Mtr Calculatins Calculating Mechanical Pwer Requirements Trque - Speed Curves Numerical Calculatin Sample Calculatin Thermal Calculatins Calculating Mechanical Pwer Requirements Physically, pwer is defined as the rate f ding wrk. Fr linear mtin, pwer is the prduct f frce multiplied by the distance per unit time. In the case f rtatinal mtin, the analgus calculatin fr pwer is the prduct f trque multiplied by the rtatinal distance per unit time. = M ω P rt P rt = rtatinal mechanical pwer M = trque ω = angular velcity The mst cmmnly used unit fr angular velcity is rev/min (RPM). In calculating rtatinal pwer, it is necessary t cnvert the velcity t units f rad/sec. This is accmplished by simply multiplying the velcity in RPM by the cnstant (2 x π) /60: ω rad / sec = ω rpm 2π 60 It is imprtant t cnsider the units invlved when making the pwer calculatin. A reference that prvides cnversin tables is very helpful fr this purpse. Such a reference is used t cnvert the trque-speed prduct t units f pwer (Watts). Cnversin factrs fr cmmnly used trque and speed units are given in the fllwing table. These factrs include the cnversin frm RPM t rad/sec where applicable. Trque Units Speed Units Cnversin Factr z-in RPM 0.00074 z-in rad/sec 0.0071 in-lb RPM 0.0118 in-lb rad/sec 0.1130 ft-lb RPM 0.1420 ft-lb rad/sec 1.3558 N-m RPM 0.1047 N-m rad/sec 1.0002 Fr example, assume that it is necessary t determine the pwer required t drive a trque lad f 3 z-in at a speed f 500 RPM. The prduct f the trque, speed, and the apprpriate cnversin factr frm the table is: 3 z in 500rpm 0.00074 = 1. 11Watts Calculatin f pwer requirements is ften used as a preliminary step in mtr r gearmtr selectin. If the mechanical pwer required fr a given applicatin is knwn, then the maximum r cntinuus pwer ratings fr varius mtrs can be examined t determine which mtrs are pssible candidates fr use in the applicatin.
Trque - Speed Curves One cmmnly used methd f displaying mtr characteristics graphically is the use f trque - speed curves. While the use f trque - speed curves is much mre cmmn in technical literature fr larger DC machines than it is fr small, irnless cre devices, the technique is applicable in either case. Trque - speed curves are generated by pltting mtr speed, armature current, mechanical utput pwer, and efficiency as functins f the mtr trque. The fllwing discussin will describe the cnstructin f a set f trque - speed curves fr a typical creless DC mtr frm a series f raw data measurements. Mtr 1624E009S is used as an example. Assume that we have a small mtr that we knw has a nminal vltage f 9 vlts. With a few fundamental pieces f labratry equipment, the trque - speed curves fr the mtr can be generated: Step One (measure basic parameters): Using a vltage supply set t 9 vlts, run the mtr unladed and measure the rtatinal speed using a nn-cntacting tachmeter (strbe, fr instance). Measure the mtr current under this n-lad cnditin. A current prbe is ideal fr this measurement since it des nt add resistance in series with the perating mtr. Using an adjustable trque lad such as a small particle brake cupled t the mtr shaft, increase the trque lad t the mtr just t the pint where stall ccurs. At stall, measure the trque frm the brake and the mtr current. Fr the sake f this discussin, assume that the cupling adds n lad t the mtr and that the lad frm the brake des nt include unknwn frictinal cmpnents. It is als useful at this pint t measure the terminal resistance f the mtr. Measure the resistance by cntacting the mt r terminals. Then spin the mtr shaft and take anther measurement. The measurements shuld be very clse in value. Cntinue t spin the shaft and take at least three measurements. This will ensure that the measurements were nt taken at a pint f minimum cntact n the cmmutatr. Nw we have measured the: n 0 = n-lad speed I 0 = n-lad current M H = stall trque R= terminal resistance Step Tw (plt current vs. trque and speed vs trque) Prepare a graph with mtr trque n the hrizntal axis, mtr speed n the left vertical axis, and mtr current n the right vertical axis. Scale the axes based n the measurements in step 1. Draw a straight line frm the left rigin f the graph (zer trque and zer current) t the stall current n the right vertical axis (stall trque and stall current). This line represents a plt f the mtr current as a functin f the mtr trque. The slpe f this line is the prprtinality cnstant fr the relatinship between mtr current and mtr trque (in units f current per unit trque). The reciprcal f this slpe is the trque cnstant f the mtr (in units f trque per unit current). Fr the resulting curves see Graph 1. Using the relatinships between mtr cnstants discussed earlier, calculate the velcity cnstant f the mtr frm the trque cnstant btained abve. By multiplying the velcity cnstant by the nminal mtr vltage, btain the theretical n-lad speed f the mtr (zer trque and n-lad speed) and plt it n the left vertical axis. Draw a straight line between this pint and the stall trque and zer speed pint n the graph. The slpe f this line is the prprtinality cnstant fr the relatinship between mtr speed and mtr trque (in units f speed per unit trque). The slpe f the line is negative, indicating that mtr speed decreases with increasing trque. This value is smetimes called the regulatin cnstant f the mtr. Fr the resulting curves see Graph 1. Fr the purpse f this discussin, it will be assumed that the mtr has n internal frictin. In practice, the mtr frictin trque is determined using the trque cnstant f the mtr and the measured n-lad current. The trque vs speed line and the trque vs current line are then started nt at the left vertical axis but at an ffset n the hrizntal axis equal t the calculated frictin trque.
Step Three (plt pwer vs trque and efficiency vs trque) In mst cases, tw additinal vertical axes are added fr pltting pwer and efficiency as functins f trque. A secnd left vertical axis is usually used fr efficiency and a secnd right vertical axis is used fr pwer. Fr the sake f simplifying this discussin, efficiency vs. trque and pwer vs. trque will be pltted n a secnd graph separate frm the speed vs. trque and current vs. trque plts. Cnstruct a table f the mtr mechanical pwer at varius pints frm n-lad t stall trque. Since mechanical pwer utput is simply the prduct f trque and speed with a crrectin factr fr units (see sectin n calculating mechanical pwer requirements), pwer can be calculated using the previusly pltted line fr speed vs. trque. A sample table f calculatins fr mtr 1624E009S is shwn in Table 1. Each calculated pint is then pltted. The resulting curve is a parablic curve as shwn in Graph 1. The maximum mechanical pwer ccurs at apprximately ne-half f the stall trque. The speed at this pint is apprximately ne-half f the nlad speed. Cnstruct a table f the mtr efficiency at varius pints frm n-lad t stall trque. The vltage applied t the mtr is given, and the current at varius levels f trque has been pltted. The prduct f the mtr current and the applied vltage is the pwer input t the mtr. At each pint selected fr calculatin, the efficiency f the mtr is the mechanical pwer utput divided by the electrical pwer input. Once again, a sample table fr mtr 1624E009S is shwn in Table 1. and a sample curve in Graph 1. Maximum efficiency ccurs at abut 10% f the mtr stall trque. Table 1. TORQUE SPEED CURRENT POWER EFFICIENCY (z-in) (rpm) (ma) (Watts) (%) ----------- --------- ------------ ---------- ------------- 0.025 11247.65 0.024 0.208 0.10 0.05 10786.3 0.048 0.399 71.87 0.075 10324.95 0.072 0.573 75.27 0.1 9863.6 0.096 0.730 74.99 0.125 9402.25 0.120 0.870 73.25 0.15 8940.9 0.144 0.992 70.78 0.175 8479.55 0.168 1.098 67.89 0.2 8018.2 0.192 1.187 64.73 0.225 7556.85 0.217 1.258 61.40 0.25 7095.5 0.241 1.313 57.95 0.275 6634.15 0.265 1.350 54.41 0.3 6172.8 0.289 1.370 50.80 0.325 5711.45 0.313 1.374 47.14 0.35 5250.1 0.337 1.360 43.44 0.375 4788.75 0.361 1.329 39.71 0.4 4327.4 0.385 1.281 35.95 0.425 3866.05 0.409 1.216 32.17 0.45 3404.7 0.433 1.134 28.37 0.475 2943.35 0.457 1.035 24.56 0.5 2482.0 0.481 0.918 20.74 0.525 2020.65 0.505 0.785 16.90 0.55 1559.3 0.529 0.635 13.05 0.575 1097.95 0.553 0.467 9.20 0.6 636.6 0.577 0.283 5.34 0.625 175.25 0.602 0.081 1.47
Graph 1. Speed r/min Efficiency Current Output Pwer 14000 1.6 12000 1.4 rpm 10000 8000 6000 4000 1.2 1 0.8 0.6 0.4 Efficiency, Pwer, & Current 2000 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Oz. In. 0 Numerical Calculatin Fr an irn-less cre, DC mtr f relatively small size, the relatinships that gvern the behavir f the mtr in varius circumstances can be derived frm physical laws and characteristics f the mtrs themselves. Kirchff's vltage rule states, "The sum f the ptential increases in a circuit lp must equal the sum f the ptential decreases." When applied t a DC mtr cnnected in series with a DC pwer surce, Kirchff's vltage rule can be expressed as "The nminal supply vltage frm the pwer surce must be equal in magnitude t the sum f the vltage drp acrss the resistance f the armature windings and the back EMF generated by the mtr.": V = Pwer supply (Vlts) I = Current (A) R = Terminal Resistance (Ohms) V e = Back EMF (Vlts) V = ( I R) + V e The back EMF generated by the mtr is directly prprtinal t the angular velcity f the mtr. The prprtinality cnstant is the back EMF cnstant f the mtr. ω= angular velcity f the mtr K e = back EMF cnstant f the mtr V = ω e k e
Therefre, by substitutin: V = I R) + ( ω k ) ( e The back EMF cnstant f the mtr is usually specified by the mtr manufacturer in vlts/rpm r mv/rpm. In rder t arrive at a meaningful value fr the back EMF, it is necessary t specify the mtr velcity in units cmpatible with the specified back EMF cnstant. The mtr cnstant is a functin f the cil design and the strength and directin f the flux lines in the air gap. Althugh it can be shwn that the three mtr cnstants nrmally specified (back EMF cnstant, trque cnstant, and velcity cnstant) are equal if the prper units are used, calculatin is facilitated by the specificatin f three cnstants in the cmmnly accepted units. The trque prduced by the rtr is directly prprtinal t the current in the armature windings. The prprtinality cnstant is the trque cnstant f the mtr. M = trque develped at rtr k M = mtr trque cnstant M = I k M Substituting this relatinship: ( M R) V = + ( ω ke) k M The trque develped at the rtr is equal t the frictin trque f the mtr plus the resisting trque due t external mechanical lading: M f = mtr frictin trque M l = lad trque M = M + M Assuming that a cnstant vltage is applied t the mtr terminals, the mtr velcity will be directly prprtinal t sum f the frictin trque and the lad trque. The cnstant f prprtinality is the slpe f the trque-speed curve and can be calculated by: M H = stall trque n 0 = n-lad speed n M = n / / 0 l f M H An alternative apprach t deriving this value is t slve fr velcity, n: Differentiating bth sides with respect t M yields: n R = M ( k ) m ke V n = k e M ( k k ) e Using dimensinal analysis t check units, the result is: -Ohms/(z-in/A) x (V/RPM) = -Ohm-A-RPM/V-z-in = -RPM/z-in It is a negative value describing lss f velcity as a functin f increased trsinal lad. m
Sample Calculatin Mtr 1624T009S is t be perated with 9 vlts applied t the mtr terminals. The trque lad is 0.2 zin. Find the resulting mtr speed, mtr current, efficiency, and mechanical pwer utput. Frm the mtr data sheet, it can be seen that the n-lad speed f the mtr at 9 vlts is 11,700 rpm. If the trque lad is nt cupled t the mtr shaft, the mtr wuld run at this speed. The mtr speed under lad is simply the n-lad speed less the reductin in speed due t the lad. The prprtinality cnstant fr the relatinship between mtr speed and mtr trque is the slpe f the trque vs. speed curve, given by the mtr n-lad speed divided by the stall trque. In this example, the speed reductin caused by the 0.2 z -in trque lad is: 0.2 z-in x (11,700 rpm/.634 z-in) = -3690 rpm The mtr speed under lad must then be: 11,700 rpm - 3690 rpm = 8010 rpm The mtr current under lad is the sum f the n-lad current and the current resulting frm the lad. The prprtinality cnstant relating current t trque lad is the trque cnstant (k M ), in this case, 1.039 z -in/a. In this case, the lad trque is 0.2 z-in, and the current resulting frm the lad must be: I = 0.2 z-in x 1 amp/1.039 z -in = 192 ma The ttal mtr current must be the sum f this value and the mtr n-lad current. The data sheet lists the mtr n-lad current as 11 ma. Therefre, the ttal current is: 192 ma + 11 ma = 203 ma The mechanical pwer utput f the mtr is simply the prduct f the mtr speed and the trque lad with a crrectin factr fr units (if required). Therefre, the mechanical pwer utput f the mtr in this applicatin is: utput pwer = 0.2 z-in x 8010 rpm x.00074 = 1.18 Watts The mechanical pwer input t the mtr is the prduct f the applied vltage and the ttal mtr current in Amps. In this applicatin: input pwer = 9 vlts x.203 A = 1.82 Watts Since efficiency is simply pwer ut divided by pwer in, the efficiency in this applicatin is: Thermal Calculatins efficiency = 1.18 Watts / 1.82 Watts =.65 = 65% A current I flwing thrugh a resistance R results in a pwer lss as heat f I 2 R. In the case f a DC mtr, the prduct f the square f the ttal mtr current and the armature resistance is the pwer lss as heat in the armature windings. Fr example, if the ttal mtr current was.203 A and the armature resistance 14.5 Ohms the pwer lst as heat in the windings is: pwer lss =.203 2 x 14.5 =.59 Watts The heat resulting frm I 2 R lsses in the cil is dissipated by cnductin thrugh mtr cmpnents and airflw in the air gap. The ease with which this heat can be dissipated is a functin f the mtr type and cnstructin. Mtr manufacturers typically prvide an indicatin f the mtr s ability t dissipate heat by prviding thermal
resistance values. Thermal resistance is a measure f the resistance t the passage f heat thrugh a given thermal path. A large crss sectin aluminum plate wuld have a very lw thermal resistance, fr example, while the values fr air r a vacuum wuld be cnsiderably higher. In the case f DC mtrs, there is a thermal path frm the mtr windings t the mtr case and a secnd between the mtr case and the mtr envirnment (ambient air, etc.). Sme mtr manufacturers specify a thermal resistance fr each f the tw thermal paths while thers specify nly the sum f the tw as the ttal thermal resistance f the mtr. Thermal resistance values are specified in temperature increase per unit pwer lss. The ttal I 2 R lsses in the cil (the heat surce) are multiplied by thermal resistances t determine the steady state armature temperature. The steady state temperature increase f the mtr (T) is given by: T inc = temperature increase I = current thrugh mtr windings R = resistance f mtr windings R th1 = thermal resistance frm windings t case R th2 = thermal resistance case t ambient 2 T = I R ( R + R ) 2 inc Fr example, a 1624E009S mtr running with a current f 0.203 Amps in the mtr windings, with an armature resistance f 14.5 Ohms, a winding-t-case thermal resistance f 8 C/Watt, and a case-t-ambient thermal resistance f 39 C/Watt. The temperature increase f the windings is given by: th1 T =.203 2 x 14.5 x (8 + 39) = 28 C If it is assumed that the ambient air temperature is 22 C, then the final temperature f the mtr windings is 50 C (22 + 28 ). It is imprtant t be certain that the final temperature f the windings des nt exceed their rated value. In the example given abve, the maximum permissible winding temperature is 100 C. Since the calculated winding temperature is nly 50 C, thermal damage t the mtr windings will nt be a prblem in this applicatin. One culd use similar calculatins t answer a different kind f questin. Fr example, an applicatin may require that a mtr run at its maximum trque withut being damaged by heating. T cntinue with the example given abve, suppse it is desired t run mtr 1624E009S at the maximum pssible trque with an ambient air temperature f 22 C. The designer wants t knw hw much trque the mtr can safely prvide withut verheating. The data sheet fr mtr 1624E009S specifies a maximum winding temperature f 100 C. Since the ambient temperature is 22 C, a rtr temperature increase f 78 C is tlerable. The ttal thermal resistance fr the mtr is 47 C/Watt. By taking the reciprcal f the thermal resistance and multiplying this value by the acceptable temperature increase, the maximum pwer dissipatin in the mtr can be calculated: P = 78 x 1 Watt/47 = 1.66 Watts Setting I 2 R equal t the maximum pwer dissipatin and slving fr I yields the maximum cntinuus current allwable in the mtr windings: I 2 R = 1.66 Watts I =.338 Amps The mtr has a trque cnstant f 1.309 z-in/a and a n-lad current f 11 ma. Therefre, the maximum current available t prduce useful trque is.327 Amps (.338 -.011), and the maximum usable trque available (M) is given by: M =.327 A x 1.309 z-in/a = 0.428 z-in th
The maximum allwable current thrugh the mtr windings culd be increased by decreasing the thermal resistance f the mtr. The rtr-t-case thermal resistance is primarily fixed by the mtr design. The case-t-ambient thermal resistance can be decreased significantly by the additin f heat sinks. Mtr thermal resistances fr small DC mtrs are usually specified with the mtr suspended in free air. Therefre, there is usually sme heat sinking which results frm simply munting the mtr int a framewrk r chassis. Sme manufacturers f larger DC mtrs specify thermal resistance with the mtr munted int a metal plate f knwn dimensins and material. The preceding discussin des nt take int accunt the change in resistance f the cpper windings as a result f heating. While this change in resistance is imprtant fr larger machines, it is usually nt significant fr small, creless mtrs and is ften ignred fr the sake f calculatin.