Motor and Drive Sizing: Method

Motor and Drive Sizing: Reflected Inertia, Move Profiles, RMS Torque and Motor Checks

Sizing a motor for a motion axis is rigid-body mechanics plus two thermal facts about electric machines: a motor may be asked for a large torque for a short time (its peak torque, set by demagnetisation or the drive current), but the heat it can reject sets the torque it may carry on average, and since the copper loss goes with the square of the current the average that matters is the root mean square. This page gives the model behind the tool, its sign conventions and what each result means. The servo rules of thumb quoted (the 10 to 1 inertia ratio, the 1/3-1/3-1/3 profile) are named as such; candidate motor data are entered by the analyst from the maker's data sheet.

The mechanism reduced to the motor shaft

The tool carries one chain for all six mechanisms: motor rotor J_M; an inertia J_c that turns with it (coupling, motor pulley, reducer input); a reducer of ratio N (motor speed over output speed) and efficiency η_g; an output shaft with inertia J_o (screw, drive pulley, pinion, drum, reducer output); and a conversion from rotation to the travel of the load, k metres per radian, with efficiency η_s: k = lead/(2π) for a screw, k = d/2 for a belt, rack, conveyor or hoist drum. A rotary load is the same chain with k = 1 and the inertia J_L in place of the mass.

Reflected inertia. A part turning at 1/N of the motor speed stores 1/N² of the kinetic energy it would store at motor speed, so its inertia counts as J/N² at the motor; the Army servomechanism handbook states the rule for every gear of a train (AMCP 706-139, par. 15-1.13). A mass m moving at v = k ω stores ½ m k² ω², the energy of the inertia m k². The reflected load inertia is therefore J_ref = J_c + (J_o + m k²)/N², and the tool lists each part. A solid cylinder has J = π ρ L d⁴/32.

Inertia ratio. J_ref/J_M is compared with an entered limit. Makers of servo drives commonly advise keeping it under about 10 (lower for belts and other compliant drives) because a light rotor coupled through a spring to a heavy load is hard to tune; it is a control guideline, not a strength check, so the tool reports it separately from the pass or fail of torque and speed.

Forces on the load. With s the sign of the load velocity, the force the transmission must put on the load besides m a is Q = F_e + s (F_f + μ m g cos θ) + m g sin θ. Friction always opposes the motion. The external force F_e and the weight component keep their direction: positive values oppose the forward move and help the return move. A hoist is the case θ = 90°. For a rotary load Q = T_e + s T_f.

Efficiency and the direction of power. A screw or reducer of efficiency η delivers η times the power put into it, whichever side is driving. When the motor drives the load (force and velocity of the same sign) the motor must supply the load-side torque divided by η; when the load drives the motor (decelerating an inertia, lowering a weight) the motor receives only η times it. The tool applies the factor stage by stage, to the whole torque that crosses the stage including the inertial part:

T_o = f_s k (m a + Q) + J_o a/k,   T = (J_M + J_c) N a/k + f_g T_o/N + s T_d,   f = 1/η when driving, f = η when driven.

T_d is a constant drag at the motor shaft. This constant-efficiency model does not represent a self-locking screw (η below about 0.5), for which the load cannot drive the screw at all; there the returned torque printed is an upper bound and the motor must in fact push the load down. At standstill the motor holds k (F_e + m g sin θ)/N, taken without efficiency credit, or nothing if a brake is selected.

Move profiles

A move of distance X in time t from rest to rest is described by its ramp times. For a trapezoid with ramps t_a, t_d and cruise t_c the peak speed is v = X/(t_a/2 + t_c + t_d/2) and the accelerations are v/t_a and v/t_d. Two special cases with equal ramps are worth knowing, both for a load that is mostly inertia. The triangle (t_a = t/2, no cruise) needs the least acceleration, a = 4 X/t², and so the least peak torque, at the price of the highest speed, 2 X/t. The 1/3-1/3-1/3 trapezoid minimises ∫a² dt, hence the RMS torque and the heating: v = 1.5 X/t, a = 4.5 X/t². Its RMS torque is 8 % below the triangle's (ratio 1.089) and its peak torque 12.5 % above. The tool also offers a trapezoid whose ramps are chosen so that the motor just reaches an entered speed: t_a = t − X/v.

S-curve. With the S-curve option the acceleration is not switched on at once: it rises linearly to twice the mean value of the ramp at mid-ramp and falls linearly back to zero, which limits the jerk. Ramp times, speed and distance are unchanged; the inertial torque peaks at twice the trapezoid's and its RMS over the ramp is 2/√3 = 1.155 times as large.

Peak torque, RMS torque and power

On every segment the torque is linear in time (constant for a trapezoid), with a kink where an efficiency factor changes direction; the tool integrates T² exactly piece by piece and reports T_rms = √(∫T² dt/t_cycle) over the whole cycle including the dwell. A dwell cools the motor only if the motor is not holding a load. The peak shaft power is the largest T ω; the continuous sizing power is T_rms ω_pk. The table of segments splits the torque into its inertial, friction, gravity and external parts, and gives the work done by the motor and the work returned to it; over a cycle the net work equals the work against friction, drag, the external force and gravity plus the loss in the transmission, because the kinetic energy is zero at both ends of a move.

Checking a motor

The rotor inertia is part of what is accelerated, so each candidate is run through the cycle with its own J_M. A motor passes when T_pk ≤ its peak torque, T_rms ≤ its rated (continuous) torque and the peak speed ≤ its maximum speed; the tool prints the three ratios demand/capability. A real motor's limits fall with speed (the voltage limit, and the continuous torque at speed is below the stall value); the tool draws the rectangle of the entered values, and the voltage limit line when the constants are given, and plots the path of the duty cycle inside it. Where the maker gives a torque-speed curve, compare the path with that curve.

Current, voltage, temperature. For a permanent-magnet DC or brushless motor the torque is proportional to the current, T = K_t I, and the steady terminal voltage is the back-EMF plus the resistive drop, V = K_e ω + I R (AMCP 706-139, eqs. 14-2 and 14-19 with a fixed field; the inductive term L dI/dt is left out). In SI units K_e in V·s/rad equals K_t in N·m/A for a DC machine; data sheets quote K_e in volts per 1,000 rpm (1 V/krpm = 0.009549 V·s/rad), and for three-phase motors the constants depend on whether currents and voltages are RMS or peak, phase or line, so K_t, K_e and R must be taken in one consistent convention. The copper loss averaged over the cycle is I_rms² R and the winding rises ΔT = I_rms² R R_th above ambient. Because copper's resistance grows by 0.393 % per °C (National Bureau of Standards copper wire tables), the loss grows with the temperature; solving ΔT = P₀(1 + 0.00393 ΔT) R_th gives ΔT = P₀ R_th/(1 − 0.00393 P₀ R_th), printed as the hot-resistance rise. Iron, friction and drive losses are not included.

Choosing a ratio

For an output-shaft acceleration α, load-side inertia J_out = J_o + m k² and a constant load torque T_L, the motor torque as a function of the ratio is T(N) = (J_M + J_c) N α + (J_out α + T_L)/(η N). The first term grows with N (the motor must spin itself up faster), the second falls. The minimum is at N_opt = √[(J_out α + T_L)/(η (J_M + J_c) α)], where the two terms are equal; with no load torque and no loss this is N = √(J_out/J_M), the inertia match, at which the reflected load inertia equals the motor's own.

The Army handbook (AMCP 706-139, par. 16-3 and its appendix, after G. C. Newton) puts the same equation as a test of a motor. With T_MP, v_MP the motor's peak torque and speed and T_LP, v_LP, a_LP the load's peak torque, speed and acceleration, the motor can drive the load at ratio R if T_MP ≥ (J_L a_LP + T_LP)/R + R J_M a_LP (eq. 1) and v_MP ≥ R v_LP (eq. 2). Eq. 1 is a quadratic in R whose roots bound the usable ratios, the lower one being eq. 4, R_min = T_MP/(2 J_M a_LP) − √[T_MP²/(4 J_M² a_LP²) − (J_L a_LP + T_LP)/(J_M a_LP)]; eq. 2 caps R at v_MP/v_LP. A motor is adequate when R_min does not exceed that cap, which the handbook restates as a least ratio of motor peak power to load peak power P_LP = v_LP(J_L a_LP + T_LP) in terms of the time constants τ_LP = v_LP/a_LP and τ_MP = v_MP J_M/T_MP (eqs. 16-1 to 16-3). The tool prints the window for every candidate, for the ramp of the profile and with the load taken at the output shaft; the handbook's equations carry no reducer efficiency.

Regeneration

While the motor torque opposes the motion the shaft power is negative and the motor works as a generator. The electrical power at its terminals is V I = T ω K_e/K_t + I² R: the winding loss is taken out of what comes back, and at low speed nothing comes back at all. What does return charges the DC bus of the drive. If the drive cannot return it to the mains, the bus capacitors must absorb it between the working voltage and the over-voltage limit, C = 2 E/(V_lim² − V_bus²), or a brake resistor must burn it: its mean power is the returned energy per cycle over the cycle time, and it must be small enough in ohms, R ≤ V_lim²/P_pk, to carry the peak returned power at the limit voltage. Drive losses and the energy the drive itself consumes are ignored, which is conservative for the resistor.

AC induction motor: torque, slip and starting time

Full-load torque. Power is torque times angular speed, so T = P/ω. One horsepower is 33,000 ft-lb per minute, which gives T (lb-ft) = 33,000 hp/(2π rpm) = 5,252 hp/rpm; in SI, T (N·m) = 9,549 kW/rpm. The synchronous speed is 120 f/p rpm for p poles at f hertz, and the slip is (n_s − n)/n_s.

Service factor. The service factor on the nameplate is the multiple of rated power the motor may carry under nameplate conditions; the U.S. Department of Energy's motor sourcebook advises against using it for continuous running. The tool compares the entered load power with the rated power and with rated power times the service factor.

Starting. Across the line the motor follows its torque-speed curve, from the locked-rotor torque through the pull-up minimum and the breakdown maximum to the running point, where the motor torque equals the load torque. Newton's law for the shaft, J dω/dt = T_m(ω) − T_L(ω), gives the starting time t = ∫ J dω/(T_m − T_L), with J the rotor inertia plus the driven inertia divided by the square of the speed ratio (WK² referred to the motor). The tool integrates along the entered curve, taken as straight lines between the entered points, against a load torque T_L = T_run[b + (1 − b)(n/n_run)^p] with p = 0, 1 or 2, up to an entered fraction of the running speed (the last part is approached asymptotically). The common hand formula uses one mean accelerating torque, t = J Δω/T_acc, or in customary units t = WK² Δn/(307.2 T_acc) with WK² in lb-ft² (the constant is 32.174 × 60/2π and is usually quoted as 308); it underestimates when the accelerating torque dips, as it does at the pull-up point of a heavily loaded start. If the motor curve falls to the load curve before the running speed the motor hangs there and the tool says so.

Rotor heating. During the start the power crossing the air gap is T_m ω_s while the shaft receives T_m ω; the difference, T_m(ω_s − ω), is dissipated in the rotor. Integrated over a start with no load torque it equals the kinetic energy the rotor and load end up with (referred to synchronous speed), whatever the shape of the curve; a load torque lengthens the start and adds to it. The DOE motor guide gives about 12 seconds as the longest acceleration a motor of 200 hp or less tolerates per start, and repeats NEMA's limits on starts per hour; the tool compares the computed time with the 12 seconds. NEMA MG 1 defines the Design A, B, C and D torque-speed classes; their limiting values are in that standard and the motor maker's curve should be entered here.

What the tool does not do

No compliance, backlash or resonance of the drive train; no critical speed, buckling or life of a screw; no belt tension or tooth strength; no thermal model beyond one resistance to ambient; no field weakening, inductance or drive current limits other than through the entered peak torque; no reduced-voltage or variable-frequency starting of the induction motor. The default axis (50 kg on a 20 mm lead ball screw, 0.5 m in 0.6 s) and its three candidate motors are illustrative values, not catalogue data.