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1 ECE470 EXAM # SOLUTIONS SPRING 08 Intructon:. Cloed-book, cloed-note, open-mnd exm.. Work ech problem on the exm booklet n the pce provded.. Wrte netly nd clerly for prtl credt. Cro out ny mterl you do not wnt rded. Nme: Problem : /0 Problem : /0 Problem : /0 Problem 4: /40 Totl: /00 P IN,ph Q IN,ph + ~ I jx ~ o + V = V 0 V ~ E = E δ E = K f ω I f Ueful Equton P IN,ph = P IN,ph = V E X n( δ = P e = P m = P out + P rot Q IN,ph = Q IN,ph = V X V E X co( δ
2 Problem (5 Pont N 4 / N / α x N / N /4 0 N /4 N / / / α The bove pcture how dtrbuted ttor wndn wth two col - nd - occupyn four ttor lot. Ech col h N / turn. A current njected nto col - nd then nto col - hown. ( Wrte four equton n the four mnetc feld ntente,,, nd 4 n the r p. = N = = N = N = = N 4 = N ( µ o ( RL = 4 = N = 0 = = 0 ( (b Solve the bove four equton or ue the prncple of uperpoton to ketch the reultnt ttor mnetc feld ntenty = (α functon of α n the r p. ( ( = + N ( + N ( + N = 0 4 = N + N = N = N 4 = N + N = N = N = N = N N N N = 0 = = N = N = 0 (c Wrte down the fundmentl component of the Fourer ere expnon of (α. (α 4 N = 4 = 4 N 4 = 4 co(α 45o + 4 N 4 co(α + 45o (co α co 45o + n α n 45 o + 4 co α N co 45o co α = 4 N N 4 (co α co 45o n α n 45 o
3 Problem (0 Pont b x f x c f b θ x b f c c x Conder the bove chemtc repreentton of two-pole three-phe ynchronou mchne wth three ttor wndn (-, b-b, nd c-c nd one rotor feld wndn (f-f. Aume tht the mnetc feld nduced n the r p by ech current vry nuodlly ccordn to the follown expreon: = (α = 4 N co α b = b (α = 4 N b co(α 0o c = c (α = 4 N c co(α + 0o f = f (β = 4 N f f co β = 4 N f f co(α θ The vrou elf nd mutul nductnce of th mchne cn be found from the follown nterl: / λ = L l + N µ o ( (α + b (α + c (α + f (α θ RLdα / 7/6 λ b = L l b + N µ o ( (α + b (α + c (α + f (α θ RLdα /6 5/6 λ c = L l c + N µ o ( (α + c (α + c (α + f (α θ RLdα /6 / λ f = L lf f + N f µ o ( (β + θ + b (β + θ + c (β + θ + f (β RLdβ /
4 After performn the nterton n the prevou pe, the elf nd mutul nductnce of th three-phe ynchronou mchne wll be of the form: λ L l + L m L m / L m / M f co θ λ b λ c = L m / L l + L m L m / M f co(θ 0 o b L m / L m / L l + L m M f co(θ + 0 o c λ f M f co θ M f co(θ 0 o M f co(θ + 0 o L lf + L mf f where L m N = L mf N f = M f N N f Leend: L l : Sttor wndn leke nductnce L m : Sttor wndn mnetzn nductnce L lf : Rotor feld wndn leke nductnce L mf : Rotor feld wndn mnetzn nductnce M f : Mxmum mutul nductnce between ttor wndn nd the rotor feld wndn M = L m : Mxmum mutul nductnce between two ttor wndn. Derve the elf nd mutul nductnce for wndn -, tht, how tht λ = (L l + L m L m b L m c + M f f co θ nd expre L m nd M f n term of µ o, N, N f,, R, nd L. Soluton: where / λ = L l + N µ o ( (α + b (α + c (α + f (α θ RLdα / = L l + N / N c / [ 4 N co α + 4 N b co(α 0o ] N f f co(α θ RLdα + 4 co(α + 0o N = L l + µ o RL [co α]/ / + µ 4 N b o RL [n(α 0o ] / / 4 N c +µ o RL [n(α + 0o ] / / + µ 4 N N f f o RL [n(α θ] / / 4 N = L l + µ o RL + µ 4 N b o RL co 4 0o + µ o = (L l + L m L m b L m c + M f f co θ 4 N L m = µ o M f = µ o 4 RL N N f RL N c RL co 4 N N f f 0o + µ o RL co θ 4
5 Problem (0 Pont ( Refer to Problem nd compute the mnetc coenery W m = W m(, b, c, f, θ of the two-pole three-phe ynchronou mchne: W m = W m(, b, c, f, θ = λ + λ b b + λ c c + λ f f = ( (L l + L m + b + c + (L lf + L mf f L m ( b + b c + c +M f f [ co θ + b co(θ + 0 o + c co(θ 0 o ] (b Show tht the electromnetc torque of the two-pole three-phe ynchronou mchne equl to T e = W m θ = M f f [ n θ + b n(θ 0 o + c n(θ + 0 o ] (c Show tht the three-phe ynchronou mchne cnnot develop n vere trtn torque t tndtll by ubttutn the follown tedy-tte quntte n the torque expreon ven n Prt (b: (t = I L,rm co(ω t + θ b (t = I L,rm co(ω t + θ 0 o c (t = I L,rm co(ω t + θ + 0 o f (t = I f,dc θ(t = θ o = rbtrry nd contnt ntl nle T e = M f I f,dc I L,rm [co(ω t + θ n θ o + co(ω t + θ 0 o n(θ o 0 o + co(ω t + θ + 0 o n(θ o + 0 o ] = M f I f,dc I L,rm [n(ω t + θ + θ o n(ω t + θ θ o + n(ω t + θ + θ o 40 o n(ω t + θ θ o + n(ω t + θ + θ o + 40 o n(ω t + θ θ o ] = M f I f,dc I L,rm n(ω t + θ + θ o T e,v = T T 0 T e (t dt = 0 N-m 5
6 Problem 4 (40 Pont A three-phe ynchronou motor wth nelble rmture retnce h the follown prmeter: p (pole/phe f (z V LL (V I L (A P rot (W P out (hp n (rpm X (Ω ( Gven tht the mchne opertn underexcted (.e., borbn vr, fll out the mn nformton n the follown tble: S IN,ph (VA P IN,ph (W Q IN,ph (VAr ω m (mech. rd/ T e (N-m E (V δ (de o V LN = V LL = 08 = 0 V S IN,ph = V LN I L = 0 0 = 600 VA P IN,ph = P e = P m = P out + P rot = = 880 W Q IN,ph = SIN,ph P IN,ph = = 60 VAr ( ( ω m = (f = (60 = 40 p 6 = 6 mech. rd/ T e = P e = 880 ω m 40 P IN,ph = V E X n( δ = E n δ = X P IN,ph V = Q IN,ph = V X =.9 N-m V E X co δ = = 96 E co δ = V X Q IN,ph = 0 60 V 0 = 48 E = = 07. V δ = 96 tn 48 = 6.4 o or Ẽ = E δ = E co δ + je n δ = 48 j96 = o V 6
7 The ynchronou mchne n the prevou problem now ued wth dfferent lne frequency f = 50 (z. (b Fll out the new mn mchne prmeter n the follown tble: p (pole/phe f (z V LL (V I L (A P rot (W P out (hp n (rpm X (Ω n = 0f 0 50 = p 6 or n = ω m 60 ( = p or n = n f = f 60 = 000 rpm (f 60 ( = 6 = 000 rpm X = ω L = (f L = (f ( X f (50 60 = 000 rpm = X f f = = 0 Ω (c Gven tht the mchne opertn underexcted (.e., borbn vr, fll out the mn nformton n the follown tble: S IN,ph (VA P IN,ph (W Q IN,ph (VAr ω m (mech. rd/ T e (N-m E (V δ (de o V LN = V LL = 08 = 0 V S IN,ph = V LN I L = 0 0 = 600 VA P IN,ph = P e = P m = P out + P rot = = 880 W Q IN,ph = SIN,ph P IN,ph = = 60 VAr ( ω m = (f = p T e = P e ω m = / P IN,ph = V E X n( δ = E n δ = X P IN,ph V = Q IN,ph = V X V E X co δ = ( 6 (50 = 00 = 7.5 N-m = 80 E co δ = V X Q IN,ph = V 0 = 05 mech. rd/ = 60 E = = 00 V δ = 80 tn 60 = 5. o or Ẽ = E δ = E co δ + je n δ = 60 j80 = o V 7
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