To calculate linear motor requirements, you must determine both the peak force and the RMS (Root Mean Square) force across your entire motion profile. The golden rule of thumb for sizing: select a motor with a continuous force rating at least 1.5 times your calculated RMS force to account for thermal derating, ambient temperature spikes, and unexpected friction. Peak force dictates the physical size and magnet mass of the motor, while RMS force dictates the coil winding gauge and thermal limits.
The Core Physics: Sizing Rule of Thumb and Worked Example
Sizing a linear motor—whether it is a true Linear Synchronous Motor (LSM), a voice coil, or a rotary motor coupled to a ballscrew—requires mapping the forces over time. A common mistake on the bench is sizing purely for the peak acceleration force and ignoring the thermal buildup during the cruise and deceleration phases.
Worked Load Example: Horizontal CNC Payload
Let us calculate the requirements for moving a 10 kg payload horizontally on a steel linear rail. We will assume a 25°C ambient environment and a friction coefficient ($\mu$) of 0.1.
- Mass ($m$): 10 kg
- Acceleration ($a$): 2 m/s²
- Gravity ($g$): 9.81 m/s²
- Friction Force ($F_{fric}$): $\mu \times m \times g = 0.1 \times 10 \times 9.81 = 9.81$ N
- Acceleration Force ($F_{accel}$): $m \times a = 10 \times 2 = 20$ N
1. Calculate Peak Force:
During acceleration, the motor must overcome both inertia and friction.
$F_{peak} = F_{accel} + F_{fric} = 20 + 9.81 = 29.81$ N.
2. Map the Motion Profile:
Assume a trapezoidal profile: Accelerate for 0.5s, cruise for 1.0s, decelerate for 0.5s.
- Accel Phase (0.5s): Force = 29.81 N
- Cruise Phase (1.0s): Force = 9.81 N (only overcoming friction)
- Decel Phase (0.5s): Force = -10.19 N (motor acts as brake, magnitude is 10.19 N)
3. Calculate RMS Force:
The formula for RMS force is $F_{rms} = \sqrt{\frac{F_1^2 t_1 + F_2^2 t_2 + F_3^2 t_3}{t_1 + t_2 + t_3}}$.
- $29.81^2 \times 0.5 = 444.3$
- $9.81^2 \times 1.0 = 96.2$
- $10.19^2 \times 0.5 = 51.9$
- Sum = 592.4. Divide by total time (2.0s) = 296.2.
- $\sqrt{296.2} = 17.2$ N.
4. Apply the Sizing Margin:
$17.2 \text{ N (RMS)} \times 1.5 \text{ (Safety Margin)} = 25.8$ N.
You must select a motor with a continuous force rating of at least 25.8 N and a peak force rating exceeding 29.81 N. According to Linear Motion Tips, failing to apply this thermal margin is the leading cause of premature coil insulation failure in custom automation rigs.
Motor Type Comparison: Which Fits Your Load Profile?
Not all linear motion architectures are created equal. Treating a stepper-driven lead screw as interchangeable with a true 3-phase linear servo is a fast track to destroyed workpieces and missed steps. Here is how the primary architectures compare when you calculate linear motor thrust requirements.
| Motor Architecture | Force Curve Profile | Control & Feedback Needs | Relative Cost & Use Case |
|---|---|---|---|
| Stepper + Lead Screw | High holding torque at zero speed; force drops sharply past 500 RPM. Prone to mid-range resonance. | Open-loop chopper drive (e.g., TMC2209). No encoder required unless running closed-loop. | Low ($20-$60). Best for low-speed, high-static-load applications like 3D printer Z-axes. |
| AC Servo + Ballscrew | Flat torque curve up to rated speed (typically 3000 RPM). Excellent dynamic response. | Closed-loop FOC drive with high-resolution absolute encoder. Requires tuning. | Medium-High ($300-$800). Standard for CNC mills and industrial pick-and-place. |
| True Linear Synchronous Motor (LSM) | Direct-drive force. Zero backlash, infinite resolution limited only by encoder. Continuous force limited by coil thermal mass. | Closed-loop FOC drive with linear encoder (e.g., Renishaw or Heidenhain). Demands high servo update rates. | High ($800-$2500+). Used in semiconductor lithography and high-speed laser cutting. |
| Voice Coil Actuator | Force is directly proportional to current. Extremely fast, but very short stroke (typically < 50mm). | Current-mode amplifier. Often paired with LVDT for position feedback. | Medium ($150-$400). Ideal for precision pressing, optical image stabilization, and valve control. |
Wiring, Terminals, and Driver Demands
If your calculation points you toward a 3-phase Brushless Linear Synchronous Motor (LSM), you cannot wire it like a standard DC motor or a bipolar stepper. The physical layout is essentially a rotary BLDC motor 'unrolled', and it demands specific terminal identification and Field Oriented Control (FOC).
Terminal Identification for a 3-Phase LSM
- Power Phases (U, V, W): These carry the main drive current. For a motor in the 25N continuous / 5A range, use 16 AWG flexible silicone or THHN wire. Keep these leads as short as possible to minimize inductance, which can cause voltage spikes during rapid deceleration.
- Hall Effect Sensors (Hu, Hv, Hw, +5V, GND): Used for initial commutation angle detection. Wire these with 24 AWG shielded cable. The shield must be grounded at the drive end only to prevent ground loops.
- Linear Encoder (A, B, Z or Sin/Cos, +5V, GND): The lifeblood of an LSM. Because there is no mechanical gearing to absorb vibration, the encoder resolution dictates your positioning accuracy. Use high-flex, double-shielded encoder cable (like Belden 8723 equivalent) routed away from the U/V/W power lines.
Failure Signatures: Diagnosing Hum, Overheat, and Stall
When your calculated linear motor setup fails on the bench, the symptoms usually manifest in three distinct ways. Recognizing these signatures saves hours of oscilloscope debugging.
- The 'Angry Hum' (Commutation Mismatch): If the motor vibrates violently in place and emits a loud hum without moving, your commutation angle is wrong. This almost always means two of your Hall sensor wires (Hu, Hv, Hw) are swapped, or the encoder A/B channels are reversed. The FOC algorithm is pushing current into the wrong stator teeth. Swap two Hall wires and re-run the calibration routine.
- Thermal Overheat (RMS Exceeded): The motor moves correctly but triggers a thermal fault after 3 minutes of operation. This indicates your RMS force calculation was flawed, or the motor's thermal path to the chassis is choked. True linear motors rely heavily on conductive cooling through their aluminum mounting base. If you mounted the coil array on a plastic bracket or used thermal paste instead of a proper thermal pad, the heat has nowhere to go.
- Hard Stall (Peak Force Exceeded or Mechanical Binding): The drive reports a 'following error' or 'stall' fault mid-move. Disconnect the power and push the carriage by hand. If it feels gritty, your linear rail is binding (mechanical issue). If it feels smooth, your load mass was underestimated, or the acceleration ramp is too aggressive, causing the peak force demand to exceed the drive's current limit.
Frequently Asked Questions
How to calculate linear motor speed from voltage and pole pitch?
For a true linear synchronous motor, the synchronous speed ($v$) is determined by the frequency of the drive current ($f$) and the pole pitch ($p$) of the magnet track. The formula is $v = 2 \times p \times f$. For example, if your magnet track has a pole pitch of 15 mm (0.015 m) and your FOC drive is outputting a 20 Hz electrical frequency, the linear speed is $2 \times 0.015 \times 20 = 0.6$ meters per second. Note that the bus voltage must be high enough to overcome the Back-EMF generated at that speed; otherwise, the motor will lose synchronism and stall.
How to calculate linear motor thrust for vertical lifting?
When mounting a linear motor vertically, gravity becomes a constant opposing force. You must add the full weight of the payload and the moving carriage ($m \times g$) to your continuous force requirement. Using our previous 10 kg example, the vertical gravity load is $10 \times 9.81 = 98.1$ N. Even when the motor is holding still (zero acceleration), it must continuously output 98.1 N just to prevent the load from falling. This drastically increases the RMS force and usually requires a motor with a much larger coil mass or a counterbalance mechanism (like a pneumatic cylinder or gas spring) to offset the static load.
How to calculate linear motor power consumption from RMS force?
Electrical power consumption in a linear motor is primarily driven by $I^2R$ (copper) losses in the windings. Once you have your RMS force ($F_{rms}$), you can find the RMS current ($I_{rms}$) by dividing the RMS force by the motor's Force Constant ($K_f$, typically provided in N/A on the datasheet). For instance, if $F_{rms} = 17.2$ N and $K_f = 4.5$ N/A, your $I_{rms}$ is 3.82 A. If the motor's phase-to-phase resistance ($R$) is 1.2 Ohms, the continuous heat dissipation (power lost as heat) is roughly $I_{rms}^2 \times R = 3.82^2 \times 1.2 = 17.5$ Watts. This is the thermal load your heatsink or chassis must absorb.
Can I use a stepper motor driver to run a 3-phase linear motor?
No. Stepper drivers (like the A4988 or TMC2209) use simple chopper circuits designed to sequence DC current through two phases (bipolar) in discrete microsteps. A 3-phase linear motor requires sinusoidal Field Oriented Control (FOC) to smoothly rotate the magnetic vector across three phases. Applying a square-wave chopper signal to an LSM will result in massive torque ripple, severe cogging, and likely a blown driver MOSFET due to inductive kickback. Always use a dedicated 3-phase BLDC/servo drive.
