Design Notes on Precision Phase Locked Speed …

1 U-113 APPLICATION NOTEDESIGN Notes ON Precision Phase Locked SPEEDCONTROL FOR DC MOTORSABSTRACTT here are a number of high volume applications for DCmotors that require Precision control of the motor s Locked loop techniques are well suited to providethis control by Phase locking the motor to a stable andaccurate reference frequency. In this paper, the small sig-nal characteristics, and several large signal effects, ofthese loops are considered. Models are given for the loopwith Design equations for determining loop bandwidth andstability. Both voltage and current motor drive schemesare addressed. The Design of a loop for a three phasebrushless motor is LOCKING GIVES Precision SPEEDCONTROLThe precise control of motor Speed is a critical function intoday s disc drives. Other data storage equipment, includ-ing 9 track tape drives, Precision recording equipment,and optical disc systems also require motor Speed the storage density requirements increase for thesemedia, so does the Precision required in controlling thespeed of the media past the read/write mechanism.

2 Oneof the best methods for achieving Speed control of a mo-tor is to employ a Phase Locked a Phase Locked loop, a motor s Speed is controlledby forcing it to track a reference frequency. The referenceinput to the Phase Locked loop can be derived from a pre-cision crystal controlled source, or any frequency sourcewith the required stability and accuracy. A block diagramof the Phase Locked loop is shown in Figure Figure 1, a Precision crystal oscillator s frequency isdigitally divided down to provide a fixed reference frequen-cy. Alternatively, the motor could be forced to track a vari-able frequency source with zero frequency error. The mo-tor Speed is sensed by either a separate Speed windingor, particularly in the case of the DC brushless motor, aHall effect device.

3 The two signals, motor Speed and ref-erence frequency, are inputs to a Phase detector. The de-tector output is a voltage signal that is a function of thephase error between the two inputs. The transfer functionof the Phase detector, K+, is expressed in volts/radian. A1/s multiplier accounts for the conversion of frequency tophase, since Phase is the time integral of the Phase detector is the loop filter. This blockcontains the required gain and filtering to set the loop soverall bandwidth and meet the necessary stability output of the loop filter is the control input to the mo-tor drive. Depending on the type of drive used, voltage orcurrent, the driver will have respectively, a VOUT/VINtransfer characteristic, or an louT/Vr~ first glance, it seems that the motor has simply re-placed the VCO (voltage controlled oscillator), in the clas-sic Phase Locked loop.

4 In fact, it is a little more complicat-ed. The mechanical and electrical time constants of themotor come into play, making the transfer function of themotor more than just a voltage-in, frequency-out block. Inorder to analyze the loop s small and large signal behav-ior it is essential to have an equivalent electrical model forthe SIMPLE ELECTRICAL MODEL FOR A DCMOTORF igure 2 is an electrical representation of a DC motor. Theterms used are defined here:LM Motor winding inductance in henrysHM Motor winding resistance in asJTotal moment of inertia of the motor in Nm-set?(Note: 1 Nm = oz-in)KT Motor torque constant in Nm/AmpKv Voltage constant (back EMF) of motor in voltage-sec/radFigure 1. Precise motor Speed control is obtained by Phase lockingthe motor to a Precision reference frequency.

5 (Note: Kv = KT in SI units)3-132 APPLICATION NOTEU-113 NODE VOLTAGEEQUATES TOMOTOR BACK EMF*N = Number of Speed sensecycles per motor revolutionFigure 2. This simple electrical model is useful for determining thesmall and large signal characteristics of the motor. Capacitor, CM is used to model the mechanical energystorage of the this model the winding inductance and resistance ele-ments correlate directly with the corresponding physicalparameters of the motor, with values taken directly off themanufacturer s data sheet. The capacitor, CM, models themechanical energy storage of the motor. Current into thecapacitor equates, via motor constant KT, to motor torque,and the voltage across the capacitor is equal to the motorback EMF. The back EMF voltage equates to motor ve-locity through the inverse of Kv.

6 In the model, the term Nis simply a multiplier equal to the number of feedback cy-cles obtained per revolution of the motor. For example, ina 4 pole brushless DC motor the commutation Hall effectdevice outputs will be at twice the rotational frequency ofthe motor, making N equal to equation for the capacitor, given in Figure 2, has theunits of Farads if J and KT are expressed in SI units. Inmodeling the overall transfer characteristic, it is importantthat the moment of inertia of the load on the motor beadded to the moment of inertia of the motor is worthwhile to note that the current into the motor, mi-nus idling current, is proportional to acceleration of themotor. This is easily seen from the model by realizing thatthe time derivative of the capacitor voltage relates directlyto acceleration. The effects of loads on the motor can bemodeled by including a current source across the capaci-tor for constant torque loads, or a resistor for loads thatare linearly proportional to motor FUNCTIONS FOR VOLTAGE ANDCURRENT DRIVEN MOTORSU sing the electrical model, the small signal transfer func-tion of the motor is easily derived.

7 Equations 1a and 1bgive the small signal frequency response for both the cur-rent and voltage driven cases transfer function given in equation (1a) describes thesmall signal response of motor Speed , OM(s), to changesin the drive current. Equation (1b) relates the dependenceof motor Speed to motor drive small signal response of the motor for the currentdriven case has a DC pole that results from the relation-ship of motor torque to velocity, that is, motor velocity isproportional to the integral of motor torque over time. Inthe current driven motor neither the winding resistancenor inductance appear in the transfer function. This is be-cause these elements are in series with the currentsource output of the driver stage. As long as the outputimpedance of the driver remains large relative to the im-pedance of these elements, the resistance and induc-tance of the motor will have a negligible effect on thesmall signal voltage driven response has a second order charac-teristic that results from the interaction of the series many cases the transfer function of the voltage drivencase can be simplified.

8 If the quality factor of the seriesRLC of the motor model is much less than one, as de-fined in equation 2, then the response of the motor canbe accurately approximated by equation THE WHOLE LOOPF igure 3 shows the complete Speed control loop for thecurrent driven case. The overall open loop response,AoLc, is easily written. N = Number of feedback cycles per motor revolutionFigure 3. In this Phase Locked loop, with current mode drive to themotor, the motor winding resistance and inductance canbe ignored as long as the current driver maintains a highoutput (5)APPLICATION NOTEU-113 For this loop, note that there are two poles in the re-For this motor, the model capacitor, CM, is calculated us-sponse at DC, , s = 0. One pole is due to the responseing the equation in Figure 2 to be equal to Farads.

9 Ifof the current driven motor, the second pole is from thewe calculate the quality factor of the series RLC, usingfrequency to Phase transformation of the Phase 2, we find it is equal to x 10 -3. This isThe 180 degrees of Phase shift this pair of poles intro-considerably less than one, and the response closely ap-duce force a Phase lead configuration of the loop filter inproximates the non-complex response of equation 3 withorder to obtain a loop Phase margin greater than at Hz and 199 complete voltage loop is shown in Figure 4, and itsopen loop response, AoLv(s), in equation loop bandwidths will fall well inside this range offrequencies. As long as this is true, the loop responsewith a voltage driven motor can be approximated by:This expression is the same as the current driven re-response equation 4, with the transconductance of the cur-rent drive stage, GPD, replaced by the gain of the voltagedrive stage divided by the motor winding resistance,*N = Number offeedback cycles permotor revolutionFigure 4.

10 With voltage mode drive to the motor the electrical timeconstant of the motor plays a part in the small signalresponse of the Speed control response has only one pole at DC, although the totalnumber of poles is three versus two for the current drivencase. For most motors, particularly those used in constantvelocity applications, this transfer function can be simpli-fied by applying the results of equations 2 and 3. This isbest illustrated by looking at an example. Consider the fol-lowing motor, (typical 3- Phase brushless for disc drive ap-plications):KT .. X 10-Z Nm/AmpKv .. X 10-Z V-sec/radJ (including platters) .. 1 X 10-s Nm-sec2RM .. THE LOOPWhen it comes to closing the loop the goal is to have astable loop with the required loop bandwidth. The vari-ables that must be considered are:1) The motor2) The power driver, type and gain3) The Phase detector gain4) Loop bandwidth5) The loop filterThe first four of the above variables are usually dictatedby conditions other than the stabilizing of the loop.