IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 6, NO.

3 , JULY 1991 463

Angle Controlled Current Regulated Rectifiers for

AC/AC Converters
Thomas G. Habetler, Member, IEEE and Deepakraj M . Divan, Member, IEEE

Abstract-Rectifier control schemes for use in ac/dc/ac volt- Required for resonant
age-sourced resonant link converters with controlled rectifiers
are discussed in this paper. Resonant link converters require
the use of discrete pulse modulation. Control of the ac to dc
converter by means of averaging methods or "duty cycle con-
trol" is not inherent with this type of modulation. However, it
is shown in the paper that the voltage-sourced rectifier cannot
be operated solely on the basis of instantaneous quantities. A
bang-bang control scheme is developed and simulated which
independently controls the angle and the magnitude of the ac
line current vector. The magnitude of the current is controlled
using a linear combination of the link voltage error and the
current magnitude error. The current reference is derived by
the use of load torque estimation. In addition, the current vec-
tor which satisfies the sliding mode criteria and results in lowest
voltage ripple is chosen to further minimize the size of the link
I ' J
capacitor.
Fig. I , ACiAC converter with controlled rectifier

I. INTRODUCTION ulator is used to control the ac line currents. The current

reference is derived by a multiplication of the dc link volt-
r" most ac/dc/ac variable speed drive applications, the
rectifier stage consists of uncontrolled diodes or, less
frequently, phase controlled thyristors. This results in high
age error with a template sinusoidal waveform. This type
of controller is shown in Fig. 2.
harmonic content in the input ac line current and gives an The bus voltage is fed back proportionally to create the
input power factor which is dependent on operating con- current reference in the scheme in Fig. 2. Proportional
ditions. Moreover, conventional uncontrolled rectifiers do feedback results in a steady state error in the capacitor
not permit bidirectional power flow. The use of a con- voltage. If integral, or proportional-integral feedback is
trolled front end rectifier allows the realization of unity used, system dynamics are severely hindered and much
power factor on the ac side, as well as reversible power higher bus ripple results as compared to proportional
flow. In addition, regulated sinusoidal ac currents can be feedback. This is due to the fact that the line current mag-
achieved at the input, as well as at the output or inverter nitude is now depends on the integral of the bus voltage
stage. An ac/dc/ac converter with controlled rectifier is error, and therefore the system response to changes in the
realized through the use of the same converter on the rec- bus voltage is hindered. In addition, with proportional
tifier (ac to dc) side and the inverter (dc to ac) side. In feedback, instability occurs when operating with high bus
this paper the ac/dc/ac converter is synthesized with a res- voltage ripple due to the multiplication of the error term
onant dc link converter [ l ] for soft switching, thereby al- with the template. This instability occurs when ( K uy)
lowing for a high switching frequency. The transistor- goes negative since the current in the link is now being
diode voltage source dc link converter with resonant link driven in the wrong direction. The quantity ( K U:") must
for soft switching is shown in Fig. 1. be positive since it is the magnitude of the line current
A comprehensive analysis of a controlled rectifier ac/ reference. If ( K U T )is negative, then positive feedback
ac system has been done by Ooi e t al. [2]-[4]. In this of the link voltage occurs, causing the link voltage to run
approach to the problem of rectifier control, a current reg- away or at the very least have large oscillations.
Lip0 and Sul [6] have introduced another method for
Manuscript received July 10, 1989; revised June 27, 1991. This paper obtaining the current magnitude reference. Load feedfor-
was presented at the 1989 IEEE Power Electronics Specialists Conference, ward is used in this scheme. Load parameters are used to
Milwaukee, WI, June 26-29.
T . G. Habetler is with the Georgia Institute of Technology, School of calculate the output power which in turn is summed with
Electrical Engineering, Atlanta, GA 30332. the integral of the link voltage error. Here again, the sys-
D. M . Divan is with the University of Wisconsin-Madison. Department tem responds slowly to changes in link voltage, yet power
of Electrical and Computer Engineering. 1415 Johnson Drive, Madison,
WI 53706. is balanced much more quickly than in the previous
IEEE Log Number 9100531. scheme due to load feedfonvard.

0885-899319110700-0463$01.OO 1991 IEEE

464 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 6. NO. 3. JULY 1991

ther reduced by using an optimum vector selection to min-

imize the ripple on the dc link voltage.

11. PRINCIPLES
OF VOLTAGE-SOURCED
CONTROLLED
t I RECTIFIERS
Even though the rectifier and inverter have the same
Regulator circuit topology, the control mechanisms are quite differ-
ent. In inverter control schemes, the input (given) quan-
Template ref tity is the dc bus voltage and the controlled quantity is the
Waveform + c
ac bus voltage or current. In the case of a rectifier, the
(cos wt)
given quantity is the ac line voltage and the controlled
Magnitu/de Probortional quantity is usually the dc bus voltage and the harmonic
of i, Gain content of the ac line currents.
Fig. 2. Block diagram of rectifier regulator proposed in [ 2 ] which uses As stated previously, the line current vector angle and
only bus voltage to derive current reference. magnitude are controlled independently. In order to ob-
tain unity power factor at the input, the line current vector
angle is controlled so as to be in phase with the input
In discrete pulse modulated systems, the switching in-
voltage vector, .: Under these operating conditions, the
stances are known points in time. This type of switching
three phase voltage-sourced rectifier is equivalent to a
is not readily adaptable to duty cycle-type control schemes
conventional boost dc/dc converter depicted in Fig. 3 . The
where pulse widths are regulated. It is important to note
problems confronted in controlling the three phase recti-
that the selection of duty cycle is akin to an averaging
fier exist in the control of the boost dc/dc converter.
process. There is no equivalent inherent averaging avail- Consider first the control of a boost dc/dc converter. A
able in a discrete pulse modulated system. Instead, a control scheme based only the output voltage, V , , is un-
choice must be made at each switching instant to turn the stable. This is true for both a discrete pulse modulated
various switching elements ON or OFF. Discrete pulse system and a duty cycle controlled system. In the discrete
switching is best suited to bang-bang (sliding mode) con- pulse case, a control scheme can be defined such that at
trol schemes. References [7], [8] present an excellent each switching instant,
sliding mode control scheme for voltage-sourced recti-
fiers. As with the method presented in this paper, the line s = [1 Vc-V;*>O
current magnitude and phase are independently con-
trolled. Once the angle of the current is regulated to be in 0 Vc-V;"<O
phase with the supply voltage, the magnitude directly re-
lates to the power transferred to the load. In [7], [8] the Although the scheme will act to control the output voltage
magnitude of the current is controlled directly by using a instantaneously, the control function is essentially a sec-
sliding line made up of a linear combination of the voltage ond order regularly sampled delta modulator which is
error and the current error. It is shown that integration of known to be unstable without additional damping.
the voltage error is not necessary to insure stable opera- If a control scheme is derived which calculates a duty
tion. However, the current error is approximated by high- cycle to control V, on a cycle by cycle basis, the system
pass filtering the line current magnitude. No load side in- is also unstable. This can be shown by considering the
formation is used to generate the current reference. There- energy balance equation for the inductor over one cycle:
fore the system responds to load transients only through
changes in the bus voltage. This results in poorer perfor-
V\i/-T = V(iL(1 - d ) T + AEL (2)
mance than in the case of load power feedforward. where A E , is the change in energy stored in L , and d is
This paper presents a method of rectifier control which the duty cycle of S :
utilizes the fundamental concepts of sliding mode rectifier
control presented in [7] and 181. That is, the ac line cur-
rent dq vector is controlled independent of the line current
vector magnitude. The magnitude of the line current vec-
tor, I , is controlled in a sliding mode fashion by consid- The output voltage can be expressed as
ering both the bus voltage error and the line current mag-
nitude error. Unlike in [7], [8], however, the scheme V,. = RiL(1 - d ) . (4)
proposed in this paper derives the current magnitude ref-
Substituting (3) and (4) into (2) results in
erence using an output torque estimator which estimates
the power delivered to the load only from motor terminal
quantities. Finally, since the rectifier has seven possible
switching states, more than one state may satisfy the con-
trol criteria. Therefore, the dc link capacitance can be fur-
HABETLER A N D DIVAN: ANGLE CONTROLLED RECTIFIERS FOR ACIAC CONVERTERS 465

tion. This can result in unstable operation for high values

of bus voltage ripple. Reference [6] solves this problem
by using load feedforward information and adding that to
the integral of the bus voltage error to eliminate steady-
state error. Controlling the rectifier according to the slid-
ing mode given by (9) is the most attractive solution since
no steady-state error exists, no integration of the voltage
Fig. 3. Boost dcidc converter schematic error term is necessary, and no template is needed to con-
trol the current vector angle.
In [8], the (1, - 1;) term in (9) is obtained by high pass
where is the desired value of the output voltage from filtering IL. That is,
which the duty cycle d is calculated. That is,

where s is the Laplace variable, and where w f is much

The actual value of the output voltage, V,., will be higher lower than the switching frequency. This type of deriva-
or lower than V y i.e., tion of the current error is not sufficient for good dynamic
response. This is especially true in ac/dc/ac converter ap-
v,.= Kv:. (7) plications since the output of the high pass filter is non-
Substituting (4) into (7) and the result into ( 5 ) yields zero only when there is first a large error in dc link volt-
age. The problem of having to wait for the system to
respond to large errors in the bus voltage before measures
are taken to correct the input current magnitude can be
If K < 1, i.e., the output voltage is less than the desired solved using load feedforward. This insures good tran-
value, the inductor current goes down and therefore the sient response and minimizes the required dc link capac-
system is not stable. Similarly for K > 1 , the current can itance.
be shown to increase indefinitely.
This implies that control of V,. alone is not sufficient to 111. ANGLE-BASED RECTIFIER CONTROLLERS
gain stable operation. If the inductor current or the duty The system which encompasses the desired perfor-
cycle is the controlled quantity then the system will al- mance traits is shown in Fig. 4. It includes current angle
ways be stable. That is, if a duty cycle is specified apriori, regulation and a torque estimator used to derive the cur-
the control is stable. Likewise, if the inductor current is rent magnitude reference. The torque estimator is based
directly controlled, the system is first order and a duty solely on inverter terminal quantities and provides the load
cycle calculator control or a bang-bang system will be sta- feedforward term for link voltage stabilization.
ble. Therefore, in order to have stable operation and reg-
ulation of the output voltage, d or iL must be controlled A. System Equcrtions and Angle Regulation
in addition to Vc. The angle regulator is realized such that the component
The same conclusion holds for the discrete sampled ac/ of the dq line currrent vector which is in quadrature with
dc rectifier considered in this paper. For the case of dis- the input voltage vector is regulated to zero. These vec-
crete pulse modulation, stable operation with bus voltage tors can be derived from three phase quantities as follows.
regulation results when a linear combination of bus volt- The input ac voltage is a balanced three phase supply, that
age error and current vector magnitude error is used as a is,
sliding surface used to regulate the ac line current vector
magnitude. As presented in reference [8], the sliding sur- E(, = E COS wt
face should be
Eh = E COS (ut - F)
The switching regulator then, at each switching instant,
causes the current vector magnitude to increase when U is
The stationar) fraine dq vector representation of the input
negative and decrease whenever U is positive.
voltage is giken by
The line current magnitude reference 1: can be obtained
in a number of ways. The control scheme presented by 2 = E cos wt + j E sin wt = Ed + jE, (12)
Ooi, etc., in [2]-[4] and by Lip0 and Sul in [6] are purely
A rotating reference frame representation of the input
current regulators. References [2]-[4] use a product of the
voltage, Fe, can be formulated by taking
- error and a template waveform which causes
bus voltage
the reference current to contain high frequency informa-
-
5
' = E t cos wt + jE, = sin wt = E (13)
466 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 6, NO. 3. JULY 1991

.-

s
Rectifier Inverter

Switcli

-
State
Angle output 4- load
I Regulator 4- Power
Estimator
Sliding
e. Mode 'out
Magnitude
I Regulator

Fig. 4 . Block schematic of rectifier regulator.

The stationary input line current vector in terms of abc

quantities is,

The rotating reference frame representation of the line

current, 1' is given by
rJ e - (Id cos u t + Zq sin u t ) + j(Zd sin u t + Zq cos u t )
= Is + jZi. (15) f
Note that I ; is simply the component of I in phase with E Ac side rectifier voltage vectors
and I ; is the component of I in quadrature with E. The
Fig. 5 . Diagram illustrating rectifier current and voltage vectors.
angle regulator is a bang-bang controller based on the sign
of I:. If I ; is negative (positive) then the rectifier is
switched to a state which advances (retards) the current the angle of - < will cause the current angle to advance
vector angle. (retard). The zero voltage vector ( k = 0) will be assumed
The rectifier ac side voltage vectors which correspond to retard the angle of I since the angle between 2 and 6
to the seven rectifier switching states at time t,, are given reduces in this case.
by
B. Current Magnitude Control and Load Feedforward
Once the angle of is controlled to be in phase with E,
I s is essentially equivalent to the magnitude of I . From
the set of vectors which are valid choices to regulate
The change in line current associated with each of the the angle of appropriate vector(s) are chosen which also
r,
seven rectifier states is given by control the magnitude of I' according to the sliding mode
- k(t) equation:
Ark(t)
L,
The current at the next switching instant t = t,l + I is,
'W,+ I) = W,) + A (t,). (18) When U , is regulated at U,*, the sliding mode control re-
For example, consider Fig. 5 . In this case I ; is nega- duces to a current regulator if the voltage term in (19) is
tive, that is, lags k. The vector s(t,) is the input voltage much smaller than the current term. If R, is chosen to be
vector at instant t,. l(t, - I ) is the current vector at the pre- large, then the impact of the voltage term on p is minimal
vious instant t, - Depending on the rectifier switching and the dc bus voltage ripple is also minimal in the steady
state chosen ( k = 0 to 6 ) the vector !(t,- ,) will change state. However, the transient response of the dc bus is
by A Ik over the next switching period. It can be seen from extremely poor. Decreasing the value of R,yimproves the
Fig. 5 that rectifier switching states 1, 2, and 3 will tend transient response of the bus voltage, but gives poor rip-
to regulate the angle of the input current since they cause ple and harmonic performance. Actual selection of R,yin-
the current vector to advance (rotate counterclockwise). volves a trade-off of transient and harmonic performance.
The actual A 1 vectors need not be calculated for the angle The value of I$* is found by estimating the inverter out-
control since any voltage vector, k , whose angle is put power delivered to the motor load by the inverter. This
greater (less) than the angle of IS and less (greater) than is accomplished by estimating the electrical torque, Test,
HABETLER A N D D I V A N . ANGLE CONTROLLED RECTIFIERS FOR ACiAC CONVERTERS 461

of the motor load to be tities must be known. In most applications, including all
those discussed in this work, the inverter dc link current
or the ac line currents are known. The control complexity
J
is increased however, since additional calculations are
where is the inverter ac side dq voltage vector and IT necessary to estimate the dc link voltage.
is the component of the inverter ac side line current in Since the switching frequency is high, it can be as-
quadrature with I . Note that in (20) the voltage drop sumed that the ripple on the input ac line current is small.
across the stator resistance is ignored. Then the power Say that the present switching instant is time tfiI.There-
delivered to the machine (the airgap power) is, fore, the dc link current out of the rectifier, idr, at the next
switching instant tr,+ can be estimated using the value
Pout = wrTr,r (21) of the line currents at the previous switching instant. The
where w, is the stator excitation frequency. For the ma- link current is given by
jority of applications, (21) can be used to find the output
power since U, is typically a known, slowly varying,
quantity. In cases where w,, is not known, it is possible to
use the dynamic content of T,,, to obtain load feedforward where S; is the j phase line switching function corre-
information. As stated earlier, high frequency informa- sponding to the kth inverter state. The current in the dc
tion in the input current vector is used to estimate ( I ; - link capacitor is,
I;*) in [8] and was shown in [8] to result in stable oper- i{ap(rri + I) i$r(rri + i ) - idl(tn+ I > . (24)
ation. Using the high frequency information in T,,,, as is
proposed in this paper, will also yield stable operation. The change in capacitor voltage is found by a linear ap-
The torque estimate method is superior to the scheme pre- proximation from icap,
sented in [8] since load change information is directly fed
forward without the need to respond only to changes in
the bus voltage.
The value of I;* in (19) can now be found from The switching state which yields the lowest value of Aut,
i.e., minimum bus voltage ripple, is selected. This pro-
cedure accomplishes the desired goals of bus voltage reg-
c
ulation, power balance, and unity input power factor. In
The choice of rectifier switching state is based on the value addition, it dynamically attempts to minimize the voltage
of p in (19). If p > 0 ( p < 0) then a state which causes ripple on the dc bus, for a given link capacitance. It is
the magnitude of to decrease (increase) and simulta- clear that such a control strategy is superior to strategies
neously satisfies the angle regulator criteria is selected. It that have been previously used.
is probable that more than one switching state will satisfy
the regulation criteria given in the previous sentence. By IV. RESULTS
examining Fig. 5 it can be seen that if I: is nearly zero, Figs. 6 shows the simulation results of the rectifier
for power flow into the rectifier, the two voltage vectors waveforms with the control scheme presented in this pa-
which enclose 2 will guarantee an increase in the mag- per, including link ripple minimization. Fig. 6 represents
nitude of -. Choosing the opposite vectors will ensure a the steady state operating conditions. Fig. 7 shows the
decrease in the magnitude of . This of course leaves three inverter output current delivered to the load. The inverter
vectors which represent possible choices which might give switching strategy is the current regulated delta modulator
better overall system performance. The only way to make [9]. change in power delivered to the load. To illustrate
a choice between all seven states is to solve (17). the transient response of the rectifier controller, Fig. 8
shows the dc bus voltage and input line current response
C. Optimum State Selection when the commanded inverter output current magnitude
As there are only two primary control criteria (angle of is instantaneously reduced by a factor of two. Since the
and p ) and seven possible switching states, there may be response time of the inverter current regulator is very fast
multiple switching states which satisfy these criterion. If (2-3 switching cycles), a very fast transient (with respect
this is true, then an additional control parameter can be to the change in bus voltage) occurs in the power deliv-
applied. From the subset of switching states which satis- ered to the motor. Note that the output of the torque es-
fies the primary conditions, the rectifier switching state is timator (Fig. 8(c)) responds very quickly to the load tran-
chosen which yields the lowest change in bus voltage, i.e., sient. It is interesting to note in Fig. 8(a) that before the
minimum bus voltage ripple. transient, the sliding mode controller (19) is essentially a
The change in dc bus voltage over a switching cycle current regulator, after the transient, the voltage error and
can be found from the dc link current out of the rectifier the current error become large due to the load feedfor-
and the dc link current into the inverter. Therefore, when ward. The line current magnitude is therefore reduced un-
using the optimum state selection method, inverter quan- til dc bus voltage returns to its reference value.
 I EEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 6, NO. 3. JULY 1991

0
5.

0-

-
4.00 1:os y;:lOIRRO;:ls u:zo r:zr d.so

Fig. 7 . Rectifier waveforms illustrating the response to load transient. (a)

DC bus voltage. (b) Input ac line current, and (c) output of torque esti-
mator.

Fig 6 Rectifier waveforms under steady-state operation (a) Input ac line

current (b) Rectifier line-line switching function (c) DC link voltage, and
(d) Input ac line current vector

In all the results given, V,, = 3.0 pu, I r I = 0.8 pu,

I ,I = 0.7 pu, Z, = 2 . 0 pu m s , X , = X , = 0.2 pu, R,
= 1 pu, and 1/(2a 60) C = 3 pu. The resonant link fre-
quency is 20 kHz. In actual units, for a 20-kW converter,
the link filter in the simulation would be 100 pf for a 300 Fig. 8. Inverter output current.
HABETLER AND DIVAN: ANGLE CONTROLLED RECTIFIERS FOR AC/AC CONVERTERS 469

Vdc bus. Under these conditions, the peak-peak ripple in J. W . Dixon, A . B. Kulkarni, M. Nishimoto, and B. T. Ooi, “Char-
the steady state is 50 V . The system could, however, acteristics of a controller current PWM rectifier-inverter link,” in
IEEE-IAS Annual Meeting Conf: Rec., 1986, pp. 685-69 1 .
function stably with even lower values of link capaci- M. Nishimoto, J. W . Dixon, A . B . Kulkarni and B . T . Ooi, “An
tance. The frequency of the input and output ac source is integrated controller-current PWM rectifier-chopper link for sliding
60 Hz, and 90 Hz, respectively. mode position control,“ in IEEE-/AS Anriual Meeting Corif. Rec.,
1986, pp. 752-757.
B . T. Ooi, J. W . Dixon, and A . B . Kulkarni and M. Nishimoto. “An
V. CONCLUSION integrated ac drive system using a controlled-current PWM rectifier/
inverter link,” in IEEE foicber Elecrronics Specialisrs Conj Rec..
This paper has addressed the issue of controlling dis- 1986. pp. 494-501.
crete pulse modulated voltage-sourced controlled recti- R . Wu. S. B . Dewan, and G . R. Slemon, “ A PWM a c t o dc converter
with fixed switching frequency,” in IEEE-/AS Annual Meeting Conk
fiers used in ac/dc/ac converters. Control of a three phase Rec., 1988, pp. 706-71 I .
ac to dc converter involves controlling the angle and mag- S. K. SUI and T . A . Lipo. “Design and performance of a high fre-
nitude of the input current along with regulation of the dc quency link induction motor drive operating at unity power factor,”
in IEEE-IAS Annual Meering Cotif: Rec.. I?88, pp. 308-313.
bus at a desired value. Control of the current angle is de- V . Ramanarayanan, A . Sabanovic, and S. Cuk, “Sliding mode con-
coupled from the magnitude control. The input current trol of a brushless dc motor.” in Rec. Platinum Jubilee Con6 on S w
magnitude is controlled indirectly by creating a sliding terns arid Signal Processinx. Indian Institute of Science, Bangalore.
India, Dec. 1986.
surface which is a combination of the dc bus voltage error R. Mahadevan, “Problems in analysis, control, and design of switch-
and an averaged current magnitude error. The reference ing inverters and rectifiers,” Ph.D.. California Institute of Technol-
value used to determine the current magnitude error is, in ogy. Pasadena, 1986.
R . D . Lorenz and D . M . Divan. “Dynamic Analysis and Experimen-
turn, based on power flow considerations. A torque esti- tal Evaluation of Delta Modulators for Field Oriented ac Machine
mator which uses inverter terminal quantities provides Current Regulators.” IEEE-/AS A n n u l Meeting Cotif: Rec., 1987,
load feedforward information from which the desired cur- pp. 196-201.
S. K . SUI and T . A . Lipo. “Field Oriented Control of an Induction
rent reference is derived. An optimal selection scheme for Machine in a High Frequency Link Power System.” in Poiivr Elec-
voltage vectors which is based on dc bus voltage is also tronics Specialist Conf. Rec., Apr. 1988.
used to minimize dc bus voltage ripple. This scheme has T . G . Habetler and D. M. Divan, “Rectifierhverter Reactive Com-
ponent Minimization,” in IEEE-/AS Annual Meerirlg Con6 Rec.,
been shown to have good transient performance, unity in- 1987, pp. 648-657.
put power factor, and low input current harmonics even R . Venkataramanan, “Sliding Mode Control of Power Converters.”
with very small values of dc link capacitance. Ph.D. thesis. California Institute of Technology, Pasadena, 1986.

ACKNOWLEDGMENT
This project has been supported through a scholarship
to Thomas Habetler from the Electro-Motive Division of
Thomas G . Habetler (S’82-M’83-S’85-M’89), for a photograph and bi-
General Motors Corporation and in part, through a grant ography. see this issue, p . 363.
from the General Electric Foundation.

REFERENCES
[ I ] D . M . Divan, “The resonant dc link inverter-A new concept in static
power conversion,” in IEEE-/AS Annual Meering Conk Rec., 1986. Deepakraj M. Divan (S‘78-M’78-S’82-M’83), for a photograph and hi-
pp. 640-647. ography, see this issue, p. 363.