Chaptr 19: Prmannt Magnt DC Motor Charactristics 19.1: ntroduction Dirct currnt (DC) motors compris on of th most common typs of actuator dsignd into lctromchanical systms. hy ar a vry straightforward and inxpnsiv mans of crating motion or forcs. Mor oftn than not, you ll find yourslf using motors to put th mch- into mchatronics. Motors ar actually complx assmblis that xploit th rlationships btwn currnt and magntic filds in ordr to crat usful torqu and do work. And, lik all ral-world componnts and complx assmblis, motors hav svral intrsting charactristics, trad-offs, quirks and vn pitfalls to avoid. Undrstanding th issus will nabl dsignrs to succssfully slct and us DC motors. 19.2: ub-fractional Horspowr Prmannt Magnt Brushd DC Motors h catgory of DC motor that is th last xpnsiv, asist to us, and thus th most popular, is th subfractional horspowr prmannt magnt brushd DC motor. ub-fractional horspowr rfrs to th thir limitd powr output, and distinguishs thm from largr varitis of motors. Prmannt magnt rfrs to th mans usd of stablishing on of th magntic filds. Brushd rfrs to th mthod of commutation (th way in which coils ar activatd to stablish usful magntic filds; which is dscribd in dtail latr in this sction). Finally, DC indicats that ths motors oprat on dirct currnt, rathr than AC, or altrnating currnt. Motors xploit th phnomnon dscribd by Maxwll s quations: a currnt flowing within a wir stablishs a magntic fild around that wir. By placing a currnt-carrying wir and its magntic fild insid anothr magntic fild, forcs ar gnratd by th intraction of ths two magntic filds. h Carryr, nny & Ohlin 10/24/03 Rv, Pag1/38
prmannt magnt DC motor s dsign uss ths forcs to crat a torqu on th motor s rotor, which is constraind by th motor s barings so that th only motion prmittd is rotation (s Figur 19.1 and Figur 19.2). Prmannt magnt DC motors ar constructd out of a numbr of componnts. h xact dsign and matrials vary with ach typ of motor and dpnd on th application and constraints, but svral lmnts ar common to most. Figur 19.1 shows a cut-away viw of a typical prmannt magnt brushd DC motor. h construction gnrally consists of a stator, which is mad up of powrful prmannt magnts that gnrat a static magntic fild; a rotor which carris th armatur (also known as th windings or coils) and th commutatator, and rotats in th barings that support it; and a housing that holds th stator, rotor baring supports and brushs in a fixd rlationship to on anothr. Figur 19.1: Prmannt Magnt Brushd DC Motor Construction, Componnts and Nomnclatur Carryr, nny & Ohlin 10/24/03 Rv, Pag2/38
n trms of gnrating torqu, th critical lmnts of th motor ar th stator and th armatur, which ar th sourcs of th two intracting magntic filds. h stator is commonly shapd lik a thick-walld tub, and th rotor and armatur fit in th hollow spac in th middl of th stator. h lins of magntic flux stablishd by th stator run from on sid of th stator to th othr. Figur 19.2 dmonstrats th magntic flux lins in a simplifid rprsntation. Figur 19.2: Prmannt Magnt DC Motor tator and Armatur Dtail Figur 19.2 also shows a singl winding of th armatur, and maks it asir to undrstand th intraction btwn th armatur and th stator. h armatur contains a larg numbr of wir loops, or coils, idntical to th singl on shown, arrangd in a radial pattrn around th rotor so that continuous torqu is gnratd as th rotor rotats. Also, th additional loops contribut additional rsulting forcs, and hnc mor motor torqu. Carryr, nny & Ohlin 10/24/03 Rv, Pag3/38
Causing currnt to flow through th loopd wir in th coil causs a magntic fild to b stablishd. Dpnding on th orintation of th coil loop in th stator s magntic fild, a forc is gnratd du to th intraction btwn th magntic fild of th stator and th magntic fild du to th currnt flow. h armatur is rigidly fixd to th rotor, and as a rsult it racts to ths forcs by rotating. f th ntir systm consistd of a singl coil loop within th magntic fild of th stator as shown in Figur 19.2, inducing a simpl, unchanging currnt flow in th coil would caus th rotor to turn until th magntic filds alignd, liminating th forcs that gnrat torqu. his will happn whn th dirction of th forc has no componnt prpndicular to th rotor s radius, and th rotor will com to rst at an quilibrium position. n ordr to spin continuously, prmannt magnt DC motors ar dsignd to switch th currnt flowing in th coils continuously, nvr coming to an quilibrium position. Whn th rotor approachs a point nar quilibrium, th dirction of th currnt flow in th coil is first stoppd, and thn rvrsd. his has th ffct of moving th quilibrium point as th rotor spins. n much th sam way that a carrot is dangld in front of a cart hors to ncourag it to pull, th quilibrium point is always bing movd out of rach of th rotor in ordr to kp it spinning continuously. h commutator and brushs togthr prform th switching of th currnt in th coils of th armatur rquird by this approach. h commutator-to-brush contact is th point whr currnt is introducd into th coils. h commutator is shapd lik a smooth annular ring with strips of conductiv matrial altrnating with strips of insulating matrial. Each of th coils of th armatur is connctd across two adjacnt conducting sgmnts of th commutator. h ovrall arrangmnt is shown in Figur 19.3. Conncting th coils in this fashion allows all coils to continuously carry currnt and contribut torqu (with th xcption that priodically on of th coils has both lads connctd to th sam brush contact, during which tim no currnt flows in that singl coil). f instad th coils wr arrangd so that thir lads wr connctd to conducting sgmnts on opposit sids of th commutator, this would limit th numbr of coils that wr conducting currnt to on or two, and thus svrly limit th ovrall torqu gnratd by th motor. Whn connctd as shown in Figur 19.3, all but on of th coils contribut torqu at all tims. Carryr, nny & Ohlin 10/24/03 Rv, Pag4/38
h commutator is part of th rotor/armatur assmbly, and rotats as th rotor spins. h stationary brushs mak contact with diffrnt commutator sgmnts as th rotor spins, and control th flow of currnt through th coils of th armatur. h brushs ar in constant contact with th commutator and provid th currnt path btwn th non-rotating, stationary housing and th spinning rotor. Brushs ar typically mad of a low friction matrial such as graphit or prcious mtals, and ar in sliding contact with th commutator as it spins. prings prss th brushs firmly against th commutator and nsur that good lctrical contact is maintaind. Figur 19.3: Elctrical connctions and layout of armatur coils, commutator and brushs. Although th commutator and brushs ar ky lmnts that mak a prmannt magnt DC motor work, thy ar also th wakst link in th systm. Brushs ar sacrificial componnts that war ovr tim, and ar th componnts of th motor that ar most likly to fail. Also, bcaus thy ar in sliding contact with th commutator, thy caus brush drag, or frictional forcs, which ar th unavoidabl rsult of dragging on matrial ovr anothr. Ovrcoming ths drag forcs rquirs torqu that is thn not availabl to do work. hs forcs ar thrfor considrd losss. n addition, sinc th brushs ar prssd against Carryr, nny & Ohlin 10/24/03 Rv, Pag5/38
th commutator with springs, this crats a dynamic systm complt with rsonancs. As th rotor and commutator spin fastr and fastr, th bushs will vntually rach a condition whr thy ar not abl to follow th contours and stay in good lctrical contact bcaus of a phnomnon known as brush bounc. Ultimatly this limits th maximum spd of th motor. ncrasing th spring forc prssing th brushs against th commutator is on solution, but this incrass frictional losss and acclrats war on th brushs. Brushs ar most commonly mad out of graphit, which is a matrial that has rlativly high lctrical rsistanc (1-10Ω, typically) and is physically dirty. Graphit brushs cast off particls and dust as war occurs. n addition, th brush-commutator intrfac is lctrically noisy as th connctions to th individual coils in th armatur ar continuously mad and brokn. hs can b th sourc of significant lctromagntic intrfrnc (EM) or nois in a systm. 19.3: Elctrical Modl Elctrically, prmannt magnt brushd DC motors can b modld as a sris of thr basic lctrical componnts: a rsistor, an inductor, and a sourc of lctro-motiv forc (EMF), or voltag (Figur 19.4). his voltag sourc is commonly calld th back-emf or countr EMF. h origins of th rsistiv and inductiv componnts ar asy to s. h rsistor in th modl is a rsult of th finit rsistanc pr unit lngth of wir usd to construct th coils in th armatur. h inductor is a rsult of coils of wir that mak up th armatur windings. All coils of wir act as inductors. h back-emf, on th othr hand, taks a littl mor discussion to clarify. Rsistor Back EMF - + nductor Figur 19.4: Elctrical Modl of a Prmannt Magnt Brushd DC Motor Carryr, nny & Ohlin 10/24/03 Rv, Pag6/38
19.4: Back-EMF and th Gnrator Effct Rcall that th torqu gnratd by a prmannt magnt DC motor is th rsult of th currnt flowing in th armatur coils in th prsnc of th stator s magntic fild. his ffct is known as th Lorntz forc law and is ultimatly dscribd by Maxwll s quations. Anothr of Maxwll s quations lads to Faraday s law, which dscribs th rsult of moving a coil of wir through an xtrnal magntic fild: a voltag is gnratd. his is th principl usd to gnrat lctricity in a hydrolctric powr gnration plant, for xampl, whr th potntial and kintic nrgy of th flowing watr is usd to spin a rotor and armatur in th prsnc of a magntic fild to gnrat lctricity. Faraday s law is th ffct that rquirs th inclusion of th back-emf componnt in our lctrical modl of a prmannt magnt brushd DC motor: th armatur is spinning insid th fild cratd by th stator. his inducs a voltag (th back-emf) across th coil as it spins. his voltag is opposd to th voltag placd across th coil that mad th rotor spin in th first plac. n short: th motor is acting as a gnrator at th sam tim that it is acting as a motor. h voltags add by suprposition, though thy hav diffrnt signs. h ffct of this voltag is to rduc th voltag drop and currnt flow in th motor s trminals whn th motor is running. 19.5: Charactristic Constants for Prmannt Magnt Brushd DC Motors As a motor turns fastr, mor back-emf is gnratd sinc th coils in th armatur ar moving fastr through th stator s magntic fild. h magnitud of th back-emf is rlatd to th rotational spd through a constant, calld th spd constant or voltag constant. E ω [ V ] Eq. 19.1 whr: E back-emf [ V ] Carryr, nny & Ohlin 10/24/03 Rv, Pag7/38
spd constant V rad s ω rotational spd [ rad s] h valu of is dtrmind by th construction, gomtry, and matrials proprtis of th motor. Quantitis lik th motor s physical dimnsions, th numbr of turns in th coil windings, and th magntic flux dnsity of th stator all contribut to th valu of. W can continu to xamin th gnrator ffct to arriv at anothr usful rlationship. n this dvlopmnt, w will ignor, for th momnt, th non-idal mchanical and lctrical losss associatd with th motor/gnrator opration. h largst of th ffcts w will assum ar ngligibl in this discussion is th torqu rquird to ovrcom friction in th motor. Bcaus of friction, th torqu gnratd by th motor may b tratd as bing mad up of two trms: frictional torqu and usabl torqu (or torqu that is availabl at th motor s output shaft and may b usd to driv a load): + [ m] M L f N Eq. 19.2 For now, w assum f 0. For most motors this is a rasonabl first approximation. f th losss ar ngligibl, thn th mchanical powr into th gnrator, ω, will qual th lctrical powr out, E. P E ω [ ] W Eq. 19.3 W can combin Eq 19.1 with Eq. 19.3 to yild: Carryr, nny & Ohlin 10/24/03 Rv, Pag8/38
ω ω [ W ] Eq. 19.4 which can b simplifid to: [ N m] Eq. 19.5 At this point, you may b wondring how w took a constant,, with units of Volts and Rotational pd tund it into on with units of orqu. h answr lis in th origin of th constant. h constant Currnt originats in th rlationship btwn currnt flow and magntic filds dscribd by Maxwll s quations. h constant can b xprssd quivalntly with units of Volts radians scond or Nwton Ampr mtr. o minimiz confusion, w trat th constant in Eq. 19.5 as diffrnt and call it, th torqu constant. h valu of is dtrmind by th sam factors that dtrmin th valu of : th construction, gomtry, and matrials proprtis of th motor. h motor s physical dimnsions, th numbr of turns in th coil windings, and th magntic flux dnsity of th stator all contribut to th valu of, just as thy did with. With th substitution of for, Eq. 19.5 taks on its mor common form: [ m] N Eq. 19.6 whr: torqu [ N m] torqu constant [ N m A] currnt [ A] Carryr, nny & Ohlin 10/24/03 Rv, Pag9/38
his distinction btwn and is particularly usful in that numrically whn compatibl units ar usd (.g., Volts radians scond and Nwton Ampr mtr ). Whn th constants ar xprssd in inconsistnt units (which is far mor common in practic), a convrsion factor must b applid to convrt btwn thm. wo of th most commonly ndd ar: [ oz in./ A] 1.3542 [ V / krpm] Eq. 19.7 [ N m / A] 9.5493 3 [ V / krpm] Eq. 19.8 Whil w hav structurd this discussion by xamining th cas of a gnrator, th dirction of torqu flow, ithr into or out of th motor shaft, has no ffct on th fundamntal intraction btwn th magntic fild and th lctrons in th conductor. As a rsult, ths quations (Eq. 19.1 and Eq. 19.6) ar tru whn th motor acts as a gnrator as wll as whn it acts as a motor. Exampl 19.1: A motor with 5.89 oz. in/a and a coil rsistanc of 1.76Ω is drivn with a supply voltag of 12V. f th motor s friction torqu is 1.2 oz. in, what is th maximum torqu availabl for driving a load? How much currnt is flowing undr ths conditions? By combining Eq. 19.2 and 19.6, w can find th torqu availabl at th output shaft: M L + f L f Carryr, nny & Ohlin 10/24/03 Rv, Pag10/38
Using Ohm s Law to substitut for currnt: V 12V 6. 82 A R 1.76Ω L L ( 5.89oz. in / A)( 12V ) V f 1.2oz. in R 1.76Ω 39.0oz. in Aftr th ffcts of friction ar subtractd off th total torqu producd by th motor undr ths conditions, 39.0 oz. in. of torqu is availabl at th motor shaft to driv a load. h currnt rquird to gnrat this torqu is 6.82A. 19.6: Charactristic Equations for Constant Voltag o mor fully undrstand th torqu and spd charactristics of a motor w can start by xamining what gos on whn w plac th motor into a circuit with a driving voltag. Motor Rsistor Back EMF V + - nductor + Figur 19.5: DC Motor circuit with driving voltag Carryr, nny & Ohlin 10/24/03 Rv, Pag11/38
W can us irchoff s laws to writ a loop quation to dscrib th stady-stat currnt flow in this circuit. V R + ω [ ] V Eq. 19.9 whr: V voltag [ V ] currnt [ A] R rsistanc of motor coils [ Ω] voltag constant V rad s ω rotational spd [ rad s] h voltag drop across th motor s coils has an R trm, as you would normally xpct, plus th ffcts of th back-emf gnratd by spinning th motor, xprssd in th trm ω. om implications of quations Eq. 19.1, Eq. 19.6 and Eq. 19.9 ar: o h highr th rotational spd of th motor, th lowr th currnt flow and thrfor th lowr th torqu. his occurs bcaus of th back-emf. o Maximum spd corrsponds to 0 currnt flow and thrfor 0 torqu (w obviously can t achiv this with a ral motor). o Whn ω 0 (a condition rfrrd to as stall ) V R and currnt and torqu will both b at a maximum. By substituting Eq. 19.9 into Eq. 19.6, w can dvlop an xprssion rlating torqu to spd. Carryr, nny & Ohlin 10/24/03 Rv, Pag12/38
V R + ω V R ω V R ω [ s] rad Eq. 19.10 Eq. 19.10 shows that, for a givn voltag V, torqu and spd for a motor ar linarly rlatd. Oftn, this is graphically rprsntd with a plot showing a family of lins rlating vs. ω for svral constant valus of voltag, V. Figur 19.6 shows a typical xampl. ω ω ω ω s1 s2 s3 Figur 19.6: ypical orqu vs. ω Curvs for a Prmannt Magnt Brushd DC Motor Carryr, nny & Ohlin 10/24/03 Rv, Pag13/38
hr ar a fw aspcts of Figur 19.6 that w should idntify and labl. h first is th y-intrcpt of ach lin. his is th maximum spd that th motor can achiv for a givn voltag, which occurs for th idalizd cas whr thr is no torqu gnratd. his is calld th no-load spd, writtn as ω, and is th V trm in Eq. 19.10. hus, for a prmannt magnt brushd DC motor: V ω [ s] rad Eq. 19.11 h slop of th lin givn by Eq. 19.10 is th multiplir on, which is R. h slop trm is also givn it own symbol, R M, and is calld th spd rgulation constant : R M R rad s Eq. 19.12 N m By substituting ω and R M into Eq. 19.10 w gt an xprssion that is mor asily idntifid as that of a straight lin: ω ω R M [ rad s] Eq. 19.13 h x-intrcpt of th constant-voltag lin rprsnts th cas whr ω 0, which occurs whn th motor is stalld. his is th point at which torqu, and thrfor currnt, ar maximizd. his is calld th stall Carryr, nny & Ohlin 10/24/03 Rv, Pag14/38
torqu, and givn th symbol ALL or. h corrsponding stall currnt is givn th symbol ALL or. f w st ω 0 in Eq. 19.13, w can also xprss ALL as: 0 ω R M ALL ALL ω R M Rcalling from Eq. 19.11 that ω : V V ALL R [ m ] M N Eq. 19.14 hn, rcalling from Eq. 19.12 that R R M and thn simplifying givs: V ALL [ m] R N Eq. 19.15 implifying furthr using Ohm s Law, w onc again obtain Eq. 19.6: [ m] N Eq. 19.6 Carryr, nny & Ohlin 10/24/03 Rv, Pag15/38
tall occurs whnvr th motor is attmpting to spin or mov against a forc that xcds th amount of torqu it can gnrat intrnally, and any tim th motor is startd from a rsting position, and any tim th motor rvrss dirction. his is an important point: stall torqu (and stall currnt) occurs any tim th motor starts from a stop or rvrss dirction. his is a critical point to considr whn dsigning circuits to driv DC motors, as covrd in Chaptr 20. Exampl 19.2: A prmannt magnt brushd DC motor will b usd to spin a cooling fan installd in a toy that would othrwis ovrhat and dform. h application rquirs a motor that can supply at last 225 mnm of torqu at 2000 rpm. Will a motor with a no-load spd of 9,550 rpm and a coil rsistanc of 2.32Ω, powrd by a 24V battry pack, b adquat for th task? tarting with Eq. 19.13 and rarranging it as an xprssion for : ω ω R M ω ω R M ( ω ω ) R COL W can dtrmin from Eq. 19.11: V 24V 2.51V krpm 9.55krpm / ω and from Eq. 19.8: Carryr, nny & Ohlin 10/24/03 Rv, Pag16/38
Nm / A 9.5493 3 V / krpm ( 2.51V / krpm) 0.0240 Nm / A 24.0mNm / A ubstituting ths valus into our xprssion for torqu: ( ω ω ) ( 24.0mNm / A)( 2.51V / krpm)( 9.55krpm 2krpm) R COL 2.32Ω 196 mnm his motor will not provid adquat torqu at 2000rpm for this application, so a diffrnt motor will nd to b valuatd. Givn that a motor can oprat anywhr btwn stall and no-load conditions, a coupl of fundamntal qustions aris: Ar all oprating points qually dsirabl and usful? Or ar som bttr than othrs? A clos look at motor powr and fficincy will hlp answr ths qustions. 19.7: Powr Charactristics For th purposs of this discussion, mchanical powr is dfind as P ω. Rcall from Eq. 19.2 that ovrall motor torqu is mad up of a friction torqu trm and a usabl torqu trm, so th full xprssion for motor powr output bcoms: Carryr, nny & Ohlin 10/24/03 Rv, Pag17/38
P ω ( f + )ω [ ] L W Eq. 19.16 For this discussion, w will assum that th friction torqu is rlativly small and may b safly nglctd. Again, for most motors this is a rasonabl first approximation. Howvr, for any of ths discussions, th ffcts of frictional torqu may b xplord by carrying th f trm in th quations through and rdvloping th rsults. As with th rlationship btwn torqu and spd, a motor s torqu and powr charactristics ar usually prsntd graphically for lins of constant voltag, and drawn as a family of curvs. Figur 19.17 shows a typical family of curvs for a rprsntativ motor. h powr output charactristic is parabolic in shap, having a maximum at ½ ALL for a givn voltag. Figur 19.7: ypical orqu vs. Powr Output Curvs for a Prmannt Magnt Brushd DC Motor o undrstand th shap of th curv and th position of th pak valu, start from th statmnt that P ω (Eq. 19.16). By substituting Eq. 19.13 for ω, this can b rwrittn as: Carryr, nny & Ohlin 10/24/03 Rv, Pag18/38
P ( ω R ) [ ] M W Eq. 19.17 hn, by combining trms and substituting ω from Eq. 19.11, w arriv at an xprssion rlating powr to torqu: V E P V 2 RM [ ] E W Eq. 19.18 aking th drivativ of Eq. 19.18 with rspct to torqu and stting th rsults qual to 0 yilds th point of maximum powr. h rsults of that xrcis ar that maximum powr output for a prmannt magnt brushd DC motor occurs whn ½ ALL. W can mak us of this by starting with Eq. 19.18, and substituting ½ ALL and ALL V R M from Eq. 19.14 to dvlop a rlationship btwn P MAX and applid voltag: P MAX 2 V V 2 2 R M RM E RM 2 ubstituting R R M from Eq. 19.12 givs: Carryr, nny & Ohlin 10/24/03 Rv, Pag19/38
P MAX V E V R 2R M 2 V 2R Finally, simplifying this givs th rsult: P MAX 4 R 2 V [ ] W Eq. 19.19 hs rsults show that P MAX is proportional to V 2, sinc th trm 4 is a constant for a givn motor. R his is an important rsult: th mchanical powr output of prmannt magnt brushd DC motors changs as th squar of th applid voltag. Changs in voltag hav a substantial impact on a motor s powr output. Exampl 19.3: A motor with a trminal rsistanc of 0.316Ω and 30.2 mnm/a is powrd by a 12V supply. Masurmnts show that th oprating rotational spd is 3616 rpm with a currnt flow of 1.79A. How much powr dos th motor gnrat undr ths conditions? What prcntag of th maximum possibl powr is this for th motor oprating at 12V? h amount of powr gnratd by th motor is givn by Eq. 19.16: P ω ( f + )ω L Carryr, nny & Ohlin 10/24/03 Rv, Pag20/38
n this xampl, w will assum that f is ngligibl, and simplify th xprssion: P ω o maintain consistnt units, w nd rotational spd in radians/sc instad of rpm: rad / s 3616 rpm 0.105 380 rad / s rpm hn from Eq. 19.6, w can dtrmin th torqu producd by th motor, sinc w know both and motor currnt, : ubstituting this back into th xprssion for powr givs th solution: P ω ( 0.0302 Nm / A)( 1.79 A)( 380 rad / s) 20. W 5 n ordr to dtrmin th maximum powr possibl for this motor at 12V, w will nd to dtrmin ALL, from which w can dtrmin ω at ½ ALL and calculat maximum powr. From Eq. 19.15: Carryr, nny & Ohlin 10/24/03 Rv, Pag21/38
V R ( 0.0302 Nm / A)( 12V ) ALL 1. 15 0.316Ω Nm For this motor powrd at 12V, th rotational spd at ½ ALL is: ω ω R M V R 1 ALL ( 2 ) V 2R ω V V 2 V 2 From w can dtrmin using Eq. 19.8:.0302 Nm / A 3.16V / krpm 9.5493 3 Nm / A V / krpm hn ω for ½ ALL is: ω V 2 2 12V ( 3.16V / krpm) 1897 rpm Finally, maximum powr is: ( 1/ 2)( 1.15 Nm)( 199 rad s) PMAX 1 / 2ALLω / Carryr, nny & Ohlin 10/24/03 Rv, Pag22/38
P MAX 114W Whn th motor is gnrating 20.5W, th prcntag of maximum powr is: P PMAX 20.5W 114W 100 18.0% 19.8: DC Motor Efficincy An additional quantity of grat intrst is that of motor fficincy, η. n this analysis, fficincy is dfind as th ratio of mchanical powr producd by th motor to lctrical powr consumd by th motor: η P P N Lω V ( ) OU M f Eq. 19.20 V ω Efficincy is maximizd at th point in th oprating curv whn a balanc is struck btwn gnrating th most usful work whil dissipating a minimum of powr as 2 R losss and friction losss. Figur 19.8 shows th typical fficincy charactristics of a prmannt magnt brushd DC motor, and its rlationship to powr, torqu, spd, and currnt. Carryr, nny & Ohlin 10/24/03 Rv, Pag23/38
Figur 19.8: Composit DC motor charactristics, showing th intrrlationship btwn fficincy η, torqu M, currnt, spd n, and powr P for a givn voltag At high torqus, with corrspondingly high currnts, 2 R loss (dissipatd through hating of th coils) ar high and thrfor fficincy is low. At vry low torqus, in spit of th high rotational spd ω, littl usful mchanical powr is producd and most or all of th powr is consumd ovrcoming friction, which scals linarly with spd: f ω. his also lads to low fficincy. Pak fficincy must thn occur somwhr in btwn. n gnral, maximum fficincy opration occurs at rlativly high ω, and low valus of torqu and currnt lading to minimal 2 R losss. h point of maximum fficincy opration varis from motor to motor. n most applications, it will b dsirabl to oprat DC motors in th rgion btwn th point of maximum fficincy and th point of maximum powr. Not that th slop of th fficincy curv falls away from th Carryr, nny & Ohlin 10/24/03 Rv, Pag24/38
maximum much mor gradually in this rgion, rathr than dropping rapidly to 0 for highr spds. Also, most motors ar not capabl of running continuously with high currnt lvls, such as occurs at torqu lvls abov th point corrsponding to maximum powr bcaus of hating. h spcifications of any givn motor should always b consultd for this typ of information. For oprating conditions that rsult in th maximum motor fficincy, a motor will gnrat th most usful torqu for a givn powr input. W will solv for th motor currnt,, that rsults in th fficincy bing maximizd. For this analysis, frictional ffcts dfinitly may not b safly nglctd, sinc thy dominat th fficincy of th motor for conditions involving low torqu/high spd opration. t will b usful to rstat fficincy in ordr to solv for th currnt,, that maximizs η. h gnral quation for motor fficincy, is statd abov in Eq. 19.20. W can rstat V using Ohm s Law as follows: V R [ ] V Eq. 19.21 sinc V, and R ar all constants. ubstituting this back into th P N trm of Eq. 19.20 givs: η ( ) M f R ω ubstituting th xprssion for ω from Eq. 19.10 givs: Carryr, nny & Ohlin 10/24/03 Rv, Pag25/38
η ( ) M f V R RM and rplacing total motor torqu producd M (Eq. 19.6) and V R (Eq. 19.21) givs: η ( ) f R R R implifying this givs: η f R ( ) R R h frictional torqu trm, f, may also b xprssd as: [ m] f N Eq. 19.22 sinc th no-load currnt,, is th amount of currnt rquird to ovrcom th forc of friction only, without gnrating any additional usful torqu th dfinition of th no-load condition. ubstituting this into our xprssion for fficincy rsults in th following: Carryr, nny & Ohlin 10/24/03 Rv, Pag26/38
η R ( ) R R Finally, sinc in consistnt units, fficincy can b xprssd as: η η ( )( R R) R ( )( ) η 1 + Eq. 19.23 1 1 η Eq. 19.24 With Eq. 19.23 and Eq. 19.24, w hav xprssions for fficincy only as functions of motor currnt (), no-load currnt ( ), and stall currnt ( ). n ordr to find th currnt that rsults in th oprating point of maximum fficincy, tak th drivativ of Eq. 19.23 with rspct to currnt (), st th rsults qual to 0 and solv for. η 1 + 0 1 2 0 Carryr, nny & Ohlin 10/24/03 Rv, Pag27/38
Eq. 19.25 ubstituting th rsult in Eq. 19.25 back into Eq. 19.24 givs us th xprssion for maximum fficincy w wr aftr: η MAX 1 1 implifying this givs th mor compact rsult: 1 MAX 2 η Eq. 19.26 Rcalling from Eq. 19.22 that f and V R from Eq. 19.21, w can rwrit 19.24 in trms that will allow us to draw a fw additional conclusions: f R 1 V 2 η Eq. 19.27 MAX his xprssion for maximum fficincy shows that incrass in friction dcras fficincy, as do incrass in rsistanc. Carryr, nny & Ohlin 10/24/03 Rv, Pag28/38
Aftr stpping through this dvlopmnt, th tradoffs btwn spd and torqu that ffct fficincy should b clar; both from th final rsults in Eq. 19.27 and th initial xprssion for fficincy givn in Eq. 19.20: o Vry high spd/low torqu opration (.g. at or nar th no-load condition) is dominatd by friction and hnc not fficint. Eq. 19.20 illustrats this point, as M gos to 0 at th no-load condition. o Vry low spd/high currnt opration (.g. nar th stall condition) is dominatd by hating of th coils through 2 R losss and hnc infficint. Eq. 19.20 also illustrats this point, as ω is 0 at stall conditions. o h point of maximum fficincy opration lis btwn ths two ndpoints, and 2 R losss typically xrt a mor dominant influnc. his rsults in maximum fficincy for rlativly high spd/low torqu opration (.g. somwhr nar th no-load condition). Eq. 19.27 illustrats th balanc btwn fficincy at stall and at no-load conditions. Exampl 19.4: f th no load currnt for th motor in Exampl 19.3 is 137 ma, what is fficincy of th motor running undr thos conditions? What rotational spd rsults in maximum fficincy opration? What is th maximum fficincy? n addition to th no load currnt, w nd to know th stall currnt,. From Eq. 19.5: 1.15 Nm 38. 0.0302 Nm / A 0 A Carryr, nny & Ohlin 10/24/03 Rv, Pag29/38
Using Eq. 19.24 to find th fficincy of th motor for ths conditions: 0.137 A 1.79 A η 1 1 1 1 A A 1.79 38 η 88.0% h currnt that rsults in maximum fficincy is givn by Eq. 19.25: ( 0.137 A)( 38 A) 2. A 28 his corrsponds to a rotational spd at 12V as givn by Eq. 19.13: ω ω R M ubstituting from Eq. 19.6, ω from Eq. 19.11 and R M from Eq. 19.12 into Eq. 19.13, and rcalling that 3.16V/krpm from Exampl 19.3 givs: ω V ( ) V R 12V ( 0.316Ω)( 2.28 A) R 3.16V / krpm ω 3567 rpm Finally, Eq. 19.26 givs th maximum fficincy for this motor at: Carryr, nny & Ohlin 10/24/03 Rv, Pag30/38
η MAX 2 0.137 1 A 1 38 A 2 η MAX 88.3% t turns out that th motor in Exampl 19.3 was running vry clos to maximum fficincy. For most practical purposs, a gain of 0.3% is ngligibl. 19.9: Garhads As shown in ction 19.8, th most fficint oprating currnts (and thrfor torqus) for prmannt magnt brushd DC motors li abov no-load currnt, and blow th condition for maximum powr, ½ ALL or ½ ALL. his rgion is spcially dsirabl if a motor is to b in continuous or high duty cycl us, as most motors ar not ratd for long priods of high torqu. h coils of th armatur will vntually rach high tmpraturs and fail. Givn that most motors hav output shaft spds of svral thousand RPM and littl torqu nar th point of maximum fficincy, and givn that many applications for prmannt magnt brushd DC motors rquir substantially lowr spds and highr torqus, garhads (also calld garboxs) may b rquird. Garhads ar an inxpnsiv and compact mans of dcrasing motor output shaft spds and incrasing availabl torqu. n thory, thy could also b constructd to incras th output shaft spd and dcras th availabl torqu, howvr this is sldom if vr usful. n practic, garhads ar usd to rduc th spd of th output shaft to mor usful rangs. h rlationship btwn th input shaft spd for a garhad and its output shaft spd is calld th gar ratio, and is givn by: Carryr, nny & Ohlin 10/24/03 Rv, Pag31/38
Gar Ratio ω N N Eq. 19.28 ω OU h numbr of tth in th gars usd to construct a garhad dtrmins th gar ratio. Gar ratios ar oftn xprssd as N:1,.g., 10:1 or 250:1. h rang of possibl gar ratios is vry wid, from nar 1:1 with spur gars to svral-thousand-to-on with plantary gars. dally, all th powr producd by th motor and introducd to th garhad at th input shaft would b availabl at th garhad output shaft. his would b th cas if th garhad wr frictionlss and 100% fficint. Unfortunatly, this is nvr th cas for ral garhads. aking losss into account, th xprssion for garhad fficincy is givn by: P ϖ η P ω OU OU OU Eq. 19.29 N N N n commrcially availabl garhads, it is not unusual to s fficincy ratings blow 50%, spcially for inxpnsiv garhads and thos with vry high gar ratios. For smallr gar rductions (4:1, say), fficincy may b in th 90% rang. n any application, garhad fficincy is a major considration and can gratly affct systm prformanc. As spd is dcrasd across a garbox, torqu is incrasd. Combining Eq. 19.28 and Eq. 29 givs an xprssion for th torqu availabl at th output shaft of a garbox: Carryr, nny & Ohlin 10/24/03 Rv, Pag32/38
OU η N Eq. 19.30 N Exampl 19.5: A mobil robot rquirs a driv motor that can supply at last 75 oz. in. torqu at 50 rpm. h garhad that has bn chosn for th application has a ratio N 24 and fficincy η 63%. inc th nxt stp will b to slct th motor that will driv th garhad, what spd and torqu oprating points will b rquird of a motor undr ths conditions? inc th ratio and th rquird output shaft spd of th garhad ar spcifid, w can dtrmin th rquird rotational spd of th motor from Eq. 19.28: ω N N ω OU ( 24 )( 50 rpm) rpm ω N N ω OU 1200 hn, sinc th fficincy and th rquird output torqu ar spcifid, w can dtrmin th torqu rquird from th DC motor from Eq. 19.30: OU η N N N OU η N 75oz. in. 4.96oz. in. ( 0.63)( 24) h motor will b rquird to produc at last 4.96 oz. in. torqu at 1200 rpm. Aftr a motor is slctd for th application, th nxt task would b to dtrmin th oprating voltag that satisfis ths spcifications. Carryr, nny & Ohlin 10/24/03 Rv, Pag33/38
19.10: UMMARY n this chaptr, prmannt magnt brushd DC motors wr dscribd and charactrizd. DC motors ar on of th most common actuators usd in mchatronics. h subsystms and componnts that compris DC motors wr numratd and dscribd, including th rotor, armatur (coil windings), housing, stator (prmannt magnts), commutator and brushs. h production of motor shaft torqu and rotation was dscribd as a rsult of th intraction btwn th magntic fild stablishd by th stator s prmannt magnts and th magntic fild cratd by currnt flowing in th armatur s coil windings. h concpt of back-emf was introducd and xplord. his is also calld th gnrator ffct, and is th rsult of th armatur spinning within th stator s magntic fild, inducing a voltag countr to th that introducd at th motor s trminals. An lctrical modl of DC motors was givn, and from this modl, th torqu constant, and, th spd constant wr dvlopd. h motor constants and fundamntal lctrical concpts wr thn usd to dvlop th charactristic quations for DC motors. Exprssions for rotational spd ω, torqu, powr, and motor fficincy wr dvlopd, and th factors that influnc thm xplord. R M, th spd rgulation constant, was introducd. y points of opration wr dscribd, including no-load conditions stall conditions, maximum powr and maximum fficincy. Finally, garhads wr introducd, and thir us in rducing spd and incrasing torqu availabl form DC motors was xplord. Garhad fficincy, and its ffct on th torqu and spd availabl at th output shaft of th garhad, was xplaind. Onc you hav mastrd th concpts containd in this chaptr, you should b abl to: Carryr, nny & Ohlin 10/24/03 Rv, Pag34/38
1) dntify th major componnts and subsystms that mak up prmannt magnt DC motors and dscrib thir function. 2) Dscrib how torqu rsults from th intractions btwn th magntic fild stablishd by th prmannt magnt stator and th rotor armatur. 3) Dscrib th charactristic constants for a DC motor, th torqu constant and th spd, and xplain th factors that influnc thm. 4) Undrstand th gnrator ffct and back-emf. 5) B abl to idntify DC motor oprating points for pak powr and pak fficincy. 6) B abl to spcify a DC motor for an application. 7) B abl to spcify and us a garhad in combination with a DC motor as rquird. Chaptr 19 Problms: 1. h mchatronic systm shown blow is dsignd to priodically hoist a 10 oz. mass abov a platform whr it is normally rsting. h spool has radius 3/8 in., and is dirctly connctd to th output shaft of th motor. f th motor has a stall torqu of 29.5 oz. in. at 15V, what is th minimum voltag rquird to hoist th mass? 2. As you stroll th isls of th local fla markt, you com across a booth stockd with surplus prmannt magnt brushd DC motors. Your ys widn with xcitmnt as you notic a particularly shiny gar motor pricd at $2.50. Whipping your trusty multi-mtr out of its blt holstr, you masur th winding rsistanc to b 18.9Ω. Nxt, you pull a small torqu wrnch out of your fanny pack and masur th stall torqu, which is 2.8 Nm whn powrd by th 12V battry you kp handy Carryr, nny & Ohlin 10/24/03 Rv, Pag35/38
for just such occasions. Finally, you us your stopwatch to tim 100 rvolutions of th garhad s output shaft and dtrmin th no-load spd to b 2 rvs/sc. h garhad is markd 100:1. h application you hav in mind for this motor rquirs th motor to dlivr 0.4 Nm at 100 rpm whn drivn at 15V. You assum that th frictional losss in th motor and garbox ar ngligibl, and dtrmin th appropriatnss of th motor by answring th following: a) Will th motor and garhad mt th rquirmnts for torqu and spd at 15V? f not, would it b possibl at a driv voltag othr than 15V? b) What is th currnt rquird to oprat at th dsign point? 3. A motor with 105 mnm/a, R COL 10Ω and ω (at 48V) 4,320 rpm will b opratd with a 48V supply. f this motor is connctd to a 12:1 garhad that has frictional torqu losss of f,garhad 2.4 mnm, what will th output shaft rotational spd b? 4. An lctrical contact is attachd to a fixd rfrnc point via a spring with spring constant 1. h spring is spoold onto th shaft of a DC motor in ordr to mov th lctrical contact from its initial location X 1, until it maks contact with a scond lctrical contact that is also attachd to a fixd rfrnc point with a spring that has a spring constant 2. h initial position of th scond contact is X 2. h first spring constant 1 100 N/m, th scond spring constant 2 15 N/m and th motor s shaft diamtr is 1/2 in. h motor is powrd at 15V, a no-load spd ω 4080 rpm at this voltag, and coil rsistanc R 9.73Ω. What is th valu of th initial distanc, X 2, that th scond contact should b placd from th contact on th spring to nsur that th scond contact is displacd by 0.1mm? Carryr, nny & Ohlin 10/24/03 Rv, Pag36/38
5. You wish to dsign a nw ultra-high quality, portabl, battrypowrd coff burr grindr for backpacking sprsso fanatics. h grinding lmnts you hav slctd (picturd) ar adjustabl so that a cours grind rsults whn th burr cons ar movd far apart from ach othr, a fin grind rsults whn th cons ar movd clos togthr, and any intrmdiat point may b slctd by th usr. h manufacturr of th burr cons claims that th torqu rquird to grind coff rangs from 0.1 Nm (cours grind) to 0.5 Nm (fin grind), and that th burr cons only function whn rotating btwn 6 to 10 rpm. You will us a motor with ω 13,900 rpm, ALL 28.8 mn m, ALL 3.55 A, coil rsistanc R 3.38 Ω, maximum continuous currnt 0.614 A, 8.11 mn m/a, and 0.847 V/krpm. hr ar 3 garhads availabl to you for this dsign: th first has a 850:1 ratio with 65% fficincy, th scond has a ratio of 1621:1 with 59% fficincy, and th third has a ratio of 3027:1 with 59% fficincy. Which garhad satisfis all th constraints? 6. A motor with 16.1 mnm/a, R COL 1.33Ω and ω (at 18V) 10,300 rpm will b opratd with a 18V supply. h maximum prmissibl continuous torqu spcification is 24.2 mnm. What rotational spd dos this corrspond to? Carryr, nny & Ohlin 10/24/03 Rv, Pag37/38
7. tarting with th xprssion for motor powr output givn in Eq. 19.18, show that maximum powr is dvlopd at ½ ALL. Carryr, nny & Ohlin 10/24/03 Rv, Pag38/38