DC Motor Design Tool: Speed, Torque and Efficiency

For constant useful flux, Faraday induction gives $E_b=K_e\omega$ and electromechanical energy conservation gives $T_{em}=K_t I_a$, with numerically equal $K_t$ in N·m/A and $K_e$ in V·s/rad. MRCET Electrical Machines I, attached pp.66–67 (printed Unit 2 pp.3–4), derives these relations. The explicit one-radian step keeps angular units consistent; its numerical value is one.

In winding mode, $a=P$ for simplex lap and $a=2$ for simplex wave. The constant is $K_t=P\Phi Z/(2\pi a)$, where $Z$ counts both active conductor sides of every turn. MRCET attached p.59 supplies independent worked EMF values for both connections.

No-load current represents a constant running loss torque $T_f=K_t I_0$, following maxon's motor-data explanation. Therefore $I_a=I_0+T_L/K_t$, $E_b=U-V_{br}-I_aR_a$ and $\omega=E_b/K_e$. At target speed, $U_{req}=K_e\omega_{target}+I_aR_a+V_{br}$.

Useful output is $P_{out}=T_L\omega$. Total input is $UI_a+P_{field}$. Losses are $I_a^2R_a$, $V_{br}I_a$, $T_f\omega$ and $P_{field}$. The displayed energy residual subtracts output and all losses from input. Efficiency is output divided by total input; at zero input and zero output it is displayed as zero by convention.

The speed–torque line is sampled at 201 equally spaced shaft torques. Its unconstrained power peak is at half the estimated stall shaft torque. Maximum sampled efficiency has 0.5% of stall-torque resolution and is not necessarily inside the continuous operating region.

For preliminary winding estimates, $I_{path}=I_a/a$ and $A_{Cu}=I_{path}/J$. From the specified effective pole-face area, $B_g=\Phi/A_{pole}$. Ampère's law gives the air-gap MMF per crossing, $F_g=B_g g_{eff}/\mu_0$, using $\mu_0\approx4\pi10^{-7}$ H/m. This last relation is derived here for uniform gap flux; it excludes iron MMF.

For linear commutation, $v_c=D_c\omega/2$, $t_c=(b_{brush}-b_{mica})/v_c$ and the average reactance-voltage magnitude is $e_r=2L_{coil}I_{path}/t_c$. MRCET attached p.35 (printed p.31) gives this current-reversal method. Its text alternates Wc and Wm for the insulation width; this tool consistently uses the mica gap width.

Brush-contact geometry assumes one brush arm per pole, equally spaced around the commutator with alternating polarity. The brush-center arc pitch is $s_b=\pi D_c/P$; the contact arc must satisfy $b_{brush}<s_b$ so neighboring opposite-polarity contacts have positive clearance. This is a geometric nonoverlap bound, not a recommended design clearance. The same assumed arm arrangement is used for both simplex winding choices; wave winding still has two electrical parallel paths. MRCET attached p.24 illustrates four brush positions on a four-pole wave winding.

• Brushed DC machine in forward steady motoring, with constant pole flux, resistance and motor constants.
• No-load current is armature current, excluding separate field current, and represents constant running loss torque. It is not a static-friction or starting-torque model.
• Resistance and any additional brush drop are entered consistently. A separate drop must not duplicate brush effects already present in measured terminal data.
• Field excitation power is supplied independently and treated as a loss. Specified pole flux does not change automatically with excitation power.
• Geometry inputs are used only in winding mode; their defaults are illustrative design assumptions, not dimensions of the maxon motor.
• Simplex lap/wave path counts are electrical approximations. Equal current sharing, uniform useful pole flux and one effective self-inductance per commutating coil are assumed.
• The animated sketch always illustrates two poles. Its rotor angle is independent of the performance calculation, and neutral-plane switching is idealized.
• In the sketch, • means current out of the page, × means into the page, and 0 marks idealized neutral-plane reversal. The split-ring insulation aligns with the brushes at the same reversal angles.
• Commutation sizing assumes P equally spaced brush arms with alternating polarity and equal contact arc lengths. Contact width is measured along the commutator surface. Other brush layouts are outside scope; the two-pole animation remains a conceptual sketch.

Use positive no-load speed and a shaft load strictly below estimated stall torque. Nonnegative shaft load is required. The current and speed messages compare the selected point with user-entered limits; they do not certify temperature or duty cycle. The target-voltage result requires a supply and controller capable of providing that voltage and current.

The plots extend into overload and stall to show the model. Stall current is prospective without current limiting; stall shaft torque extrapolates a running loss torque and does not predict breakaway friction. Neither stall torque nor the unconstrained power maximum is a continuous rating. Continuous-current capability depends on mounting, ambient temperature, cooling and duty cycle.

Expect differences from measured data. For maxon RE 25 118752 at its nominal 26.3 mN·m load, the model gives about 8695 rpm versus the published 8330 rpm, while current is about 1.161 A versus 1.16 A. The reference case explicitly allows a 5% speed difference. Predicted no-load speed and peak efficiency are also approximate. Brush nonlinearity and speed-dependent losses are not fitted away.

The field calculation supplies only gap MMF per crossing, not complete field turns or permanent-magnet sizing. Add all gap crossings and iron, leakage and saturation effects in a complete magnetic circuit. Conductor area excludes insulation, slot fill, cooling and end-turn resistance. Resistance is not synthesized from the winding geometry.

Commutation results are average inductive estimates, not spark-free limits. Mutual inductance, simultaneous coil commutation, contact resistance, brush dynamics, armature reaction and interpoles need separate analysis. The animated split-ring sketch is illustrative. Winding pitch, equalizers, dummy coils and closure feasibility are outside scope.

The metric/US switch converts supported physical fields and chart axes. Drawing annotations explicitly retain SI units.

Brush contact arc length must be strictly below the brush-center pitch πD/P. Equality means the neighboring contacts touch and is rejected. This check prevents geometric overlap only; practical clearances, insulation and commutation performance require separate design verification.