# Optimal Motor Sizing

This technical note describes optimal motor sizing process for a servo motor. A motor model is provided, and thermal issues are discussed. Torque speed curves and typical load torques are reviewed. Both motion velocity and torque profiles are presented. The calculation of peak and continuous torque is outlined. The paper concludes with a worked example.

## Introduction

To ensure optimal servo control, an appropriately sized motor and drive must be selected. Under sizing results in inferior performance and motor overheating. An oversized motor/drive adds unnecessary cost. The system should be sized to provide the required torque and speed with sufficient margins to allow for variations in load parameters.

To select a correct drive/motor, the load must be characterized, and the motion profile defined. In some applications, the maximum allowable motor temperature is the dominant parameter.

A frameless motor or a framed motor that does not connect to the load via a flexible shaft/coupling is assumed. Load/motor inertia ratio manipulation to mitigate mechanical resonance is therefore not a factor. For more information, see our technical note, **Frameless Motor Advantages**.

## Motor Model & Fundamental Equations

A motor can be modelled as shown in figure 1. R and L represent the resistance and inductance of the motor windings. EMF represents the generator action of the motor as it rotates. Back-EMF increases with speed and opposes the bus voltage applied to the motor windings.

Figure 1

Two key equations define the characteristics of the servomotor and drive:

Torque = K_{T} x I

Back-EMF = K_{E} x speed

Both K_{T} and K_{E} are proportional to rotor magnet strength, magnetic path reluctance, number of stator coils, length of coil turns, number of rotor poles. A motor designed to deliver high torque per amp also generates a high back-EMF. For a given motor the magnet strength and magnetic path reluctance are fixed. Motor manufacturers, however, typically offer high K_{T} / K_{E} windings for maximum torque per amp when back-EMF is not an issue (high bus voltage and/or low speed) and low K_{T} / K_{E} windings when back-EMF is a factor (high speed and/or low bus voltage).

## Motor Losses

When current (I) is driven through the motor, the winding resistance (R) generates heat defined by the power loss equation, I^{2}R. Typically referred to as copper losses, I^{2}R losses are independent of speed.

As the rotor turns, the magnetic field on the stator rotates to maintain optimal orientation with the rotor magnetic field. The changing magnetic field induces eddy currents in the motor iron which in turn cause further heat losses. As the stator magnetic field rotates, the magnetic field at a specific point is constantly changing polarity. The reorientation of polarity in the motor iron requires energy and some of that energy is lost in the form of intermolecular friction. These losses are known as hysteresis losses. Often referred to as core losses, both eddy current and hysteresis losses, are proportional to speed.

Temperature increase due to copper/core losses depends on the thermal resistance (ability to transfer heat) of the motor components and mounting surface. The motor power output is constrained so that the maximum allowable temperature of the motor windings is not exceeded.

## Torque-Speed Curves

Before sizing a servo system, it is necessary to understand the capabilities of the motor and drive. These can be characterized in a torque-speed curve. An example is shown in figure 2.

Figure 2

Continuous Duty |
Continuous duty zone is defined by the copper/core losses and thermal characteristics of the installation. The upper limit decreases with speed. As core losses increase with speed, copper losses must be reduced to compensate. Less current can supplied to the windings, reducing torque. The RMS (root-mean-square) value of the torque must be in this region. |

Stall Torque |
The maximum continuous torque available when eddy current and hysteresis losses can be neglected and losses are dominated by the power equation, I^{2}R. Stall torque is typically specified at zero speed. |

Intermittent Duty |
The motor can operate in the intermittent zone during acceleration and deceleration or to overcome a load disturbance. |

Peak Torque |
The peak torque available is defined in different ways. A motor supplier defines it as the maximum torque before magnetic saturation. A drive/motor supplier defines peak torque as the motor torque constant, K_{T}, times the peak current available from the drive. |

Back-EMF Rolloff |
The motor back-EMF increases with speed. At rolloff, the generated voltage sufficiently opposes the bus voltage to reduce the current which can be driven through the motor windings. This causes the torque to rolloff. When the back-EMF voltage equals the bus voltage, current and hence torque is zero. |

Rated Speed |
The rated speed is the speed at which the output power is a maximum. Since power = torque x speed, the rated speed is approximately at the point when the continuous torque experiences back-EMF rolloff. |

Torque-speed curves typically assume a 20-25°C ambient temperature and a motor heatsink approximating a “standard plate” (25 x 25 x 0.6 cm aluminum). If the motor is operated outside these parameters, the torque-speed curves must be derated. In this case a thermal analysis of the installation is recommended.

When comparing torque/speed curves of different motors be sure to check the maximum rated temperature of each motor. Derating is proportional to the difference between the maximum and ambient temperatures. A motor with a higher maximum temperature rating will require less derating.

## Load Parameters

Four load parameters must be specified at the motor shaft. If there is a gearbox, leadscrew or belt between the motor and load, the mechanical advantage and efficiency of these devices should be considered. See our technical note, Frameless Motor Advantages for more details.

Inertia (J) |
Inertia is the resistance of an object to acceleration. It is the rotational equivalent of mass. Newton’s law states that the force required to accelerate a mass is the mass times the acceleration (F = m.a). For rotation, the equivalent equation is T = J.α. This torque will be referred to as T_{J}. |

Friction Torque (T_{F}) |
Friction torque is the torque required to overcome static friction between mechanical components. Friction torque is independent of speed. |

Viscous Torque (T_{V}) |
Viscous torque is the torque required to move an object through a fluid. Viscous torque is proportional to speed. There is a small amount of viscous torque related to bearing lubrication, but it is not relevant here. |

Load Torque (T_{L}) |
Load torque is the torque required to overcome gravity in vertical applications or tension in web handling. |

## Motion Profiles

To execute a move to a new position, a velocity profile must be defined. There are tradeoffs between acceleration time and run speed to consider. A good place to start is to use a trapezoidal move with equal times for acceleration, run speed and deceleration as shown in figure 3. The total distance is the area under the curve. Run speed is easy to calculate as it is half the move distance divided by one third of the total move time. An alternative approach, for very short moves, is the triangular profile with lower acceleration and higher run speed. The trapezoidal profile is recommended.

The run speed defined by the motion profile must be less than the rated speed of the torque-speed curve. If not, the profile can be modified to accelerate/decelerate faster to a lower speed. Run speed time is longer. It may now be possible to move the required distance in the desired time. As acceleration is increased, care should be taken not to exceed the peak torque rating. Sometimes it is not possible to move the target distance in the desired time. For more on this go to, *Velocity Profiles Revisited*.

Figure 3

A trapezoidal profile has a step function acceleration profile as shown in figure 4. Rate of change of acceleration is known as jerk. High jerk, infinite in this case, causes wear and can excite mechanical resonance. An S-curve velocity profile can be employed to limit jerk. Up to twice the peak torque is required to accelerate in the same time. Most controllers can shape the acceleration to trade off jerk and peak torque.

Figure 4

## Peak Torque Calculation

The peak torque is defined by the torque required to accelerate and decelerate. These torques may be different for equal acceleration and deceleration times. For example, TF opposes acceleration but aids deceleration. The following equations can be used to calculate accel/decel torque. Note that the convention is positive torque for clockwise motion.

Move CW T_{accel} = T_{J} + T_{F} + T_{V } T_{decel} = = -T_{J} + T_{F} + T_{V }

Move CCW T_{accel} = -T_{J} – T_{F} – T_{V} T_{decel} = = +T_{J} – T_{F} – T_{V}

Move Up T_{accel} = T_{J} + T_{F} + T_{V} + T_{L} T_{decel} = = -T_{J} + T_{F} + T_{V} + T_{L}

Move Down T_{accel} = -T_{J} – T_{F} – T_{V} + T_{L} T_{decel} = = +T_{J} – T_{F} – T_{V} + T_{L}

T_{F}, T_{V}, and T_{L} are measured or calculated. T_{J} depends on the inertia and the velocity profile. T_{J} is equal to the inertia (J) times the angular acceleration/deceleration (α). α is equal to the angular velocity divided by the time to accelerate/decelerate. If S-curve profiles are used T_{J} should be increased accordingly.

The calculated peak torque must fall inside the peak torque line on the torque/speed curve. A safety margin should exist to accommodate changes in load torques as the machine ages. The magnitude of the margin should be based on observations of the specific application.

## Torque Profiles

Figure 5 shows a repeated move in the same direction. Friction torque is present which increases the torque to accelerate and reduces the torque to decelerate. Friction torque is present at run speed but not during the hold time between moves. A move in the opposite direction simply inverts the torque profile.

Figure 5

Torque profiles of moves in opposite directions become more complicated when gravity is involved.

Figure 6

In figure 6 frictional torque has been replaced by a gravity torque of equal magnitude. For a move against gravity torque must now be supplied at hold time to maintain position. Because of the accelerating effect of gravity, the subsequent downward move requires less torque to accelerate, the same positive torque at run speed, and the same decelerating torque as was required during acceleration in the opposite direction.

## Continuous Torque (T_{RMS}) Calculation

The T_{RMS} formula for a repeated move executed by a trapezoidal velocity profile is:

The cycle time, t_{c}, incorporates all the move profile times plus the hold time.

There may be a sequence of moves rather than a single repeated move. The calculation must then include all the T^{2}t segments. The cycle time is then the time of the entire sequence.

T_{RMS} must fall in the continuous region of the torque/speed curve. As with peak torque, an application dependent safety margin should be included.

## Velocity Profiles Revisited

In most positioning applications load torques are minimized by design. T_{RMS} is therefore typically dominated by accel/decel torques. The velocity profile clearly plays a key role in motor sizing. When the profile does not fit any torque/speed curve alternatives there are limited options when using a drive/motor system. The drive usually operates from AC power, so bus voltage selection is limited. When using drive motor components, it is possible to create a custom torque/speed curve by selecting a bus voltage, and motor K_{t}/K_{e} which are optimal for the application.

## Units

It is critical to be consistent with units of measure in motor sizing. This paper uses the following units:

Torque |
T | Nm |

Angular Distance |
θ | radians (rad) |

Angular Velocity |
ω | radians per second (rads-1) |

Angular Acceleration |
α | radians per second per second (rads-2) |

Inertia |
J | kilogram meter squared (kgm2) |

#### Example

Problem Statement Direct Drive Rotary

Move: 2 revs in 600 msec

Hold time (t_{d}): 800 msec

Load inertia (J_{L}): 0.2 kgm^{2}

Friction (T_{L}): 0.5 Nm

Bus voltage (V_{BUS}): 325 VDC

## Solution

Try 3-step trapezoidal velocity profile: t_{a} = t_{r} = t_{d} = 0.2 s

Run speed = half move distance / t_{r} = 1 / 0.2 = 5 revs/sec

Angular velocity (ω) = 5 x 2 x _{π} = 31.42 rads^{-1}

Angular acceleration (α) = ω / t_{a} = 31.42 / 0.2 = 157.1 rads^{-2}

Torque to accel/decel inertia T_{J} = J x α

Only J_{L} is known. The load inertia is high, and the goal is a direct drive solution. For now, assume that the motor is no more than 1 tenth of J_{L}.

J = 0.2 + 0.02 kgm^{2}

T_{J} = 0.22 x 157.1 rads^{-2} = 34.56 Nm

Load torque due to friction (T_{L}) opposes acceleration and aids deceleration

T_{a} = T_{J} + T_{L} = 34.56 + 1 = 35.56 Nm

T_{d} = T_{J} – T_{L} = 34.56 – 1 = 33.56 Nm

As there is no gravity or tension, the torque during hold time T_{h} = 0 Nm

**T _{PEAK} = 35.56 Nm**

**T _{RMS} = 18.5 Nm**

Now check on the back-EMF at rated speed to ensure sufficient current can be driven though the windings to generate the peak torque.

V

_{EMF}= K

_{e}x Speed

Maximum current = (bus voltage – back-EMF / winding resistance)

In conclusion, the methodology behind sizing a motor is straightforward. Step two is more complex, and it may be necessary to use a best approximation.

- Define the motion profile.
- Determine load parameters by measurement, technical data, or calculation.
- Calculate the accel/run/decel/hold torques.
- Calculate RMS torque.
- Check the back-EMF at run speed to ensure sufficient current can be applied.
- Be sure to leave a safety margin and double check units of measure.

For optimal motor sizing, sometimes the calculated torques and run speed do not fit any of the published torque/speed curves. It is normal to use an iterative process to get satisfactory results. It may be necessary to consider a custom bus voltage and/or motor winding to meet performance requirements.