:

The H-bridge of Set 17 covers all four with no additions — which is exactly what it was built for, and why that set's four-quadrant chopper was worth working through before any machine appeared.

Where the recovered energy goes, and the complication nobody mentions:

The first row catches most people. A drive fed from an ordinary diode bridge cannot send power back — the 9.75 kW has nowhere to go, so it charges the link capacitor until the drive trips on overvoltage. A braking resistor with its own chopper dumps it as heat, which is standard on general purpose drives and wastes every joule of the 5426.

The firing angle:

A large firing angle, and Set 14 established what that implies for the supply: a displacement factor of only 0.393, and a substantial reactive burden. Chopper-fed drives avoid this entirely — their input rectifier runs uncontrolled.

The ripple voltage, and its dependence on \(\alpha\). The dominant component is the sixth harmonic of the supply at 300 Hz:

A factor of 5.5. This is the fact that makes controlled-rectifier drives awkward: the ripple is smallest where the output is largest, and grows dramatically as the drive is turned down — so the worst ripple occurs at low speed, where the machine is least able to filter it and where the mean current is smallest.

The inductance required. Conduction stays continuous while the mean current exceeds half the peak-to-peak ripple:

Three times the armature's own inductance, added purely to keep the current continuous at light load. That inductor is physically large — it carries 50 A — and it is a real fraction of the drive's cost.

What happens without it. When the current becomes discontinuous the whole analysis changes:

The same nonlinearity Set 13 found in a semiconverter and Set 20 in a buck-boost — a converter in DCM is a different plant, with a different gain and a different pole. A speed loop tuned in CCM becomes underdamped or unstable when the drive drops into DCM, which happens exactly when the load is removed.

Chopper against controlled rectifier:

The fourth row is decisive for performance. A phase-controlled converter can only correct its output once per 60° of the supply — six times per cycle — so its current loop bandwidth is capped near 30 Hz however good the controller. A 5 kHz chopper reaches 500 Hz, which is why servo-quality DC drives are always chopper-fed.

Why cascade rather than one loop. The armature current is what must be limited, and it is also what produces torque:

Putting the current in an inner loop makes limiting it trivial — simply clamp the reference — and it linearises the plant seen by the speed loop, which then controls torque directly rather than voltage. Every modern drive of every type uses this structure.

The current-loop plant:

A single pole at 6.1 Hz. The back EMF also feeds in, but it varies at mechanical speed — hundreds of times slower than the current loop — so it is treated as a slowly varying disturbance that the integrator rejects.

Pole cancellation. Place the PI's zero on the plant's pole:

Cancelling the pole leaves a pure integrator, whose phase margin is 90° before other delays are counted — the PWM transport delay and the sampling take perhaps 25° of that, leaving a comfortable 65°. Set 23 built the same argument for a voltage regulator.

The speed-loop plant, with the current loop treated as unity:

Read that gain again. It demands 119 A for every rad/s of error, against a machine rated at 50:

So any speed change larger than four revolutions per minute saturates the current reference immediately. The linear design describes only the last few rpm of any transient; everything before that is the drive accelerating at its current limit.

Which is exactly the intended behaviour:

Time from standstill to 1500 rpm is therefore 1.19 s, set entirely by the current limit and the inertia — not by any controller gain. The speed loop's job is to place the limit and then land accurately, not to shape the acceleration.

The one thing this structure must handle:

With the current reference clamped for over a second, the speed integrator accumulates error the whole time and then must unwind, producing a large overshoot. Anti-windup — freezing or back-calculating the integrator while the output is limited — is not an optional refinement in a cascade drive; it is required for the loop to behave at all.

The mechanical requirement:

Referred to the motor:

A 15 kW machine at about 1150 rpm — and note that this is the worst-case rating. A properly counterbalanced elevator, with the counterweight at car plus half payload, spends most of its life at a fraction of it.

Now the quadrants, which is where an elevator becomes interesting:

All four quadrants, in ordinary passenger service — not as an occasional braking event but as half of every journey. An elevator is the textbook four-quadrant load, and it is why hoisting applications drove four-quadrant converter development.

Which fixes the converter:

The last row is a safety requirement rather than a control one. The mechanical brake must not release until the drive is already producing holding torque, or the car drops before the loop catches it — the characteristic lurch of a badly commissioned lift.

Why the front end matters so much here. Unlike a fan or a conveyor, an elevator regenerates constantly:

The last row is worth noting. In a building with several lifts on a shared DC bus, a descending full car directly supplies an ascending one — the energy never touches the supply at all, and no active front end is needed to capture most of it.

Why a modern lift is not DC. Everything above works, and almost nobody builds it:

A DC machine needs brush inspection and commutator servicing, in a machine room at the top of a building, for the life of the installation. A gearless permanent-magnet synchronous machine under vector control delivers the same four-quadrant torque with no brushes, no gearbox and higher efficiency — which is Set 36. The DC drive taught the control structure; the structure outlived the machine.