Handbook of Small Electric Motors Episode 6 ppsx

Flashover and ring fire may be observed on starting some motors yet have very little effect on their life because it is intermittent and not seen under normal opera- tion. It is obvious from the formulas that the best way to control flashover on motors with variable loads is to limit T pc .This usually means more armature slots and larger commutators with more bars. For a more detailed explanation of flashover, see Puchstein (1961) and Gray (1926). 4.4 PMDC MOTOR PERFORMANCE This section is intended to give the PMDC motor designer a method to calculate PMDC motor performance given the material magnetic and electrical properties and physical dimensions. The basic construction of a PMDC motor is as shown in Fig. 4.78.To calculate the performance for this motor, one must predict the air gap flux and calculate the no- load speed and current and the stall torque and current. A straight line drawn between the no-load speed and the stall torque represents the speed-torque curve of the motor. A straight line drawn between the no-load current and the stall current represents the current-torque performance curve. Examples of such curves are shown in Figs. 4.79 and 4.80. 4.96 CHAPTER FOUR FIGURE 4.78 PMDC motor construction. 4.4.1 Predicting Air Gap A typical approach to predicting air gap flux is described here.The magnets are gen- erally attached to a magnetically soft steel housing.When they are charged, they set up a nearly constant flux in the air gap between the magnets and the armature. In order to determine the motor performance, you need to know the amount of air gap flux linking the armature conductors, the number of conductors, the number of DIRECT-CURRENT MOTORS 4.97 FIGURE 4.79 PMDC motor performance curves. FIGURE 4.80 PMDC motor performance varying with voltage. poles,and the current in the armature. Figure 4.81 shows the direction of the flux due to the magnets.You can determine the flux in the air gap by the following procedure: ● Finding the permeance coefficient of the magnetic circuit ● Determining the flux density in the magnet ● Finding the total flux ● Factoring out the leakage flux 4.98 CHAPTER FOUR FIGURE 4.81 Flux due to magnets. The remaining flux interacts with the armature conductors and produces the motor torque. The permeance coefficient is determined by the geometry of the cross section of the motor: P c = ΂΃ (4.378) where σ=flux leakage factor (typically 1.05 to 1.15) R f = reluctance factor (typically 1.1 to 1.3) L mr = radial length (thickness) of the magnet L g = length of the air gap in the radial direction A m = area of the magnet A g = area of the air gap L mr A g ᎏ A m L g σ ᎏ R f The area of the magnet is not necessarily the same as the area of the air gap, because the magnets typically overhang the armature. A g is usually smaller than A m . The permeance coefficient determines the load line of the magnet on its normal demagnetization curve. These curves are commonly supplied by the magnet manu- facturer.A typical curve is shown in Fig. 4.82;it is the upper left quadrant of the hys- teresis loop shown in other chapters of this handbook. The H and B axes must be appropriately scaled for this technique to be accurate. DIRECT-CURRENT MOTORS 4.99 FIGURE 4.82 Finding magnet load point. To plot the load line, take the arctangent of the permeance coefficient, calculate the angle ψ=tan −1 P c , and plot the line as shown in Fig. 4.82.The flux density in the magnet B m can be found by finding the intersection of the load line and the normal curve and reading the induction from the right vertical axis. The flux in the magnet is found by multiplying the flux density by the area of the magnet: φ m = B m A m (4.379) The air gap flux can then be found by dividing the magnet flux by the leakage factor: φ g = (4.380) This is a ballpark approach used by Ireland (1968) and Puchstein (1961). The effects of magnet overhang should be included as they add some additional flux. The method used here also predicts the permeance coefficient, but the effect of slots in the armature on the magnetic air gap length is accounted for using Carter’s coefficient. φ m ᎏ σ The effects of magnet overhang are predicted by calculating the permeance coef- ficient of each section of the overhang using methods developed by Roters (1941). The magnetic air gap length L gl is determined by Carter’s method, but first you must calculate the circumferential width of the armature slot, as follows. Given the following dimensions (Figs. 4.83 and 4.84): R a = outside radius of the armature lamination W ast = width of the armature slot top in straight-line distance N at = number of armature teeth R pole = radius of the permanent-magnet pole face 4.100 CHAPTER FOUR FIGURE 4.83 PMDC motor cross-section. To find the tooth pitch angle, in degrees, between two consecutive tooth centers: θ tp = (4.125) To convert to radians: θ tpr =θ tp =θ tp (0.017453) (4.126) Tooth pitch, the circumferential distance between two tooth centers, in inches: t p = R a θ tpr (4.127) π ᎏ 180 360° ᎏ N at Width of the armature slot along the circumference, in inches: W asc = 2R a sin r −1 ΂΃ (4.128) Mechanical length of the air gap L g , in: L g = R pole − R a (4.381) The magnetic length of the air gap is longer than the mechanical length because the lines of flux fringe through the armature slot toward the armature teeth.The fol- lowing is a mathematical method for determining Carter’s coefficient K C , which accounts for this fringing. λ= ΄ tan r −1 ΂΃ − ln ΂ 1 + ΃΅ (4.382) K C = (4.383) Effective magnetic air gap length, in inches: L gl = L g K C (4.384) Next, calculate the leakage factor σ. This is determined by finding the total per- meance factors, including the leakage permeance factors at the ends of the magnet and along the edges of the magnet. Now that you know the magnetic air gap length, determine the magnetic areas of the air gap between the armature and the magnet. t p ᎏᎏ t p −λW asc W 2 asc ᎏ 4L 2 g L g ᎏ W asc W asc ᎏ 2L g 2 ᎏ π W ast ᎏ 2R a DIRECT-CURRENT MOTORS 4.101 FIGURE 4.84 Permeances for stack, corner and end. Geometric mean radius of the magnet, in inches: R mm = 0.707͙R 2 hi + R ෆ 2 pole ෆ (4.385) where R hi is the radius of the inside of the housing. The effective magnet area is the arc distance of the pole at the geometric mean radius times the mechanical stack length, in square inches: A ms =θ pole R mm L stk (4.386) where θ pole = magnet pole arc, degrees L stk = length of armature lamination stack The average air gap radius is the distance from the center of the armature to the center of the air gap, in inches: R gm = (4.387) Air gap area over the stack, in square inches: A gs =θ pole R gm L stk (4.388) The permeance factor (or permeance path) for the area between the magnet and the armature stack is the ratio of the area of the air gap over the stack to the mag- netic air gap length, in units of inches (these units may not make sense at this time, but they cancel out when calculating the leakage factor σ): P gs = (4.389) To account for the flux at the corner, as shown in Figs. 4.84 and 4.85, the effective air gap length at the corner, in inches, has been empirically determined to be L gc = 1.021L gl (4.390) The effective radius of the air gap at the corner is the distance from the center of the armature to the pole face minus 65 percent of the magnetic air gap length, in inches: R mc = R pole − 0.65L gl (4.391) The circumferential length of the gap at the corner,as shown in Fig.4.86, is the arc length of the radius of the air gap at the corner swung along the arc of the magnet, in inches: L c =θ pole R mc (4.392) Area of the gap at the corner, in square inches, A gc = 0.76L c L gl (4.393) π ᎏ 180 A gs ᎏ L gl π ᎏ 180 R pole + R a ᎏᎏ 2 π ᎏ 180 4.102 CHAPTER FOUR Permeance factor of the gap at the corner: P gc = (4.394) Area of the magnet at the corner, in square inches: A mc =θ pole R mm L g (4.395) The flux at the end of the magnet segment will now be accounted for. The over- hang length per end L moe contributes flux to the armature stack depending on the lengths of the armature radius and the air gap. π ᎏ 180 A gc ᎏ L gc DIRECT-CURRENT MOTORS 4.103 FIGURE 4.85 Permeances for corner and end. FIGURE 4.86 Permeances of corner and mean flux path. L moe = (4.396) If the overhang per end is equal to or greater than the armature radius R a plus the air gap length L g , the mean flux path radius R mf , in (Fig. 4.85) is R mf = ͙L g (L g ෆ + R a ) ෆ (4.397) This formula for R mf assumes that the magnet overhang in excess of R a + L g con- tributes a negligible amount of useful flux to the armature stack. If the overhang is less than R a + L g , then use Eq. (4.398): R mf = ͙L g L mo ෆ e ෆ (4.398) The radius from the center of the armature to the mean flux path of the end R me , in, is R me = R a + L g − R mf (4.399) The circumferential length of the mean flux path L re , in (Fig. 4.85), is L re =θ pole R me (4.400) To determine the area of the gap at the end, you must find the length of magnet overhang which produces the flux you are accounting for. Figure 4.85 shows L om , in: L om = L moe − L g (4.401) Area of the gap at the end, in square inches: A ge = L om L re (4.402) Length of the mean flux path for the end, in inches: L me = (90°)R mf (4.403) Area of the magnet at the end, in square inches: A me =θ pole R mm L om (4.404) Permeance factor for the gap at the end: P ge = (4.405) Now, consider the flux leakage at the ends of the magnet and the sides of the magnet. The permeance factor for the flux leakage at the axial end of the magnet (leakage off the end of the overhang) is P le = (0.0181511)K ml θ p R mm (4.406) where K ml is the magnetic leakage constant based on the material type and orienta- tion. This constant accounts for energy product BH max of the magnet and its ability to hold orientation. A ge ᎏ L me π ᎏ 180 π ᎏ 180 π ᎏ 180 L ma − L stk ᎏᎏ 2 4.104 CHAPTER FOUR The permeance factor for the leakage along the axial length of the magnet accounts for the flux which leaks off the sides of the straight edges of the magnet and into the housing: P lma = (1.04)K ml L ma (4.407) The leakage factor is the ratio of the sum of all permeance factors to the sum of all of the useful flux permeance factors: σ= (4.408) Permeance coefficient over the stack: P cs = (4.409) where R f is the reluctance factor (initially set equal to 1.5 and calculated later). Flux density over the stack, in gauss: B ps = (4.410) where M x is the slope of the demagnetization curve. The flux over the stack is the flux density in gauss converted to lines per square inch times the area of the magnet over the stack. Flux over the stack, in lines: φ ms = (6.4516)B ps A ms (4.411) In a similar manner to the method for determining flux over the stack, you deter- mine the useful flux supplied by the corner segment of the magnet to the armature. The permeance coefficient for the corner is P cc = (4.412) Flux density of the magnet segment at the corner, in gauss: B pc = (4.413) where M x is the slope of the demagnetization curve. The useful flux supplied by the magnet at the corner is the flux density in gauss converted to lines per square inch times the area of the magnet over the corner.Flux over the corner, in lines: φ mc = (6.4516)B pc A mc (4.414) Determine the flux supplied by the magnet at the end as follows: P ce = (4.415) B pe = (4.416) B r ᎏᎏ 1 + (M x /P ce ) L mr A ge σ ᎏᎏ L me A me R f B r ᎏᎏ 1 + (M x /P cc ) L mr A gc σ ᎏᎏ L gc A mc R f B r ᎏᎏ 1 + (M x /P cs ) L mr A gs σ ᎏ A ms L gl R f (P gs + P gc + P ge + P le + P lma ) ᎏᎏᎏ (P gs + P gc + P ge ) DIRECT-CURRENT MOTORS 4.105 [...]... 4.58 68 .49 7.08 82.19 12.10 95.89 32 .66 101.37 57.13 109.59 134 .62 112.33 170.50 115.07 233.24 117.81 295.98 13.70 27.40 41.10 54.79 68 .49 82.19 95.89 101.37 109.59 111.21 113.92 1 16. 63 1.75 0 .60 2.44 0.84 3.29 1.13 4.58 1.57 7.08 2.43 21.10 4.15 32 .66 11.20 57.13 19.59 134 .62 46. 16 155.01 53.15 2 06. 89 70.94 269 .00 92.24 15. 06 1.79 0 .68 30.12 2 .66 1.02 45.18 3.44 1.32 60 .24 5 .63 2. 16 75.30 9. 26 3.55... 3.55 90. 36 19.72 7. 56 105.42 83. 16 31.98 111.45 158.04 60 .61 120.48 384.42 147.42 123.49 502.12 192. 56 1 26. 51 64 6. 76 248.03 129.52 837.81 321.30 1.01 1.02 1.03 1.04 1.05 1.05 1.07 1.08 1.09 1.09 1.09 1.10 Stator yoke Flux density Field intensity MMF drop 15.21 1.79 0 .69 30.72 2.71 1.04 46. 52 3.51 1.35 62 .63 5.90 2. 26 79.03 10.45 4.01 95.19 30.57 11.72 112.78 180.75 69 .32 119.95 363 .61 139.44 130. 86 922.71... 112.78 180.75 69 .32 119.95 363 .61 139.44 130. 86 922.71 353. 86 134.53 1788.8 68 6.03 138.23 2947.1 1130.2 141.94 4111.5 15 76. 7 Stator section 1 Flux density Field intensity MMF drop 7.94 1.54 1.31 16. 03 1.81 1.54 24.29 2.21 1.88 32 .69 2.87 2.44 41. 26 3.30 2.80 49 .69 3.84 3. 26 58.87 5.47 4 .61 62 .62 5.90 5.01 68 .31 7.02 5. 96 70.23 7 .64 6. 49 72. 16 8. 26 7.01 74.09 8.88 7.54 Armature teeth Apparent flux density... 3. 36 0.33 44. 16 3.41 0.34 45.37 3.45 0.34 46. 59 3.52 0.35 Stator section 4 Flux density Field intensity MMF drop 4.94 1.11 0.08 9.97 1 .62 0.12 15.10 1.79 0.13 20.32 1.94 0.14 25 .65 2.31 0.17 30.89 2.73 0.20 36. 60 3.09 0.23 38.92 3.21 0.24 42. 46 3.34 0.25 43 .66 3.39 0.25 44. 86 3.43 0.25 46. 06 3.47 0. 26 Stator section 5 Flux density Field intensity MMF drop 3 .60 0.81 0.03 7.28 1.52 0.05 11.02 1 .66 0. 06. .. 7.2 15.0 47.8 11 .6 13.8 5.05 0.01 5.41 2 .6 0.999 119.88 114.83 461 31.5 1 .68 13.41 −11.73 N/A N/A N/A 1.00 1.04 14.3 30.0 95.7 23.1 27.4 8.10 0.02 10.81 5.2 0.9 96 119.51 111.41 2 260 6.8 6. 66 7.27 −0 .61 N/A N/A N/A 1.50 1.57 21.5 45.0 143.5 30.3 3 96. 0 11.15 0.03 16. 21 7.8 0.991 118.90 107.75 166 28.1 13.14 5.71 7.43 91.40 178.35 51.25% 2.00 2.09 28.7 60 .0 191.3 32.5 38 .6 14.20 0.03 21 .61 10.4 0.984 118.04... 13.52 149.48 2 36. 08 63 .32% 2.50 2 .61 35.8 75.0 239.2 34.7 41.2 17.25 0.03 27.01 13.0 0.974 1 16. 92 99 .67 13429.9 25.08 4.88 20.20 200.80 292.30 68 .70% 3.00 3.13 43.0 90.0 287.0 35.7 42.4 20.30 0.03 32.40 15.7 0. 963 115.45 95.24 12471.8 30.97 4 .63 26. 34 243.13 3 46. 63 70.14% 3.50 3 .65 50.1 105.0 334.9 36. 6 43.4 23.35 0.03 37.80 18.4 0.949 113.89 90.54 115 86. 3 36. 97 4.39 32.57 279.32 398 .62 70.07% 4.00... 0.14 11.13 1 .66 0.18 16. 86 1.83 0.20 22.70 2.10 0.23 28 .64 2.54 0.28 34.50 2.98 0.33 40.88 3.29 0. 36 43.48 3.38 0.37 47.43 3 .61 0.40 48. 76 3.74 0.41 50.10 3.88 0.43 51.45 4.02 0.45 Stator section 3 Flux density Field intensity MMF drop 4.125 5 Stator section 2 Flux density Field intensity MMF drop 4.99 1.13 0.11 10.08 1 .62 0. 16 15.27 1.79 0.18 20. 56 1.95 0.19 25.94 2.33 0.23 31.25 2. 76 0.27 37.02 3.11... θB = 20° TPCf = 140 turns Lstk = 1.25 in Fstk = 0.95 Lstke = 1.1875 in frequency = 60 Hz Ram = 2 .63 4 Ω Rf = 3. 466 Ω Rtot = 6. 1 Ω Vbr = 2.0 V Xtot = 10.79 V Kf = 0.002493 oz и in и s Tfi = 1.37 oz и in 5.00 5.22 71 .6 150.0 478.4 38.0 45.2 32.50 0.03 53.98 26. 7 0.893 107.17 74 .67 9182 .6 54. 96 3.77 51.19 347.89 535. 86 64.92% ... 0.21 60 .89 81.85 104. 46 1 36. 65 245.24 360 .18 69 9.89 1088.3 161 0.0 2155.3 Total mmf drops (using BH curves at 60 Hz) 21.89 41.44 Note: Air gap flux versus mmf excitation, 1-inch stack, BH curve at 60 Hz, where: Lgap = 0.0187 Lsy = 1.4305 Laym = 0.3835 Lpm = 0.0893 Ls1 = 0.8495 Ls2 = 0.1108 Ls3 = 0.0987 Ls4 = 0.0742 Ls5 = 0.0335 Wgap = 1 .61 72 Wsy = 0.50 06 Way = 0.3320 Wpm = 0.8 861 Ws1 = 0.3180 Ws2 =... 54.37 12 367 3877 72.50 15459 48 46 90 .62 18551 5815 108.75 2 164 2 67 84 1 26. 87 22879 7172 134.12 24734 7754 144.99 25353 7948 148 .62 25971 8141 152.24 265 89 8335 155.87 Armature tooth tips Flux density Field intensity MMF drop 4.124 Component or factor 3.99 0.90 0.07 7.98 1.54 0.12 11.97 1 .69 0.13 15. 96 1.81 0.14 19.95 1.91 0.15 23.93 2.19 0.17 27.92 2.48 0.19 29.52 2 .62 0.20 31.91 2.81 0.22 32.71 2.88 . length of the gap at the corner,as shown in Fig.4. 86, is the arc length of the radius of the air gap at the corner swung along the arc of the magnet, in inches: L c =θ pole R mc (4.392) Area of the. width of the armature slot top in straight-line distance R a2 = radius of the circle on which the arc segments of the bottoms of the slots are centered φ T ᎏ σ 4.1 06 CHAPTER FOUR N at = number of. length of the air gap in the radial direction A m = area of the magnet A g = area of the air gap L mr A g ᎏ A m L g σ ᎏ R f The area of the magnet is not necessarily the same as the area of the