{"doc":{"type":"doc","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 1: Introduction, Objective, and Given Data"}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","text":"Introduction:"},{"type":"hardBreak"},{"type":"text","text":"In alternating-current electrical machine design, concentric winding configurations with graded turn distributions are widely employed in fractional-horsepower and integral-horsepower induction motors to approximate a sinusoidal air-gap magnetomotive force (MMF). By varying the coil pitch and turn counts across concentric groups, space harmonic content can be suppressed at the expense of a slight reduction in the fundamental winding factor."}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Given Data:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","marks":[{"type":"bold"}],"text":" Machine type: Three-phase, four-pole consequent-pole induction motor"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","marks":[{"type":"bold"}],"text":" Rated mechanical power: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"P_{\"out\"} = 10\\ \"hp\""}],"marks":[{"type":"bold"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","marks":[{"type":"bold"}],"text":" Rotor diameter: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"D = 5.7\\ \"in\" = 0.14478\\ \"m\""}],"marks":[{"type":"bold"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Active core length: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"L = 6\\ \"in\" = 0.1524\\ \"m\""}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Radial air gap length: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"g = 0.0185\\ \"in\" = 0.0004699\\ \"m\""}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Total stator slots: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"S = 48"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Number of poles: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"P = 4"}]},{"type":"text","text":" (pole pairs "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"p = 2"}]},{"type":"text","text":" )"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Number of phases: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"m = 3"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil group arrangement: 2 groups per phase, 6 concentric coils per group"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil turn distribution: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"{N_1, N_2, N_3, N_4, N_5, N_6} = {6, 12, 18, 18, 12, 6}\\ \"turns\""}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil spans in slot pitches: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"{W_1, W_2, W_3, W_4, W_5, W_6} = {17, 15, 13, 11, 9, 7}\\ \"slots\""}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Total turns per coil group: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"N_{\"group\"} = sum_(i=1)^6 N_i = 72\\ \"turns\""}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Total series turns per phase: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"N_{\"ph\"} = 2 xx 72 = 144\\ \"turns\""}]}]}]}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Objective:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","marks":[{"type":"bold"}],"text":" Determine the stator winding geometry, slot electrical angle, and individual coil pitch angles"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","marks":[{"type":"bold"}],"text":" Calculate the turns-weighted fundamental pitch factor "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_{p1,\"eq\"}"}]},{"type":"text","text":", the distribution factor "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(d1)"}]},{"type":"text","text":", and the overall fundamental winding factor "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(w1)"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Derive the expression for the fundamental air-gap MMF and quantify its reduction relative to an equivalent uniformly distributed, full-pitch winding"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Analyze the impact of the non-uniform concentric coil arrangement on air-gap space harmonics"}]}]}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Explanation: The approach involves determining the winding geometry from the slot and pole data, computing electrical angles and pitch factors for each concentric coil, and then using weighted averages to find equivalent winding factors. The fundamental MMF is derived from the winding factor and turns, and harmonic content is assessed through the harmonic pitch factor expression."}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Step 2: Winding Geometry and Slot Electrical Angle"}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Slots per Pole: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"tau_s = (S)\/(P) = (48)\/(4) = 12\\ \"slots per pole\""}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Electrical Angle Between Adjacent Slots: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"alpha = (180^deg xx P)\/(S) = (180^deg xx 4)\/(48) = 15^deg\\ \"electrical\""}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Slots per Pole per Phase: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"q = (S)\/(m xx P) = (48)\/(3 xx 4) = 4\\ \"slots\/pole\/phase\""}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Mechanical Pole Pitch:  "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"tau_m = (pi D)\/(P) = (pi xx 0.14478)\/(4) = 0.1137\\ \"m\" = 4.477\\ \"in\""}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Explanation: The slot electrical angle is the phase shift between adjacent slots in electrical degrees. The slots per pole per phase q determines the distribution factor. The mechanical pole pitch is the physical arc length per pole."}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Step 3: Coil Electrical Pitch Angles and Individual Pitch Factors"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":"Coil Electrical Pitch Angle: "},{"type":"text","marks":[{"type":"bold"}],"text":" "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"beta_i = W_i xx alpha"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 1: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"W_1 = 17 \\Rightarrow beta_1 = 17 xx 15^deg = 255^deg"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":"Coil 2: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"W_2 = 15 \\Rightarrow beta_2 = 15 xx 15^deg = 225^deg"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 3: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"W_3 = 13 \\Rightarrow beta_3 = 13 xx 15^deg = 195^deg"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 4: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"W_4 = 11 \\Rightarrow beta_4 = 11 xx 15^deg = 165^deg"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 5: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"W_5 = 9 \\Rightarrow beta_5 = 9 xx 15^deg = 135^deg"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 6: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"W_6 = 7 \\Rightarrow beta_6 = 7 xx 15^deg = 105^deg"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Fundamental Pitch Factor of Each Coil: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(p1,i) = sin((beta_i)\/(2))"}]},{"type":"hardBreak"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 1: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(p1,1) = sin(127.5^deg) = 0.7934"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 2: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(p1,2) = sin(112.5^deg) = 0.9239"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 3: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(p1,3) = sin(97.5^deg) = 0.9914"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 4: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(p1,4) = sin(82.5^deg) = 0.9914"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 5: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(p1,5) = sin(67.5^deg) = 0.9239"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil 6: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(p1,6) = sin(52.5^deg) = 0.7934"}]}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Explanation:The electrical pitch angle is the coil span converted to electrical degrees. The pitch factor quantifies the reduction in induced EMF due to short-pitching or over-pitching of the coil."}]}]}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Step 4: Turns-Weighted Equivalent Fundamental Pitch Factor"}]},{"type":"paragraph","attrs":{"textAlign":"left"}},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Turns-Weighted : "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"sum_(i=1)^6 N_i k_(p1,i) = 6(0.7934) + 12(0.9239) + 18(0.9914) + 18(0.9914) + 12(0.9239) + 6(0.7934)"}]}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"sum_(i=1)^6 N_i k_(p1,i) = 4.7604 + 11.0868 + 17.8452 + 17.8452 + 11.0868 + 4.7604 = 67.3848"}]}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":"  "}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Equivalent Pitch Factor: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_{p1,\"eq\"} = (sum_(i=1)^6 N_i k_(p1,i))\/(sum_(i=1)^6 N_i) = (67.3848)\/(72) = 0.9359"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Explanation:The equivalent pitch factor is the turns-weighted average of individual coil pitch factors. It represents the effective fundamental pitch factor of the entire concentric winding group."}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Step 5: Fundamental Distribution Factor"}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Distribution Factor: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(d1) = (sin(q(alpha)\/(2)))\/(q sin((alpha)\/(2)))"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Substitution: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(d1) = (sin(4 xx (15^deg)\/(2)))\/(4 sin((15^deg)\/(2))) = (sin(30^deg))\/(4 sin(7.5^deg))"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Calculation: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(d1) = (0.5)\/(4 xx 0.130526) = (0.5)\/(0.522104) = 0.9577"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Explanation: The distribution factor accounts for the phase spread of coils across multiple slots. It is always less than or equal to unity and reduces the resultant EMF compared to concentrated winding."}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Step 6: Overall Fundamental Winding Factor"}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Overall Winding Factor:  "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(w1) = k_{p1,\"eq\"} xx k_(d1)"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Substitution:  "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(w1) = 0.9359 xx 0.9577 = 0.8963"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Explanation: The overall winding factor is the product of the pitch factor and distribution factor. It represents the total reduction in fundamental EMF due to winding layout."}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Step 7: Peak Fundamental Air-Gap MMF"}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Peak Fundamental MMF:  "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"F_(1m) = (3sqrt(2))\/(pi) \\frac{k_(w1) N_{\"ph\"} I_{\"ph\"}}{p}"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Substitution:  "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"F_(1m) = (3sqrt(2))\/(pi) \\frac{(0.8963)(144) I_{\"ph\"}}{2}"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Calculation:  "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"F_(1m) = (3sqrt(2))\/(pi) (64.5336) I_{\"ph\"} approx 87.1 I_{\"ph\"}\\ \"A-turns\""}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Explanation: The peak fundamental MMF is derived from the three-phase winding distribution. The factor "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"3sqrt(2)\/pi"}]},{"type":"text","text":" arises from the summation of three-phase sinusoidal currents."}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak","marks":[{"type":"bold"}]},{"type":"text","marks":[{"type":"bold"}],"text":"Step 8: Fundamental Rotating MMF Expression and Air-Gap Flux Density"}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","text":"Rotating MMF Expression:  "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"F_1(theta_m, t) = F_(1m) cos(2theta_m - omega t) = 87.1 I_{\"ph\"} cos(2theta_m - omega t)\\ \"A-turns\""}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Peak Air-Gap Flux Density:  "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"B_(1m) approx (mu_0 F_(1m))\/(g)"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Substitution:  "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"B_(1m) = \\frac{(4pi xx 10^(-7))(87.1 I_{\"ph\"})}{0.0004699} approx 0.233 I_{\"ph\"}\\ \"Tesla\""}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Explanation: The rotating MMF wave travels at synchronous speed. The air-gap flux density is obtained from the MMF divided by the air-gap length, assuming negligible iron reluctance."}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Step 9: Comparison with Full-Pitch Reference Winding"}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"MMF Ratio:   "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"\\frac{F_(1m)}{F_{1m,\"full\"}} = \\frac{k_{p1,\"eq\"} k_(d1)}{1.0 xx k_(d1)} = k_{p1,\"eq\"} = 0.9359"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Percentage Reduction:   "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"\"Reduction\" = (1 - 0.9359) xx 100\\% = 6.41\\%"}]},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Explanation:"},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"A full-pitch reference winding has a pitch factor of unity. The reduction in fundamental MMF is solely due to the equivalent pitch factor of the concentric winding."}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Step 10: Space Harmonic Content Analysis"}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Harmonic MMF Amplitude:  "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"F_(nm) \\propto (|k_(wn)|)\/(n) = \\frac{|k_{pn,\"eq\"} k_(dn)|}{n}"}]}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Harmonic Pitch Factor : "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(pn,eq)= \\frac{sum_{i=1}^{6} N_i\\sin\\left(n \\frac{\\beta_i}{2}\\right)}{\\sum_{i=1}^{6} N_i}"}]}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Explanation:"},{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"The non-uniform turn grading acts as spatial low-pass filtering. The sinusoidal stepping of turn counts attenuates low-order space harmonics (5th, 7th, 11th), minimizing parasitic torques and core losses."}]},{"type":"paragraph","attrs":{"textAlign":"left"}},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer :"}]},{"type":"heading","attrs":{"textAlign":"left","level":2},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"1. Stator Winding Geometric Arrangement:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Slots per pole:"},{"type":"text","marks":[{"type":"bold"}],"text":" "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"tau_s = 12\\ \"slots\""}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Slot electrical angle: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"alpha = 15^deg\\ \"electrical\""}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Slots per pole per phase: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"q = 4"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Mechanical pole pitch: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"tau_m = 4.477\\ \"inches\""}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Coil pitch angles: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"beta = {255^deg, 225^deg, 195^deg, 165^deg, 135^deg, 105^deg}"}]}]}]}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"2. Fundamental Winding Factors:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","marks":[{"type":"bold"}],"text":" "},{"type":"text","text":"Turns-weighted pitch factor"},{"type":"text","marks":[{"type":"bold"}],"text":": "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_{p1,\"eq\"} = 0.9359"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Fundamental distribution factor: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(d1) = 0.9577"}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Overall fundamental winding factor: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"k_(w1) = 0.8963"}]}]}]}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"3. Fundamental Air-Gap MMF:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Peak fundamental MMF: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"F_(1m) approx 87.1 I_{\"ph\"}\\ \"A-turns\""}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Rotating MMF expression: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"F_1(theta_m, t) = 87.1 I_{\"ph\"} cos(2theta_m - omega t)\\ \"A-turns\""}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","text":" Peak air-gap flux density: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"B_(1m) approx 0.233 I_{\"ph\"}\\ \"Tesla\""}]}]}]}]},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"hardBreak"},{"type":"text","marks":[{"type":"bold"}],"text":"4. Effect of Winding Arrangement on Air-Gap MMF:"}]},{"type":"paragraph","attrs":{"textAlign":"left"}},{"type":"paragraph","attrs":{"textAlign":"left"},"content":[{"type":"text","marks":[{"type":"bold"}],"text":" Fundamental MMF reduction relative to uniformly distributed full-pitch winding: "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"6.41\\%"}]},{"type":"hardBreak"},{"type":"text","text":" Space harmonic suppression: The sinusoidal stepping of turn counts "},{"type":"inlineMath","attrs":{"mathType":"asciimath"},"content":[{"type":"text","text":"{6, 12, 18, 18, 12, 6}"}]},{"type":"text","text":" attenuates low-order space harmonics (5th, 7th, 11th), minimizing parasitic torques and core losses."}]}]},"timestamp":1792229918079,"source":"Master Aristotle Editor"}