Development of a High Bandwidth, High Power Linear Amplifier for a Precision Fast Tool Servo System

Transcription

1 Development of a Hgh Bandwdth, Hgh Power near Amplfer for a Precson Fast Tool Servo System S. Rakuff 1, J. Cuttno 1, D. Schnstock 2 1 Dept. of Mechancal Engneerng, The Unversty of North Carolna at Charlotte, Charlotte, NC Dept. of Mechancal Engneerng, Tulsa Unversty, Tulsa, OK Introducton A longrange fast tool servo (FTS) has been developed at UNCCharlotte that s currently able to delver 1mm lnear motons at frequences as hgh as 50 Hz. To date the accuracy s 0.1 µm for a 250µm snusodal oscllatory moton at 20Hz. The FTS conssts of a DSPsystem, a voce col drven flexure carryng the damond cuttng tool, and a laser nterferometer, whch s used for poston feedback. A PID algorthm runnng on the DSP card s used for poston control. The controller sgnal s amplfed wth a hgh performance analog amplfer to drve the voce col. The hgh power output and bandwdth requrements of the amplfer mean that t must be a carefully desgned, ntegral part of the FTS system. The amplfer n ths paper features the APEX PA03 lnear operatonal amplfer. The paper descrbes the desgn technques appled durng development of the amplfer crcut and ponts out how classcal control technques can be used. 2. Desgn objectve The amplfer s desgned to operate n a frequency span from DC up to 50Hz. Ths dstngushes the amplfer from commercally avalable audo amplfers, whch are typcally desgned as band pass flters. A requrement for the FTS was to be able to hold the flexure at a commanded fxed poston wth a constant force, whch corresponds to a nomnally constant current through the voce col. However, typcal sgnal frequences wll le around 40 Hz, whch s close to the fundamental frequency of the flexure. Operatng the system close to ts natural frequency helps to conserve power but s also harder to control. Because of the low pass flter characterstcs of the amplfer, desgn technques from classcal controls may be optmally appled n the desgn process. In ts smplest form, the voce col can be modeled as a resstor n seres wth an nductance. Ths gves a transfer functon between the load current and the load voltage of Fgure 1: Frequency response of the voce col I 1 TF (1) s R where 2.5mH ± 30% and R 2.4Ω 12.5% taken from the data sheet of the voce col. Fgure 1 shows the Bode plots of the voce col for and R varyng wthn ther uncertantes. The 3dB bandwdth s only about 1000rad/s (159Hz), whch would also be the bandwdth of an open loop amplfer confguraton. The bandwdth can be ncreased f a properly desgned controller and feedback network s mplemented.

2 3. oltage controlled current sources The amplfer s desgned as a voltage controlled current source, or CCS. The amplfer converts the poston command voltage from the DSP card nto a current flow I through the voce col to move the flexure. The voltage to current converson factor s set to 2.2A/. The voce col s assumed to behave lke an deal actuator havng a lnear realatonshp between the current and the output force. The force constant s N K f (2) A To generate a snusod force wth an ampltude of 100N at a frequency of f50hz (ω314.2rad/, the requred current ampltude I s 4.68A. From fgure 2 t can be seen that the voltage across the load mpedance R and s the dfference between v and v S. If the feedback resstor R f s large compared to the sense resstor R S, then t can be assumed that all current flowng through the load wll eventually flow to ground, so that s approxmately equal to S. The voltage drop accross the load can then be wrtten as d v ( R R S ) (3) dt where d dt ( t) ωi sn( ωt) (4) The magntude of d /dt s maxmzed at tmes t 0, π/ω, 2π/ω, and so forth. The magntude of the voltage s about 12. As shown n fgure 3, the voltage wll lead the current by a phase angle φ. CCS F v R (t), v(t) v (t) cos(ωtφ) (t) Icos(ωt) v S R F S R S t Fgure 2: oltage controlled current source (CCS) Fgure 3: Current voltage relatonshp If the frequency of the current ncreases, so does the magntude of the phase angle. The varyng phase angle between load voltage and load current does not affect the PID poston controller of the FTS system because the load current s drectly proportonal to the force exerted by the voce col and normally n phase wth the controller sgnal gong nto the amplfer. In equaton (3) v conssts of the nductve voltage drop and the voltage drop across the resstors R and R S. At 50Hz the load s stll more resstve than nductve and the phase shft s relatvely small. The power dsspaton n the load wth small or zero phase angle s calculated by P I cos( ) (5), rms rms φ I A cos( φ ) cos(0) 28 W 2 2 where,rms and I rms are the root mean square values of the load voltage and load current respectvely. Ths amount s also a ballpark value for specfyng the power supples needed. 4. Senstvty of the amplfer due to modelng errors A good amplfer desgn should be relatvely nsenstve and robust to parameter varatons. The resstance R and the nductance of the load have hgh uncertantes of 10 and 30 percent respectvely and the gans ntroduced by the op amps may change some when the system ages. These parameter varatons can be modeled as

3 unstructured addtve uncertantes. Another source of errors arses from unmodeled plant dynamcs. If G denotes the dfference between the actual plant G and the nomnal modeled plant G then G ( (6) Fgure 4 shows a smple open loop amplfer wth a proportonal controller, nput voltage ( and output current I (. The plant represents the voce col and amplfcaton gans of the amplfer. In the openloop confguraton, the percentage error of the load current equals the percentage error of the model. I ( (7) I ( G ( Equaton (7) shows how senstve the openloop confguraton s to modelng errors alone. If the modelng error s apprecable the resultng system output error mght be unacceptably hgh. If nformaton about I s fed back through a feedback gan H nto the nput termnal of ( I ( the amplfer as ndcated n fgure 5, then the relatonshp between Fgure 4: Open loop amplfer percentage output error and percentage modelng error s expressed by I ( 1 (8) I ( 1 H ( G ( If the return dfference 1HGK were large, then a sgnfcant percentage error n modelng the plant would result only n a small percentage error n the output. Therefore, t s desred to maxmze HGK n equaton (8). It can be seen that wth a feedback network nstalled, command followng s less senstve to modelng errors. ( H( D( I ( N( Fgure 5: Closed loop amplfer wth dsturbance and nose 5. Dsturbance rejecton and nose response of the amplfer The amplfer also has to be nsenstve to nternal nose N and compensate for external dsturbances D as shown n fgure 5. The expected dsturbances are assumed to be low frequency sgnals representng mechancal nput nto the tool. Nose s generated n the crcut and s usually of hgh frequency. The output current I can be expressed n terms of nput, nternal sgnal nose N, and external dsturbances D by I S) H ( 1 ( ( N( D( (9) 1 H ( K ( 1 H ( 1 H ( To nvestgate the nfluence of dsturbances D on the amplfer assume that nose N and nput equal zero. Then I s related to D by 1 I ( D( (10) 1 H( For good dsturbance rejecton, the output I n equaton (10) should be close to zero snce the nput was assumed to be zero. Ths mples that the return dfference 1HGK has to be large at lower frequences where the dsturbances have ther major energy content. To analyze the effects of nose, assume that the dsturbances D and the nput voltage are zero. From equaton (9) the load current then equals H( I ( N( (11) 1 H( Agan, for good nose mmunty the output I n equaton (11) should be close to zero because the nput was assumed to be zero. Snce nose N s a sgnal wth energy contents at hgh frequences, the transfer functon HGK has to be small at hgher frequences. Thus, the openloop frequency response of the op amps used to buld the amplfer should look lke a low pass flter wth hgh gan at low frequency and low gan at hgh frequency. Output

4 feedback n the amplfer s mportant to reduce the effects of nose and modelng errors as well as to ncrease command followng propertes. 6. Desgnng an ntegratortype feedback controller The amplfer s bandwdth s requred to be hgher than the open loop bandwdth of 160Hz determned by the voce col. Two cascaded feedback networks are used to enhance the frequency response of the amplfer. A crcut dagram s gven n fgure 6 and represented n block dagram form n fgure 7. The block dagram contans several transfer functon blocks, whch correspond to elements n the amplfer crcut. Table 1 summarzes the transfer functons of the varous elements wth a bref descrpton. n R K K f G 0 G 1 G 2 I ( R C K K f G 0 G 1 G 2 I ( 0 H R H H S H R H H s R S Fgure 6: Crcut dagram dvded up nto TF blocks Fgure 7: Block dagram representaton of amplfer Table 1: Transfer functons of varous elements n the amplfer crcut Name Transfer functon Unts Descrpton R I 1 A Input resstor of 5.1kΩ o 5100 K ol o 76dB 12kHz Open loop TF of power op amp used to buld the controller (AD826). Open loop response taken from data sheet. s 12kHz s G 0 o kHz Close loop TF of power op amp AD826 set to a gan of 1.6 s 12kHz s G 1 o Closed loop TF of power op amp PA03 set to a gan of 7.5 s G 2 I 1 1 A TF of the load (voce col) s R s 2.4 K f Io 1 Cs A TF of the nner feedback path. The values of R and C are to be determned by tunng the controller R 1 RCs 1 Cs H R I 1 A Resstor of 5.1kΩ n the outer feedback path after op amp o 5100 H H S o kHz s 12kHz s I o 0.02 A Gan ntroduced by an op amp (AD826) n the outer feedback path set to a gan of 20.8 Sense resstor of 0.02Ω to convert the load current nto a voltage The closed loop frequency response G 1 and the open loop frequency response A PO of the PA03 are shown n fgure 8. G 1 has a pole at about rad/s. Snce the gan s 7.5, the closed loop frequency response of the power op amp G 1 s o G1 (12) s

5 The open loop response of the AD826 s obtaned the same way by notng that the open loop gan gven n the data sheet s 76dB wth a bandwdth of 12kHz. Hence, the open loop response can be expressed as o 76dB 12kHz K (13) ol s 12kHz s These fast poles of the op amps could be neglected n the calculatons and K ol set to 6310/. R, H s, and H R are resstors that convert voltages nto currents n the block dagram representaton. At the summng pont all currents sum to zero accordng to Krchhoff s current law. Magntude (db) A 120 PO G k 10k 0.1M 1M 10M Fgure 8: Open and closed loop response of the PA03 Frequency (rad/ K and K f n fgure 7 can be combned to an equvalent block C contanng a strctly proper thrd order transfer functon. By selectng values for the resstor and capactor contaned n block K f the amplfer can be tuned. The equvalent transfer functon can be used to smulate the frequency response of the amplfer for dfferent values of the capactor and resstor n the ntegraltype feedback controller. Fgure 9 shows an array of Bode plots for a constant capactance of 0F and R rangng from 5kΩ to 500kΩ. Increasng R Increasng R 7. Frequency response measurements of the amplfer Fgure 9: Response of the model for ncreasng R and C0 The actual frequency response of the amplfer wth no ntegral gan was measured wth a sgnal analyzer. The frequency response was computed by dvdng the cross spectrum between the nput sgnal sequence and the output sgnal sequence by the power seres spectrum of the nput sgnal sequence. Therefore, the frequency response s a drect representaton of the characterstcs of the true system. Fgure 10 shows the frequency response of the amplfer/voce col assembly resultng from a snusodal nput voltage wth ampltude 0.5. Unfortunately, the maxmum frequency of the measurements was lmted by the bandwdth of the current probe that was used to measure the output of Fgure 10: Frequency response of actual amplfer the amplfer. There are stll some dscrepances between the expermental and the theoretcal models. Fgure 10 shows some dynamcs n the real system that have not yet been dentfed. For example, the dynamcs at 48Hz are apparently caused by the frst mode of the flexure that s attached to the voce col. The bode plot shows the presence of multple poles and zeros nteractng wth each other, as evdenced by the dp at 45 Hz, rse shortly thereafter (ndcatng a pole/zero par), followed by a gentle 10 db/decade slope that begns to roll off further at 1 khz. It s beleved that the response wll reach a 40 db/decade slope and 180 degree phase beyond 1 khz, but the dscrepances are stll beng nvestgated.

6 8. Nonlneartes Durng the process of tunng the PID controller for the FTS system, the amplfer sometmes saturated because of very large and fast controller output sgnals. Ths means that the FTS system represents a dffcult nonlnear system. One possblty to descrbe nonlneartes s to ft a hgher order lnear transfer functon to nonlnear frequency response data about some operatng pont. These transfer functons may be used to desgn the PID controller for the FTS system. Matlab and Smulnk were used to create an amplfer model (fgure 11) that contans slew rate lmts, saturaton blocks, and backemf (fgure 12). The backemf subsystem calculates the velocty of the flexure by dfferentaton of the dsplacement and a constant relatonshp between velocty and backemf voltage generated. rtual test nputs equvalent to those from the sgnal analyzer were fed nto the model to calculate the frequency response n the lnear regon. Efforts are currently under way to verfy the model. Fgure 11: Smulnk block dagram of actual amplfer 9. Concluson Ths work shows how technques from classcal control theory can be appled to desgn and analyze the performance of an amplfer for a precson moton mechansm utlzng a voce col. It also ponts out many of the dffcultes n gettng the theory to accurately model the actual response. Work s contnung to dentfy the prncpal components requred to develop accurate models of amplfers used for precson applcatons. 10. References Fgure 12: Backemf subsystem for model 1. Za A. Yamayee, Juan. Bala, Jr., Electromechancal Energy Devces and Power Systems, John Wley & Sons, Inc., Rchard C. Dorf, Robert H. Bshop, Modern Control Systems, 7 th edton, Addson Wesley, APEX Mcrotechnology catalog v8.5, Applcaton notes, Wllam J. Palm III, Modelng, Analyss, and Control of Dynamc Systems, 2 nd edton, John Wley and sons, Inc