# Rotating DC Motors Part I

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1 Rotting DC Motors Prt I he previous lesson introduced the simple liner motor. Liner motors hve some prcticl pplictions, ut rotting DC motors re much more prolific. he principles which eplin the opertion of liner motors re the sme s those which eplin the opertion of prcticl DC motors. he fundmentl difference etween liner motors nd prcticl DC motors is tht DC motors rotte rther thn move in stright line. he sme forces tht cuse liner motor to move right or left in stright line cuse the DC motor to rotte. his chpter will emine how the liner motor principles cn e used to mke prcticl DC motor spin Electricl mchinery Before discussing the DC motor, this section will riefly introduce the prts of n electricl mchine. But first, wht is n electricl mchine? An electricl mchine is term which collectively refers to motors nd genertors, oth of which cn e designed to operte using AC (Alternting Current) power or DC power. In this supplement we re only looking t DC motors, ut these terms will lso pply to the other electricl mchines Physicl prts of n electricl mchine It should e pprent tht the purpose of n electricl motor is to convert electricl power into mechnicl power. Prcticl DC motors do this y using direct current electricl power to mke shft spin. he mechnicl power ville from the spinning shft of the DC motor cn e used to perform some useful work such s turn fn, spin CD, or rise cr window. he rottion of the DC motor is ccomplished y the force which is developed on current-crrying conductor in mgnetic field. he current-crrying conductor is connected to the shft which is le to rotte reltive to the sttionry ody of the DC motor. With this simple understnding, we cn divide ny motor into two physicl prts; one prt which rottes clled the rotor nd one prt which doesn t clled the sttor. Figure 1 shows simple digrm of n electricl mchine showing the rotor nd sttor.

2 ttor Rotor hft Figure 1: Lyout of ypicl Rotting Mchine DC motors use DC current nd voltge to power the motor. For resons which will e mde cler elow, it is necessry to chnge the direction of the DC current tht is pplied to the current-crrying conductor within the rotor. his is ccomplished y utilizing segmented metl ring, clled commuttor. A commuttor is directly connected to the current-crrying conductor, so it will rotte with the rotor. he commuttor mintins electricl contct with its eternl DC electricl power source y using metl or hrd cron rushes. A more detiled description of commuttion will e given elow Functionl prts of n electricl mchine As previously stted electricl mchines re divided into two physicl prts, rotor nd sttor. Electricl mchines cn lso e divided into two functionl prts. One functionl prt is the mgnetic field, simply clled the field, nd the other functionl prt is the conductor, which is clled the rmture. In given mchine, one functionl prt is ssocited with one physicl prt, nd the other functionl prt is then ssocited with the other physicl prt. o there re two possile configurtions for electricl mchines: 1) the field rottes with the rotor nd the rmture is on the sttor, or 2) the rmture rottes with the rotor while the field is on the sttor. Any electricl mchine cn e designed in either configurtion, lthough for prcticl resons one design tends to dominte DC mchines. he DC motor tht we will study in EE301 will use rotting rmture inside mgnetic field, which is developed within the sttor s shown in figure 2.

3 16.2 DC Motor Opertion Figure 2. Rotting DC motor In simplest terms, DC motor opertes y using the force descried y the Lorentz force lw presented in Lesson 15. DC voltge is pplied cross the rmture. he currentcrrying rmture is in the mgnetic field generted in the sttor. In Figure 2, the sttor is represented y permnent mgnet with its North nd outh Pole s shown. he current-crrying wire in mgnetic field results in force which will rotte the rotor. Figure 3 is simple digrm of DC motor, showing the mgnet poles s the sttor nd the direction of the current flowing through wire of the rmture. ince the mgnetic force ( F = IL B ) is cross product, we cn see tht the mgnetic force cts to pull the top conductor ( ) of the rmture loop towrds the left, nd cts to pull the lower conductor ( ) towrds the right. hese two forces rotte the rmture tht is ttched to the rotor. N F Current out of pge X Current in to pge W F Figure 3.

4 You my e le to see tht if the rmture current is lwys in the sme direction, the conductor ( ) shown on the top in Figure 3 will lwys e pulled towrds the left nd the conductor ( ) shown on the ottom in Figure 3 will lwys e pulled towrds the right. At est, this motor would only rotte through one-hlf (180º) of rottion nd would stop when the conductor is in the 9 o clock position nd the conductor is in the 3 o clock position. his is where the commuttor comes into the picture. Recll tht the function of the commuttor is to reverse the direction of the current flowing through the rmture in the rotor; this is precisely wht hppens. Agin, the Lorentz force eqution tells us tht if the current direction is reversed, the sign of the cross product is lso reversed. his mens tht the Lorentz force is now directed to pull the conductor towrds the right nd the conductor towrds the left, llowing the motor to rotte through more thn one-hlf of rottion. By chnging the current direction every hlf-rottion (when the conductors re in the 3 nd 9 o clock positions), the Lorentz force is lwys cting to keep the motor spinning 360º in one direction. Wht out torque production? We pply voltge to the rush terminls. his cuses current to flow into the top rush of the mchine. Let s cll this current I (suscript stnding for rmture ) nd consider it for the loop/commuttor positions shown in Figure 4. Note, while the rushes contct oth commuttor segments (positions 1 nd 3), the current is diverted from the loop nd will flow directly from the top rush to the lower rush (this short-circuit condition will go wy s we move to more prcticl mchine). In position 2, the current flows into the top rush nd then must flow into coilside nd ck out coil-side. he current then eits through the ottom rush. In position 4, the current enters the top rush ut this time the commuttor routes this current to coil-side. he current trverses the loop nd returns to the lower rush vi coil-side. hus, the commuttor ensures tht the top conductor of the rmture coil is lwys crrying current INO the pge while the ottom conductor of the rmture coil is lwys crrying current OU OF the pge. he importnce of this comes into ply when we consider the production of force nd recll our previous results regrding the Lorentz force: F = IL B he eqution used to determine torque is = R F, where R is the vector which gives the rmture position reltive to the centrl is of the motor nd F is the Lorentz force vector. If we pply the Lorentz force eqution to the mchine in position 2 (Figure 4), the top conductor will eperience force to the LEF ( I is directed IN, while B is DOWN, so there is nturl 90 degrees etween them) while the ottom conductor will eperience force to the RIGH ( I is OU, while B is DOWN, gin 90 degrees etween them). ince these forces re tngentil to the rotor, the cross-product in the torque epression ecomes multipliction nd we rrive t the developed torque of d = BLI R + BLI R = 2BLI R

5 orque hs mimum vlue when R is norml to F, nd its minimum vlue, 0, when R nd F re prllel. his is shown in figure 4. In position 1, the rmtures wires re locted hlf-wy etween the two mgnets, so R is prllel to F. No torque is developed y the motor until something mkes the motor rotte slightly. Mimum torque, s shown in position 2 is developed when the rmture wires re locted t the top nd ottom.

6 N I + V out = O e = O N I + V out = O e = 2 BLI R N + V out = O I e = O N + V out I e = 2 BLI Figure 4: Current Flow nd orque Production in Rotting Loop R

7 Agin ssuming rdilly-directed B-field, s the loop nd commuttor turns, the developed forces will remin tngentil. he resultnt torque wveform is sketched in Figure 5. Note the lrge vrition in torque when the rushes short-circuit the terminls of the loop. e 2BLI R π 2 π 3π 2 2π Figure 5: Developed Electromgnetic orque for ingle Loop Mchine o develop more prcticl DC motor (less vrition in the developed electromgnetic torque), we need to: 1. Add nd sptilly distriute more rotting loops 2. Increse the numer of commuttor segments As noted in the section , the rotting loops re clled the rmture of our DC motor (hence the suscript tht we hve een using). Up to this point the rmture shown in Figures 2, 3, nd 4 hs een simply single loop of wire. If we clculted the mimum torque generted on such n rmture using typicl current nd mgnetic field vlues where = RF = 2RLIB, the result would e very smll numer. he mimum torque could conceivly e incresed y using higher currents or stronger mgnetic fields, ut these re usully not prcticl solutions. A more prcticl solution is to construct the rmtures using coils of wire rther thn single loop. his multiplies the mimum torque y the numer of wire loops, N, (usully clled the numer of turns) in the rmture. hen the eqution for mimum torque ecomes: = NRF = 2NRLIB. he role of the commuttor in multiple rmture motors is lso worth noting. Recll tht the commuttor in single-loop rmture motor simply chnged the direction of the

8 current through the rmture. In multiple-loop rmture motors it still does this, ut it lso performs the dditionl function of delivering current only to the rmture loop which is est positioned to develop mimum torque. Ail View N ide View trtor ttor pring N Pole Brush holder Commuttor Brush End Iron ell frme hft Bering End-turns Pigtil Armture winding Figure 6: Lyout for Prcticl Permnent Mgnet DC Mchine

9 16.3 A Relistic DC Mchine A relistic DC mchine is shown in Figure 6. In the il (hed-on) view, we oserve tht the rmture conductors in the top hlf of the plne ll crry current into the pge, while the conductors on the ottom hlf of the plne ll crry current out of the pge. he sptil distriution of current, enforced y the ction of the commuttor, ensures tht ech of the top conductors will eperience force to the left while ech of the lower conductors will eperience force to the right (hence the mchine will rotte counterclockwise). he north nd south poles shown on the sttionry prt of the mchine my e derived from permnent mgnets or they could e generted from n electricllyecited sttionry coil clled the field windings. he side view illustrtes some more prcticl mchine issues. Note the ctul length of the mchine consists of the length of the rmture, end-turns of the rmture windings, the commuttor, nd the iron frme of the sttor. he rushes re shown spring-loded so they mke dequte contct with the commuttor. he shft runs through the rmture nd commuttor, nd is supported in the sttor iron frme y erings. he height of the mchine depends on the dimeter of the rmture, the poles of the field mgnet, nd the sttor iron frme. Here, we consider mchine with two poles. A more compct nd efficient design would utilize more poles Permnent Mgnet Mchines In deriving n epression for the developed electromgnetic torque ( d ), we hve sid very little out the ir-gp mgnetic flu density, B. (he ir-gp is the smll spce etween the rotting rmture nd the sttionry field poles.) We hve ssumed tht it hs een creted somehow, perhps with coil wrpped out the sttionry core. We hve lso een it remiss in not commenting on how the ir-gp field is influenced y the mgnetic field creted y the rmture windings. For now, we will ssume tht this rmture rection hs een compensted for y our mchine designer. he ir-gp flu density cn lso e creted y using permnent mgnets ttched to the sttor core. he permnent mgnet coupled with fied ir-gp will estlish reltively constnt vlue of B-field in the ir gp. o simplify our nlysis nd focus on one clss of DC mchine, we will only consider permnent-mgnet mchines (you re free to cheer t this point). If the B-field is fied, then the term N2BLR is simply constnt (recll, L is the il length of the mchine nd R is the rdius of the rmture). We cn replce this constnt y new prmeter, K ν, so tht: d = ( N 2BLR) I = Kν I (K v is clled the DC motor constnt, or the mchine constnt, nd hs units of volt-seconds, V-s) he mechnicl dynmics of the DC motor re governed y Newton s Lws. he developed electromgnetic torque is lnced y n inertil torque, lod torque, nd rottionl loss torque:

10 dω d = Kν I = J + Lod + Loss dt In the stedy-stte, the ngulr velocity ω is constnt (thus dω = 0 ) nd therefore the dt electromgnetic torque must lnce only the lod nd loss torques. Wht is the rtionle for including Loss? Even unloded, the mchine will hve to produce torque to overcome friction nd windge (ir resistnce) effects. Further, there re other secondry losses in the mchine iron tht re not ccounted for in the rmture resistnce tht we lump into this prmeter. We cn evlute the no-lod losses t vrious speeds. If this power loss chrcteristic is liner versus speed, then it is resonle to model the etr loss component y constnt torque (since P = ω ). If the no-lod power vries more s qudrtic with speed, then we would replce Loss y term like C ω where C is constnt. It is interesting (to someody) to oserve tht the mechnicl power developed due to the electromgnetic torque is found to e P = ω = Kν I ω d d If we ignore rottionl losses so tht Loss = 0, this developed power ecomes the output mechnicl power nd we cn evlute the mchinery efficiency s η P Pd 100 = 100 ( = 0) P OU = loss PIN in Emple 1: A DC motor is used to rise nd lower window in n utomoile. he motor drws 8.1A nd runs t 800 rpm while rising the window. he mchine constnt is V-s. If the torque required to lower the window is 80% of tht required to lift the window (grvity helps), find the current required to lower the window. d, up d, up = K d, up ν I = (0.095N m / A)(8.1A) = N m hen: d, down d, down = 0.8 d, up = N m = 0.8(07697N m) d, down N m I, down = = = 6. 48A K 0.95N m / A ν

11 16.5 Bck EMF In the previous sections the opertion of rotting DC motors ws introduced. It ws shown tht the force eerted on current-crrying conductor in mgnetic field cn e used to turn motor electricl energy is converted into mechnicl energy. In lter lesson, genertors will e introduced. It will e shown tht genertors use mechnicl energy to crete electricl energy. hey do this y forcing conductor to move through mgnetic field. his forced motion of wire through mgnetic field cuses voltge to e developed on the wire tht cuses current to flow. his developed voltge ws clled E induced in the lesson on liner motors. his genertion of n induced voltge eists nytime conductor is moved through mgnetic field, including such motion in motor. In other words, the intended rottion of the rmture in motor lso cretes n unintended genertion of voltge. his is the induced voltge descried y Frdy s Lw in the sections on liner motors. We will use the symol E from now on to represent the induced voltge. his generted voltge on the rmture is referred to s induced electromotive force, or, more commonly, the ck EMF. As the lel ck EMF implies, the polrity of the ck EMF is opposite of the polrity of the pplied voltge which cuses the rmture current to flow, just s we sw for the liner motor. he eqution used to clculte ck EMF is: E = Kvω where K ν is the DC motor constnt, which depends on the construction of the motor, nd ω represents the ngulr velocity of the rmture (in rdins per second). Greter ck EMF cn e developed y turning the rotor fster. As mentioned previously, the eqution for torque using the DC motor constnt is: K v I =. It should e cler from this eqution tht greter torque cn e generted y incresing the rmture current.

12 Here is summry of the rotting motor equtions: Rotting Motor Equtions Electricl Mechnicl P in P elec loss = P d If no lod, then P out nd out = 0, nd P d = P mech loss, d = mech loss P d P mech loss = P out Power eqution (P out = P lod ) d mech loss = lod orque eqution ( out = lod ) energy conversion here ( = P / ω, or, P = ω) ( d suscript mens developed ) P in = V DC I P elec loss = P = = I 2 R P d = E * I = d ω = K v I ω E = K v ω d = K v I η = P out / P in

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