INTRODUCTION
Speed control of ac motor, which requires concomitant variation of the frequency
and stator voltage to keep the air gap flux to its rated value, was mainly implemented
by PWM controlled inverters. In the last two decades, a more flexible power
dc to ac power conditioning technique, space vector PWM (Broeck et al.,
1988), has became increasingly popular in high performance ac servo drives applications.
Unlike conventional PWM control strategies, where the determination of the switching
pattern of the inverter power switches is in essence an analog signal processing
problem, SV PMW, in contrast, deals with the three phase inverter as a unique
space vector using the Clark’s Transformation. More importantly, because
of the placement of the target voltage space vector, representing the desired
inverter output voltage, is an anticipative process, this technique is easily
implemented using dedicated microcontrollers and DSP processors (ADCorp/AN401,
2000; Texas Instrument/SPRA524, 1999). Because of the availability of dedicated
timers for SV PWM pulse pattern generation, within most of present DSP and software
routines for implementing reference frame transformations, field oriented control
of ac drives can be directly formulated in terms of space vector theory (Vas,
1999; Leonhard, 1997). Performance assessment of sv pwm control strategies through
theoretical investigation and modelling is another present field of interest
(Pinheiro et al., 2003). Although several models have been reported in
different publications, they are either formulated in terms of Sfunction, state
variable model (Pinheiro et al., 2003) or given as SUMILINK black boxes.
Besides the inevitable mathematic formulation of these models, they do neither
give the physical understanding of the model nor provides the flexibility for
parameter variation for performance analysis. This paper proposes a new space
vector pwm Simulink modulator which emulates DSP implementation of a space vector
modulator.
SPACE VECTOR CONTROL STRATEGY
In the 180° square wave VSI inverter, shown in Fig.
1, the three phase output voltages are synthesised according to a
predetermined switching pattern of the semiconductor switches.

Fig. 1: 
Topology of the VSI 
Within one sixth of a period the output voltage is constant because the
conduction times of any semiconductor switch is held constant over 180°
electrical degrees. By shifting the gating signals from each other by
60°, the resulting output phase voltage is a six steps approximation
of a sinusoidal wave. These discrete operating states of the inverter
can be described by a space vector
which represents the stator phase voltages
and two additional null vectors mapped into the origin of the stationary
reference frame. In contrast, the space vector PWM control strategy consists of elaborating
a switching pattern of the power semiconductor switches, within a sector,
by varying the conduction time of the semiconductor switches in order
to impose a discrete sinusoidal variation of the output phase voltage
between two consecutive states of the six step inverter. By switching
at high frequency the power switches according to a predetermined pattern
the tip of the reference space vector, which represents the desired approximation
of the output voltage, is forced to follow a smooth circular path. This
objective is met by building the target reference space vector from the
time weighted combination of two adjacent sampled vectors. In terms of
voltsecond, the averaging process within a sampling period is
Hence
where t_{1} and t_{2}, for instance, are the time to
be spent in the active states 100 and 110.
According to Fig. 2, which shows the reference vectorfor
a particular angular position θ, into the stationary stator reference
frame αβ
The active state durations are finally found from the previous relation
as

Fig. 2: 
Inverter operating states 

Fig. 3: 
Switched sequence required to syntheses the target space
vector in the first sector 
where
is the modulation index.
Although infinite vector samples can be obtained within a sector, their
components are always linearly dependent on the base vectors delimiting
a given sector. For that reason, only two adjacent switching states say
(100 and 110) are required to synthesise the current reference vector.
Figure 3 shows the switching pattern required to synthesise
the target vector in sector one.
To reduce the switching frequency of the inverter, the next position
of the space vector is obtained by changing the conducting state of only
a single power switch. A transition is made by including a null state between every two target vectors. The duration of the null vector is
the remaining time from the switching period.
Therefore a switching sequence includes always a starting null vector,
the required switching pattern, finally ending by a null vector; the reverse
sequence is then applied for the next sampling period. If the time application
of the null vector is divided equally Texas Instrument/SPRA524, 1999),
a symmetrical space vector modulation is obtained.
The maximum output phase voltage is obtained when the reference space
vector lies in the middle of the first sextant. In that case
therefore t_{0} = 0, no time is then available for a null vector
insertion to enable smooth transition between two consecutive target vectors.
Overmodulation, is reached for
i.e., for m>0.866, is used in some applications to boost of the inverter
output voltage. Although two different overmodulation modes are available,
only mode I, suitable for Simulink implementation, has been used, the
active switching durations of which are calculated according to
Simulink model of the SV PWM VSI: By comparison to the space vector
modulator proposed by Math Works (2004). which is made up from seven blocks,
the present models include only four blocks connected in cascade.
SWITCHED MODE INVERTER
Active state computation block: In ac drives, the control effort
signal of the inner current loop represents the reference voltage, whereas
in an open loop configuration this signal is derived from the Clark’s
transformation of the desired three phase output voltage.
The simulation of the on time duration is carried out, according to relations
4 and 8, by sampling, at fixed frequency the reference space vector.

Fig. 4: 
Sector number deternimation 
Since the time duration values remain the same within an interval of
60°, a modulus block is used to divide, a time period of V^{*},
into six intervals.
Figure 4 shows some features of this block. Depending
on the sign of t_{0}, the linear or over modulation mode is selected;
through a selector bloc, the resulting output is fed to the next block,
the task of which is to compute the duty time cycle of each inverter leg.
Duty cycle computation block: Its purpose is to assign the on
duty time duty cycles T_{aon} T_{bon} and T_{con},
to the right inverter leg; such a task requires the knowledge of the sector
number in which the reference vector is currently lying. Two techniques
have been used to simulate the sector number as a sub block. The first,
used with DSP (Texas Instrument/SPRA524, 1999), is not suitable for Simulink
programming as it can be seen from Fig. 4b. The second
relies on the sampled magnitude and position of the reference vector,
carried out in the previous block, to retrieve the sector number stored
in a two dimensional look up table. In the first sector the on duty time
cycles T_{aon} T_{bon} and T_{con}, shown in Fig.
3, are related to the inverter the active state duration and through
the matrix formulation.
The element of the matrix can be found from the required gating signal
(Texas Instrument /SPRA524, 1999), applied to the upper power switches
of the six steps inverter. To each sector corresponds a well defined matrix
(Texas Instrument/SPRA524, 1999). In the simulation, the on time duty
cycle are normalised with respect to the sampling period of the reference
space vector.
Switched mode inverter block: The final block consists of the
control and the six steps inverter blocks. By comparing the on time duty
signals generated by the previous block with a 10 KHz frequency triangular
carrier wave, the gating pulses of the inverter power switches are produced
using a relay block and then fed to the inverter block. The power inverter
has been implemented in terms of the switching function g_{i}
associated with each power switch. The switching function g_{i}
of a given power switching can assume either 1 or 0 according to its conducting
state. Since, two power switches of the same leg can not be on simultaneously,
the switching function of the phase α, for instance, is defined as
The output phase voltages, in terms of the switching functions, are

Fig. 5: 
Normalized pole voltage at f = 25 Hz 

Fig. 6: 
Normalized pole voltage at f = 50 Hz 

Fig. 7: 
Normalized pole voltage at f = 140 Hz 
Hence the output stage of the switched mode inverter in modelled merely
by a standard matrix multiplication bloc.
Simulation of the switched mode inverter: Figure
5 to 7 show the inverter pole voltage for the operating
frequencies of 25, 50 and 140 Hz at or the maximum value of the modulation
index m = 0.866 which corresponds to the upper limit of the linear mode
of the space vector modulation technique.
As it can be expected from relation 12, the output voltage at the inverter
pole is made up from a voltage pulses which result from the switching
action of the semiconductor switches as shown in the top curves of each
figure. The continuous curves represent the corresponding pole voltage
after filtering the chopped voltage using a low pass filter. It can be
seen that for the frequency of 25 Hz, this signal is the typical signature
of space vector modulation. Figure 6 shows that the
switched mode inverter still behaves satisfactory in accordance with the
space modulation technique up to 50 Hz.

Fig. 8: 
Pole voltage waveforms in the first sector 
In contrast, at frequency of 140 Hz, the switched mode inverter model
is no longer able to reproduce this typical waveform owing to the high
harmonic generation due to the high frequency commutation of the inverter
power switches. To overcome this obvious limitation as far as the frequency
response of the inverter is concerned another inverter model has been
developed.
CONTINUOUS MODE INVERTER
The formulation of the sine mode or continuous SV PWM inverter is derived
from the computation of the average pole voltage of each phase.
From Fig. 8, the average pole voltage of phase in sector
one can be found as
where is
the on time duty cycle of the semiconductor switch S_{A} of the
inverter.
Similar relations can be derived for the remaining phases.
The offset voltage v_{NO} is found from the previous relations
bearing in that the load voltage should form a balanced three system voltage
Finally, the phase voltages are

Fig. 9: 
Comparison between actual and simulated pole voltage
waveforms 

Fig. 10: 
Simulated pole voltage waveforms at f = 400 Hz for linear
and over modulation mode 
Examination of relation 12 and 16 shows that the mathematical formulation
of both inverter models is the same except for that the first is expressed
in terms of a switching function while the second is expressed in terms
of the pole duty cycle. Both models share the same computational blocks,
they differ only from the inverter model formulation.
The switched mode inverter will reproduce at its output the pulse train
of the switching function whereas the sine mode inverter is expected to
produce continuous voltage waveform since the on time duty cycle d_{aon},
d_{bon}et d_{con} are a piecewise continuous function
of times.
Simulation of the sine mode inverter: The validity of the sine
mode inverter model has been checked by comparing the results of the simulation
with the experimental data available in reference (Analog Device Corp/AN401,
2000).
Simulations of the sine mode inverter have been carried out for the linear
and over modulation mode as shown in Fig. 9. Figure
10, which corresponds at a frequency of f = 400 Hz, has been included
to show the stiffness response of the model to frequency demand. This
comparison gives a full measure of the reliability of the sine mode model
which emulates faithfully the DSP space vector implementation with the
ADMC401 (Analog Device Corp/AN401, 2000).
CONCLUSIONS
Two space vector pwm modulated inverter models have been presented; their
respective performances have been assessed by comparing their frequency response. The reliability
of the sine mode inverter model has been confirmed through comparison
with the experimental results kindly provided by the Mixed Signal DSP
Group of Analog Devices Corporation.