Servo Loop Bandwidth, Motor Sizing and Power Dissipation Mark 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) Celera Motion Senior Engineer, Motion Control Specialist. Automation software and hardware, motor modeling\analysis software, supporting customers with their motion control applications 2016-Present Modeling control of thrust actuation devices onboard kinetic warheads (AeroJet Corp, 2009-2011)
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. Typical Servo Block Diagram PID Amplifier Motor Mechanics Sensor There is not a unique solution for each gain that results in a targeted bandwidth.
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.
Phase (deg.) Magnitude (db) Higher Resolution Sensor Example- Bode Plot 1 3 Loop TF w/nominal sensor gain Loop TF w/10x sensor gain 0 db. Freq. 5 With 10x sensor gain 4 Gain Margin requirement is violated. Loop Gain must be reduced. 6 db of required Gain Margin. 2 Phase Margin -180 degrees Frequency (Hz) With 10x Sensor gain, stability Gain Margin rule is violated. Loop Gain must be reduced by 10x to maintain stability.
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; P t = I t 2 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).
Force (N) Duty Cycle and Continuous Force (RMS) Duty Cycle is calculated by taking time spent applying force and dividing by the whole time. F1 Time Accel\Decel F2 DC% = (t1 + t2)/(t1+t2+dwell)*100. F rms = F c = F12 t1+f2 2 t2+ Total Time When F1 = F2, etc. F rms = DC F1
Acceleration (m/s^2) Move and Settle Application Bandwidth and Trajectory 50 Hz vs. 100 Hz bandwidth create similar motion. Duty Cycle = Time accelerating/total Time = ~74% F1 Acceleration vs. Time Black- Accel Cmd Red- 50 Hz BW Blue- 100 Hz BW t1 t2 t3 Acceleration is dominated by the command trajectory, not the servo bandwidth. 8.5 m/s^2 rms 8.3 m/s^2 rms Time (sec.) -F1
Move and Settle Power Calculation Dissipated Power (Pd) Pd = ( DC F1)2 Km 2 = Force 2 RMS, Km 2 where; F RMS = A RMS mass (from example, assume mass of 1 Kg Pd = (8.5 1)2 Km 2 Km = 72 Pd 2 Solve for Km based on your known Pd requirement.
Disturbance Tracking Application Example of a Tracking System Z2 (t) M2 (1Kg) Focus Signal Goal minimize Z2(t), while M1 has motion Z1(t). M1 (20 Kg) Z1(t) = 16 um RMS (uniform vibe spectrum up to 1 khz)
Motor Force (N) Phase (deg.) Mag. (db) Disturbance Tracking- Continuous Force Plant and Loop Transfer Functions 1 50 Hz Frequency (Hz) Red Solid- Plant Blue Dash- Loop The loop is showing ~50 Hz 0 db crossing servo bandwidth. 2 Motor Force Blue 50 Hz Red 100 Hz Motor force RMS 50 Hz case Frms = 18.7 Nrms 100 Hz case Frms = 46.7 Nrms 3 The higher bandwidth case requires ~2.5x more force to track the higher frequencies. Time (sec.)
Disturbance Tracking- Power Calculation Dissipated Power Dissipated power in the system is below and goes with the ^2 of RMS Force; Pd = Frms2 Eq. 1 Km 2 where Km is the motor constant in ( N W ), Km is calculated using; Km = 2 3 Kf 2 R ph In our example, Kf = 50 N/amp and Rph is 2 ohms and Km is calculated to be 29 N/sqrt(W) Eq. 3 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. In this example, a 2x increase in bandwidth results in a 6x increase in dissipated power.
Z2 (m) Z2 (m/s^2) Z1 (m) Z2 (m) Disturbance Tracking- Acceleration vs. Bandwidth Pcmd * CLTF * w^2 = Acceleration of target mass 1000Hz Yellow 100 Hz case Green 75 case Red 50 Hz case. RMS acceleration is the area under the acceleration spectrum. Higher bandwidths create a broader acceleration spectrum.
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 Holcomb Senior Engineer, Motion Control Specialist Celera Motion 125 Middlesex Turnpike Bedford, MA 01730-1409 USA Tel: 781-266-5200 Telephone: 916-751-5588 Email: mark.holcomb@celeramotion.com www.celeramotion.com