Download Simulation of Voltage-Fed Converters for AC

ECE 8830 - Electric Drives
Topic 8: Simulation of Voltage-Fed
Converters for AC Drives
Spring 2004
Converter Models for AC Drives
Goals:
- To describe voltage-fed power converter
in terms of switching functions
- To convert switching functions into d-q
reference frame variables
- To illustrate space vector methods for
converter model analysis
Basic Three-Phase VSI Inverter
The basic circuit of a 3 VSI inverter is
shown below.
Ref: D.W. Novotny and T.A. Lipo, “Vector Control and
Dynamics of AC Drives”
Six Connection Possibilities of
VSI Inverter
There are six possible connections when a
VSI inverter is connected to an induction
motor as shown below:
Ref: D.W. Novotny and T.A. Lipo, “Vector Control and
Dynamics of AC Drives”
Line-line and Line-neutral Voltages
for 6-step VSI Inverter
Ref: D.W. Novotny and T.A. Lipo, “Vector Control and
Dynamics of AC Drives”
VSI-Switching Function Formulation
Ref: G. Venkataraman, Dynamics and Control of AC Drives,
Univ. of Wisconsin Course Notes April 2002
Switching Functions for ABC Pole
Voltages and DC bus Current
Ref: G. Venkataraman, Dynamics and Control of AC Drives,
Univ. of Wisconsin Course Notes April 2002
Phase Currents
In the absence of a neutral connection,
the phase currents in the windings of the
induction motor must sum to zero, i.e.
ias + ibs + ics = 0
This further implies that for a balanced
load (e.g. wye connected induction motor)
the line-neutral voltages must sum to
zero, i.e.
vas + vbs + vcs = 0
Neutral Voltages
The neutral voltages are the voltages at
the neutral of the motor windings w.r.t.
the negative side of the dc bus. With a
balanced load, the neutral voltages may
be expressed as:
1
Vsn  [Van  Vbn  Vcn ]
3
Vdc

[ha  hb  hc ]
3
Phase Voltages

Phase A voltage
Vdc
Vdc
vas  Vas  Vsn  haVdc 
[ha  hb  hc ] 
[2ha  hb  hc ]
3
3

Phase B voltage
Vdc
vbs 
[2hb  ha  hc ]
3

Phase C voltage
Vdc
vcs 
[2hc  ha  hb ]
3
Phase Voltages - 612 Connection
Example
Ref: G. Venkataraman, Dynamics and Control of AC Drives,
Univ. of Wisconsin Course Notes April 2002
Phase Voltages - 123 Connection
Example
Ref: G. Venkataraman, Dynamics and Control of AC Drives,
Univ. of Wisconsin Course Notes April 2002
d-q Modeling of VSI Inverter
For analysis of motor drives, it is useful
to model the inverter in the same
reference frame as the induction motor,
i.e. in terms of d,q,0 components.
This type of modeling is particularly
useful when simulating the combined
performance of the motor with the
inverter.
d-q Modeling of VSI Inverter (cont’d)
The d,q model for a VSI inverter is
obtained by simply applying the d,q
transformation seen earlier to the
inverter equations on a mode by mode
basis.
d-q Modeling of VSI Inverter (cont’d)
The transformation from the abc axes to
the dq0 axes in the stator reference frame
is given by:
2
1
1
f  f as  fbs  f cs
3
3
3
s
qs
1
1
f 
f cs 
fbs
3
3
s
ds
1
f  ( f as  fbs  f cs )
3
s
0s
where f represents voltage, v, current, i or
flux linkage, .
d-q Modeling of VSI Inverter (cont’d)
Let us consider the example of the 612
mode for the 6-step VSI inverter. We just
saw that the phase voltages are given
by:
2
1
vas  Vdc ; vbs  vcs   Vdc
3
3
Applying the d,q transformation gives:
2
v  Vdc ; vdss  v0s s  0
3
s
qs
d-q Modeling of VSI Inverter (cont’d)
Thus, simulation of the inverter during
this connection mode can be achieved by
applying 2/3 Vdc to the q-axis equivalent
circuit of the induction motor while
shorting the d- and 0- axis circuits.
d-q Modeling of VSI Inverter (cont’d)
In the 612 connection, the a-phase
current, ias = ii, the instantaneous current
supplied by the inverter from the dc link,
and since ias+ibs+ics = 0, the d,q
transformation of the currents yields:
i  ii
s
qs
;
1
i 
(ics  ibs ) ;
3
s
ds
i 0
s
0s
Performing this analysis for the other five
modes of operation yields the relations
shown on the next slide. Note: i0s,v0s=0 in
all cases and are not included in the table.
d-q Modeling of VSI Inverter (cont’d)
Ref: D.W. Novotny and T.A. Lipo, “Vector Control and
Dynamics of AC Drives”
d-q Modeling of VSI Inverter (cont’d)
The d-q voltages for different switch positions
can be expressed as space vectors as shown
below:
Ref: G. Venkataraman, Dynamics and Control of AC Drives,
Univ. of Wisconsin Course Notes April 2002
d-q Modeling of VSI Inverter (cont’d)
The d,q relations for the VSI inverter can be
conveniently described by defining two
switching functions to express the constraint
equations given two slides ago. The switching
functions are shown on the next slide and are
described by the following equations:
v 
s
qs
2

Vdc g
s
qs
; v 
s
ds
2

Vdc g
s
ds
;

3
ii  iqss g qss  idss g dss
These expressions relate the instantaneous
inverter input quantities Vdc and ii to the
instantaneous d,q output quantities.
d-q Modeling of VSI Inverter (cont’d)
Ref: D.W. Novotny and T.A. Lipo, “Vector Control and
Dynamics of AC Drives”
d-q Modeling of VSI Inverter (cont’d)
The choice of  / 3 and  3 / 6 as the
s
s
g
g
amplitudes of qs and ds is arbitrary and
have been chosen so that the fundamental
in the Fourier series for the two functions
is unity. The complete Fourier series for
these two switching functions are given
by:
1
1
s
g qs  cos  e  cos 5 e  cos 7 e  ...
5
7
1
1
s
g ds   sin  e  sin 5 e  sin 7 e  ...
5
7
d-q Modeling of VSI Inverter (cont’d)
The complex vector forms of these d,q
equations take on simple forms and are
useful for visualizing and manipulating
the equations. The complex d,q equations
for the VSI voltages become:
v qds  vqss  jvdss 
s
2

Vdc ( g qss  jg dss ) 
2

s
Vdc g qds
s
s
g
g
From the Fourier series
of qs and ds the
s
complex function g qds is given by:
g qds  e
s
j e t
1  j 5et 1 j 7et
 e
 e
 ...
5
7
d-q Modeling of VSI Inverter (cont’d)
This represents rotating vectors in
alternating directions at speeds of e
and multiples of e. The complex
vector form of the current equation is
given by:

s
s
†

ii  Re i qds ( g qds )


3
d-q Modeling of VSI Inverter (cont’d)
An alternative and very useful form of the
d,q complex vector voltage equation can
be obtained for each of the modes
described earlier.
In mode 1:
v
In mode 2:
v qds
s
qds
s
2
 Vdc e j 0
3
2  1
3 2
j / 3
 Vdc    j

V
e

dc
3  2
2  3
Repeating for all six modes yields the
result shown on the next slide.
d-q Modeling of VSI Inverter (cont’d)
Ref: D.W. Novotny and T.A. Lipo, “Vector Control and
Dynamics of AC Drives”
d-q Modeling in the Synchronous
Reference Frame
Until now we have considered d-q
modeling in stationary reference frame.
However, the synchronous reference
frame is more useful for induction motorinverter simulation. The switching
functions in the synchronous reference
frame can be derived as (see handout):
2
2
g  1  cos 6 et 
cos12 et  ...
35
143
12
24
g dse  sin 6 et 
sin12 et  ...
35
143
e
qs
d-q Modeling in the Synchronous
Reference Frame (cont’d)
In these two equations we have assumed
that the rotating d,q axes have been
synchronized with the fundamental
frequency of the inverter output voltage,
i.e.  e   et . The below figure shows the
time functions that generate these series.
Ref: D.W. Novotny and T.A. Lipo, “Vector Control and
Dynamics of AC Drives”
d,q Model for PWM Operation
In a PWM inverter the various modulation
techniques will alter the switching functions
s
g qs and g dss resulting in unique switching
functions for each modulation type.
d,q Model for PWM Operation
One important difference between PWM
and modulated VSI operation is the
existence of two additional zero voltage
states (giving a total of 8 states).
Ref: D.W. Novotny and T.A. Lipo, “Vector Control and
Dynamics of AC Drives”
d,q Model for PWM Operation
The below table shows the various inverter
switching states and the corresponding
space vectors.
Space-Vector PWM
The space vector method is a d,q model
PWM approach. Let us first consider the
linear or undermodulation region. The
modulating command voltages are
sinusoidal and correspond to a rotating
space vector V*. This vector rotates at a
speed e. The figure on the next slide
shows the rotating space vector in terms
on the complex plane together with the
inverter switching state space vectors.
Space-Vector PWM (cont’d)
Space Vector PWM (cont’d)
A convenient way to generate the PWM
output is to use the adjacent vectors V1
and V2 of sector 1 on for part of the time
to meet the average output required. The
V* can be resolved into:


 
V * sin      Va sin  
3

3
 
V sin   Vb sin  
3
*
i.e.
Va 
2 *


V sin    
3
3

2 *
and Vb  V sin 
3
Space Vector PWM (cont’d)
During the period TC where the average
output should match the command,
vector addition can be used to write:
ta
tb
t0
*
V  Va  Vb  V1  V2  (V0orV7 )
Tc
Tc
Tc
or
V *Tc  V1ta  V2tb  (V0orV7 )t0
Va
where ta  Tc ,
V1
Vb
tb  Tc , and t0  Tc  (ta  tb )
V2
Space Vector PWM (cont’d)
The below pulse pattern satisfies the
equations on the previous slide.
Space Vector PWM (cont’d)
Two overmodulation modes.
Overmodulation mode 1 starts when the
reference voltage V* exceeds the hexagon
boundary. Where V* exceeds boundary, loss
of fundamental voltage. To compensate for
this loss, a modified trajectory partly on the
circle and partly on the hexagon is selected
as shown in the next slide. Circular part of
trajectory has larger radius (Vm*) and
crosses hexagon at angle  - the crossover
angle.
Space Vector PWM (cont’d)
Space Vector PWM (cont’d)
Overmodulation mode 1 ends when the
trajectory is fully on the hexagon
(trajectory comprises only line segments).
Overmodulation mode 2 starts when V* is
increased further. The trajectory is again
modified so that the output fundamental
voltage matches the reference voltage. In
this case the voltage is partly held at the
hexagon corner for a holding angle h and
partly by tracking the sides of the hexagon
(see next slide).
Space Vector PWM (cont’d)
Space Vector PWM (cont’d)
Details of how to calculate  and h are
given in the textbook. The implementation
of the SVM algorithm is shown in the
below figure.