Solutions Manual for Principles of Electric Machines and Power Electronics

Solutions Manual Principles of Electric Machines and Power Electronics Third Edition PC Sen CONTENTS 14a F Hc lc Hs lg Bc Bj 05 T lg 015 x 102 m lc 4 x 8 015 x 102 03185 m For cast steel Hc 3D Am for Bz 05 T CHAPTER 1 a H Ni l 25D x 100 50000 Atm 05 Airgap H 11 4π 10⁷ 875 10⁵ Atm Fgap 2 01 10² 875 10⁵ 1750 A Pole2 For cast steel H 800 Atm for B 11 T Fpole2 2 10 10² 800 160 At Yoke crosssection of yoke is half of that of armature Hence flux density is same in yoke as in the armature MMF for upper yoke is same as the MMF for lower yoke Fyoroke 800 160 10² 640 At Total mf required is F 55 1750 160 640 2605 At Ampereturns per pole 13025 Armature flux Φ BA 11 400 10⁴ 0044 μWb B μ₀H 4π x 10⁷ x 50000 0062832 T b L N φ NBπr² l Ni πr² l 250² x 4π x 10¹⁷ x 31416 x 25 x 10²² 05 30843 μH a Rthick 70 x 10² 20004 x 10⁷ x 15 x 10¹⁴ 1856803 Bv 06 T Bh 15 06 09 T lv 7 2 14 cm lh 75 2 15 cm Hv 400 06 300 Am Hh 400 1000 400 700 Am F Hh kh Hv kv 700 015 300 014 147 A F Γm N I2 Nx 2 200 Iz 100 147 A I2 253 A F 05 20 196 100 296 A Obviously flux density will be higher now Solve by Trial Error Bv Bh Hv Hh F Hh kh hv kv 240 56 296 A Φ Bv A 08 15 1022 018 mWB Rthin 80 x 10² 20004 x 10⁷ x 10 x 10¹⁴ λ N Φ N Bh dA N Bh t dh μ N i2 2π dh h2 μ N2 i2 2π dh h1 μ N2 i2 2π lnh2 h1 μ 10 100 L 102 100 3 102 ln6 4 1936 mH b Inner panifying of the core will saturate first Hence H 100 Am at r 4 cm i H l N 100 2 π 4 102 100 02513 A c This will occur if H 100 Am around the order panifying Imini 100 2 6 102 100 0377 A 3183091 Emax 444 N f A Bmax 444 N f A Bmax 127 T Hmax 112 i max 9549 90 102 200 04297 A Imin 04297 sin 377 c E2 N2 N1 E 400 151973 303946 V Rthick Rthin 5039894 CHAPTER 2 21 a E1 NI x emf induced per turn 4600 NI x 10 NI 460 turns N2 460 10 46 turns b From equation 140b E1 444 fN Φmax 444 fN Bmax AC 4600 444 x 60 x 460 x 085 AC AC 0044 m2 22 0 t 1 60 sec e N dΦ dt 400 x 12 x 103 60 288 V 2 60 t 4 60 sec e 400 x 2 x 12 x 103 2 60 288 V Φ 500 x 1 5039894 6009921 Wb 123 When core is unsaturated core reluctance is zero and coil inductance is infinite When core is saturated core reluctance is infinite and coil inductance is zero Φrot Brot x A 15 x 2 x 104 3 x 104 Wb Flux swing ΔΦ 2 Φrot 6 x 104 Wb Time t1 to swing from negative to positive saturation Et1 N ΔΦ 1000 x 6 x 104 1 180 μsec All the input voltage will be across the coil infinite inductance during 0 to t1 and all the input voltage will be across the resistance during t1 1120 sec zero inductance of coil during this period b Bthick 0009921 150 x 10⁴ 06614 T 124 Operating point before the keeper was inserted was Bn 095 T Hm 42 kAm Now μrec 4 x 47 x 107 The new operating point is B2 095 16 π 106 42 103 116 T 125 From Fig 124 product of HmBn is maximum at approximately Bm 045 T Hm 350 kAm HmBn max 1575 x 103 Jm3 b lm 04 x 106 x 08 000 7276 m 07276 cm Am 08 x 25 x 104 44444 cm2 045 Vsam lm Am 07276 x 44444 3238 cm3 VAtm 606 x 2105 127567 cm3 Vsam 32388 127583 02535 126 a From Fig 124 HmBn product is maximum at approximately Bm 06 T Hm 480 kAm HmBn max 288 x 103 Jm3 b lm 04 x 106 x 08 0005305 m 005 305 cm Am 08 x 25 x 104 m2 32339 cm2 c Vwood 05305 x 0999 17683 cm3 Vwood 17683 127563 01386 Bthin 0009921 100 x 10⁴ 09921 T 23 VI EI 444 Nf phimax phimax frac444 imes 800 imes 601000 467 imes 103 Wb Bmax frac467 imes 10350 imes 104 0934 T c Xm frac1205 24 Ω d Im frac242 pi f N 637 mH 26 Pout 4000 W Eff frac40004220 imes 100 9479 Ni 28 Load voltage VL frac1a cdot fracj11j11 imes 100 7071V Load power PL fracVL2RL RL 05Ω a 141 VL 50V PL 5000W Φ The equivalent circuit referred to the primary side is Z 3 j5 j20 3 j3 424 Ω I1 20 424 472 A V2 472 x 3 1415 V Actual load voltage V2 100 x 1415 1415 V Rthick VH rated 1000 V IH rated 100 x 103 1000 100 A Vrated 100 V Irated 100 x 103 100 1000 A From open circuit test RCL 1002 400 25 Ω ICL 100 25 4 A IML 62 42 447 A XML 100 447 2237 Ω Turn ratio a 1000 100 10 From short circuit test ReqH 1800 1002 018 Ω ZeqH 50 100 05 Ω XeqH 052 0182 04665 Ω Equivalent circuit referred to HV side VH V1 I1 ZeqH 10000 10053018 j04665 974425 VR 9744 1000 1000 x 100 25 Rthin From the open circuit test RCL 2202 680 7446 Ω ICL 226 744 295 A IML 952 2952 903 A XML 220 903 2436 Ω a 220 440 05 From the short circuit test ReqH 930 55 0314 Ω ZeqH 375 55 068 Ω XeqH 0682 03142 06032 0779 μΩ Base quantities VbH 440 V IbH 25000 440 5682 A ZbH 440 5682 7744 Ω VbL 220 V IbL 25000 220 11364 A ZbL 220 11364 1936 Ω Req 0314 7744 00405 μΩ Xeq 068 7744 00878 μΩ RCL 7446 3846 μΩ XML 2436 9936 1258 μΩ 219 a i IHVNL 11000 57600 07 1634 A IHVrated 300 x 103 11 x 103 2727 A IHVNL in 07 2727 x 100 257 216 a n 11000 2200 5 XL1 377 x 008 3016 Ω XL2 5 x 377 x 00032 3016 Ω XM1 377 x 160 6032 kΩ RC1 125 kΩ R1 60 Ω R1 5 x 028 70 Ω 218 a I1rated 3000 240 125 A fullload current a 240 120 2 I2rated 125 x 2 25 A fullload current RLV 005 x 2 02 Ω XL1 018 x 22 072 Ω Pcore 20172 60 x 103 0678 w Pca 22732 x 255 1314 w V 11255 V VR 11000 11255 11255 x 100 227 Eff tan115 4 5134 X 06455 I₁ 1087 A R 27501 KVA 229 a Ip j15 b Equivalent primary yvoltage 2300 1328 V Equivalent secondary yvoltage 460 2656 Equivalent turns ratio a 290 3 a Zeq1Δ 0045 j016 5 Zeq1Y 113 j40 038 j133 Ω Equivalent ip circuit is 05 j15 038 j133 Ip Is V 1328 b 215 c Is 300 x 103 3 x 460 37654 A Φ ej085 318 Ip 37654 5 753 318 A V 1328 0 753 318 08 j 283 14974 566 V d I1 753 3 4348 A I2 37654 3 2174 A 29 230 a Maximum secondary line current IL 250 x 103 54348 460 3687 b P1 250 cos 30 3687 248205 kW P2 250 cos 30 3687 98205 kW P 248205 98205 34641 kW c Ip 54348 x 460 200 10895 A d P 750 x 08 600 kW increase 600 34641 34641 x 100 732 30 231 a The voltage is a 3rd harmonic voltage b Voltages induced in primary windings have the same waveform Ratio of phase voltages 10 Ratio of line voltages 10 c VinN V3 1200 V VAN 4000 2310 V VP Van2 Vnn2 Van1 Vn Vin2 Van 12002 2310 113 Van 113 x 2310 2610 V 30 232 a Base Voltage Vb High voltage 2100 V Low voltage 210V Base voltamperes Sb 200 kVA 200 kVA Base current Ib Sb Vb 9524 A 4534 A Base impedance Zb Vb Ib 2205 0205 Ω b Zeq 025 j 15 2205 001134 j 0068 c Im 025 j0075 9524 001743 pu d Pcore 0025 x 2102 11025 W 11025 200000 000551 pu Pcu 001124 pu Ploss Pcore Pcu 001685 pu 31 Eff 1x085 x1001x085 0015 001 9714 I 1318 0015 j004 I V₁ 10 13180015 j004 1034124 pu VR 1034 10 x 100 34 For east plate Fig 17 Hc 350 Atm at B 05T Ib 10000250 40A Zb 25040 625Ω ii Arms move slowly Therefore it remains essentially constant dWc i dλ i N A dB ZeqLV 0015 j006 x 625 00988 j0375Ω ii i 125 x 2 x 103 79577 A RcLV 625 x 60 375Ω L 45 180 µH k₅ 065 x 10⁶ Nmrad T 12 i² dLdt 12 i² x 18 x 10⁶ x 9 x 10⁶ i² Tₗᵛ 9 x 10⁶ i² nm In the equilibrium position 065 x 10³ θ 9 x 10⁶ Iₑₘ θ 1385 x 10³ Iₑₘ Deflection θ is proportional to square of the rms current Scale on the ammeter will be nonlinear θ 1385 x 10³ x 10² 1985 radians 18π x 1385 7935 L 45 18 1385 mH 2943 mH Z 0015 j 377 x 2943 x 10⁶ 0015 j 0011 Z 00186 Ω V 00186 x 10 0186 V XmLV 625 x 20 125Ω fₘ SSD B²2μ₀ pole area B SSD x 2 x 4 x 10⁷2 x 5 x 5 x 10⁴ 05258 T Nᶜ Hₗ g Bμ₀ g I 05258 x 2 x 1 x 10³4π x 10⁷ x 400 209 A V 5 x 20 1045 V dc Wₓ B²2μ₀ x Volume B²2μ₀ x 0 055 J or Wₓ 12 λ I i linear system 12 NABI 12 x 400 x 5 x 10⁴ x 05258 x 209 055 J b L N²Rₗ N²2gμ₀Aₑ β β² x 4 x 10⁷ x 5 x 5 x 10⁴2 x 10³ Z 5² 2π 6 x 02513² 9487 Ω fₘ Bᵣ Bᵣ Iᵣ Hence same rms current is required to produce the same average torque Iᵣ 209 A Vₘₐₓ 209 x 9487 1983 V V₁ 250V T 12 i² dLdt 12 i² x 18 x 10⁶ x 9 x 10⁶ i² Tₗᵛ 9 x 10⁶ i² nm In the equilibrium position θ 1385 x 10³ Iₑₘ Deflection θ is proportional to square of the rms current θ 1285 x 10³ x 10² 1985 radians L 45 18 1385 μH 2943 μH 00186 j 011 V 0186 x 10 0186 V I₂ 250 0938 j0375 j5 5404A 310 For 𝑤ₘ 0 average torque are produced when 𝑤ₘ 377 radsec or 𝑤ₘ 𝑤ₗ 𝑉 Tₐ𝑟𝑔 15 𝑝𝑎𝑖𝑛 26 Tₘₐ𝑥 15 Nm Pₘₐₓ 15377 5655 W For 𝑤ₘ 1885 rads Tₐ𝑟𝑔 20 𝑝𝑎𝑖𝑛 48 Tₘₐ𝑥 20 Nm Pₘₐₓ 201885 3770 W Note At lower speed more torque but less power is available C For 𝑤ₘ 0 Tₐₗ 30 𝑝𝑎𝑖𝑛 26 40 𝑝𝑎𝑖𝑛 45 For maximum torque dTdδ 0 0 60 cos 2δ 160 cos δ δ 2586 Tₘₐ𝑥 30 𝑝𝑎𝑖𝑛5172 40 𝑝𝑎𝑖𝑛 10344 6245 mm V₂ 5404 x 5 2702V 310 a ϕ 𝑢 𝑑𝑡 1N 5120𝑝𝑎𝑖𝑛𝑡 5120 cos 𝑤𝑡200 𝑤𝑡 5120213103 cos 𝑤𝑡 𝜙𝑠 cos 𝑤𝑡 b 𝑖 𝜙𝑓𝑅 𝑖 𝜙𝑅²N² d𝑙d𝑡 𝑁² d₂d𝑡 𝑁² d𝑅d𝑑 T₁ 12 𝑖² d𝑙d𝑡 12 𝜙²𝑝²𝑅² N² d𝑅d𝑑 C R 210¹⁵ C 𝜙 𝜃 Rₐ Rₐ cos 4𝜙 T 12 𝜙 d²Rdо 12 𝜙² cos 𝜙 4 Rₐ 𝑝𝑎𝑖𝑛 4𝑤𝑚𝑡𝑑 𝜙𝑚 𝑅ₐ sin 4𝑤𝑚𝑡d cos 2 𝑐𝑜𝑠 𝑤𝑚𝑡4𝜙 𝜙𝑚 𝑅ₐsin 4𝑤𝑚𝑡 𝑑 ½ sin22𝑤𝑚𝑤 𝑡 4𝜙 VR 250 27022702 x 100 748 311 a e₂ d𝑓2d𝑡 d𝑓dthsvs Let θ 𝑤𝑚𝑡 δ 360060 2π𝑡 δ 120π𝑡 δ i 25 𝑝𝑎𝑖𝑛𝑡 707 𝑝𝑎𝑖𝑛 2π𝑡 707 𝑝𝑎𝑖𝑛 120 t e₂ ddt 008 cos120π𝑡 δ 707 𝑝𝑎𝑖𝑛 120 t 2132 e₁240π𝑡 δ 21508 cos240πt δ 1508 V and f₁ 120 H₂ b T 𝑖² 𝑖𝑓 dLₕr 𝑖² dLᵗdθ 𝑖² 𝑣𝑥5 𝑝𝑎𝑖𝑛𝑡 T 50 𝑝𝑎𝑖𝑛² 𝑤𝑡 008 𝑝𝑎𝑖𝑛 𝑤𝑚𝑡 δ 4 1 e²ᵗ 𝑝𝑎𝑖𝑛wₘt δ 2 𝑝𝑎𝑖𝑛𝑤ₘ 𝑡 δ 𝑝𝑎𝑖𝑛𝑤ₘ 2𝑤𝑡 δ 𝑝𝑎𝑖𝑛𝑤ₘ 2 𝑤𝑡 δ For average torque 1 𝑤𝑚 0 Tₐ𝑟𝑔 2 𝑝𝑎𝑖𝑛 δTₘₐ𝑥 2 Nm 2 𝑤ₘ 2𝑤 240 radsec PF 10000 00000 00000 PF 06000 00000 00000 a Ifmax 120100 12 A Eamax 125 x 15001200 15625 V Ifmin 120250 048 A Eamin 92 x 15001200 115 V 01000 09073 06000 00000 00000 02000 09485 04000 00000 00000 03000 09617 02000 00000 00000 Iamax will occur at Rfe 0 Draw field resistance line for Rf Rfw 150Ω Eamax 222V b Iarated 20000200 100A Vtrated 200V Rf 200125 160Ω Rfc 160 150 10Ω c Ea Vt IaRa 200 100 01 210V Pdc EaIa 210 100 21000 W Wm 180060 2π 1885 radsec T EaIa 21000Wm 11141 Nm 04000 09675 01000 08550 00000 Earesidual 10V b Rfcrit 20005 400Ω c Rf 25016 15625Ω Rfc 15625 133 2325Ω d Ea n Draw magnetization curve for 800 rpm Field Resistance line for 133Ω intersects this mag curve at Ea 192V e Magnetization curve at n rpm will intersect the field resistance line for 133Ω at Ea 200V Therefore If 15A At 1000 rpm for If 15A Ea 245V n 2001000 245 8163 rpm 05000 09702 03000 09378 09280 IaFL 24000240 100A Ea Vt IaRa 225 100 012 237V Wm 100060 2π 10467 radsec T EaIa 237 10010467 22643 Nm b EaNL 240V EaFL 237V ΔEaAR 240 237 3V c MMF required at fullload 600 22 1320 At MMF provided by shuntfield winding 600 18 1080 At MMF provided by series field winding NsrIsr NsrIa 1320 1080 240 At Nsr 240100 24 turnspole 06000 09713 05000 09530 09343 a Ea 106V Vk 106 20 03 100V 07000 09715 06000 09332 09528 08000 09695 07000 09517 09470 09000 09633 08000 09286 09364 200 Kaφ x 1800 x 2π Kaφ 1061 Ia T Kaφ 100 1061 9425 A a Ea 220 9425 x 01 210575 V n 210575 x 60 rpm 1061 x 2π 18952 rpm b Ea 220 9425 x 01 229425 V n 229425 x 60 rpm 1061 x 2π 20649 rpm Rfc 0Ω If 200 150 13337 A From the magnetization curve Ea 218 V n 200 1800 n 16514 rpm Rfc 200Ω If 200 150 200 03714 A From the magnetization curve Ea 150 V n 200 1800 n 2400 rpm Note If the field current decreases the speed increases 10000 09621 09000 09292 09270 a Ia FL 20 x 1000 200 100 A Ea 200 100 x 01 190 V n 190 218 x 1800 15688 rpm b 218 Kaφ x 1800 190 Ka09φ n n 196 218 x 1800 17431 rpm Note Speed increases due to armature reaction effect a Rfc 50Ω If 200 150 50 1 A Ea 1800 100 A Ia FL 100 A If off If Ns Nf Ia 1 5 1200 x 100 14167 A From the magnetization curve the corresponding Ea is Ea 1800 220 V The actual Ea is Ea 200 10001 005 185 V Thus n 185 220 x 1800 15136 rpm 1 PF 10 PF 08 PF 06 PF 04 a Same as problem 420 n 1800 rpm b If off 1 5 x 100 1200 05833 A From the magnetization curve the corresponding Ea is Ea 153 V The actual Ea is 185 V problem 420 Thus n 185 153 x 1800 21765 rpm a Ia 50 A The equivalent If is If 5 1200 x 50 02083 A The corresponding Ea at 1800 rpm Ea 54 V The actual Ea is Ea E0 200 5001 005 1925 V 54 Kaφ x 1800 60 x 2π Kaφ 02865 T 02865 x 50 143239 Nm b Ia 100 A If 5 x 100 1200 Ea 112 V from the magnetization curve Ea 1800 200 10001 005 185 V n 185 x 1800 112 29732 rpm Kaφ 112 x 60 1800 x 2π 05942 T 05942 x 100 5942 Nm Note In series motor low torque high speed and high torque low speed Efficiency 098 096 094 092 Vt Ea Ia into the machine motor CHAPTER 3 From noload condition rotational loss is Prot EaIa 240 20 x 05 x 20 240 1 x 20 4780 W Wf idλ λ32 25λx12 dλ 25λ57 25λ2x12 429 contd Rfc 200xI f 200 05714A Ea1800 150 KaΦ 150 1800 x 260 07958 fm dWfλxdxλ 25 λ2 x 2 x1 25 λ2 T Ea Ia Wm1 2355 x 6 1200 x 2π 60 x06m i λ2 q122 Wf idλ λ2122 dλ q122 x gλ32 fm dWfλgd gλ λ3g12 x 3 439 i T k1 x Φ x i k2 x i2 T1T2 05 i2i2 i2 05 x 752 28125 i 5303 A ii Eal 600 75 x 05 600 375 5625 V E2 600 5303 x 05 600 2652 5735 V n2 5735 x 755625 x 5303 144 n2 144 x 500 72097 rpm For i 2A and g 10cm λ 12 x 2 i10 x 102 1697 wbturn 441 a Ra Rsr 55 1 Ia ksr s2 Ksr 525 02 T 02 x 102 50 Nm b Ea ksr Iaωm 02 x 10 x 30060 x 2π 6283 V 120 6283 101 Rext Rext 120 628310 1 472 Ω 442 a Ea 230 40 025 01 216 V P Ea Ia 216 x 40 8640 W ωm 120060 x 2π 12566 radsec T 864012566 6876 Nm Ea ksr Iaw Klan Ea 230 20 025 01 223 V 216 40 x 1200223 20 x n n 2478 rpm ωm 2595 radsec P 223 x 20 4460 W T 44602595 1719 Nm Note a 1200 rpm 6876 Nm low speed high torque b 232466 rpm 1719 Nm high speed low torque fm 1697 x 2 x 01 i 12 x 3 22625 N Total resistance required after switching out the first register R1 RT1 200100 025 Ω 400 Ra2 025 015 01 Ω With Ia reduced to 200 A Ea 200 025 x 200 150V Total resistance required after switching out the second resistance R2 RT2 200150 0125 Ω 400 This is less than Ra and therefore Ia will not increase to 400 A Thus two resistances are required in the starter box R1 035 01 025 Ω R2 R2 01 Ω 444 a Full load Ia 5000250 20 A Lowest speed 200 rpm Ifm 08 A Ia 20 A From curve at 1200 rpm Ifm 08 Eam 250 V Eam800 rpm 250 x 200 4167V Eaq 4167 20 2x 05 6167V From curve Ifq 0103 A Highest speed 1200 rpm Ifm 08 A Ia 20 A Eam1200 250V Eaq 250 20 2 x 05 270V From curve Ifq 12 A Hence required range is 0103 A Ifq 12 A Wf xdλ 12 i2g di 12g x 23 i32 b Ifq 10 A Ifm 02 A From curve Eaq 262 V Ifq 10 A Eam 262 20 2 x 05 242 V From curve Eam at 1200 rpm and Ifm 02 A 120 V Hence n 1200 x 242 120 2420 rpm 445 a Vt 2 x 265 π 1 cos30 22257 V Ea 22257 40 x 025 21257 V N 21257018 118093 rpm b T 6875 Nm c P 22257 x 40 89028 W 446 a Vt 3 6 x 277 2π 1 cosα 324 1 cosα d 0 Vt 324 1 cosα 648 V Ea 648 165 x 00874 6466 V No 6466 033 1959 rpm fm 2Wfg i g iconstant 12 x 2 x 312 i32 g 22625 N α 30 Vt 324 1 cos30 6046 V Ea 6046 165 x 00874 60316 V No 60316 033 18278 rpm b Ea 033 x 1800 594 V Vt 6084 V 608 324 1 cosα α 288 c 218 447 a Tload Tmotor KIa² 200 Nm If torque is constant Ia and flux remain constant Vt αV 05 x 400 200 V Ea 200 40 x 075 170 V Ea Kaφωm T KaφIa Kaφ TIa 200φ 5 Wm 170s 340 radsec 3248 rpm Pout HP EaIa 170 x 40746 912 P1 170 x 40 40² x 075 6800 1200 8000 W Eff 6800 8000 x 100 85 b Wm 34 radsec Ea 170 V Ia Rext Ra Rsv Vk Ea Rext 075 400 170 40 575 Rext 5 Ω HP 912 Eff 6800 6800 6800 40² x 575 6800 6800 9200 425 CHAPTER 5