S.P. Srivastava∗
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Horley, The general theory of alternatingcurrent machines (Chapman and Hall, 1975).List of SymbolsVd DC input voltage (p.u.)Ed DC voltage applied to inverter (p.u.)R DC link resistancexdc DC link inductance (p.u.)αcs Trigging angle of converter (CS)V rms value of armature emfu Overlap anglevdo, vqo Steady-state values of motor voltage in d andq axesvd, vq Instantaneous d and q axis motor voltageid, iq Instantaneous d and q axis motor currentido, iqo Steady-state d and q axis motor currentId DC input current (p.u.)γ Actual leading angle of commutationγo Leading angle of commutation derived fromnoload emfxd, xq Per unit d axis and q axis reactances (p.u.)xd(p) d and q axis operationalxq(p) reactances (p.u.)G(p) Operational impedanceif Field excitation current (p.u.)Vs Supply rms voltages (p.u.)Kω, KI Proportional controller’s gainsTω, TI Integral time constantsr Per phase resistance of armature (p.u.)ωmo Angular speed of rotation (p.u.)xk Per unit reactance at rated frequencycorresponding to mean value of d and q axissub-transient reactancesf Base frequency (p.u.)Tdd-axis transient short circuit time constantTdd-axis sub-transient short circuittime constantTdo d-axis sub-transient open circuittime constantTqq-axis sub-transient short circuittime constantTqo q-axis sub-transient open circuittime constantp differential operatorψd, ψq d and q-axis flux linkagesxmd direct axis magnetizing reactanceψdo, ψqo d and q-axis steady state flux linkagesM Inertia constant (p.u.)Appendix IThe overall system can be expressed as,GA(p)∆ωr0−∆T2=GI(p) ·idoId+ C11(p)GI(p) ·iqoId+ C12(p)GI(p) · Gω(p)+ C13(p)C21(p) C22(p) C23(p)C31(p) C31(p) C33(p)×∆id∆iq∆ωM312C11(p) = A1 · ido + vdo(A2A7 + A3A9) − vqo(A2A9− A3A9)C12(p) = A1 · iqo + vdo(A3A8 − A2A10) − vqo(A2A8+ A3A10)C13(p) = ψdo(A2vqo + A3vdo) − ψqo(A2vdo − A3vqo)C21(p) = vdo(A7A6 + A4A9 − 1) − vqo(A9A6 − A7A4)+ ido(A5 − A7) − A9iqoC22(p) = vdo(A4A8 + A10A6) − vqo(A6A8 + A4A10 − 1)+ ido(A5 − A8) − A10idoC23(p) = ψdo(A6vqo + A4vdo − ido) − ψqo(A6vqo − A4vdo− 1) + ido(A5 − A8) + A10idoC31(p) = iqoxd(p)ψdoC32(p) = ψdo − idoxd(p)C32(p) = −MpA1 = (R + xdcp + (πxk/6))/Id,A2 =Ed −πxkid6/V 2,A3 =πxk · p · Id6V 2,A4 =π6V 2· pxk · I2d ,A5 =EdId +π6· xk · I2d/I2d ,A6 =EdId −π6· xk · I2d/V 2,A7 = r + pxd(p),A8 = r + pxq(p),A9 = ωmxd(p),A10 = ωmxq(p),A11 =π6· xk · Id,A12 =π6· xk · I2d ,Xd(p) = ((1 + pTd) + (1 + pTd)xd)/((1 + Tdop)+ (1 + Tdop))Xq(p) = (1 + Tqp)/(1 + Tqop)xqThe characteristics equation can be expressed as,GI(p) ·idoIdGI(p) ·iqoIdGI(p) · Gω(p)+C11(p) +C12(p) +C13(p)C21(p) C22(p) C23(p)C31(p) C31(p) C33(p)= 0The D-partition boundary is plotted using the aboveexpression in terms of speed controller parameters andultimately for speed controller design. The characteristicsequations in terms of controller parameters can be writtenas,K1A(p) + K2B(p) + C(p) = 0where K1 =1Kωand K2 =1TωA(p) = x(p) · px(p) =idoId+C11(p)GI(p)[C22(p) · C33(p)− C23(p) · C23(p)] +idoId+C12(p)GI(p)[C23(p) · C31(p) − C21(p) · C33(p)]+C13(p)GI[p][C21(p) · C32(p) − C31(p) · C22(p)]GI(p) = KI1+1TrpB(p) = C21(p) · C32(p) − C22(p) · C31(p)C(p) = [C21(p) · C32(p) − C22(p) · C31(p)]pAppendix IIThe system with only current controller can be expressedas,GI(p)∆idr0 =GI(p) ·idoIdGI(p) ·iqoId+ C11(p) + C12(p)C21(p) C22(p)∆ id∆ iqThe explanation of different component of matrix isgiven in Appendix I. With this the characteristics equationscan be derived for a system having only current feedbackloop as,GI(p) ·idoId+ C11(p) GI(p) ·iqoId+ C12(p)C21(p) C22(p) = 0This equation can be written as,K1 · A(p) · p + K2 · B(p) · p + C(p) = 0whereK1 =1KIand K2 =1TIA(p) = [C11(p) · C22(p) − C21(p) · C12(p)]pB(p) = [C22(p)ido − C21(p)ido]IdC(p) = [C22(p)ido − C21(p)iqo] · p/Idand is used for current controller design by D-partitiontechnique.313Appendix IIIParameter used during analysis for controller designs are,rating of machine: 3 KVA, 400 V, 4 poles, Y connected,50 Hz.R 0.008 Tdo 73.79 ms xdc 2.00xmd 0.715 Td23.236 ms M 3440.00xk 0.150 Tdo 14.58 ms π 3.14r 0.086 Td7.536 ms xd0.13xq 0.33 Tq0 10.77 ms xq0.168xd 0.74 Tq6.90 msAt steady state ωm = 0.804 p.u., Vd = 0.80 p.u., γ =0.965 m, u = 0.107 and Id = 0.80 p.u. The field excitationsis constant if = 2.00 p.u.
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