A Vector Modulated Three-Phase Four-Quadrant Rectifier - Application

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    A Vector Modulated Three-Phase Four-Quadrant
      Rectifier – Application to a Dc Motor Drive
                                   Matti Jussila, Mika Salo, Lauri Kähkönen, and Heikki Tuusa


   Abstract–This paper introduces a theory for a space                      freewheeling diode drive can operate only in the first
vector modulation of a three-phase four-quadrant PWM                        quadrant as presented in [3]. In [3] the two-quadrant
rectifier (FQR). The presented vector modulation                            operation is achieved with four-quadrant dc-dc converter in
method is simple to realize with a microcontroller and it                   field excitation circuit, which increases the complexity of
replaces the conventional modulation methods based on                       the control system and also delays the control.
the analog technology. The FQR may be used to supply                           With the FQR the four-quadrant operation of the dc drive
directly a dc load, e.g. a dc machine. The vector                           is achieved using the vector modulation with a constant field
modulated FQR is tested in simulations supplying a 4.5                      excitation. Thus, the FQR is superior in dynamic response
kW dc motor. The simulations show the benefits of the                       when compared to systems of thyristor bridges using field
vector modulated FQR against thyristor converters: the
                                                                            excitation or mechanical switches to achieve reverse
supply currents are sinusoidal and the displacement
                                                                            rotational speed.
power factor of the supply can be controlled.
                                                                               The FQR is also a more supply friendly solution than a
Furthermore the load current is smooth.
                                                                            conventional combination of two antiparallel connected six-
  Index Terms–PWM rectifier, dc motor drive, space                          pulse converters and simpler than a four-quadrant dc-dc
vector modulation                                                           chopper needing a separate rectifier, both of which are
                                                                            presented in [10]. Furthermore, with the IGBTs with reverse
                          I. INTRODUCTION                                   voltage blocking capability presented in [11] the number of

I  N recent years research on current source PWM rectifiers                 conducting semiconductor switches in series in the FQR is
   (CSR), presented in Fig. 1, has become popular [1], [2],                 reduced to one.
[3]. The polarity of the output voltage of the CSR can be
reversed but the direction of the output current cannot. Thus,                                                               idc    p
it cannot act in all four quadrants when it is supplying a dc
load without an additional converter bridge or mechanical
systems. A four-quadrant operation can be achieved with a
three-phase four-quadrant PWM rectifier (FQR) presented                         a
in Fig. 2. It is also possible to use the FQR as a line bridge                                                                vdc
                                                                                b                                                       M
of an indirect matrix converter (IMC) studied in [4], [5], [6],
[7], [8]. Instead of using the FQR as a PWM rectifier the                       c
same topology may be also used as a three-to-single-phase
matrix converter, when the dc motor in Fig. 2 can be
replaced with a single-phase system as in [9].                                                                                      n
   Most research on FQR has been done using it as a
rectifier stage of the IMC with triangular wave voltage                                   Fig. 1. Current source PWM rectifier.
command modulation [4], [5], which is suitable for analog
technology but is complex when digital technology is used.                                                                   idc    p
With modern digital technology space vector modulation is
a less complex and more effective method. In this paper, the
space vector modulation method for the FQR is introduced.
                                                                                               iia
   The possible application to the vector modulated FQR is a                    a
dc machine drive. When compared to the conventional                                             iib
solutions better efficiency, supply currents and controllable                   b                                             vdc
                                                                                                                                        M
displacement power factor are achieved as presented in [3],                                           iic
where the CSR has been used. However, the CSR can                               c
produce only positive armature current. Thus it can operate
with constant field excitation only in the first and the fourth
quadrants if a freewheeling diode is not included. With                                                                             n

   The authors are with the Institute of Power Electronics, Department of                  Fig. 2. Four-quadrant PWM rectifier.
Electrical Engineering, Tampere University of Technology, P.O. Box 692,
FIN-33101 Tampere FINLAND, Tel: +358-3-3115-4257, Fax: +358-3-
3115-2088, e-mail: matti.jussila@tut.fi
                                                                                                                                                                                                                2



   The theory for the space vector modulation of the FQR is
                                                                                            i2(0,1,-1)                                             i5(0,-1,1)
presented in Section II, which is derived from the space
                                                                                                                                                                                                         iδ
vector modulation of the IMC, presented in [8], [12], and                 i3(-1,1,0)                              i1(1,0,-1)     i6(1,-1,0)                            i4(-1,0,1)
                                                                                                                                                                                              dδ iδ
the space vector modulation of the CSR, presented in [1],                                   III       II                                           III       II
                                                                                                                                                                                                       ii,ref
[2]. The control system of a FQR supplied separately                                   IV
                                                                                                  0
                                                                                                           I                                  IV
                                                                                                                                                         0
                                                                                                                                                                  I
                                                                                                                                                                                                θi
excited dc motor is presented in Section III and its                                        V         VI                                           V         VI                       dγ iγ

simulation model and results in Section IV. Finally,                      i4(-1,0,1)                             i6(1,-1,0)      i1(1,0,-1)                           i3(-1,1,0)
                                                                                                                                                                                                       iγ
conclusions are drawn in Section V.                                                         i5(0,-1,1)                                             i2(0,1,-1)
                                                                                                               ix(swa,swb,swc)
                                                                          (a)                                                    (b)                                                (c)
II. SPACE VECTOR MODULATION OF FOUR-QUADRANT PWM
                     RECTIFIER                                         Fig. 3. Input phase current sectors and current vectors of switching states
                                                                       of the vector modulation of the FQR (a) when idc > 0, and (b) when idc < 0;
   Space vector modulation of the FQR can be derived from              (c) forming a reference vector of input current.
the modulation of the CSR [1], [2] or the modulation of the
line bridge of the IMC [8], [12].                                      the switching vectors are
   The space vector of the input current of the FQR ii may be
                                                                                             ⎛π     ⎞                                                                                                 (8a)
defined [10]:                                                                     Tγ = mi sin⎜ − θi ⎟Ts
                                                                                             ⎝ 3    ⎠
                       (
       i i = (2 3) iia + aiib + a 2iic        )                  (1)
                                                                       and
where a = e j2π/3.                                                                Tδ = mi sin(θi )Ts ,                                                                                                (8b)
   When each common-emitter connected IGBT pair in Fig.
2 is treated as a single bidirectional switch, the values of           where the modulation index mi = |ii,ref | / |idc|. Then the
phase switching functions (swa, swb, swc) can be defined to            duration of a zero vector is
be 1, 0 or –1, denoting the on-state of the upper switch, the                    T0 = Ts – Tγ – Tδ.                                                                                                   (8c)
off-state of both switches and the on-state of the lower
switch in one phase leg respectively. Because only one input              Equations (8) are identical to those presented in [2], but
phase may be connected to each bar at a time, it is possible           the difference is that now the link current is allowed to be
to write:                                                              positive or negative.
                                                                          The ratio between the output dc voltage and the input ac
      swa + swb + swc = 0                            (2)
                                                                       voltage of the FQR may be derived as follows: If the power
  Phase currents in (1) can be presented with the help of              losses in the converter are neglected it can be assumed the
switching functions and dc current:                                    ac and dc powers of the converter are equal [10]:
                       (                              )
       i i = (2 3) swa idc + aswb idc + a 2 swc idc = sw i idc   (3)
                                                                                   p = (3 2 ) Re{v i i * } = vdc idc ,
                                                                                                       i
                                                                                                                                                                                                       (9)
where swi is the space vector of switching functions of input          where vi is the space vector of the input phase voltages and
current:                                                               the superscript (*) denotes a complex conjugate. Substituting
                            (
       sw i = (2 3) swa + asw b + a 2 swc .       )              (4)   first (3) to (9) and then dividing by idc the dc voltage is
   Substituting permissible value combinations of swa, swb,            found to be
and swc into swi it is possible to derive the input current                       v dc = (3 2 ) Re{v i sw * }.
                                                                                                          i
                                                                                                                                                                                                      (10)
vectors, which are presented in Fig. 3(a) when idc > 0 and in
                                                                          When the angle of vi is ω t the angle of ii is ω t – ϕ if,
Fig. 3(b) when idc < 0. The ix(swa,swb,swc) denotes current
                                                                       which is the same as the angle of swi. Thus, by (10) the
vectors produced by permissible switching states. E.g. for
                                                                       instant link voltage vdc in polar coordinates is found to be
current vector i1 swa = 1, swb = 0, and swc = –1, when ia = idc,
ib = 0, and ic = –idc, i.e. phase a is connected to the p-bar and                                                   {
                                                                                  vdc = (3 2) Re | v i | e j(ωt −ϕ if ) | swi | e− jωt                                     }
phase c to the n-bar.                                                                                                                         {
                                                                                        = (3 2 ) | v i | ⋅ | sw i | ⋅ Re e j( ωt − ωt −ϕ if ) .                        }                              (11)
   As presented in Fig. 3(c) it is possible to produce an input
current reference vector ii,ref in one switching period as a           According to (3) |swi| = |ii,ref|/|idc| = mi and (11) can be
sum of two weighted current vectors [12]:                              written as
       ii,ref = dγiγ + dδiδ                                      (5)              vdc = mi (3 2) | v i | cos(ϕ if ) .                                                                                 (12)
                                                                       From (12) it is possible to solve the voltage ratio of the
where e.g. iγ = i6 and iδ = i1 when ii,ref lies in sector I and idc    FQR:
> 0. In (5) the weighting coefficients dγ and dδ are the
                                                                              vdc
relative durations of current vectors:                                              = mi (3 2 ) cos(ϕif )                   (13)
                                                                             | vi |
              | i i,ref |       ⎛π      ⎞
       dγ =                 sin ⎜ − θ i ⎟ ,                      (6)
                                                                          The maximum current ratio mi in (13) is unity in linear
                 idc            ⎝3      ⎠
                                                                       modulation. Thus, the maxima of the ratio vdc/ |vi| are ±3/2,
and
                                                                       which are achieved when ϕ if = 0 and ϕif = π. Thus,
              | i i,ref |
       dδ =                 sin (θ i ) .                         (7)   |vdc|/ |vi,line-to-line| ≤ 3 / 2 and the FQR is a buck converter,
                 idc
                                                                       which can produce average voltage up to ±487 V when the
  If the switching period is marked with Ts the durations of           rms value of the line-to-line supply voltage is 400 V.
                                                                                                                                                                                 3



             III. CONTROL SYSTEM OF DC MACHINE DRIVE                                                              antiparallel diodes of the IGBTs. In steady state operation it
                                                                                                                  means overlapping in switchings may be used in the same
   As presented in Fig. 2 the FQR may supply the dc
                                                                                                                  way as in the CSR. When the dc current reference value is
machine directly. The block diagram of the control system
                                                                                                                  reversed and the dc current is reaching a zero value, the
of the FQR supplied dc motor drive is presented in Fig. 4.
                                                                                                                  modulation is stopped. After a short dead time period the
Its outer control loop is speed control and its inner loop is
                                                                                                                  modulation of the IGBTs conducting to the reverse direction
armature current control. The feedback signals for the
                                                                                                                  when compared to the previous steady state is started and
controllers are the angular rotor speed ω r and the armature
                                                                                                                  the switchings are performed again using overlapping. The
current iAr.
                                                                                                                  result of this method is equal to the two-step commutation
   The speed control of the motor is based on [10]. It
                                                                                                                  used in matrix converters [13].
produces the reference value of the armature current iAr,ref
and it is executed every 800 µs.                                                                                                                 IV. SIMULATIONS
   The current control of the FQR is based on [2]. The
current control and modulator are executed every 100 µs in                                                           The simulations were performed with Matlab Simulink®
                                                                                                                  software. A selected dc machine in simulations is a 4.5 kW
a 200 µs switching period, i.e. the switching frequency is 5
                                                                                                                  separately excited dc motor, whose parameters are presented
kHz. Both the current and the speed controllers are discrete
                                                                                                                  in Table I. The model of the separately excited dc motor is
time anti-windup PI controllers.
                                                                                                                  based on the equations:
   The current controller produces the reference value of the
dc voltage of the FQR vdc,ref from which the value of the real                                                                                   diAr
                                                                                                                         v Ar = RAr iAr + LAr         + cψω r ,               (18)
component of the input current iix,ref in a supply voltage                                                                                        dt
oriented reference frame is generated using (9) when vi = vix                                                            t = cψ iAr,                                          (19)
+ jviy and ii* = iix – jiiy:
                                                                                                                  and
                                {
         p = (3 2 ) Re (vix + jviy ) (iix − jiiy ) .                       }                            (14)
                                                                                                                         t = t load + J
                                                                                                                                          dω r                                (20)
                                                                                                                                           dt
In the supply voltage based reference frame the real axis is
tied to the supply voltage vector, thus viy = 0 [2]:                                                              where cψ is a constant when excitation current iexcitation and
                                                                                                                  excitation inductance are constants.
         p = (3 2 ) vix iix = vdcidc ,                                                                  (15)
                                                                                                                     In simulations the control system of the drive was based
from which we get for reference values:                                                                           on the block diagram presented in Fig. 4. The modulation of
                        2                                                                                         the FQR was performed as presented in Section II with ideal
        iix, ref =          v dc,ref idc = kvdc,ref idc .                                               (16)
                                                                                                                  switches, i.e. the commutation method presented in Section
                       3vix
                                                                                                                  III is not needed. The supply voltage (400 V, 50 Hz)
   The imaginary component iiy,ref of the input current in the                                                    waveform in simulations was purely sinusoidal and the LC
input-voltage-oriented reference frame may be generated                                                           filter type supply filter consisted of 2.3 mH inductors and Y-
from the definition of the reactive power as presented in [2]:                                                    connected 10 µF capacitors, so the resonance frequency of
                                                                                                                  the filter was 1049 Hz.
                              { }
        q = (3 2) Im v i i * = −(3 2) vix iiy ,
                           i
                                                                                                        (17)
                                                                                                                     The simulation results at nominal speed with load torque
i.e. iiy,ref may be used to control the reactive power of the                                                     step from nominal value to negative nominal value and back
input. In this study it is set at zero, i.e. ϕ if = 0, and by (13)                                                are presented in Fig. 5, i.e. the working of the motor drive in
the maximum dc voltage vdc may be achieved.                                                                       the first and the second quadrants is proved. The dc current
   The modulation of the FQR may be performed as                                                                  sign is reversed very fast, which leads to 180° phase change
presented in Section II. In practice switchings of the FQR                                                        of input currents. It produces together with current reference
should be performed such that a dc current path is not cut                                                        oscillation small non-sinusoidal periods to the load steps.
and the supply is not shorted. This can be done when the                                                          The oscillation of the dc current reference is caused by
FQR is considered to consist of two CSRs connected                                                                speed controller, which is tuned concerning mainly on
antiparallel, which series diodes are at the same time the                                                        transients situation of speed reference.

                                                                idc= iAr                                                                          TABLE I
   va          LC-                                                                                                                          DC MOTOR PARAMETERS
   vb                                FQR            vdc = vAr                  M
   vc         filter                                                                          ωr
                                                                                                   -
                                                                                                                        Nominal shaft power, Pn                    4.5 kW
                                                                     iAr                            +
                                                                                                         ωr,ref
                                                                                                                        Nominal rotational speed, nn               1000 rpm
        θi                      Modulator
                                                                                           Speed                        Nominal torque, Tn                         42.6 Nm
                                                                                         controller
                           iiD,ref            iiQ,ref                                                                   Nominal armature voltage, VAr,n            440 V
                                     x,y ->                                                                             Nominal armature current, IAr,n            13.7 A
                                     D,Q
                                                                                                    iAr,ref             Nominal excitation current Iexcitation,n   1.33 A
                                                                            Dc current   -
                                                            k       vdc,ref controller
                                                                                          +
                          iiy,ref       iix,ref                                                                         Armature inductance, LAr                   121 mH
                                                                                                                        Armature resistance, RAr                   9.5 Ω
Fig. 4. Separately excited dc motor drive supplied by vector modulated
four-quadrant rectifier.
                                                                                                                                                        4



   The speed reversal with constant load torque is presented                       Steady state waveforms of the supply and load currents
in Fig. 6, thus the working of the FQR drive in the fourth                      and voltages with the supply current spectrum are presented
quadrant is also proved. As can be seen from the presented                      in Fig. 7. The total harmonic distortion (THD) of the supply
waveforms the FQR drive can operate both with positive                          current includes harmonics up to 2 kHz. As can be seen
and negative dc current and voltage producing sinusoidal                        from the presented waveforms the FQR produces sinusoidal
input current during the transients as well.                                    supply currents and a smooth armature current.




                                        (a)                                                                           (a)




                                        (b)                                                                           (b)




                                        (c)                                                                           (c)




                                        (d)                                                                           (d)




                                        (e)                                                                           (e)
Fig. 5. Simulation results of load torque steps when nn = 1000 rpm: (a)         Fig. 6. Simulation results of speed steps when T = 42 Nm: (a) mechanical
mechanical speed n; (b) electrical torque T; (c) armature current iAr (black)   speed n; (b) electrical torque T; (c) armature current iAr (black) and its
and its reference (red) ; (d) armature voltage vAr; (e) supply current ia.      reference (red) ; (d) armature voltage vAr; (e) supply current ia.
                                                                                                                                                      5



                             V. CONCLUSION
   In this paper the novel modulation method for the direct
four-quadrant PWM rectifier (FQR) has been introduced.
The vector modulated FQR supplying a dc motor was tested
in simulations. The simulation results show that the vector
modulated FQR can supply a dc machine.
   The FQR produces sinusoidal supply currents with low
distortion and offer an opportunity to control input power
                                                                                                                    (a)
factor. In addition the load current of the FQR is smooth.
Thus, the FQR may be used when dc drive with sinusoidal
supply currents is required. Especially the FQR is a suitable
solution when an integration of a converter on a dc machine
is desired.


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                                                                                                                    (f)
                                                                              Fig. 7. Simulation results of the FQR drive in steady state, when T =
                                                                              42 Nm, nr,n = 1000 rpm: (a) supply voltage va; (b) supply current ia; (c)
                                                                              armature current iAr; (d) armature voltage vAr; (e) more detailed iAr; (f)
                                                                              spectrum of supply current (f1=50 Hz).