KEVOS® MAKEOS · Power Transmission
Permanent-Magnet Control Motor Constants (Torque and Voltage Constants, Damping, Time Constant, Resonance)
From the rated voltage, winding resistance, stall torque, no-load speed and inertia of a permanent-magnet DC control motor, what are its torque and voltage constants, motor constant, damping and mechanical time constant, what torque does the amplifier current limit allow, and does the torsional resonance with my load fall inside the control bandwidth?
Derived parameters of a permanent-magnet DC control motor by the definitions the handbook gives for its idealised straight-line torque–speed characteristic: stall current V ÷ R, torque constant K_T = T_STo ÷ I_STo, voltage constant K_E = V ÷ ω_NL, motor constant K_M = K_T ÷ √R, damping D_M = T_STo ÷ ω_NL and mechanical time constant T_M = J_M ÷ D_M, with the torque at the amplifier current limit and at rated current, the speeds at those torques on the characteristic, the time constant with the load inertia, and the torsional resonance f_n = √(K ÷ J_L) ÷ 2π compared with the control bandwidth.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Motor data | ||
| Rated voltage | V | Rated voltage of the motor, from its data sheet. |
| Winding resistance | Ω | Resistance of the armature winding, terminal to terminal, at the working temperature. |
| Stall torque with no current limiting | N·m | Torque at stall with no current limiting, T_STo, the top of the straight-line characteristic. |
| No-load speed | rpm | Speed at no load, ω_NL, the end of the straight-line characteristic. |
| Motor (rotor) inertia | kg·m² | Rotor inertia of the motor, J_M. |
| Amplifier | ||
| Current limit set in the amplifier | A | Current limit I_L set in the amplifier; it caps the motor torque. |
| Rated (continuous) current | A | Rated current I_R the motor may carry continuously. |
| Load and control | ||
| Load inertia referred to the motor shaft | kg·m² | Inertia of the load referred to the motor shaft (load inertia divided by the square of any gear ratio). |
| Torsional stiffness between motor and load | N·m/rad | Torsional stiffness of the shafts and coupling between motor and load. |
| Bandwidth of the control system | Hz | Bandwidth of the closed-loop control system. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
Motor data
What this means: Rated voltage
- What it is
- Rated voltage of the permanent-magnet DC motor, from its data sheet.
- Units and format
- Enter a number in V: volts. The form accepts 0.000001 V to 2000 V. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 48 V as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the stall current and the voltage constant.
- Get it right
- Use the voltage the amplifier actually supplies at full output.
What this means: Winding resistance
- What it is
- Resistance of the armature winding, measured terminal to terminal.
- Units and format
- Enter a number in Ω: Ω. The form accepts 0.000001 Ω to 10000 Ω. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.2 Ω as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the stall current and the motor constant.
- Get it right
- Use the hot resistance; copper resistance rises about 0.4 % per kelvin.
What this means: Stall torque with no current limiting
- What it is
- Stall torque of the motor with no current limiting.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0.000001 N·m to 100000 N·m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the torque constant, the damping and the characteristic.
- Get it right
- Take it from the data sheet; it is the top of the straight torque–speed line.
What this means: No-load speed
- What it is
- Speed of the motor at no load and rated voltage.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 200000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4580 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the voltage constant and the damping.
- Get it right
- Take it from the data sheet; it is the end of the straight torque–speed line.
What this means: Motor (rotor) inertia
- What it is
- Rotor inertia of the motor itself, from its data sheet.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0.000001 kg·m² to 1000 kg·m². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.00012 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the mechanical time constant.
- Get it right
- Convert from g·cm² by dividing by 10⁷, or from oz·in·s² with the maker’s factor.
Amplifier
What this means: Current limit set in the amplifier
- What it is
- Current limit set in the servo amplifier for peak torque.
- Units and format
- Enter a number in A: amperes. The form accepts 0.000001 A to 10000 A. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 A as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the torque available to accelerate the load.
- Get it right
- Read it from the amplifier settings; it is usually above the rated current.
What this means: Rated (continuous) current
- What it is
- Rated current the motor may carry continuously without overheating.
- Units and format
- Enter a number in A: amperes. The form accepts 0.000001 A to 10000 A. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 6 A as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the rated torque and the speed at it.
- Get it right
- Take it from the motor data sheet.
Load and control
What this means: Load inertia referred to the motor shaft
- What it is
- Inertia of the driven load referred to the motor shaft.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0.000001 kg·m² to 1000000 kg·m². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.0004 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Lengthens the time constant and lowers the torsional resonance.
- Get it right
- Divide the load inertia by the square of the gear ratio between motor and load.
What this means: Torsional stiffness between motor and load
- What it is
- Torsional stiffness of the shafts and coupling between the motor and the load.
- Units and format
- Enter a number in N·m/rad: N·m/rad. The form accepts 0.000001 N·m/rad or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 N·m/rad as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A stiffer drive raises the torsional resonance.
- Get it right
- Add the compliances of each shaft and coupling in series; take couplings from the maker’s data.
What this means: Bandwidth of the control system
- What it is
- Bandwidth of the closed control loop driving the motor.
- Units and format
- Enter a number in Hz: hertz (cycles per second). The form accepts 0.000001 Hz to 100000 Hz. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 Hz as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the resonance check.
- Get it right
- Take it from the tuned loop’s response; the velocity loop is usually the relevant one.
Method and formulas
- ω_NL = no-load speed × 2π ÷ 60 rad/s; I_STo = V ÷ R.
- K_T = T_STo ÷ I_STo (N·m/A); K_E = V ÷ ω_NL (V·s/rad); K_M = K_T ÷ √R.
- D_M = T_STo ÷ ω_NL (N·m·s/rad); T_M = J_M ÷ D_M seconds, and (J_M + J_L) ÷ D_M with the load.
- Torque at a current I = K_T × I (the torque limit T_L at I_L, never above T_STo; T_R at I_R); speed on the straight-line characteristic ω = ω_NL (1 − T ÷ T_STo).
- Peak acceleration at the torque limit = T_L ÷ (J_M + J_L); torsional resonance f_n = √(K ÷ J_L) ÷ 2π Hz.
- Example: a 48 V motor with 1.2 Ω and 4 N·m stall torque has K_T = 0.1 N·m/A and a time constant of 14.388 ms; with the load the torsional resonance is 355.88 Hz against a 40 Hz bandwidth.
Limitations
- The handbook’s characteristic is idealised as a straight line from stall torque to no-load speed; real motors depart from it near stall and at high temperature.
- Peak torque may be limited by demagnetisation in motors without rare-earth magnets; check the maker’s peak torque before using the current limit.
- Duty-cycle and thermal checks (the rated current over the cycle) are not covered.
- The resonance treats the motor as fixed and the load as one inertia on one stiffness; several modes need a fuller model.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Work out the constants a motor data sheet does not list
- Check the torque a current limit leaves for acceleration
- Check whether the drive’s torsional resonance falls in the control bandwidth
Who it is for
- Controls engineers
- Machine designers
- Automation technicians
What you get
- Stall current I_STo = V ÷ R
- Torque constant K_T and voltage constant K_E
- Motor constant K_M = K_T ÷ √R
- Motor damping D_M and mechanical time constant T_M
- Torque and speed at the current limit and at rated current
- Time constant with the load and torsional resonance against the bandwidth
Interpreting the result
- The current limit, not the stall torque, sets the torque available to accelerate the load.
- If the torsional resonance lies inside the control bandwidth, self-sustained oscillation is likely: stiffen the drive, reduce the load inertia or filter the loop (a notch at the resonance).
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 3413–3414. Topic reference; no source tables or recommended settings are reproduced.
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