DC Machines

DC motors and generators convert energy using a commutator. Learn back-EMF, the torque and speed equations, shunt/series/compound characteristics, starting current, speed control and why brushed DC still matters.

A DC machine is a rotating coil (the armature) inside a magnetic field. Push current through it and it turns (motor); turn it mechanically and it produces voltage (generator). A mechanical switch called the commutator reverses the armature current every half-turn so the torque always acts in the same direction.

DC machines taught engineers everything about electromechanical energy conversion. Brushed DC motors still run car wipers, toys, power tools and cranes, and their equations are the model that modern drive control imitates for AC motors.

An analogy

A DC motor is like pushing a merry-go-round: each time a handle comes round, you push it (the commutator times the push). The faster it spins, the harder it is to keep up — that 'pushing back' is the back-EMF.

Construction

Stator: field poles made by permanent magnets or field windings. Rotor: the armature, a laminated iron core carrying coils. Commutator + brushes: a segmented copper ring that connects the rotating coils to the stationary circuit and flips their current at the right moment.

Motor and generator are the same machine

A conductor carrying current in a magnetic field feels a force F = B·I·L (motor action). A conductor moving in a field has a voltage induced, e = B·L·v (generator action). Every DC motor also generates a back-EMF that opposes the supply; every generator also feels a braking torque.

Back-EMF controls the current

At standstill E_b = 0, so the only thing limiting current is the small armature resistance: I = V/R_a, often 10–20× rated current. As the motor speeds up E_b rises and the current falls. That is why large DC motors need starters (series resistance switched out in steps) or current-limited drives.

Shunt, series and compound machines

How the field winding is connected shapes the speed–torque characteristic.

• Shunt: field across the supply → nearly constant speed (drops a few % with load). Machine tools, fans.
• Series: field in series with armature → huge starting torque, speed rises dangerously at no load. Traction, cranes, starter motors. Never run unloaded.
• Compound: both windings → a blend of the two. Rolling mills, elevators.
• Permanent magnet: fixed φ, behaves like a shunt motor. Most small motors.

Speed control

From N ∝ (V − I_a R_a)/φ: armature-voltage control gives constant-torque operation from zero to base speed; field weakening (reducing φ) gives constant-power operation above base speed. Modern drives use a four-quadrant chopper (H-bridge) with PWM to vary V smoothly and even regenerate energy while braking.

Armature reaction and commutation

Armature current creates its own flux that distorts and weakens the main field, shifting the magnetic neutral plane and causing sparking at the brushes. Interpoles and compensating windings cancel it. Brush wear, sparking and commutator maintenance are why brushless machines replaced DC in most high-power uses.

Formulas

• EMF equation: E = P·φ·Z·N / (60·A)
• Motor voltage equation: V = E_b + I_a·R_a
• Generator voltage equation: V = E − I_a·R_a
• Torque: T = E_b·I_a / ω = k·φ·I_a, ω = 2πN/60
• Speed: N ∝ (V − I_a R_a) / φ

Worked example: Generated EMF

Given: 4-pole, lap-wound (A = P = 4); Z = 480 conductors; φ = 0.025 Wb; N = 1000 rpm

Find: Generated EMF.

• E = PφZN / (60A)
• = 4 × 0.025 × 480 × 1000 / (60 × 4)
• = 48 000 / 240 = 200 V

Answer: E = 200 V

Worked example: Motor torque and starting current

Given: V = 220 V; R_a = 0.5 Ω; I_a = 20 A at 1000 rpm

Find: Back-EMF, developed torque, and direct-on-line starting current.

• E_b = V − I_a R_a = 220 − 20 × 0.5 = 210 V
• P_dev = E_b I_a = 210 × 20 = 4200 W; ω = 2π × 1000/60 = 104.7 rad/s
• T = 4200 / 104.7 = 40.1 N·m
• At start E_b = 0 → I_start = 220 / 0.5 = 440 A (22× rated!). To limit it to 40 A: R_total = 220/40 = 5.5 Ω → add 5 Ω starting resistance.

Answer: E_b = 210 V, T ≈ 40 N·m, 440 A DOL start → 5 Ω starter

Common mistakes

• Wrong: Running a series motor with no mechanical load (belt removed). — Right: A series motor can overspeed destructively at no load — always keep it coupled to a load.
• Wrong: Opening the field circuit of a running shunt motor. — Right: Losing field flux makes the speed soar (N ∝ 1/φ). Field-loss relays trip the drive.

Applications

• Automotive: Starter motors are series motors for enormous starting torque; wipers, windows and seats use small permanent-magnet DC motors.
• Traction heritage: Trams and locomotives used series DC motors for a century; modern ones use AC motors controlled to imitate the DC characteristic.
• Robotics & toys: Brushed DC gearmotors are cheap, simple to drive with an H-bridge and easy to speed-control with PWM.

Key takeaways

• E = PφZN/(60A); motors obey V = E_b + I_a R_a.
• Back-EMF limits current; at start it's zero, so starters are needed.
• Torque ∝ φ·I_a; speed ∝ (V − I_a R_a)/φ.
• Shunt ≈ constant speed, series = high torque but dangerous at no load.

Going further

• Four-quadrant operation: Motoring forward, braking forward, motoring reverse and braking reverse map to the four quadrants of the V–I plane. An H-bridge chopper covers all four; during regenerative braking E_b > V and energy flows back to the supply.
• The DC machine as the control template: Torque ∝ φ·I_a with decoupled flux and torque makes DC easy to control. Field-oriented control of AC machines mathematically transforms the AC motor into this DC-like picture (d-axis flux, q-axis torque).
• Losses and efficiency: Losses are copper (I²R in armature and field), iron (hysteresis + eddy), brush contact drop (~2 V) and friction/windage. Maximum efficiency occurs when variable (copper) losses equal constant losses.