Educational material — not for design use. Every number cites its source; see provenance & disclaimers.

DC Motor (Permanent Magnet)

torque-powerkinematics

Verified build 5 relations · 4 identities proven · 3 modeling steps · 6 parity samples

Almost everything else in this catalog is a load — shafts that get wound, beams that get bent, flywheels that get spun. This page is the thing that does the driving. A permanent-magnet DC motor at a fixed supply voltage has the simplest personality in machine design: its entire steady-state behavior is one straight line in the torque–speed plane,

T=Tstall(1−ωω0)P=T ωPmax=Tstall ω04T = T_{stall}\left(1 - \frac{\omega}{\omega_0}\right) \qquad P = T\,\omega \qquad P_{max} = \frac{T_{stall}\,\omega_0}{4}

Three things to notice, one per formula:

The two configurations run the same line in opposite directions. Pick a speed hands you the torque and power at that point. The load picks the torque is how a motor actually lives — the machine it drives demands a torque, and the line answers with the speed the motor settles at. Ask for more than TstallT_{stall} and the widget refuses rather than report the fiction of a backwards-spinning “operating point”: a stalled motor delivers its stall torque and no more. Drive the speed knob past ω0\omega_0 and the delivered torque and power go negative with a warning — the machine is braking its load and generating, a real regime, but no longer motoring.

This THING exists to head a chain. In the chain builder’s headline example the motor’s delivered torque TT and shaft speed ωout\omega_{out} wire into a planetary gearset’s sun, the gearbox multiplies the torque at the expense of speed, and the output simultaneously winds a driveshaft and spins up a flywheel — with the gear ratio trading spin-up time against shaft stress while the power PP you put in survives the wiring end to end. The gearbox is the impedance matcher between what a motor is good at (speed) and what loads usually want (torque); the chain lets you feel that bargain with numbers that keep their citations.

Try it

Inputs
N·m
Delivered torque
— N·m
Delivered shaft speed
—
Shaft power
—
Peak available power
—
Peak-power speed
—

Governing relations

T=Tstall(1−ωω0)T = T_{stall}\left(1 - \frac{\omega}{\omega_0}\right)

Assumes: permanent-magnet (constant-flux) brushed DC machine at a FIXED supply voltage — with the field constant, torque is proportional to current and back-EMF to speed, so the torque–speed characteristic is a straight line from stall to no-load; steady state — electrical (winding inductance) and mechanical (rotor inertia) transients have died out; the point (ω, T) sits ON the line, not near it; constant armature resistance; brush voltage drop, armature reaction, and rotational losses neglected (real no-load speed is set by friction, slightly below V/k) · Valid while: Past the no-load speed the machine is being driven faster than its back-EMF allows for this supply voltage: current reverses, delivered torque and power go NEGATIVE, and the "motor" is braking its load and generating. The line still describes it — but it is no longer motoring. No motoring operating point exists at a negative shaft speed. Demanding more torque than T_stall solves to exactly this state — the only "solution" on the extended line has the load back-driving a stalled motor backwards — and a negative speed entered or wired in directly lands in the same plugging regime this model does not cover. The evaluation refuses rather than report a fictitious operating point.

Source: Hughes, A., & Drury, B., Electric Motors and Drives: Fundamentals, Types and Applications, 5th ed., Newnes/Elsevier, 2019 (ISBN 978-0-08-102615-1) — ch. 3 "D.C. Motors": §3.3 motional e.m.f. and equivalent circuit (T = kI, E = kω, with k_t = k_e = k in SI); §3.4.1 no-load speed ω₀ = V/k under the stated constant-flux assumption; §3.4.3 "Behaviour when loaded", eq. (3.10) — the straight torque–speed line; §3.4.6 "Maximum output power" — peak mechanical power at half the no-load speed.

P=T ωP = T\,\omega

Assumes: steady rotation — torque and speed constant at the operating point

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §3-12 (power transmission, P = Tω).

Pmax=Tstall ω04P_{max} = \frac{T_{stall}\,\omega_0}{4}

Assumes: the largest mechanical power ANY point on the line can deliver — attained at half the no-load speed, where the motor gives exactly half its stall torque

Source: Hughes, A., & Drury, B., Electric Motors and Drives: Fundamentals, Types and Applications, 5th ed., Newnes/Elsevier, 2019 (ISBN 978-0-08-102615-1) — ch. 3 "D.C. Motors": §3.3 motional e.m.f. and equivalent circuit (T = kI, E = kω, with k_t = k_e = k in SI); §3.4.1 no-load speed ω₀ = V/k under the stated constant-flux assumption; §3.4.3 "Behaviour when loaded", eq. (3.10) — the straight torque–speed line; §3.4.6 "Maximum output power" — peak mechanical power at half the no-load speed.

ωp=ω02\omega_p = \frac{\omega_0}{2}

Assumes: the vertex of the power parabola P(ω) = T_stall·ω(1 − ω/ω₀) — machine-proven by completing the square in the derivation below

Source: MIT 2.007 (Design and Manufacturing I) motor tutorial, "D.C. Motor Torque/Speed Curve", MIT Center for Innovation in Product Development, 1999 — §3.1–3.2: the linear torque–speed equations and the explicit statement that maximum output power occurs at half the stall torque and half the no-load speed.

ωout=ω\omega_{out} = \omega

Assumes: rigid output shaft and coupling — whatever the motor drives turns at rotor speed and receives the delivered torque T (no slip, no compliance, no gearbox inside the case)

Source: Norton, R. L., Design of Machinery, 6th ed., McGraw-Hill, 2020 — §2.19 "Motors and Drivers", permanent-magnet DC motors: the speed–torque curve from stall torque at zero speed to zero torque at no-load speed; the contrast with shunt-, series-, and compound-wound curve shapes; the warning that motors cannot tolerate a full-current zero-speed stall for more than minutes without overheating.

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

Ia=Vs−kmωRaI_{a} = \frac{V_{s} - k_{m} \omega}{R_{a}}

1. The modeling step — one loop of circuit. A DC supply V_s pushes current through the armature resistance R_a against the back-EMF k_m·ω that the spinning rotor generates (a motor is also a generator, always). Kirchhoff's voltage law gives the armature current: whatever voltage the back-EMF doesn't cancel, the resistance converts to current. At stall (ω = 0) ALL of V_s drives current; at ω = V_s/k_m none is left. — circuit model: KVL with back-EMF modeling step

T=IakmT = I_{a} k_{m}

2. The electromechanical coupling: torque is proportional to armature current, and — in SI units — the torque constant (N·m per ampere) is numerically THE SAME k_m as the back-EMF constant (volt-seconds per radian). One constant couples the electrical and mechanical sides in both directions; this is where the machine physics enters, by citation. — torque constant = back-EMF constant (SI) modeling step

Iakm=Vskm(1−kmωVs)RaI_{a} k_{m} = \frac{V_{s} k_{m} \left(1 - \frac{k_{m} \omega}{V_{s}}\right)}{R_{a}}

3. Eliminate the current. Substituting the loop equation into T = k_m·I_a and factoring out the stall value: torque falls LINEARLY with speed, from k_m·V_s/R_a at ω = 0 to zero at ω = V_s/k_m. The straight line is not an approximation of this model — it IS this model. — eliminate I_a — the line, in circuit constants

T=Tstall(−ωω0+1)T = T_{stall} \left(- \frac{\omega}{\omega_{0}} + 1\right)

4. Name the intercepts. The two corners of the line are exactly what a datasheet quotes: stall torque T_stall = k_m·V_s/R_a (all the supply's current, none of its back-EMF) and no-load speed ω₀ = V_s/k_m (back-EMF eats the whole supply, no current left to make torque). Measure those two numbers and the entire behavior at this voltage is known — the THING's knobs are the datasheet. This identification of the knobs with the circuit constants is a naming step, not a machine-checkable identity (T_stall and ω₀ are free knobs); it is proven independently against the circuit model in the pipeline's cross-check test. — name the datasheet intercepts modeling step

P=Tstallω(−ωω0+1)P = T_{stall} \omega \left(- \frac{\omega}{\omega_{0}} + 1\right)

5. Mechanical power is torque times speed, P = Tω — zero at BOTH ends of the line (at stall there is torque but no motion; at no-load, motion but no torque), so the power a motor can deliver must peak somewhere between. — P = Tω along the line

P=Pmax−Tstall(ω−ωp)2ω0P = P_{max} - \frac{T_{stall} \left(\omega - \omega_{p}\right)^{2}}{\omega_{0}}

6. Complete the square. The power parabola in vertex form: P peaks at ω_p = ω₀/2 — half the no-load speed, where the motor delivers half its stall torque — and every rad/s away from that point costs (T_stall/ω₀) · (distance)² of power, symmetrically on either side. No calculus required: the vertex IS the maximum. — complete the square — vertex form

Pmax=Tstallω04P_{max} = \frac{T_{stall} \omega_{0}}{4}

7. Read off the vertex: the most mechanical power this motor can EVER deliver at this voltage is one quarter of stall torque times no-load speed. This is the number that sizes a drivetrain — and note the price of collecting it: peak power lives at half speed and half stall torque, deep in the high-current half of the line. — peak available power

How it fails

The straight line is honest about where a PM DC motor can operate, and silent about for how long. Real motor failures are mostly about heat, and the line’s most tempting corners are exactly the ones the thermal reality forbids.

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    • kinematics
    • torque-power
  • Four-Bar Linkage (Position)

    Four pinned links — ground, crank, coupler, rocker — and the oldest mechanism in the book. Spin the crank and the rocker answers through pure geometry. Every position has TWO valid assemblies (open and crossed): the first THING in the catalog where one input has two honest answers, and the widget lets you pick the circuit.

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    • dynamics
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  • Flywheel (Solid Rotating Disk)

    The machine that stores work as spin — and loads itself doing it. Centrifugal self-loading grows with ρω²R², so the energy a flywheel can hold per kilogram is capped not by its size but by one material index: strength over density.

    • energy-storage
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    • mass-cost
  • Planetary (Epicyclic) Gearset

    Three coaxial members — sun, ring, planet carrier — share one gear mesh law. With two degrees of freedom, it has no single "ratio": fix a different member and the same hardware becomes a different transmission.

    • kinematics
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  • Axial Disk Clutch / Brake (Uniform Wear vs Uniform Pressure)

    The torque an axial plate clutch can pass depends on an assumption you cannot see: how the contact pressure is distributed across the friction annulus. A new, rigid clutch presses uniformly; a worn-in one wears until pressure ∝ 1/r, concentrating load at the inner edge. This page shows both torque predictions side by side — never picking a winner — with the worn-in model always giving the smaller (safe) number, and the r_i = r_o/√3 that squeezes the most torque from a given lining.

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Chains with

Outputs whose SI dimension and quantity kind match another THING's input — the only wires the planner's connectionLegal accepts (invariant 2, computed at build time, not hand-listed). Wire these on the chaining demo.

+ 8 more THINGs its outputs can legally feed (showing the first 8 in course order).

Sources