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?