Lecture 9: Three-Phase Transformer Models, Schemes and Mind Maps of Basic Electronics

Download Lecture 9: Three-Phase Transformer Models and more Schemes and Mind Maps Basic Electronics in PDF only on Docsity! ECE 5984: Power Distribution System Analysis Lecture 9: Three-Phase Transformer Models Reference: Textbook, Chapter 8 Instructor: V. Kekatos 1 Distribution system transformers 2 • Found at the substation and in-line substation transformer pole-mounted transformer pad-mounted single-phase transformer Three-phase connections 5 1) Grounded Wye – grounded Wye 2) Delta – grounded Wye (step-up) 3) Delta – grounded Wye (step-down) 4) Ungrounded Wye – Delta (step-down) 5) Ungrounded Wye – Delta (step-up) 6) Grounded Wye – Delta (step-down) 7) Delta-Delta 8) Open Wye – open Delta 9) Open Delta – open Delta • Transformer connections • Multi-phase transformers usually implemented by connecting 1ϕ transformers Voltage conversions (review) 6 • LN-to-LL conversion • LN voltages v := 2 4 Va Vb Vc 3 5 ṽ := 2 4 Vab Vbc Vca 3 5LL voltages • LL voltages are zero-sum: singular matrix! 1>ṽ = Vab + Vbc + Vca = 0 • Given LL voltages, recover equivalent LN voltages v = Wṽ, W := 1 3 2 4 2 1 0 0 2 1 1 0 2 3 5 even for unbalanced conditions ṽ = Dfv, Df := 2 4 1 1 0 0 1 1 1 0 1 3 5 • Vector of LN voltages represents – line-to-ground for grounded Wye – line-to-neutral for ungrounded Wye – ‘equivalent’ line-to-neutral for Delta connections Current conversions (review) 7 • Phase to line conversion • Line currents phase currents • If line currents exit triangle (delta source), vector i gets negative sign or • Given line currents, recover equivalent delta currents IA IC IB i := 2 4 Ia Ib Ic 3 5 ĩ := 2 4 Iab Ibc Ica 3 5 • Singularity (shift-invariance) can be waived by fixing the sum of delta currents ĩ = W>i • Textbook follows a different derivation and finds matrix Lwith a zero column i = 2 4 Iab Ica Ibc Iab Ica Ibc 3 5 = 2 4 1 0 1 1 1 0 0 1 1 3 5 ĩ = D> f ĩ <latexit 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sha1_base64="1V4LjUNEWi+iedUAKnu14X47mxE=">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</latexit> <latexit sha1_base64="HGMBGWaCbKRUS5lLHR34VaYkf4Y=">AAACgHicbZFLa9wwEMe17ivdPrJJjr2YLoUcwtYuDS2BQqCX5pZCdxNYmWUsj70ismyk8dJF+Jv1i/Taa/shKq99aJMMCP2Yp/SftFbSUhT9HAUPHj56/GTv6fjZ8xcv9ycHhwtbNUbgXFSqMtcpWFRS45wkKbyuDUKZKrxKbz538asNGisr/Y22NSYlFFrmUgB512qy4CRVho6XQOs0d7Jtzz5xhTkteYqF1A6MgW3rRDu+WLkUWs47EOkA4AMcdTbkcSOLNSWryTSaRTsL70I8wJQNdrk6GB3yrBJNiZqEAmuXcVRT4ruSFAr9jMZiDeIGClx61FCiTdxOgDZ84z1ZmFfGH03hzntvxUm2kbXt0W6KHgi8Won73jf7t85Bae22TE8K1N3tJ3U62ds5nfO+2LKh/GPipK4bQi36h+aNCqkKu22EmTQoSG09gDDS/zUUazAgyO9s7FWMb2t2FxbvZvHpLPr6fnoeDXrusVfsNTtmMfvAztkXdsnmTLAf7Bf7zf4EQXAcvA3iPjUYDTVH7D8Lzv4Cza7GfQ==</latexit> ĩ := 2 4 Iba Icb Iac 3 5 10 + – – Vab Vbc + + Vag IcIbIa ICIBIA In X1a nt X2b X3c H1A H2B H3C g Zta Ztb Ztc VCA IAB – –+ + VBC IAB IBC Ic IaIb ICA ICA Vtb Vtc Vta Vab IA VABIBC VAB VBC Vta – – –+ + + VtcVtb – • Backward model • Forward model • Sanity check (for ideal transformer) • Voltage and current transformations phasor diagrams assume balanced conditions <latexit sha1_base64="fN/Uprx+oTjLP1NgxacuXRmaOlQ=">AAACiHicbZHNatwwEMe1Tj/S7dcmPfZiuhR6MEbebLPJoRDaSw+hpNDdBNaukeWxIyLLRhovWYzfLi/RV+i1fYDK64W2SQaEfvw1M9L8lVRSGKT0x8DZefDw0ePdJ8Onz56/eDna21+YstYc5ryUpb5ImAEpFMxRoISLSgMrEgnnydWn7vx8BdqIUn3DdQVRwXIlMsEZWikefW/CTZOlzpOoof7x0eFkeuhRn9JZMAk6mMymB9 NWxfghzDTjzSIOEa6xOT31Ki0Kptdt+1f84hngpUo3chuPxl2rLty7EGxhTLZxFu8N9sO05HUBCrlkxiwDWmHUMI2CS2iHYW2gYvyK5bC0qFgBJmo2M7TuW6ukblZquxS6G/XeCi9dicr0aFZ5D8isa1Fz3Tf7t65hhTHrIvFyUN1ubyoYXprbOZ1439myxuwoaoSqagTF+4dmtXSxdLtfcVOhgaNcW2BcCzuryy+ZdRvt3w2ti8Ftz+7CYuIH7336dTo+8bd+7pLX5A15RwIyIyfkMzkjc8LJDflJfpHfztChzsw57lOdwbbmFfkvnI9/AJCDxg8=</latexit> nt = VLL,primary VLN,secondary <latexit sha1_base64="c1c6qwr8XykZ9TyhxO2v4QHB6+c=">AAACh3icbVFNT+MwEHXDspTC7pZy5BJthbQH1E3Q8nFBAsGBI0hbQGqykeNMioXjRPakorLy6/gV/IS9Ln8ApymCpYxk+XnmvbHnOS4E1+h5jy1n6dPy55X2amdt/cvXb92N3pXOS8VgyHKRq5uYahBcwhA5CrgpFNAsFnAd353W9esJKM1z+RunBYQZHUueckbRpqJuGGQUb+PU8CqSRy+HsypK/wSYFwFykYB5Jc1YqaLM+JWREVaLkteGWdTtewNvFu4i8OegT+ZxEW20ekGSszIDiUxQrUe+V2BoqELOBFSdoNRQUHZHxzCyUNIMdGhmPlTuts0kbporuyS6s+yHip1kwgvdQD0ZNwCpNS00902ztzpDM62nWbwzBlnv9qZ6SP2eUyc/qo1KTA9Dw2VRIkjWPDQthYu5W3+Km3AFDMXUAsoUt7O67JZal9F+Xce66L/3bBFc7Q78vYF3+at/7M39bJMt8p38ID45IMfknFyQIWHkgfwl/8iTs+r8dPadw4bqtOaaTfJfOCfPvQHLIg==</latexit> in = D> f ĩn = 1 nt D> f im <latexit sha1_base64="4Q08WpyQvwDCY4EqxSC4wFScnSI=">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</latexit> vm = vt dg(z)im = 1 nt Dfvn dg(z)im • How can we determine it is a step-up without knowing the turns ratio? <latexit sha1_base64="Vxdw6Wk2nRSL5oU+G9JehbiQWLg=">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</latexit> ĩH n ṽn = ✓ 1 nt iH m ◆ (ntvt) = iH m vm <latexit sha1_base64="I+NGHpkqETLOtgAwnma4fo2wtKk=">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</latexit> 2) Step-up YG connection <latexit sha1_base64="GL7GOtSXv755MZkyEFrQMiZDyhE=">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</latexit> ṽn = ntvt and ĩn = 2 4 IAB IBC ICA 3 5 = 1 nt im 11 + – – Vab Vbc + + Vag IcIbIa In X1a nt X2b X3c H1A H2B H3C g Zta Ztb Ztc VCA VCA – – –+ + + VBC VBC Vtb Vtc Vta Vab VAN VABVAB Vta – – –+ + + VtcVtb – • Voltage transform where permutation matrix • Key properties <latexit sha1_base64="nZueLlnGjMX/5zOouI1xFpX4wUA=">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</latexit> Av := 2 4 0 1 0 0 0 1 1 0 0 3 5 <latexit sha1_base64="HYC/ML3pDJiazu7pq6T4zvOGRfc=">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</latexit> p1) A1 v = A> v p2) A> v Df = D> f why step-down? • Forward model <latexit sha1_base64="b3ece7MteyzGPMQdqp/ULLFoJ7w=">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</latexit> vm = vt dg(z)im = 1 nt A> v ṽn dg(z)im = 1 nt A> v Dfvn dg(z)im = 1 nt D> f vn dg(z)im <latexit sha1_base64="xkhqB19O4XNiNbGvn2KHTt9cg94=">AAACR3icdVBNbxMxEPWmBdrw0UCPXKymSFRqol2gkN4iUakciyBNUHYbeZ1JYsVrr+zZttFq/x0Xrr31L/TSQxHiiPMlpQhGsvT03swbz4tTKSz6/rVXWlt/8PDRxmb58ZOnz7Yqz1+cWp0ZDi2upTadmFmQQkELBUropAZYEktox+OPU719DsYKrb7iJIUoYUMlBoIzdFSvcpaHM5OuGcYRzf26P6v9FXDgB4fvgyJEuMT87R79gpDW+vpC0d3wCCQyWqPfenP5uNilXCsFfGpfFL1KdelEV8DckgYLpkoWddKrXIV9zbMEFHLJrO0GfopRzgwKLqEoh5mFlPExG0LXQcUSsFE+O6GgrxzTpwNt3FNIZ+zqRM4SaydJ7DoThiP7tzYl/6V1Mxw0olyoNENQfL5okEmKmk5DpX1h3MVy4gDjRri/Uj5ihnF00ZddCMtL6f/B6Zt6cFD3P7+rNhuLODbIS7JDXpOAfCBN8omckBbh5Du5IXfkp/fDu/V+eb/nrSVvMbNN7lXJ+wPtg61n</latexit> 3) Step-down YG connection <latexit sha1_base64="eHiB/DMXoNMJ5pN65rKsY7QRfs0=">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</latexit> ṽn = 2 4 VAB VBC VCA 3 5 = ntAvvt 12 + + – – Vab Vbc + Vag IcIbIa In nt X1a X2b X3c H1A H2B H3C g Zta Ztb Ztc IAC IBA ICB Ia Ib Ic IBA ICB Ia Ic Ib IAC IA IBIA IC – • Backward model • Current transformation Step-down YG connection <latexit 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sha1_base64="/oONkvrRTZU9LfRyyemrj868kHY=">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</latexit><latexit sha1_base64="/oONkvrRTZU9LfRyyemrj868kHY=">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</latexit><latexit sha1_base64="/oONkvrRTZU9LfRyyemrj868kHY=">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</latexit> <latexit sha1_base64="fMolKIKe6kZZqzj7PhECagbf4uU=">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</latexit> ĩn = 2 4 IBA ICB IAC 3 5 = 1 nt Avim <latexit sha1_base64="ocv2IlP0Y4UyuMEF4aY0ex2fg4Q=">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</latexit> in = D> f ĩn = 1 nt D> f Avim = A> v Df > im = 1 nt Df im 15 • Similar analysis observing • Backward model • Forward model ungrounded Y <latexit sha1_base64="o/Vh+qFkuSB7Phdawq3PWmv6pNY=">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</latexit> 5) Step-up Y connection <latexit sha1_base64="0ititJc8jhLH8qjIuCH8H1ilZx4=">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</latexit> vm = 1 nt WAvvn Wdg(z)Lim key connection difference with step-down? <latexit sha1_base64="5lj9xX04BrlOL36YJ19STF49PPc=">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</latexit> vn = ntA > v ṽt and in = 1 nt A> v ĩm where ĩm = 2 4 Iba Icb Iac 3 5 <latexit sha1_base64="NAI0qmUfF0abtCBeKDSYFErhSf4=">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</latexit> in = 1 nt A> v Lim 16 [Gonen] Discussion on Y connection <latexit sha1_base64="HfwPeldm30D139DwOdfaiyyElDk=">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</latexit><latexit sha1_base64="HfwPeldm30D139DwOdfaiyyElDk=">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</latexit><latexit sha1_base64="HfwPeldm30D139DwOdfaiyyElDk=">AAACPXicdVA7SwNBEN7zbXxFLW0Wo2Chcic+O0ELSwXjg9wR9jaTuGRv79idE8Nxf8zG/2BnZ2OhiK2te0mEKDrswsc3M9/MfGEihUHXfXKGhkdGx8YnJktT0zOzc+X5hQsTp5pDlccy1lchMyCFgioKlHCVaGBRKOEybB8V+ctb0EbE6hw7CQQRaynRFJyhperl88zvitR0Kwxo5m663VgfADuud7Dr5T7CHWbHwvDUFHLUvpXrDf8YJLIVymOlgBeieV4vV7776QDoCVGvz1RIP07r5Ue/EfM0AoVcMmNqnptgkDGNgkvIS35qIGG8zVpQs1CxCEyQdRfP6aplGrQZa/sV0i472JGxyJhOFNrKiOGN+Z0ryL9ytRSb+0EmVJIiKN4b1EwlxZgWVtKG0PZi2bGAcS3srpTfMM04WsNL1oTvS+n/4GJr07P4bLtyuN+3Y4IskWWyRjyyRw7JCTklVcLJPXkmr+TNeXBenHfno1c65PR7FsmPcD6/AClcqyM=</latexit><latexit sha1_base64="HfwPeldm30D139DwOdfaiyyElDk=">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</latexit> mid-tap of one secondary is grounded and connected to secondary neutral conductor 17 – – + + – + + – + – + – + – + – + – + – + –– + VAG VBG VCG IB IC IA VAN VCN VBN IA+ IB+ ICIN Ztab Iba Iac Icb Vtab Vtbc Vtca Ztca Ztbc Ia Ic Ib Vca Vab Vbc Zg • Challenging part is to relate delta currents to secondary line currents • Exploit the fact that LL voltages across the delta loop are zero-sum <latexit sha1_base64="wQjuBOaprnBD+3PDbor6x32W2dY=">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</latexit> 6) Step-down YG connection 20 Step-down YG connection <latexit sha1_base64="SYTvEYXZp/Xgqmx+1gVhjMyRqJY=">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</latexit><latexit sha1_base64="SYTvEYXZp/Xgqmx+1gVhjMyRqJY=">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</latexit><latexit sha1_base64="SYTvEYXZp/Xgqmx+1gVhjMyRqJY=">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</latexit><latexit sha1_base64="SYTvEYXZp/Xgqmx+1gVhjMyRqJY=">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</latexit> • Substitute (7) into (1) to get the backward update with • Substitute (7) into (6) to get the forward update with • This is the only connection with a Cmatrix! <latexit sha1_base64="l+koI7UCX9L7LqQts/Zcd47CV+8=">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</latexit> E = W ✓ 1 nt I ✓ dg(z) + zg nt 11> ◆ k31 > ◆ F = W ✓ dg(z) + zg nt 11> ◆ [k1 k2 0] • Matrix C usually multiplies secondary voltages vm. However, this equation can still be used during backward sweep as primary voltages are known and fixed during this update. <latexit sha1_base64="7IhQtZaDY0BljevjsxHUu+4Xt5g=">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</latexit> in = C0v̄n + Fim where C0 = 1 nt k31 > and D = 1 nt [k1 k2 0] Discussion on Yg — A connection Three-phase four-wire 4 primary b 210° angular displacement ; 240 V Three-phase three-wire A secondary [Gonen] 21 22 • Voltage transformations • Backward model • Use zero-sum voltage drop across delta to relate phase to line currents <latexit sha1_base64="z4/O4GB+V2o/0O/5rhgdr3crOOA=">AAACN3icdVDLSgMxFM34tr6qLt0Eq6CgZUZ87gRduBIFq0JnKJn0tg1mMkNyRyzD/JUbf8Odblwo4tY/MH0IVfRCyOHce+7jhIkUBl33yRkaHhkdG5+YLExNz8zOFecXLk2cag4VHstYX4fMgBQKKihQwnWigUWhhKvw5qiTv7oFbUSsLrCdQBCxphINwRlaqlY8zfxuk6puhgHN3LLbjY0BsON6B7te7iPcYba3Tlf8Y5DINnvfCuWxUsA77fK8Vix9K+kA6LWgXp8pkX6c1YqPfj3maQQKuWTGVD03wSBjGgWXkBf81EDC+A1rQtVCxSIwQdZdOaerlqnTRqztU0i77KAiY5Ex7Si0lRHDlvmd65B/5aopNvaDTKgkRVC8N6iRSoox7ZhI60Lbi2XbAsa1sLtS3mKacbRWF6wJ35fS/8HlVtnbKbvn26XD/b4dE2SJLJM14pE9ckhOyBmpEE7uyTN5JW/Og/PivDsfvdIhp69ZJD/C+fwCTkaoFw==</latexit> 7) connection <latexit sha1_base64="udUvoTBLFuDtH4id3hcDbTvRgOI=">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</latexit> ĩn = 1 nt ĩm ) D> f ĩn = 1 nt D> f ĩ ) in = 1 nt im • Stack previous equation with two from <latexit sha1_base64="khgVUlOIYZB+UzoVaRm/axVtZis=">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</latexit> im = D> f ĩm = 0 <latexit sha1_base64="FmJENwNrhc3cv8drHgr41MHQM1w=">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</latexit> ṽm = ṽt dg(z)̃im = 1 nt ṽn dg(z)̃im ) 1>ṽm = 1>ṽt 1>dg(z)̃im ) z> ĩm = 0 • Forward model <latexit sha1_base64="8By7Q4OU67eW3pgt2iV3hOzFzW8=">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</latexit> vm = Wṽm = Evn Fim where E = 1 nt WDf and F = Wdg(z)[k1 k2 0] <latexit sha1_base64="q1g32qZOy6cz7sFwhsVLvZvF7Jc=">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</latexit>2 4 Ia Ib 0 3 5 = K1 ĩm where K1 = 2 4 1 0 1 1 1 0 zab zbc zca 3 5 and K = [k1 k2 k3] 25 • Secondary voltages –+ –+ +– +– + – +– – + – + Vab Vbc Vca Ib IcIa IBIA X1-a X2-b X3-c nt H1-A H2-B Ztab Ztbc VAB Vca1 VBG1 VAB1 VAG1 Vab1 Van1Vbc1 Vtab VAG VBG Iba Icb Vtbc • Backward model <latexit sha1_base64="C7SpYum4RLCytyi331uLw5ESJXA=">AAACnnicbVFNa9wwEJXdr8T92qTHXkSXlh7SRQ4tyaUQKJReSlPoJgsrY2Tt2Csiy0YaL12Mf2j7ayqvfdgmGRj0NDNPo3mT1Vo5ZOxPED54+Ojxk4PD6Omz5y9eTo6Or1zVWAlzWenKLjLhQCsDc1SoYVFbEGWm4Tq7+dLnrzdgnarML9zWkJSiMCpXUqAPpRPkpcB1lreqS81nnlsh27hrTYod15DjkmdQKNMKa8W2a6WUXRTTd5T1znnERvwh3ruwiINZjRxuVbHGZK9PmU6mbMZ2Ru+CeARTMtplehQc81UlmxIMSi2cW8asxsR3QCU1dBFvHNRC3ogClh4aUYJL2p08HX3rIyuaV9a7QbqL3ss4WW1U7QboNsUAUHgtk/b38Ng+rxWlc9syOynA9Kfv1A/pbtf0wftyywbz86RVpm4QjBw+mjeaYkX7XdGVsiBRbz0Q0io/K5Vr4VeEfqORVzG+rdldcHU6iz/N2M+P0ws26nlAXpM35D2JyRm5IN/IJZkTSf4GJDgMopCGX8Pv4Y+hNAxGzivyn4WLf68Cygs=</latexit> in = 1 nt 2 4 1 0 0 0 0 1 0 0 0 3 5 im <latexit sha1_base64="jbP06HHjSPexSHp0WF6Dw2MYA18=">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</latexit> ṽm = 1 nt 2 4 1 0 0 0 1 0 1 1 0 3 5vn + 2 4 zab 0 0 0 0 zbc zab 0 zbc 3 5 im• Collecting in matrix form • Forward model <latexit sha1_base64="bA3414e8Hb4Zpsb4E9MWj8Ro9jQ=">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</latexit> vm = Wṽm = 1 3nt 2 4 2 1 0 1 1 0 1 2 0 3 5vn 1 3 2 4 2zab 0 zbc zab 0 zbc zab 0 2zbc 3 5 im <latexit sha1_base64="gE6NcURHzO3+1TMZ7kiuTwHtW60=">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</latexit> Open Y – open connection (cont’d) <latexit sha1_base64="NYVujsioiazkAfhOTjYuIzLdCRI=">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</latexit> Vab = Vt,ab Iazab = 1 nt VAN Iazab Vbc = Vt,bc + Iczbc = 1 nt VBN + Iczbc Vca = Vab Vbc 26 • Secondary LL voltages • Backward model Ztab Ztbc A B C + – + – + – + – VAB VBC IA IB IC Ia Ia Ic Ib + – – + – Vca Vbc + Vab – + VCA Vtab Vtbc c a b IA IC Ic 3-line primary <latexit sha1_base64="3JlwJI+HHL5nQ9lcGoicBsMRj7w=">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</latexit> 9) Open – open connection <latexit sha1_base64="JaXzPrsUixX4aCFJU3HRZoM9jFM=">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</latexit> in = 1 nt 2 4 1 0 0 1 0 1 0 0 1 3 5 im <latexit sha1_base64="yRw1Lx7NhF4weAVtRVoIfdjosHc=">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</latexit> vm = Wṽm = 1 nt W 2 4 1 0 0 0 1 0 1 1 0 3 5Dfvn W 2 4 Zab 0 0 0 0 Zbc Zab 0 Zbc 3 5 im <latexit sha1_base64="VKc0R3IotczkZiOJVXwpyKPY9k4=">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</latexit> Vab = 1 nt VAB Iazab Vbc = 1 nt VBC Iczbc Vca = Vab Vbc • Forward model 27 • Used in emergency or when load is expected to grow [Gonen] Discussion on open or V connection <latexit sha1_base64="4ozD7wibOROibthT7gCE+S9GFy0=">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</latexit><latexit 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sha1_base64="4ozD7wibOROibthT7gCE+S9GFy0=">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</latexit><latexit sha1_base64="4ozD7wibOROibthT7gCE+S9GFy0=">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</latexit>