Transcript Microelectromechanical Devices

ECE 8830 - Electric Drives
Topic 5: Dynamic Simulation of
Induction Motor
Spring 2004
Stationary Reference Frame
Modeling of the Induction Motor
We now consider how the model of the
induction motor that we have
developed can be used to simulate the
dynamic performance of the induction
motor.
We will consider the model of the
motor in the stationary reference
frame.
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Consider a 3, P-pole, symmetrical
induction motor in the stationary reference
frame with windings as shown below:
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Consider first the input voltages for the
given neutral connections of the stator
and rotor windings shown.
The three applied voltages to the stator
terminals vag, vbg, and vcg need not be
balanced or sinusoidal. In general, we can
write:
vas  vag  vsg
vbs  vbg  vsg
vcs  vcg  vsg
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Therefore,
3vsg  (vas  vbs  vcs )  (vag  vbg  vcg )
In simulation, the voltage vsg can be
determined from the flow of phase currents
into the neutral connection by:
d
d

vsg  Rsg (ias  ibs  ics )  Lsg (ias  ibs  ics )  3  Rsg  Lsg  i0 s
dt
dt 

where Rsg and Lsg are the resistance and
inductance between the two neutral points.
Of course, if s and g are shorted, vsg=0.
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Now consider transformation of stator abc
phase voltages to qd0 stationary voltages.
With the q-axis aligned with the stator aphase axis, the following equations apply:
2
1
1
2
1
1
v  vas  vbs  vcs  vag  vbg  vcg  vsg
3
3
3
3
3
3
s
qs
1
1
v 
(vcs  vbs )
(vcg  vbg )
3
3
s
ds
1
1
v0 s  (vas  vbs  vcs )  (vag  vbg  vcg )  vsg
3
3
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Transformation of the abc rotor winding
voltages to the qd0 stationary reference
frame can be done in two steps.
First transform the “referred” rotor abc
phase voltages to a qd0 reference frame
attached to the rotor with the q-axis
aligned to the axis of the rotor’s a-phase
winding.
In the second step, transform the qd0
rotor quantities to the stationary qd0
stator reference frame.
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Step 1 ->
2 ' 1 ' 1 '
2 ' 1 ' 1 '
v  var  vbr  vcr  van  vbn  vcn  vrn'
3
3
3
3
3
3
'r
qr
1 '
1 '
'
'
v 
(vcr  vbr ) 
(vcn  vbn
)
3
3
'r
dr
1 '
1 '
'
'
'
v  (var  vbr  vcr )  (van  vbn
 vcn' )  vrn'
3
3
'r
0r
where vrn’ = voltage between points r and
n and the primes indicate voltages
referred to the stator side.
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Step 2 ->
vqr' s  vqr' r cosr (t )  vdr' r sinr (t )
vdr' s  vqr' r sinr (t )  vdr' r cosr (t )
t
 r (t )   r (t )dt   r (0)
0
where r(t) = rotor angle
at time t, r(0)= rotor angle
at time t=0, and r(t) =
angular velocity of rotor.
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
The qd0 voltages at both the stator and
rotor terminals, referred to the same
stationary qd0 reference frame, can be
used as inputs along with the load torque
to obtain the qd0 currents in the
stationary reference frame. These can
then be transformed to obtain the phase
currents in the stator and rotor windings.
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
The inverse transformation to obtain the
stator abc phase currents from the qd0
currents is given by:
ias  iqss  i0s
1 s
3 s
ibs   iqs 
ids  i0 s
2
2
1 s
3 s
ics   iqs 
ids  i0 s
2
2
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
The abc rotor currents are obtained by
a two-step inverse transformation
process. Step 1 transforms the
stationary qd0 currents back to the qd
frame attached to the rotor. Step 2
resolves the qd rotor currents back to
the abc rotor phase currents.
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Step 1 ->
i  i cosr (t )  i sinr (t )
'r
qr
's
qr
's
dr
i  i sinr (t )  i cosr (t )
'r
dr
's
qr
's
dr
Step 2 ->
i  i i
'r
ar
's
qr
'
0r
1 'r
3 'r '
i   iqr 
idr  i0 r
2
2
'r
br
1 'r
3 'r '
i   iqr 
idr  i0 r
2
2
'r
cr
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
The model equations can be rearranged
into the form of equations (6.112) to
(6.117) in Ong’s book (provided in
separate handout).
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
The torque equation is:
3 P
Tem 
( dss iqss  qss idss )
2 2b
(eq. 6.118)
The equation of motion of the rotor is given
by:
d rm
J
 Tem  Tmech  Tdamp
dt
where Tmech is the externally-applied
mechanical torque in the direction of the
rotor speed and Tdamp is the damping
torque in the opposite direction of rotation.
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Normalized to the base (or rated speed) of
the rotor b is given by:
2 J  b d ( r /  b )
 Tem  Tmech  Tdamp (eq. 6.120)
P
dt
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Stationary Reference Frame Modeling
of the Induction Motor (cont’d)
Saturation of Mutual Flux
See Ong text.
Linearized Model
Solving the nonlinear equations by
numerical integration allows
visualization of the dynamic
performance of a motor. However, in
designing a control system, we would
like to use linear control techniques.
For this application we need to develop
a linearized model of the induction
motor.
Linearized Model (cont’d)
To develop a linearized model for the
induction motor, we select an
operating point and perturb the
system with small perturbations over a
linear regime.
Linearized Model (cont’d)
The general form of the behavior of the
induction motor may be described by
the function:
f( x , x, u, y) =0
where x is a vector of state variables
'
'

,

,

,

( qs qr ds dr ,r / b ); u is the vector of
e
e
input variables ( vqs , vds , Tmech ); and y is
the vector of desired outputs, such as
e
e
iqs , ids , Tem .
Linearized Model (cont’d)
When a small perturbation  is applied
to each of the components of the x, u,
and y variables, the perturbed variables
will satisfy the equation:
f( xx=x0+x , x0+ x , u0+u, y0+y) =0
where the 0 subscript denotes the
steady state value about which the
perturbation is applied.
Linearized Model (cont’d)
In steady state,
xx=x0
Neglecting higher order terms and
regrouping some of the terms in the earlier
equations, the linear equations including
perturbations can be re-written as:
x  Ax  Bu
y  Cx  Du
Linearized Model (cont’d)
See Ong text to learn how to use
Matlab/Simulink to solve for the [A B C D]
matrix.
See handout from Krishnan’s book for
more detailed description of small signal
analysis of induction motor.