High-Speed Current Vector Control of PWM Inverter
Minimizing Current Error at Every Sampling Point
Toshihiko Noguchi, Hirokazu Kodachi

Isamu Saito

Nagaoka University of Technology
1603-1 Kamitomioka, Nagaoka,
Niigata 940-2188, JAPANtnoguchi@vos.nagaokaut.ac.jp, kodachi@stn.nagaokaut.ac.jp

Toshiba Corporation
Information and Industrial Systems & Services Company
1-1-1 Sibaura, Minatoku, Tokyo 105-8001, JAPANisamu.saito@toshiba.co.jp

Abstract – This paper proposes a novel digital current control
technique of a PWM inverter. The key feature of this method is
minimization of a current error vector norm at every sampling
point with no predetermined current error tolerance, such as a
spatial circular area or hysteresis bands. In order to make thecurrent vector trajectory until the next sampling point as close
as possible to a predicted current command vector, one of the six
non-zero-voltage vectors is appropriately selected in the inverter.
A zero-voltage vector is inserted once a sampling period toadjust the current vector velocity and to move the current
vector to the closest position to the predicted current command
vector. Although manipulating variables in the current loop
during the sampling period are limited by the discrete voltage
vectors as described above, the proposed method achieves adead-beat response in a two-dimensional space without thepredetermined current error tolerance.
Also, on-line
identification of a load inductance is introduced to the system toovercome degradation of the response caused by the parametermismatch. Resultant data obtained from computer simulations
and experiments prove feasibility of the proposed technique.

vector norm at every sampling point with no predeterminedcurrent error tolerance, such as a spatial circular area orhysteresis bands. The proposed method is a kind of dead-beatcontrol. However, the proposed control is achieved in a
two-dimensional space, regarding all control variables as
instantaneous spatial vector quantities, while theconventional technique was based on the dead-beat control
with respect to time. Also, on-line identification of the load
parameter is introduced to the controller to solve problems
associated with a parameter mismatch. In this paper, atheoretical aspect of the proposed technique is described andseveral computer simulation and experimental results arepresented to show feasibility of the method.

I. INTRODUCTION
Three-phase current-controlled inverters have extensively
been used in many applications such as ac servomotor
controllers, switched ac-dc rectifiers, active filters and soforth [1][2]. Basically, these kinds of inverters have current
feedback loops to regulate phase currents detected with
Hall-effect CTs. The most common and conventional
techniques to regulate the currents are a proportional-integral(PI) element based scheme and a hysteresis comparator based
scheme. The former has an advantage over the latter in termsof generation of optimum pulsewidth-modulated (PWM)
patterns but has drawbacks in a current response and asteady-state error. The latter has both advantages anddisadvantages vice versa; hence it is rather difficult for thesetechniques to satisfy all requirements at the same time
without sacrificing any of the performance indexes describedabove. In order to overcome such dilemma, several proposals
have been made on the basis of sophisticated but complicatedapproaches. One of the approaches is a method using aspatial circular area as a current error tolerance, which can
restrict the current error vector within the predeterminedcircular area [3]. Another is a dead-beat control basedalgorithm, which achieves an excellent current response with
only one sampling time delay [4]. Also, other several papers
have reported modified hysteresis comparator based solutions,e.g., an adaptive hysteresis bandwidth approach and an
inverter switching sequence optimisation approach [5][6].
This paper proposes a novel digital current vector control
technique, which features minimization of a current error

0-7803-7474-6/02/$17.00 ©2002 IEEE

II. PRINCIPLE OF PROPOSED CURRENT VECTOR CONTROL
A. Current-Controlled PWM Inverter
Fig. 1 shows a schematic diagram of the current control
system discussed in this paper. It is assumed that a balanced
three-phase inductive load with a back electromotive force(e.m.f.) is connected to the inverter and that load resistance isnegligibly small. As shown in Fig. 2(a), the inverter can
generate six non-zero-voltage vectors v1 - v 6 and two
zero-voltage vectors v 0 and v 7 , according to its switchingstate S u , S v and S w . The inverter output voltage vector v n
is mathematically expressed as follows:v n (k ) = vn α (k ) + jvn β (k ),(1)
= 2 3 Vdc S u (k ) + S v (k )e j2π 3 + S w (k )e j4π 3[]where n = 0 - 7 and S u , S v , S w = 0 or 1 . As can be seen in (1),v1 - v 6 have constant magnitude, which is proportional to the
dc-bus voltage Vdc , and discrete directions as shown in Fig.
2(a), while v 0 and v 7 have no magnitude.
Since manipulated variables of this current controller are
discrete vectors, it is rather difficult to keep the current error
vector norm as small as possible during every switching stateof the inverter. A continuous time mathematical expressionof the system is as follows:
Sw

Sv

Su

1

1

1

iu

0

iv

V dc
0

0

iw

L
ve u

L

ve v

Fig. 1. Current-controlled PWM inverter and load with back e.m.f.

ve w

L

β

v n = (S u S v S w )i ( k + 1)

i v 2 ( k + 1)βi err (k )

∗

v6

v2
v3

v7
v0

v1

β

i err β = − i err α 2

v4

3
α

v 2 (k )v (k )4α1∗i (k )

i v e ( k + 1)v5

2i (k )v e (k )0

5

6

i err β = i err α 2
α

(a)
(b)
Fig. 2. Inverter output-voltage vectors and current vector trajectory.
(a) Voltage vectors of inverter. (b) Current vector trajectory.

i err α = 0
Fig. 4. Spatial sectors to determine current error vector direction.

β

TABLE I SELECTION OF NON-ZERO-VOLTAGE VECTORi ∗ ( k + 1)

i v 2 ( k + 1)

i v 6 ( k + 1)

Sectori err (k )i ∗ (k )i v 3 ( k + 1)

i v e ( k + 1)
v e (k )i v 4 ( k + 1)
i (k )

0i v 1 ( k + 1)

αi v 5 ( k + 1)

Fig. 3. Possible reachable destinations of current vector.

v n + v e = L di dt .
(2)
In the above equation, v e is a back e.m.f. vector of the load.
Therefore, the current vector i can be expressed as anintegral of sum of v n and v e . Although v e is a rotating
voltage vector with specific frequency and amplitude, it canbe regarded as a constant vector during a sampling period
because frequency of v e is much lower than a sampling rate.
Fig. 2(b) shows an example of a current vector trajectoryduring a sampling period Ts . As can be seen in this figure, if
v 2 is selected throughout Ts , i proceeds in the direction ofv = v 2 + v e at a constant velocity, which is determined by theswitching state, the dc-bus voltage, and the amplitude and thephase of the back e.m.f.
B. Proposed Current Control Algorithm
In many applications of three-phase systems, sinusoidal
current waveforms are generally required; thus the inverter
discussed in this paper is operated with three-phase sinusoidal
current commands given. On this assumption, the proposedcurrent control is executed according to the proceduredescribed below.
1) Acquisition of current command and feedback vectors

Vector to
selectConditions

1

ierr α ≥ 0−ierr α 2 < ierr β ≤ i err α 2v42ierr α ≥ 0ierr β > ierr α 2v63ierr α < 0ierr β ≥ − ierr α 2v24ierr α < 0ierr α 2 ≤ ierr β < − ierr α 2v35ierr α < 0ierr β < ierr α 2v1

6ierr α ≥ 0ierr β ≤ − ierr α 2

v5

It is assumed that the following current vector is sampled ata sampling point k :i (k ) = iα (k ) + ji β (k ).
(3)
= 2 3 iu (k ) + iv (k )e j2π 3 + iw (k )e j4π 3[]

On the other hand, the current command vector at the next
sampling point can be predicted as follows:i ∗ (k + 1) = iα∗ (k + 1) + jiβ∗ (k + 1)
, (4)= 2 3 iu∗ (k + 1) + iv∗ (k + 1)e j2π 3 + iw∗ (k + 1)e j4π 3[]where the three-phase current commands are calculated on
the assumption that they are sinusoidal, e.g.,iu∗ (k + 1) = I ∗cosω ∗ (t + TS )
 ∗
∗
∗
.
iv (k + 1) = I cos ω (t + TS ) − 2π 3∗∗∗iw (k + 1) = I cos ω (t + TS ) − 4π 32) Detection of load back e.m.f. vector
It is necessary to select an optimum non-zero-voltage
vector that allows the current vector i (k ) to approach thepredicted current command vector i ∗ (k + 1) the most closely.
However, the back e.m.f. vector of the load given by (5)prevents i (k ) from proceeding in the direction of theselected v n (k ) .
v e (k ) = ve α (k ) + jve β (k )(5)
= 2 3 v e u ( k ) + v e v ( k ) e j 2π 3 + v e w ( k ) e j 4π 3

[
[

[

]
]

]

This back e.m.f. information is usually obtained from sensors,e.g., a rotor position sensor of an ac servomotor andpower-source voltage sensors of a switched ac-dc rectifier.
As described previously, movement of i (k ) is determined by

β

β
∗

i ( k + 1)

i v 2 ( k + 1)∗

i (k )i dest (k )i dest ( k + 1)

i (k )rorSc∗i (k )

d
di
m

α

t

em

en

0

la c

ce

sp

la

α

0

i ∗ ( k − 1)i ( k − 1)dei (k )spv e (k )dii v e ( k + 1)

ee

le d

oc

du

Pr

he

2

Er

v (k ) v ( k )entFig. 5. Optimum destination of current vector.

v (k )v e (k )v (k )

Fig. 8. Principle of inductance identification.
D irection of current vector
N on-zero-voltage vectorv 1− 6v0, 7v 1− 6

T1 − 6 / 2

T0 , 7

T1 − 6 / 2

Inverter output
Zero-voltage vector

Ts
Fig. 6. Switching sequence in one sampling period.
βi.e., T0 , 7 = 0 and T1−6 = Ts , v 2 (k ) allows i (k ) to reach thedestination of i v 2 (k + 1) . On the other hand, if the sampling
period is occupied by T0, 7 , i.e. T1− 6 = 0 and T0 , 7 = Ts , thedestination is i ve (k + 1) , which is determined by v e (k ) .
Therefore, a combination of v 2 (k ) and v 0 (k ) that satisfies(7) allows the destination of the current vector after Ts to be
on a straight line between i v 2 (k + 1) and i ve (k + 1) .
Equation (2) can easily be transformed to a discrete-time
expression as follows:i dest (k + 1) = i (k ) + [v n (k ) + v e (k )]Ts L̂
= i (k ) + v e (k )Ts L̂ + v1− 6 (k )T1− 6 L̂ .

∗i ( k + 1)
i dest ( k + 1)

v (k )
T1 − 62v e (k )v (k )

T0 , 7∗i (k )

T1 − 620i (k )
αFig. 7. Scheduled trajectory of current vector.

the sum of v n (k ) and v e (k ) shown by (6). If the inverter
output is one of v 0 and v 7 , the direction and the velocity ofi (k ) are uncontrollable because v e (k ) solely governs
movement of i (k ) . This situation is inherently unavoidable
because applying either v 0 or v 7 is indispensable to control
the velocity of i (k ) .
v (k ) = v n (k ) + v e (k ) = vα (k ) + jv β (k )(6)
= v n α (k ) + v e α (k ) + j[v n β (k ) + v e β (k )]3) Selection of optimum non-zero-voltage vector
Every sampling period Ts is divided into two durations,
i.e., duration T1− 6 of a non-zero-voltage vector and duration
T0, 7 of a zero-voltage vector; hence Ts can be expressed as
Ts = T1− 6 + T0 , 7 .
(7)As illustrated in Fig. 2(b), if the non-zero-voltage vectorv 2 (k ) is applied to the load throughout a sampling period,(8)= i ve (k + 1) + v1− 6 (k )T1− 6 L̂
The above equation represents that the destination of thecurrent vector i dest (k + 1) is determined only by v1− 6 (k )because i ve (k + 1) is regarded as a constant vector during thesampling period. Therefore, possible reachable destinationsof the current vector can be plotted as shown in Fig. 3,
depending on selection of the non-zero-voltage vectors.
From the coordinates of i ve (k + 1) , an optimum
non-zero-voltage vector should uniquely be selected, which
produces a current vector trajectory closest to the current
command vector. For example, if the current error vector
i err (k ) = i ∗ (k + 1) − i ve (k + 1) is closer to i v 2 (k + 1) thani v 6 (k + 1) as illustrated in Fig. 3, v 2 (k ) is the mostappropriate non-zero-voltage vector that makes the current
vector trajectory the closest to i ∗ (k + 1) . Fig. 4 and TABLE I
indicates selection manner of the optimum non-zero-voltage
vectors. As shown in Fig. 4, a two-dimensional reference
frame of which origin is placed at i ve (k + 1) is divided into
six sectors to detect the direction of i err (k ) . The sector wherei err (k ) belongs can easily be detected by evaluating ierr α (k )and ierr β (k ) , and the only one appropriate non-zero-voltage
vector is selected according to the sector where i err (k )belongs as described in TABLE I.
4) Calculation of optimum destination of current vector
After selecting a non-zero-voltage vector, e.g., v 2 (k ) , on
the basis of Fig. 4 and TABLE I, an optimum destination of
the current vector, which is the closest to the predicted current
command vector i ∗ (k + 1) , is calculated. The destination*iα , iα
4 (A/div)
0*iα , iα2 (A/div)
0vuv
200 (V/div)0^L
20 (mH/div)veu
100 (V/div)00Time 10 (ms/div)(a)Time 5 (ms/div)

(a)

∆ iβ

∆ iβ

1 (A/div)

1 (A/div)

0

0

∆iβ

1 (A/div)
0∆iα

1 (A/div)
0 ∆i

0

0 ∆i

α

(b)

α

1 (A/div)

1 (A/div)(b)(c)
Fig. 9. Simulation result of inductance identification and current error vectorloci.
(a) Waveforms of currents and estimated inductance. (b) Current-error vector
locus at initial condition. (c) Current-error vector locus after convergence of
estimated inductance.

i dest (k + 1) is a spatial point that the current vector isscheduled to reach in one sampling period and is a foot of
perpendicular from i ∗ (k + 1) on the straight line between
i v 2 (k + 1) and i ve (k + 1) as illustrated in Fig. 5. The
following expressions give the coordinates of i dest (k + 1) :i dest (k + 1) = i dest α (k + 1) + jidest β (k + 1) ,
(9)
where
i dest α (k + 1)∗∗2
= vα (k )iα (k + 1) + vα (k )v β (k ) i β (k + 1) − ive β (k + 1) ∗
∗ + v β2 (k )ive α (k + 1) vα2 (k ) + v β2 (k )i(k + 1)
 dest β = v β2 (k )i β∗ (k + 1) + vα (k )v β (k ) iα∗ (k + 1) − i ve α (k + 1) ∗
∗ + vα2 (k )ive β (k + 1) vα2 (k ) + v β2 (k ) .


{

[

{

[

}[

]
]

}[

]
]

The current vector reaching i dest (k + 1) makes a current error
vector norm minimum at every sampling point.
5) Adjustment of switching sequence and duration
To make the current vector reach i dest (k + 1) from i (k ) ,the durations T1− 6 and T0 , 7 are calculated as follows:[i (k + 1) − ive α (k + 1)]2 + [idest β (k + 1) − ive β (k + 1)]2
T1−6 = dest αL̂
2 3Vdcve α ( k )[idest β (k + 1) − iβ ( k )]− ve β ( k )[idest α (k + 1) − iα (k )]L̂=vα (k )ve β (k ) + vβ ( k )ve α ( k )T0 , 7 = Ts − T1−6

(10)
The inverter output is switched twice in one sampling periodih
10 (dB/div)

Frequency 5 (kHz/div)(c)Fig. 10. Simulation result of proposed current controller.
(a) Waveforms of currents, inverter line-to-line voltage and load back e.m.f.
(b) Current-error vector locus. (c) Frequency spectra of current.

and a zero-voltage vector is inserted once a period as shownin Fig. 6. Adopting this switching sequence, the current
vector proceeds from i (k ) to i dest (k + 1) along the scheduled
trajectory shown in Fig. 7. The calculation of the durations isaffected by variations of the inverter dc-bus voltage and theinductance value set in the controller. The former can becompensated by detecting the real voltage, while the lattermust be estimated with an identification mechanism
described below.
C. On-Line Identification of Load Inductance
As shown in Fig. 8, identification of the load inductance isperformed by evaluating an error between a scheduled
destination i dest (k ) and an actually reached point i (k ) .
Since the error depends on an inductance mismatch between
the controller and the load, the inductance can adaptively be
estimated by the following equation, which is based on an
integral algorithm with a gain K I :L̂ = K I ∑ [i dest (k ) − i (k )] .
(11)
The above equation represents that the estimated inductanceconverges to a true value when the error between i dest (k ) andi (k ) becomes zero.

iu

iv

*

iα , iα2 (A/div)

L
ve u

V dc

ve w

L

ve v
L

iw

G ate
signals
6bit
CPLD(Lockout)
A/D
converterve u , ve w

14bit

D SP
T M S320C6711

12bit

^
L
20 (mH/div)0

A/D
converters

A/D
converters

3bit

16bit

0

}

D /A
converters

AnalogmonitorFig. 11. Schematic diagram of proposed current controller.

Time 10 (ms/div)
(a)

∆ iβ

∆ iβ

1 (A/div)

1 (A/div)

0

0

0 ∆i

α

III. COMPUTER SIMULATIONS AND THEIR RESULTS
A. Test Conditions of Computer Simulation
Several computer simulations have been conducted to
examine basic controllability of the proposed method under
the condition of R = 1.0 (Ω) , L = 50 (mH) , Ts = 100 ( µs) ,and v e = 160 (V peak, line to neutral) . Execution delay timeof the proposed algorithm is assumed to be 10 ( µs) taking
actual implementation into account. The inverter’s dc-bus
voltage is kept constant at Vdc = 350 (V) . A lockout circuit
function to prevent a short circuit across the dc-bus is
introduced in the simulations and its lockout-time is 2.0 ( µs) .
The three-phase current commands are obtained from thedetected load back e.m.f. and are given to the controller tomake the currents in phase with the back e.m.f.
B. Simulation Results of Inductance Identification
Fig. 9(a) shows a characteristic of the inductanceidentification. The identification was started from an initialvalue of 10 (mH) and the estimated value asymptoticallyconverged to 53 (mH) approximately after 70 (ms) . It isfound that the current controllability is remarkably improved
as the estimated value converges to the true one in theidentification process. This improvement can be verified by
the current error vector loci shown in Figs. 9(b) and 9(c).
C. Simulation Results of Current Control
Fig. 10 shows several waveforms of the proposed current
controller, where a step response of the current amplitude isincluded. As can be seen in Fig. 10(a), the phase current
follows its current command with a minute error regardless ofa sudden change in the current amplitude, which can also be
proven by the current error locus shown in Fig. 10(b). The
locus indicates a small hexagonal-star-shaped distribution of
the current error around the origin although the proposed

1 (A/div)0 ∆iα1 (A/div)(b)(c)
Fig. 12. Experimental result of inductance identification and current-error
vector loci.
(a) Waveforms of currents and estimated inductance. (b) Current-error vector
locus at initial condition. (c) Current-error vector locus after convergence of
estimated inductance.

controller does not have a predetermined current error
tolerance. A switching frequency of the inverter is 4 (kHz) ,which is specified by the sampling period, i.e., Ts = 100 ( µs) .
As shown in Fig. 10(c), however, the current waveform has
conspicuous harmonics at 10 (kHz) and its multiples. This
profile of the frequency spectra is an advantage to reduce thecurrent harmonics when low-speed switching devices such as
GTOs are employed in the inverter.
IV. EXPERIMENTAL SET-UP AND ITS RESULTS
A. Experimental System Set-up
Fig. 11 shows a system configuration of an experimental
set-up. The controller consists of full digital hardware
employing a digital signal processor (DSP; TMS320C6711)
and high-sampling-rate A/D converters (four LTC1414s andone LTC1604). Feedback quantities in this set-up are thephase currents, the load back e.m.f. and the dc-bus voltage ofthe inverter. The sampling period is set at 100 ( µs) using one
of internal timers in the DSP. Also, another timer is utilized
to control timings of the inverter output-voltage vectors,
which makes it possible to generate accurate PWM patterns.
As described previously, it takes 10 ( µs) for the DSP tocomplete the whole calculation needed for the proposedcurrent control. Resultant switching state of the inverter isdirectly generated by the DSP as 3-bit logic signals and isdelivered to an IGBT-based power circuit via a digital lockout
circuit integrated with a complex programmable logic device(CPLD), of which lockout-time is 2.0 ( µs) .

*iα , iα
4 (A/div)0

around the origin, which means that the steady-state phaseand amplitude errors are minute. Fig. 13(c) shows frequency
spectra of the phase current measured in the experiment and aconspicuous peak can be found at 10 (kHz) , of which result
coincides with the frequency spectra profile obtained in thesimulation. Also, there can be seen no irregular pulses in
PWM patterns of the inverter, which is excellent as well as
those of sub-harmonic modulation techniques or space vector
modulation techniques. However, the proposed methodfeatures dispersed frequency spectra of the phase current
regardless of the constant switching frequency of 4 (kHz) .

vuv
200 (V/div)0veu
100 (V/div)0Time 5 (ms/div)

(a)

∆iβ

1 (A/div)
0∆iα

1 (A/div)
0

(b)

ih
10 (dB/div)

Frequency 5 (kHz/div)(c)Fig. 13. Experimental result of proposed current controller.
(a) Waveforms of currents, inverter line-to-line voltage and load back e.m.f.
(b) Current-error vector locus. (c) Frequency spectra of current.

B. Experimental Results of Inductance Identification
Fig. 12 shows an experimental result of the inductanceidentification. Reactors used in the experiment have similarparameter values with those used in the simulation tests. It isobserved that the identification is completed approximately in
70 (ms) and that the converged value of the inductance is
61 (mH) . Also, the current error is diminished around theorigin forming a hexagonal star shape as the estimatedinductance converges to the true value as shown in Fig. 12(c).
It can be found that the current controllability is also
improved as the estimated inductance gradually approaches
to the true value.
C. Experimental Results of Current Control
Experimental tests have been conducted under same
conditions as those of the simulations. Fig. 13 shows theexperimental results.
As can be seen in Fig. 13(a), excellent controllability of thephase current is observed and this fact can be confirmed by
the current error vector locus shown in Fig. 13(b). From thisfigure, it is found that the maximum current error isapproximately 0.3 (A) and the current error concentrates

V. CONCLUSION
This paper described a novel digital current control strategy,
which makes it possible to minimize a current error vector
norm at every sampling point. The most unique point of the
proposed method is to restrict automatically the current error
vector within a minute spatial area with no predetermined
error tolerance, e.g., a hysteresis band for each phase current
or a two-dimensional tolerant area such as a circle. The
method achieves optimum PWM patterns, a quick current
response, and a small steady-state current error at the sametime. Also, it has been clarified that the most conspicuous
spectrum of the current harmonics can be observed at afrequency of 2.5-times of the inverter switching frequency.
Not only a basic operation of the proposed technique has been
examined with computer simulations, but also excellent
performances have been confirmed with a prototype throughexperimental tests.
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[2] M. P. Kazmierkowski, and L. Malesani, “Current
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[3] J. Holtz, and S. Stadtfeld, “A Predictive Controller for
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Switched Voltage Source,” Conference Record of 1983
IPEC-Tokyo, 1665-1675.
[4] T. Kawabata, T. Miyashita, and Y. Yamamoto, “DeadBeat Control of Three-Phase PWM Inverter,” IEEE
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[5] B. K. Bose, “An Adaptive Hysteresis-Band Current
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5, 402-408, 1990.
[6] K. Ohyama, H. Nakazawa, M. Daijou, T. Yoshida, and
Y. Tsunehiro, “A Novel Current Control Technique for
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