f-rUj 10 Name: Spring, 2008 EE423 Test 3 (25) 1. Consider the induction machine depicted below. The stator winding self-inductance and the maximum stator-to-rotor mutual inductance are both equal to 3 H. The leakage inductances are zero. OS = Llr = (a) Express the stator and rotor flux linkages in the following form, i.e. fill in the matrix elements with numerical values or trigonometric functions. O *• = 3 0 -Sin dr. 3 O (b) Consider the transformations depicted below. fdr fbr Express the transformations of stator and rotor variables in the following form, i.e. fill in the matrix elements with appropriate functions. Jqr /*_ *. fa fdr_ (c) Express the stator flux linkages in the following form, i.e. fill in the matrix elements with appropriate values or trigonometric functions. Note: a derivation is not required! \s _v -\to v +r 7 3 J >. . O 0 ' O 3 \ >. (25) 2. Consider the equations of a two-phase induction machine expressed in the arbitrary reference frame. "* (1) ~~ (2) (3) (4) 0 =r i 0 = V'rfr (5) ~ (® ~ ®r) (6) Te =PoWhen deriving the equations that describe field oriented control, it was assumed that the reference frame speed is selected such that A, = 0. Suppose, instead, we assumed that the reference frame speed is selected so that \ = 0. (a) Establish, for steady-state operation, the speed of this reference frame in terms of the stator currents / , Ids, the rotor speed co r , and the machine parameters (no other variables allowed in final expression - parameters OK). Hint: cross out all variables that are or evaluate to zero in equations above. (b) Show that in the steady state, the electromagnetic torque is proportional to Iaslds • (25) 3. The parameters of a two-pole two-phase induction machine are: Lls = Llr - 1 mH , rs = 0 Lms = 5 mH , and rr = 0.1 Q. The machine is supplied by a field-oriented drive system. The relevant block diagrams are shown below. K^ = [K^] T ' rrLms ms l qs = co e -co, CO, lds •rr/Lrr -1 •*ni * '« T f 'bs xr «/* p + rr/Lr Assume /|L = 6 A and /^ = 10 A for a long period of time. Also, assume that the actual currents are equal to the commanded currents, f -- iqs eqs and ds = ieds . The rotor speed is cor = 200 rad/s . Establish the steady-state values for the following variables. Specifically, fill in the blanks with numerical values. Assume all values in SI units (A, V, sec, N, m, etc.). (a) A. (b)A^ (c)co e - 0 u .goS <o Q ($Te (e) I* = _Q 10 (25) 4. The A, - / characteristics of an 8-6 switched reluctance motor are depicted in the figure below. (a) Consider the trajectory starting from A to B to C back to A along path indicated above. Based upon the above diagram, approximate the energy supplied by the electrical source: (i) from A to B, bWe = #•/ J /4/m (ii) from B to C, (iii) from C to A, AFFg = ~O*I J (b) How much of the net energy in (a) is converted to mechanical energy? (c) How far does the rotor move from B to C?