Electronics Guide

Construction

Stator: Frame, Poles, and Field Windings

The frame, or yoke, is a steel shell that carries flux from one pole to the next and supports the machine. Bolted inside it are the main poles, an even number, alternately north and south, each ending in a pole shoe that spreads the flux over the air gap. A shunt field coil has many turns of fine wire and carries a small current; a series field coil has a few turns of heavy conductor and carries the full armature current.

Larger machines add two windings that carry armature current. Interpoles, or commutating poles, are narrow poles midway between the main poles. Compensating windings lie in slots in the faces of the main poles. Both control armature reaction and commutation, described below.

Armature and Its Windings

The armature is a laminated steel cylinder with slots that hold the armature coils. Each part of its core passes alternately under north and south poles and so carries alternating flux, which is why the core is laminated. The coils are connected in a closed loop, and each junction is tapped to one commutator segment, so the number of segments equals the number of coils.

The coil connections set the number of parallel paths, a, through the winding between brushes of opposite polarity:

• Lap winding. Each coil connects to the next coil under the same pair of poles. A simplex lap winding has as many paths as poles, a = P, and suits low-voltage, high-current machines.
• Wave winding. Each coil connects to a coil under the next pair of poles, so the winding travels around the whole armature before it closes. A simplex wave winding has two paths, a = 2, whatever the number of poles, and suits higher-voltage, lower-current machines.

Commutator and Brushes

The commutator is a cylinder of copper segments insulated from one another by mica; on most industrial machines the mica is undercut slightly below the copper so that the brushes ride on copper alone. Brushes of carbon, graphite, or metal-graphite are pressed against it by springs in brush holders fixed to the frame.

Each armature conductor must carry current in one direction under a north pole and in the other under a south pole, yet the terminal current is steady. In a generator the commutator rectifies the alternating coil voltages; in a motor it inverts the direct supply current into coil currents synchronized with rotation. Either way it holds the pattern of armature current fixed in space relative to the poles. The brushes are placed so that the coils they short-circuit during commutation lie in the neutral zone midway between main poles, where those coils cut almost no flux and generate almost no voltage.

Generated Voltage and Torque

A conductor of length l carrying current I across flux density B feels the force BIl developed in Magnetostatics, and the same conductor moving at speed v across the field generates the motional voltage Blv.

The Voltage Equation

Let the machine have P poles, a flux Φ per pole, an armature of radius r and active length l, Z active conductors, and a parallel paths, and let it turn at ω radians per second. The flux of one pole spreads over one pole pitch, an area 2πrl/P, so the average flux density is

B_avg = PΦ / (2πrl)

A conductor moving at surface speed ωr generates on average B_avglωr = PΦω/(2π). Each path holds Z/a conductors in series, and the armature voltage is the voltage of one path:

E_a = [PZ/(2πa)] Φω = KΦω, with K = PZ/(2πa)

The machine constant K is fixed by construction. With the speed n in revolutions per minute, ω = 2πn/60, and the classical form follows:

E_a = PZΦn / (60a)

The Torque Equation

Each conductor carries one path's share of the armature current, I_a/a, and feels an average tangential force B_avglI_a/a at radius r. Summing over all Z conductors gives the electromagnetic torque developed at the air gap:

T = ZB_avglrI_a/a = [PZ/(2πa)] ΦI_a = KΦI_a

One Constant for Motor and Generator

The electrical power converted is E_aI_a = KΦωI_a, and the mechanical power developed is Tω = KΦI_aω. The two are equal, as energy conservation requires of a coupling that neither stores nor dissipates energy. With the field held constant, KΦ is a single number, and in SI units it is both the torque constant K_t in N·m/A and the back-EMF constant K_e in V·s/rad of the motor model in Control System Modeling and Block Diagrams. That model's back EMF e_b, resistance R, and inductance L are this article's E_a, R_a, and L_a.

For example, a four-pole armature with 480 active conductors and 20 mWb per pole, turning at 1,500 rpm, generates 240 V if simplex lap-wound (a = 4, K = 76.4) and 480 V if simplex wave-wound (a = 2, K = 152.8). The lap-wound machine develops 76.4 N·m at 50 A.

The Equivalent Circuit

In steady state the armature is the generated voltage E_a in series with the armature circuit resistance R_a, which includes the interpole and compensating windings; a series field adds its own resistance R_s. The shunt or separate field is a resistance R_f carrying the field current I_f. The two circuits interact only through the flux, which sets E_a and T.

Motor and Generator Conventions

The sign of the armature current decides the direction of energy flow, so each equation needs a stated convention. For a motor, take I_a positive flowing into the positive terminal:

V_T = E_a + I_aR_a (motor)

For a generator, take I_a positive flowing out of the positive terminal:

V_T = E_a − I_aR_a (generator)

A motor whose generated voltage rises above its terminal voltage, because a load drives it faster or its supply voltage falls, reverses its current and becomes a generator with no change in connection. Regenerative braking rests on exactly this.

Carbon brushes add a contact voltage drop that rises only gradually with current, so it is better represented as a small voltage V_b than as a resistance: V_T = E_a + I_aR_a + V_b for a motor. Dynamic behavior adds the armature inductance, v = R_ai_a + L_a di_a/dt + KΦω, which with the mechanical equation J dω/dt = KΦi_a − bω − T_L forms the model whose time constants Control System Modeling and Block Diagrams derives.

The Magnetization Curve

The link between field current and flux is the magnetization curve, measured as the open-circuit characteristic: drive the machine at constant speed with the armature open and record E_a against I_f. Because E_a = KΦω, the curve is the flux curve scaled by speed, and a curve measured at one speed converts to another by multiplying every voltage by the ratio of the speeds. It starts at a small residual voltage left by the remanent magnetism of the poles, rises nearly linearly along the air-gap line while the iron is unsaturated, and bends over as the poles and armature teeth saturate.

Several results below assume linear magnetics, flux proportional to field current. The assumption is reasonable below the knee and optimistic above it, where a change in field current changes the flux less than in proportion.

Field Connections and Their Characteristics

How the field connects to the armature decides how the flux changes with load, and that decides the machine's character.

Separately Excited and Shunt Motors

With constant flux, the motor and torque equations combine into the speed equation:

ω = V_T/(KΦ) − R_aT/(KΦ)²

The first term is the no-load speed; the second is the droop caused by armature resistance. The torque-speed characteristic is a straight line that falls little from no load to full load, because R_a is small. Its slope is summarized by the speed regulation:

speed regulation = (n_{no load} − n_{full load}) / n_{full load} × 100 percent

The motor of the worked example has a regulation of 6.4 percent. Drives that vary the armature voltage excite the field separately, since a shunt field would weaken with it.

Losing the field is the dangerous fault. With the flux down to its residual value, the speed equation predicts that a lightly loaded motor accelerates toward a destructive speed, while a heavily loaded one slows or stalls and draws a very large armature current. Field-failure protection disconnects the armature when field current is lost.

Series Motors

In a series motor the field carries the armature current, so the flux grows with load. Assuming linear magnetics, Φ = K_fI_a, where K_f is a constant of the field winding, the torque is

T = KK_fI_a²

and the motor equation V_T = KK_fI_aω + I_a(R_a + R_s) gives the speed as a function of torque:

ω = V_T/√(KK_fT) − (R_a + R_s)/(KK_f)

Torque proportional to the square of current gives the series motor high starting torque per ampere, which suited it to traction, hoists, and engine starters. Speed varies roughly as the inverse square root of torque, so a load that falls to a quarter of its torque roughly doubles the speed. As torque approaches zero, the first term grows without limit: unloaded, the motor runs away until friction, windage, or mechanical failure ends the acceleration. A series motor must therefore be permanently coupled to its load, by gears or a solid coupling rather than a belt that could come off.

A real machine departs from the table at both ends. At high current the iron saturates and torque rises more nearly in proportion to current; at low current the residual flux keeps the speed finite, though still far above a safe value.

The square law also lets a series motor run on alternating current. When the supply reverses, armature and field current reverse together and the torque keeps its direction, pulsating at twice the supply frequency. A series motor built for that duty, with a laminated field, is the universal motor of vacuum cleaners, power tools, and kitchen appliances, covered in Appliance Motor Drives.

Compound Motors

A compound motor has both a shunt and a series field. In a cumulative compound motor the series field aids the shunt field: the speed falls more steeply with load than a shunt motor's, the starting torque is higher, and the shunt field still sets a safe no-load speed. A shunt motor with only a light series winding, enough to offset the flux loss caused by armature reaction, is called a stabilized shunt motor.

In a differential compound motor the series field opposes the shunt field. The flux falls as load rises, the speed can rise with load, and operation tends to be unstable; under heavy overload or at starting the series field can overpower the shunt field and reverse the torque. Differential compounding is rarely used for motors.

DC Generators

DC generators once supplied lighting systems, electrochemical plants, and the field windings of large alternators. Semiconductor rectifiers have replaced them almost everywhere.

Separately Excited Generators

With its field supplied independently, a generator's no-load voltage KΦω is set by field current and speed. As load current rises, the terminal voltage falls by I_aR_a, and by a little more where armature reaction weakens the flux. The terminal characteristic is a gently drooping line, and raising the field current with load holds the voltage constant.

Self-Excitation and Voltage Build-Up

A shunt generator feeds its own field and must build its voltage from almost nothing by positive feedback: the residual flux generates a small voltage, which drives a small field current, which raises the flux and the voltage. Build-up stops where the magnetization curve crosses the field-resistance line V = R_fI_f, because there the generated voltage exactly sustains the field current that produces it. Four conditions must hold:

1. The poles must retain residual magnetism.
2. The field must be connected so that its current aids the residual flux. Connected the other way, the field current cancels the remanence and the voltage falls instead of building.
3. The field circuit resistance must be less than the critical resistance, the slope of the air-gap line at the running speed.
4. The speed must exceed the critical speed. Because the magnetization curve scales with speed, running slower flattens the air-gap line until its slope falls below R_f.

Shunt, Series, and Compound Generators

A shunt generator's voltage falls faster with load than a separately excited generator's, because a falling terminal voltage also reduces its field current. Past a certain load the characteristic turns back on itself, and a short circuit at the terminals removes the field supply, so the voltage collapses and the sustained current is only what the residual flux can drive through the armature.

Compound generators combine the two fields. A cumulative compound generator can be over-compounded, with full-load voltage above no-load voltage to offset the drop along a feeder; flat-compounded, with the two equal; or under-compounded. A differential compound generator's voltage falls steeply with load, a drooping characteristic once used in arc-welding generators, which must limit the current when the electrode touches the work.

The DC tachogenerator, a small permanent-magnet generator whose output voltage KΦω measures speed, survives in drives; Motor Feedback Devices describes it.

Armature Reaction and Commutation

Armature Reaction

The armature current produces a magnetomotive force whose axis lies along the brush axis, at right angles to the main field. This cross-magnetizing field strengthens the flux under one tip of each pole and weakens it under the other, with two consequences. First, the magnetic neutral plane, where the resultant flux crosses zero, shifts away from the geometric neutral: in the direction of rotation in a generator and against it in a motor. The coils being commutated then lie in flux and generate voltage, which causes sparking. Second, the strengthened pole tip saturates, so it gains less flux than the weakened tip loses, and the net flux per pole falls with load.

The loss of flux lowers a generator's voltage under load. In a shunt motor it raises the speed under load, which offsets the resistance droop and, if strong enough, produces a speed that rises with torque, a characteristic that is unstable with many loads. The light series winding of the stabilized shunt motor restores a falling characteristic.

The Commutation Process

While a coil's two segments pass under a brush, the brush short-circuits the coil, and in that interval the coil current must reverse from +I_c to −I_c, where I_c = I_a/a. The time available, the commutation period t_c, is roughly the brush width divided by the commutator's surface speed. The coil's effective inductance L_c opposes the reversal with an average reactance voltage

e_r ≈ L_c × 2I_c/t_c

which grows with both load current and speed. Unopposed, it leaves the reversal unfinished when the segment leaves the brush, and the remaining current breaks as an arc. Mersen's data sheet TDS-01 calls this under-commutation, with sparks at the brush's trailing edge, and describes reversal completed too early, over-commutation, as sparking at the leading edge.

Brush Shifting, Interpoles, and Compensating Windings

• Brush shifting. Moving the brushes to the magnetic neutral plane restores sparkless commutation at one load. The plane moves with every change of load and reverses between motoring and generating, so shifted brushes suit only steady loads with one direction of power flow.
• Interpoles. Interpoles on the geometric neutral, wound in series with the armature, cancel the armature field in the commutating zone and induce in the commutating coils a voltage opposing the reactance voltage. Both effects scale with armature current, so the series connection keeps the correction right at every load. In a generator each interpole takes the polarity of the next main pole in the direction of rotation; in a motor, that of the main pole just passed.
• Compensating windings. Conductors in slots in the main pole faces, in series with the armature and carrying current opposite to the armature conductors beneath them, cancel the armature magnetomotive force under the poles, removing both the flux distortion and the loss of flux. They also prevent flashover, an arc that spreads around the commutator from brush to brush when distortion raises the voltage between segments under the pole tips. Such windings are expensive and appear in large machines and in machines subject to heavy overloads, rapid load changes, or deep field weakening. By canceling the armature field they also reduce the armature inductance, which speeds a drive's current response.

Starting

Why Starting Current Is Large

At standstill the generated voltage is zero, so only the armature circuit resistance limits the current. That resistance is kept small for efficiency, and the stall current V_T/R_a is many times rated current: sixteen times in the worked example. Such a current overloads the commutator, causes heavy sparking, and delivers a torque shock to the load.

Sizing a Starting Resistor

A resistance R_ext in series with the armature limits the initial current to a chosen I_max:

R_ext = V_T/I_max − R_a

As the motor accelerates, the generated voltage rises and the current falls, so the resistance is cut out in steps. Suppose each step is taken when the current has fallen to I_min and is sized to return the current to I_max. If armature inductance is neglected, so that the current changes instantly, and the speed does not change during switching, each total resistance is the previous one multiplied by I_min/I_max. Starting from R₁ = V_T/I_max and ending at R_a, a starter with N steps satisfies

R_a = R₁(I_min/I_max)^N

so the current ratio fixes the number of steps, and the resistances form a geometric series. The derivation assumes constant flux, which means starting with the shunt field at full strength; that also gives the most torque per ampere.

Starters and Current-Limited Drives

The classic manual starter is a faceplate rheostat with a spring-return arm held in the run position by an electromagnet. In the three-point starter the holding coil carries the shunt field current, so the arm returns to off if either the supply or the field circuit fails. The four-point starter connects the holding coil across the supply, so a weakened field cannot release the arm, but it gives up protection against an open field.

An electronic drive dispenses with the resistor: it raises the armature voltage from zero while a current loop holds the current at its limit, the method described under Motor Drive Architectures.

Speed Control

Rewritten as ω = (V_T − I_aR_a)/(KΦ), the speed equation offers three handles: resistance in the armature circuit, the voltage applied to the armature, and the flux.

Armature Resistance Control

Resistance in series with the armature steepens the speed droop without changing the no-load speed. It is simple and wasteful. At constant torque, the fraction of input power lost in the added resistance approximately equals the fractional reduction in speed.

Armature Voltage Control

Varying the armature voltage at full field moves the torque-speed line up or down without changing its slope. With the flux and the permitted armature current both fixed, the available torque KΦI_a is the same at every speed, so this is the constant-torque region. It extends from standstill to base speed, reached at rated armature voltage and full field.

This is the method of Ward Leonard's patent. In the system that bears his name, a generator driven at constant speed feeds the motor armature directly, and adjusting the generator's small field current varies the motor voltage smoothly through zero and into reverse. Phase-controlled thyristor rectifiers later replaced the rotating generator, varying the average output voltage by delaying the firing angle; Thyristors and Controlled Rectifiers describes the devices. From a battery or other DC source, a transistor chopper varies the average armature voltage by pulse-width modulation.

Field Weakening

Above base speed the armature voltage can rise no further, but reducing the field current lowers KΦ and raises the speed for the same generated voltage. With armature current held at rated value, the converted power E_aI_a stays roughly constant while the available torque falls in proportion to the flux, and so in inverse proportion to the speed. This is the constant-power region.

The range is limited. A weaker main field makes armature reaction relatively stronger, which distorts the flux, worsens commutation, and can destabilize the speed. The field winding's many turns give it a long time constant L_f/R_f, so field control responds slowly. Even so, the flux can fall faster than the speed can rise, and the armature current then surges.

Operating Quadrants

On a plot of speed against torque, the first and third quadrants are forward and reverse motoring and the second and fourth are forward and reverse braking, in which the machine generates. The machine itself works in all four; its supply decides which it can reach. A diode rectifier can deliver power but not accept it, a single thyristor converter cannot reverse current, and a battery-fed H-bridge reaches all four. The circuits are covered under Motor Drive Architectures and in Motor Drivers and Controllers.

Braking

A DC motor can be stopped electrically in three ways. Each makes the machine a generator, and they differ in where the energy goes. None holds a load at rest, so holding still requires a mechanical brake.

Dynamic Braking

Dynamic, or rheostatic, braking disconnects the armature from the supply and connects it across a resistor R_db while the field stays excited. The generated voltage drives the current I_a = −E_a/(R_a + R_db) in the motor convention, producing a braking torque proportional to speed and turning the kinetic energy into heat. With constant flux, negligible armature inductance, and no load torque or friction, J dω/dt = −(KΦ)²ω/(R_a + R_db), so the speed decays exponentially with time constant

τ = J(R_a + R_db)/(KΦ)²

A series motor needs its field connection reversed for dynamic braking, because otherwise the reversed armature current would oppose the residual flux and the machine would fail to excite itself.

Regenerative Braking

Regenerative braking keeps the machine connected and makes its generated voltage exceed the applied voltage, so the current reverses and energy flows back to the supply. A separately excited or shunt motor regenerates by itself when an overhauling load, such as a descending vehicle or a lowering hoist, drives it above no-load speed. A drive can command regeneration down to low speed by lowering the armature voltage below E_a or by strengthening the field. The supply must be able to accept the energy; otherwise a braking resistor must absorb it. A plain series motor cannot regenerate, because reversing its current also reverses its field; series motors that regenerate have their fields separately excited while braking.

Plugging

Plugging reverses the armature connections while the motor runs. The generated voltage then adds to the supply voltage, and the current (V_T + E_a)/R, with R the total armature circuit resistance, would reach nearly twice the stall current without added resistance. Near full speed E_a is close to V_T, so the plugging resistor must be roughly twice the starting resistor. The braking torque persists all the way down, and at standstill the motor starts in reverse unless a zero-speed switch or timer disconnects it.

Worked Example: A 240 V Shunt Motor

A shunt motor runs from a constant 240 V supply. Its armature circuit resistance, including interpoles and, as an approximation, the brush drop, is R_a = 0.30 Ω, and its field circuit resistance is R_f = 240 Ω. Unloaded, it draws 2.0 A of armature current at 1,800 rpm, and its rated armature current is 50 A. The data are illustrative. The calculation neglects armature reaction, so the flux does not change with load; takes the rotational losses as constant at their no-load value; and neglects stray load losses.

Speed, Regulation, and Efficiency

1. Machine constant. The field current is 240/240 = 1.0 A. At no load, E_a = 240 − 2.0 × 0.30 = 239.4 V, and 1,800 rpm is 188.5 rad/s, so KΦ = 239.4/188.5 = 1.270 V·s/rad.
2. Full-load speed. At 50 A, E_a = 240 − 50 × 0.30 = 225.0 V, so ω = 225.0/1.270 = 177.2 rad/s, or 1,692 rpm.
3. Speed regulation. (1,800 − 1,692)/1,692 = 6.4 percent.
4. Torque and power. The developed torque is 1.270 × 50 = 63.5 N·m, and the converted power is 225.0 × 50 = 11.25 kW, equal to Tω.
5. Efficiency. The rotational loss (friction, windage, and core loss) is taken from the no-load test as 239.4 × 2.0 = 479 W, so the output is 11.25 − 0.48 = 10.77 kW. The input is 240 × (50 + 1.0) = 12.24 kW, and the efficiency is 88.0 percent. The losses are 750 W in the armature circuit, 240 W in the field, and 479 W rotational.

Starting

1. Without a starter. At standstill the current is 240/0.30 = 800 A, sixteen times rated current.
2. With one resistor. Limiting the current to twice rated, 100 A, requires a total of 240/100 = 2.40 Ω, so R_ext = 2.10 Ω. The starting torque is 1.270 × 100 = 127 N·m, twice full-load torque.
3. With a stepped starter. The ratio R₁/R_a is 2.40/0.30 = 8. Four steps give I_min/I_max = 8^−1/4 = 0.595, so the current swings between 100 A and 59.5 A. The sections cut out are 0.973, 0.579, 0.344, and 0.205 Ω, totaling 2.10 Ω, at 732, 1,167, 1,425, and 1,579 rpm.

Field Weakening and Braking

1. Field for 2,000 rpm. At rated armature current E_a is still 225.0 V, and 2,000 rpm is 209.4 rad/s, so KΦ must fall to 225.0/209.4 = 1.074 V·s/rad, 0.846 of its full value. Assuming linear magnetics, the field current must fall to 0.846 A, which requires a field circuit resistance of 283.7 Ω, or 43.7 Ω of field rheostat. A saturated machine needs a larger reduction in field current.
2. Torque at 2,000 rpm. 1.074 × 50 = 53.7 N·m. The converted power is still 11.25 kW: torque has fallen in the ratio that speed has risen.
3. A sudden field change. If the field fell to its new value instantly at 1,692 rpm, E_a would drop to 190.3 V before the speed could change, and the armature current would rise toward (240 − 190.3)/0.30 = 166 A, 3.3 times rated, with a torque of 178 N·m. Field changes must be ramped or the armature current limited.
4. Plugging from full-load speed. Holding (240 + 225.0)/R to 100 A requires R = 4.65 Ω, an added 4.35 Ω, about twice the 2.10 Ω starting resistor.
5. Dynamic braking from full-load speed. Limiting the initial current to 100 A requires R_a + R_db = 225.0/100 = 2.25 Ω, so R_db = 1.95 Ω, with an initial braking torque of 127 N·m. The braking time constant is 2.25/1.270² = 1.39 s for each kg·m² of total inertia.

Losses and Efficiency

Where the Losses Arise

IEC 60034-2-1:2024, Rotating electrical machines, Part 2-1: Standard methods for determining losses and efficiency from tests (excluding machines for traction vehicles), applies to DC machines as well as to synchronous and induction machines. Its loss categories map directly onto the DC machine:

• Constant losses. Iron losses and friction and windage losses, with brush friction included.
• Excitation circuit losses. The field winding losses, plus any exciter losses.
• Load losses. The I²R losses in the armature circuit and the electrical brush losses, contact loss included.
• Additional load losses. Losses from stray fluxes under load in iron and conductors, and additional brush losses caused by commutation.

Each part of the armature core reverses its flux at f = Pn/120, with n in rpm. At a given flux density, hysteresis loss rises roughly in proportion to that frequency and eddy-current loss roughly with its square.

Brush Contact Drop

The electrical brush loss is the contact voltage drop multiplied by the armature current. Mersen's technical guide Carbon Brushes for Motors and Generators explains that the drop depends on the brush grade and on everything that alters the commutator film, including temperature, humidity, pressure, speed, and impurities. The guide groups its grades by the drop summed over both polarities, measured on a copper commutator at 10 A/cm², 12.5 m/s, 18 kPa, and 65 to 70 °C.

Mersen describes its electrographitic grades as having a medium drop and its metal-graphite grades, used in low-voltage machines, as having a very low drop. A drop of 2 or 3 V is negligible on a 500 V machine but costly on a 12 V motor.

The Condition for Maximum Efficiency

Divide the losses into a constant part P_c, the field and rotational losses, and a variable part I²R_a. For a motor at constant terminal voltage, with the small field current neglected so that the line current I equals the armature current,

η = 1 − (P_c + I²R_a)/(V_TI) = 1 − P_c/(V_TI) − IR_a/V_T

Setting dη/dI = 0 gives P_c/I² = R_a, or

I²R_a = P_c

Efficiency peaks at the load at which the variable losses equal the constant losses. A loss proportional to current, such as the brush electrical loss, adds only a constant term to η and does not move the peak. The result holds only while P_c really is constant, as it nearly is for a shunt or separately excited motor at nearly constant speed and flux.

In the worked example P_c = 240 + 479 = 719 W, so the peak falls at √(719/0.30) = 49.0 A, just below rated current; an exact calculation that keeps the field current in the input puts it at 48.0 A. The curve is flat near its peak: 88.0 percent at rated current, 87.6 percent at 125 percent of rated current, and 85.5 percent at half.

Permanent-Magnet DC Motors

Small brushed motors commonly replace the wound field with permanent magnets, which supply constant flux with no field loss and no field supply. The motor then behaves as a separately excited machine at fixed field, described by a single constant KΦ, written K in this section and quoted on data sheets as K_t or K_e.

Magnets and Armature Reaction

Arc-shaped magnets inside a steel housing, which serves as the flux return, face the armature. Ferrite magnets are inexpensive; neodymium iron boron gives the most flux for its size; aluminum-nickel-cobalt (AlNiCo) is thermally stable but easily demagnetized. Magnetic Materials compares these materials.

Armature reaction is the permanent-magnet motor's particular hazard. The armature field opposes the magnet over part of each pole, and at stall, during a hard start, or when the motor is plugged it can exceed what the magnet withstands and lower K permanently. Ferrite loses coercivity as it gets colder and NdFeB as it gets hotter, so their worst cases lie at opposite ends of the temperature range. A drive for such a motor must limit peak current as well as average current.

Iron-Core and Ironless Armatures

An iron-core armature is pulled toward positions of minimum reluctance set by its teeth and the magnets, a cogging torque felt when the unpowered shaft is turned by hand. Ironless, or coreless, armatures avoid it: the winding is a self-supporting cylinder turning in the air gap while the magnet and flux return stay stationary. With no rotor iron there is no cogging and no rotor iron loss, and inertia and inductance are low. The manufacturer maxon lists for its ironless winding no magnetic cogging, low inductance, low electromagnetic interference, and an efficiency of up to 90 percent.

The Linear Characteristic

At constant voltage V, a motor with armature resistance R and constant K follows a straight torque-speed line from the no-load speed V/K to the stall torque KV/R, reached at the stall current I_s = V/R. Neglecting friction, the output (V − IR)I peaks at half the stall current, half the stall torque, and half the no-load speed, where

P_max = V²/(4R)

at an efficiency of exactly 50 percent. The motor of Control System Modeling and Block Diagrams, with V = 24 V, R = 1 Ω, and K = 0.05 N·m/A, has a no-load speed of 480 rad/s, a stall torque of 1.20 N·m at 24 A, and a maximum output of 144 W at 240 rad/s.

Without friction the efficiency, E_a/V, would rise all the way to no-load speed; friction and core losses pull the peak back. Represent them by a constant no-load current I₀, equivalent to a constant friction torque KI₀. The efficiency (I − I₀)(V − IR)/(VI) then peaks at I = √(I₀I_s), where

η_max = [1 − √(I₀/I_s)]²

If the example motor draws 0.24 A at no load, 1 percent of its stall current, its peak efficiency is (1 − 0.1)² = 81 percent, at 2.4 A and 432 rad/s. This optimum does not satisfy the equal-loss rule, because the friction loss KI₀ω changes with speed.

Brushes, Wear, and Electrical Noise

Brush Materials

In Mersen's guide, electrographitic grades, graphitized above 2,500 °C, serve stationary and traction DC machines at 8 to 12 A/cm² in steady operation and peripheral speeds up to 50 m/s. Copper-graphite grades carry 10 to 30 A/cm² steadily in low-speed, low-voltage machines, and silver grades serve tachometer generators. Small motors often use precious-metal brushes instead: maxon's November 2014 catalog lists them as typical for small motors in continuous operation at low current and for DC tachometers, noting their low, constant contact resistance even after long standstill, and lists graphite brushes for larger motors, high currents, start-stop and reversing duty, and PWM supply.

The Commutator Film

A healthy commutator carries a uniform, light-to-dark brown film that Mersen describes as a mix of metal oxides, carbon, and water from the air. Several conditions upset it:

• Dry air. Robert H. Savage of the General Electric Research Laboratory showed in 1948 that graphite's low friction depends on adsorbed vapors, chiefly water, and that in vacuum, graphite brushes wear rapidly to fine dust. The effect had surfaced as what D. Ramadanoff and S. W. Glass of the National Carbon Company called, in 1944, the high-altitude brush problem of aircraft generators. Mersen gives 8 to 15 g of water per cubic meter of air as the best range for the film and 2 g/m³ as the critical lower threshold, possible at altitude, in dry-gas enclosures, in totally enclosed motors, and in desert or arctic conditions.
• Light load. Mersen warns that a low current density can be more harmful to brushes and commutator than a high one, and lists underloaded brushes among the causes of a streaky film.
• Contamination. Corrosive vapors destroy the film even at low concentration; Mersen names chlorine compounds, ammonia, hydrogen sulfide, sulfur dioxide, and products of the hot distillation of silicones.
• Spring pressure. Mersen's data sheet TDS-11 explains that too little pressure lets electrical wear from sparking dominate and too much lets mechanical wear dominate. Its guide recommends about 18 to 25 kPa for stationary machines and about 34 to 49 kPa for machines under heavy vibration such as traction motors.

Service Life and Noise

For its small motors, the same maxon catalog states that life can exceed 20,000 hours under favorable conditions, fall below 100 hours under extreme ones, and reach roughly 1,000 to 3,000 hours under average requirements, shortened by higher current, higher speed, extreme start-stop or reversing duty, and harsh environments.

Every commutation interrupts current in an inductive coil, and the arcs and current steps at the brushes conduct and radiate broadband noise. A 2024 maxon article by Angelica Perzan describes maxon's CLL (capacitor long life) design, whose RC networks between commutator segments suppress brush sparking. Conventional suppression places capacitors across the terminals and from each terminal to the frame, with series inductors or ferrite beads in the leads. EMI/EMC in Power Electronics treats coupling paths and filters.

Where DC Machines Remain in Service

Most new industrial and traction drives use inverter-fed AC machines, which have no commutator to limit speed and voltage and no brushes to replace; AC Induction Motor Drives and BLDC and PMSM Drives describe the alternatives. DC machines persist where their strengths still count:

• Small brushed motors. A permanent-magnet motor runs from a battery with no electronics, and a transistor or H-bridge adds speed control cheaply, so brushed motors remain common in vehicle accessories, toys, tools, pumps, and instruments. Transistor Switching Circuits shows how one transistor drives such a motor from a logic signal.
• Engine starters. A starter must deliver very high torque for a few seconds from a 12 V or 24 V battery, which suits a series or permanent-magnet DC machine.
• Installed industrial and traction machines. Mersen's guide still tabulates brush grades for rolling-mill, paper-mill, mine-winder, forklift, and traction commutator machines, and older DC motors can be kept in service by replacing only their converters. Railway Traction Systems describes rail traction's move to AC.

Summary

Two equations carry most of DC machine theory. The generated voltage E_a = KΦω and the developed torque T = KΦI_a share the constant K = PZ/(2πa), and with the armature circuit equation they predict speed, torque, and current for any connection. The field connection sets the character: nearly constant speed for separately excited and shunt machines, high starting torque and a risk of runaway for series machines, and a compromise for compound machines.

The practical limits come from the armature current. Its field distorts and weakens the main flux and hinders commutation, which interpoles and compensating windings correct, and its size at standstill demands a starter or a current-limited drive. Speed control follows from the equations: armature voltage below base speed at constant torque, field weakening above it at roughly constant power.

Brushes and commutator are the DC machine's defining feature and its weakness. Their wear and noise explain why AC machines have taken over most new drives. The separately excited DC motor nonetheless remains the model those drives imitate.

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