3.2 Dynamic and Static Characteristics of Induction Machine Drives Drives
3.2.3 Reduction of a Mathematical Model to an Equivalent
A dynamic system, such as electric drive described with ordinary differential equations for given initial conditions and input functions, is characterized with a specific trajectory of the motion. This trajectory represents the history of all vari- ables in a system. The steady state of such a system occurs when the trajectory is represented by a fixed point, that is
{ }ϕt(q) ={ }q
or by a periodic function with the period of T, when
T T
t ϕ
ϕ + = (3.106) For an electric drive this occurs when variables in a system forming the vector of generalized coordinates q are either constant functions or periodically variable ones. In a induction motor drive we can assume in an idealized way that the steady state occurs when the angular speed is constant, i.e. Ωr=θr =const and the elec- tric currents which supply the windings are periodic functions with the period in conformity with the voltages enforcing the flow of the currents.
One can note that the history of both the supply voltages and the resulting cur- rents is relative to the transformation of the co-ordinates of the system, as presented in the models of the motor in α,β, d,q or x,y axes (3.61 – 3.65). In a x,y system rotating with the speed ωc = ωs = pωf the symmetric system of sinusoidal voltages supplying phase windings as a result of transformation (3.37) is reduced to constant voltages. In such coordinate system the steady state literally means a fixed point on the trajectories of all variables. The situation will be different for a steady state in the case of asymmetry of the supply voltages or cyclically variable load torque. In such a case the steady state will be characterized by periodically variable waveforms of electric currents and angular speed, while in the speed waveform the constant component will form the predominant element. The ac- quaintance with steady states is relevant for the design and exploitation of a drive since it provides information regarding its operating conditions and, hence, forms the basis for the development of strategies regarding methods of drive control. The familiarity with the steady states makes it possible to determine the characteristics of the drive, i.e. functional relations between variables that form the sets of con- stant points on a trajectory and ones that are time invariable. For the reasons given here the steady state of the induction motor drive can be conveniently described in axial coordinates x,y. Therefore, we will take as the starting point the transformed equations (3.60) in current coordinates, which after the substitution ωc = ωs gives:
⎥⎦
⎢ ⎤
⎣
⎥⎡
⎦
⎢ ⎤
⎣
−⎡
⎥⎦
⎢ ⎤
⎣
⎥⎡
⎦
⎢ ⎤
⎣ + ⎡
⎥+
⎦
⎢ ⎤
⎣
⎥⎡
⎦
⎢ ⎤
⎣
=⎡
⎥⎦
⎢ ⎤
⎣
⎡
rxy sxy r
m m s rxy sxy r m
m s
rxy sxy r s sxy
sX sX
X X L
L L L dt d
R R
i A
i A i
i i i 0
u
2 2
0 0
(3.107)
where: Xs=ωsLs Xr=ωsLr Xm =ωsLm while
s r
s p
s=ω −ω Ω (3.108) - is the slip of the rotor speed in relation to the rotating magnetic field (see 3.63).
We assume that the steady state forms the fixed point of the trajectory ϕt(q)=q, hence, it denotes the constant angular speed Ωr = const and the constant slip s = const. This condition is possible due to the constant values of currents isxy, irxy, and, as a result, the constant electromagnetic torque Te. This requires the constant sup- ply voltages after the transformation of x,y, which take the following form in ac- cordance with (3.37):
⎥⎦
⎢ ⎤
⎣
= ⎡
⎥⎦
⎢ ⎤
⎣
=⎡
γ γ sin 2 cos 2 3
ph sy
sx
sxy U
u
u u (3.109)
The form of voltages (3.109) suggests the introduction of complex values:
ph j
ph sy
s usx ju U e U
U = + = 3 γ = 3 (3.110) where: Uph - is the RMS value of the voltage supplying the phase of the motor.
Subsequently, we can substitute:
ry rx r r sy sx s
s =I =i + ji i =I =i + ji
i (3.111)
In the following transformations of equations (3.107) the latter of the equations in each pair is multiplied by the imaginary unit j and is added to the first of the equa- tions, thus giving the equations for a complex variable. For the stator we obtain:
m r s s
s s
s R I jX I jX I
U = + + (3.112) Here we have applied: Is =0
dt
d , Ir =0 dt
d , which results from the steady state and
⎥⎦
⎢ ⎤
⎣
⎡
= −
⎥⎦
⎢ ⎤
⎣
⎥⎡
⎦
⎢ ⎤
⎣
⎡
= −
sx sy sy
sx
sxy i
ji ji
i 1
1
2i A
which after addition of row vectors leads to:
sxy⇒−jis
i
A2 .
As we perform similar operations for the other pair of equations, i.e. rotor’s equa- tions, and dividing this equation by slip s, we obtain:
r r s m r
r I jX I jX I
s
R + +
=
0 (3.113) This makes it possible to develop an equivalent circuit for an induction motor in the steady state as a result of merging equations (3.112, 3.113) in the form of a two port, using a common magnetizing reactance term jXm. The equivalent circuit in the form in Fig. 3.26, beside the voltage and current relations presented in every two port, also realizes in an undisturbed manner the energetic relations occurring in the steady state. This comes as a result of the application of orthogonal trans- formations that preserve scalar product and quadratic forms in the transformation of equations.
Fig. 3.26 Equivalent circuit of an induction motor for the steady state
In this circuit we have to do with a resistance term Rr /s, which realizes in the energetic sense both Joule’s losses in the rotor windings and the mechanical out- put of the drive transferred via the machine’s shaft as the product of torque Te and the angular speed of the shaft Ωr. Hence the resistance term can be divided into two terms: Rr, Rr(1-s)/s, which realize the losses of the power in the stator’s wind- ings and mechanical power Pm, as it is presented in Fig. 3.27. The following components of the electric power are encountered in the equivalent diagram:
Fig. 3.27 Equivalent circuit of induction motor with physical interpretation of electric power components
s s Is
U
P1= cosϕ input power
s s
els I R
P = 2 Joule’s losses in stator windings
s R I
Pf = r2 r/ air gap field power (3.114)
r r
elr I R
P = 2 Joule’s losses in rotor windings
s R s I Pm= r2 r1−
mechanical power
f
els P
P P1= +
The energy balance for a 3-phase machine is preserved due to the fact that
ph
s U
U = 3 , hence, the transformed power is three times higher than the power of a single phase. In the analysis of the expression for the mechanical power out- put of an induction motor drive we can distinguish the following areas of opera- tion:
1. for 0<s<1 Pf >0, Pm>0 - motor regime 2. for s<0 Pf <0, Pm<0 - generating regime 3. for s>1 Pf >0, Pm<0 - braking regime 4. for s=1 Pf >0, Pm=0 - stall of the motor 5. for s=0 Pf =0, Pm=0 - idle run
From the expression for the mechanical power we can calculate the motor’s torque in the steady state:
R s p I
T P r r
s r m e
2 1
=ω
=Ω (3.115) The equivalent circuit can additionally be useful in the calculation of the stator and motor currents:
r r
m s
s
s s
jX s R jX X R I U
+ + +
=
/
2 (3.116)
r r
m s
r R s jX
I jX
I =− +
/ (3.117) It would be valuable to present the currents in the standardized parameters (3.57) since as a consequence of such presentation it is possible to depart from the par- ticular design of an induction motor. The standardized parameters assume values in the ranges presented in Table 3.1. In this case the relations (3.116-3.117) take the form:
r s
s r
s r
r s s s s
s s
s s
j js k js I I
j s j s
U I X
α ω ω
α
ω σ
ω ω α ω
− +
=
− + − +
= 2(1 )
(3.118)
Currents I ,r Is represent symbolic values of stator and rotor currents for steady state sine curves. The electromagnetic torque in the steady state can be derived from relation (3.45)
) Im(
)
(syrx sxry m s r
m
e pL i i i i pL I I
T = − = (3.119)
Using relations (3.116, 3.117) presenting stator and rotor currents we obtain:
2 2
2
1 ) (1
1 )
( s
r r s z r r m s
r
s s e
k R s R X k
k s R X
R
s R T pU
+ +
= −
ω (3.120)
where: Xz=σXs (3.121) - is a blocked-rotor reactance.
The electromagnetic torque can also be presented using standardized parame- ters (3.59), and takes this form:
2 2
2
) (
) (
) 1 (
s r s r s s
r
s s e
s s
s X
T pU
α α ω α σ ω α σ α
+ +
−
= − (3.122)
The expressions (3.120), (3.122) representing electromagnetic torque relative to supply voltages and motor parameters are frequently subjected to certain simplifi- cations in order to simplify the analysis of these expressions. The basic procedure applies disregarding of the resistance of the stator winding Rs and, subsequently, αs in some or all terms of this expression. A detailed analysis of this type of sim- plification will be conducted later on during the determination of the characteris- tics of the drive regime.