Servo Loop Bandwidth, Motor Sizing and Power

DissipationMark Holcomb

Senior Engineer, Motion Control Specialist

Celera Motion

Professional Background

University of Buffalo, 1994

MS ME

Active Systems product development.

(CSA\Moog, 1994-1997)

Owner- Dynamic Systems Engineering supported KLA-

Tencor with servo modeling\tuning, vibration characterization, data

acquisition(1997-2009, 2011-2016)

Modeling control of thrust actuation devices onboard

kinetic warheads (AeroJet Corp, 2009-2011)

Celera MotionSenior Engineer, Motion Control Specialist.Automation software and hardware, motor

modeling\analysis software, supporting customers with their motion control applications

2016-Present

Overview

How are servo loop bandwidth, motor sizing and power dissipation all related?

• Background loop gain, bandwidth and power basics.

• Move and Settle Application servo bandwidth is not related to motor sizing.

• Disturbance Tracking Application servo bandwidth directly affects motor sizing.

In each case, power dissipation is ultimately the primary driver for motor sizing.

What is Loop Gain?

• Loop gain is the multiplication of each gain block in the loop (the plant, sensor, motor, motor drive and control gain).

• Higher loop gain higher bandwidth.

• Once the maximum loop gain is identified (stability met with minimal margin), any individual gain can be lower or higher, as long as the total gain is conserved.

There is not a unique solution for each gain that results in a targeted bandwidth.

MechanicsMotorAmplifierPID

Sensor

Typical Servo Block Diagram

Loop Gain and Stability Example

Higher resolution sensors and higher bandwidth• Sensor gain for a system is 1 count per micron.

• The PID control is optimized meets gain and phase margin requirements with minimal margin (see next slide).

• The sensor gain is then changed to 10 counts per micron. • PID gain set must be reduced by 10x to meet stability requirements (see next

slide ).

• In cases where bandwidth is limited by stability, increasing sensor resolution will not result in higher bandwidth.

Increased sensor resolution does not always improve bandwidth.

Higher Resolution Sensor Example- Bode Plot

Gain Margin requirementis violated. Loop Gain mustbe reduced.

-180 degrees

0 dB. Freq.

PhaseMargin

With 10x sensor gain

With 10x Sensor gain, stability Gain Margin rule is violated. Loop Gain must be reduced by 10x to maintain stability.

Loop TF w/nominal sensor gain

Loop TF w/10x sensor gain M

agn

itu

de

(dB

)

Frequency (Hz)

Ph

ase

(deg

.)6 dB of required Gain Margin.

1

23

4

5

“Strong Motors”

• High Km motors do not always result in higher bandwidths.

• Higher bandwidths are often limited by stability, not by the motor’s Km value.

• Stability is limited by mechanical resonances or phase delays in the sensor\actuator path.

• Higher Km motor is more efficient (Nm/sqrt(w)).

• Higher bandwidths higher frequencies higher accelerations higher required motor forces higher dissipated power higher required Km.

Bandwidth can be limited by stability or by allowable power dissipation.

Motor Power

• Total Power = Thermal + Mechanical for motor sizing we only consider thermal power.

• Power on its own is instantaneous value, meaning at any point in time, the system has certain power in watts (J/sec).

• The thermal power at any point (t) during a move is; 𝑃𝑡 = 𝐼𝑡2 ∗ 𝑅.

• This is not the power loss over a duty cycle, which is typically how motors are sized.

• The term “Average Power” is a commonly used term for sizing motors.

• Average Power is based on the current needed to supply the RMS force (which is time weighted value).

Duty Cycle and Continuous Force (RMS)

• Duty Cycle is calculated by taking time spent applying force and dividing by the whole time.

• DC% = (t1 + t2)/(t1+t2+Dwell)*100.

• 𝐹𝑟𝑚𝑠 = 𝐹𝑐 =𝐹12∗𝑡1+𝐹22∗𝑡2+ …

𝑇𝑜𝑡𝑎𝑙 𝑇𝑖𝑚𝑒

• When 𝐹1 = 𝐹2, etc. 𝐹𝑟𝑚𝑠 = 𝐷𝐶 ∗ 𝐹1

Forc

e (N

)

F1

F2

Time Accel\Decel

Move and Settle Application

Bandwidth and Trajectory

• 50 Hz vs. 100 Hz bandwidth create similar motion.

• Duty Cycle = Time accelerating/Total Time = ~74%

Acceleration is dominated by the command trajectory, not the servo bandwidth.

8.5 m/s^2 rms

8.3 m/s^2 rms

t1t2

Acc

eler

atio

n (

m/s

^2)

Time (sec.)

Black- Accel CmdRed- 50 Hz BWBlue- 100 Hz BW

Acceleration vs. Time

F1

-F1

t3

Move and Settle Power Calculation

Dissipated Power (Pd)

𝑃𝑑 =( 𝐷𝐶∗𝐹1)2

𝐾𝑚2 =𝐹𝑜𝑟𝑐𝑒𝑅𝑀𝑆

2

𝐾𝑚2 ,

where; 𝐹𝑅𝑀𝑆 = 𝐴𝑅𝑀𝑆 ∗ mass

(from example, assume mass of 1 Kg

Pd = (8.5 ∗1)2

𝐾𝑚2

𝐾𝑚 = 72

𝑃𝑑2

Solve for Km based on your known Pd requirement.

Disturbance Tracking Application

Example of a Tracking System

M1(20 Kg)

M2 (1Kg) Goal minimize Z2(t), while M1 has motion Z1(t).

Z1(t) = 16 um RMS (uniform vibe spectrum up to 1 kHz)

Z2 (t)Focus Signal

M1(20 Kg)

Disturbance Tracking- Continuous Force

• The loop is showing ~50 Hz 0 dB crossing “servo bandwidth”.

• Motor force RMS• 50 Hz case Frms = 18.7 Nrms• 100 Hz case Frms = 46.7 Nrms

The higher bandwidth case requires ~2.5x more force to track the higher frequencies.

50 Hz

Plant and Loop Transfer Functions

Red Solid- PlantBlue Dash- Loop

Time (sec.)

Mag

. (d

B)

Ph

ase

(deg

.)M

oto

r Fo

rce

(N)

Frequency (Hz)

Motor Force Blue – 50 HzRed – 100 Hz

1

2 3

Disturbance Tracking- Power Calculation

Dissipated PowerDissipated power in the system is below and goes with the ^2 of RMS Force;

Pd =Frms2

𝐾𝑚2

where Km is the motor constant in (𝑁

𝑊),

Km is calculated using;

Km =2

3

Kf

2 ∗ 𝑅𝑝ℎ

In our example, Kf = 50 N/amp and Rph is 2 ohms and Km is calculated to be 29 N/sqrt(W)

Substituting Km into Eq. 4, and the rms force values for the 50 and 100 Hz cases, yields dissipated power of 0.4 watts for the 50 Hz case and 2.4 watts for the 100 Hz case.

Eq. 1

Eq. 3

In this example, a 2x increase in bandwidth results in a 6x increase in dissipated power.

Disturbance Tracking- Acceleration vs. Bandwidth

Yellow 100 Hz caseGreen 75 caseRed 50 Hz case.

RMS acceleration is the area under the acceleration spectrum.Higher bandwidths create a broader acceleration spectrum.

Pcmd * CLTF * w^2 = Acceleration of target mass

1000HzZ1 (

m)

Z2 (

m)

Z2 (

m)

Z2 (

m/s

^2)

Case Study Summary

• Move and Settle Application• Motor sizing is independent of servo loop bandwidth.

• Motor sizing is a function of motion trajectory.

• Moving mass (or inertia) and acceleration drive power dissipation.

• Disturbance Tracking Application• Motor sizing is largely driven by the servo bandwidth.

• Higher bandwidth higher frequencies higher accelerations higher motor forces higher dissipated power.

• Example 2x increase in servo bandwidth 6x more dissipated power.

In each case power dissipated in the motor is the ultimate driving factor for motor sizing.

Conclusions

• Consider the application of your motor and do the appropriate analysis to predict motor power dissipation.

• Move and settle applications focus on motion trajectory and duty cycle

• Disturbance tracking focus on servo bandwidth

• In both cases, the acceleration and mass are critical parameters

Allowable power dissipation is the primary factor for motor sizing.

Thank You

Mark HolcombSenior Engineer, Motion Control Specialist

Celera Motion125 Middlesex Turnpike

Bedford, MA 01730-1409 USA

Tel: 781-266-5200

Telephone: 916-751-5588

Email: mark.holcomb@celeramotion.com

www.celeramotion.com