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ARCHIVES OF ELECTRICAL ENGINEERING VOL. 66(3), pp. 533-545 (2017)
DOI 10.1515/aee-2017-0040
Three-phase diode rectifier with current modulator
in DC circuit based on multi-channel converter
MICHAL KRYSTKOWIAK, MICHAL GWÓŹDŹ
Poznan University of Technology
e-mail: Michal.Krystkowiak@put.poznan.pl
(Received: 16.11.2016, revised: 08.03.2017)
Abstract: In the paper the 3-phase power diode rectifier system with quasi-sinusoidal input
(power grid) current is presented. In order to benefit from it, the current modulation in the
DC circuit of a rectifier is used. The essential part of the current modulator is the wide-band
power electronics controlled current source based on a multi-channel converter. The control
algorithm of the current modulator respects impact of aliasing phenomena at system stabil-
ity by using the method proposed by the authors. The paper includes the rectifier system de-
scription and rules of operation of the current modulator. Also, some results of research on
both the rectifier simulation model and the laboratory prototype of a current modulator are
presented.
Key words: control methods of electrical systems, diode rectifier, generalized sampling
expansion, multi-channel converter, PWM
1. Introduction
The power rectifiers belong to a very widely utilized group of power electronics convert-
ers. Unfortunately, a standard diode and thyristor rectifier operation is the reason for distortion
of currents and voltages in a power grid. It has caused a very serious problem for the energetic
system for many years thus, improving quality of input (power) grid current of a rectifier is
often necessary. One of the solutions to obtain it is to apply a rectifier based on a passive or
active filter at the input [1-5]. The following way relies on the use of an active rectifier, which
is built with a transistor inverter switching at frequency, being several times higher than fre-
quency of voltage in a power grid [3, 6]. Other proposed solutions use the PFCs based on
multi-level converters [7, 8]. Also, to correct the waveform of the power grid current the recti-
fier built with diodes (or thyristors) being interconnected by means of coupled inductors can
be used [9]. Another solution of the rectifier system includes the voltage modulator in the DC
circuit of a rectifier. The modulator is built with thyristors and a special inductor with several
taps. This has already been considered before, e.g. in [10, 11].
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All the mentioned solutions are relatively expensive ones. In the paper another possible so-
lution, with the ability to shape the waveform of a power grid current towards the sinusoidal
waveform is presented (Fig. 1). This one is based on the current modulation in the DC circuit
[7, 8]. The rules of such system’s operation are described in details in e.g. [11]. Therefore,
only a brief description of this one is given here for reader’s convenience.
The system is built with two 6-pulse diode rectifiers, which are supplied by two 3-phase
transformers with “star-star” and “star-delta” connections. In this way 30 el. deg. phase shift
of transformers output voltages is obtained. Additionally, in the DC circuit the special power
electronics converter called the “current modulator” is placed. The modulator is responsible
for shaping rectifier input currents. In fact the “star-star” connected transformer is not neces-
sary for proper operation of the modulator although it helps to scale the voltage levels of both
rectifiers in the laboratory prototype of the system [10].
Fig. 1. General block diagram of 3-phase diode rectifier with current modulator in DC circuit
The rectifier system consists of the following blocks: two transformers connected in the
star-delta (TR1) and star-star (TR2) manner, two standard 6-pulse diode rectifiers (DRCT1
and DRCT2) with DC circuits connected in parallel, the active (transistor) rectifier (TRCT)
and the current modulator block (CMB). The CMB is connected to the DC circuit across the
pulse transformer (PT). Rectifiers power a common load, expressed by L
Z. Similarly like the
TR2 transformer, the transistor rectifier is not necessary in the rectifier circuitry too. Its role
depends on control of the voltage level in the DC link of the modulator only – without any
impact on the modulator output current [10].
The presented solution of the rectifier system makes it possible to improve the quality of
rectifier power grid currents iA, iB and iC. That is to say, waveforms of these are close to the
sinusoidal shape. The current modulator works as a power electronics controlled current
source being connected to the DC circuit of rectifiers through a wide-band pulse transformer
with two taps on a primary site. The iCM output current of the modulator is added (with a sign
of “plus” or “minus”), with the aid of the PT, to the output current iD1 and iD2 of each rectifier
in terms of the following equations:
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2112 )( NiNii CMDD =− , )(
2
1
1
2
1CMLD i
N
N
ii −= , )(
2
1
1
2
2CMLD i
N
N
ii += , (1)
where: N1/N2 – the PT windings turn ratio, L
i – the load current and CM
i – the CMB output
current.
This is the way in which waveforms of input currents of two rectifiers are modified. In
consequence, resultant waveforms of power grid currents of the rectifier system (iA, iB, and iC)
are modified as well. The power of each, the active rectifier and the current modulator is only
2÷3% of the total power at the output of the DC circuit. This is a great advantage of this idea
of the rectifier system.
In order to obtain the sinusoidal shape of the power grid current, the modulator current iCM
should satisfy the following equation [11]:
()
() ( )
()
2
1
6
π
ω6cosω6cos
6
π
ω6cosω6cosωcos
π
6
N
N
tktk
tktkt
Iti
k
gg
k
ggg
LCM ∑
∑
⎥
⎦
⎤
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛++−
⎥
⎦
⎤
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛++−
=. (2)
However, the complex current form of (2) can be replaced by a current with triangular
shape with only a small deterioration of the THD factor of a rectifier input current [11]. The
fundamental frequency of the modulator current has to be equal to 6-times of power grid fre-
quency, i.e. 300 Hz:
()
()
(
)
(
)
⎥
⎦
⎤
⎢
⎣
⎡−+−= ...
t t
tIti gg
gLCM 22 5
ω65sin
3
ω63sin
ω6sin
π
4, (3)
where Tg is the fundamental frequency of the voltage in the power grid.
With such replacement, assuming the power grid is a symmetrical one, a THD factor of
a power grid current is equal to about 1% [11]. Thus, the current modulator has to include in
its own structure a wide-band power electronics controlled current source which would be able
to match precisely an output (modulator) current in the reference signal. This feature of the
modulator is essential for proper working of the rectifier system because it determines directly
the quality of its input current.
Initially, in the current modulator the standard (i.e. one-channel) inverter has been imple-
mented. This one has been a part of the laboratory prototype of rectifier system [10]. The
presented article is focused mainly on a wide-band power electronics controlled current source
based on a multi-channel (interleaved) converter, being utilized in a current modulator.
The whole text is divided into 6 sections. The first one deals with a general conception of
rectifier construction. The second one shows a basic description of the wide-band power elec-
tronics controlled current source based on a multi-channel converter. In the third section, an
issue of controlled source stability is considered. The fourth section presents mainly, among
other items, a simulation model of the rectifier system. In the fifth section selected results of
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research on a laboratory prototype of the current modulator are presented. The last part is dedi-
cated to conclusions.
2. Wide-band power electronics controlled current source
Dynamic changes of parameters of both energy source and load are the reasons for de-
creasing exactitude of output signals towards reference signals. In order to improve these
parameters more advanced solutions of power electronics converters are often necessary. They
can be exemplified by a wide-band power electronics voltage-controlled voltage source
(VCVS) or voltage-controlled current source (VCCS). Such a converter should match an out-
put signal precisely in a reference waveform so that both a modified electrical structure of
a converter and an effective control algorithm are necessary. It has many applications in power
electronics equipment.
In Fig. 2 a general structure of a VCCS is shown. It is based on the conception of a multi-
channel converter where a iL total output current is proportional to the sum of currents iL,i:
i = 0, 1,..., M-1 in individual channels of a converter.
The VCCS is a system which works in a closed, current type, negative feedback loop. In-
verters in an execution block of the VCCS are controlled in PWM mode with the constant
value of carrier frequency [3, 12, 13]. One of the fundamental blocks of the VCCS is a passive
low-pass filter at the output of power stage consisting of a set of connected in parallel induc-
tors. This filter has two basic tasks to do, namely it obtains the suitable value of the output
impedance of a converter and minimizes magnitude of PWM carrier components in the output
current, making it possible for the converter to meet requirements of EMC.
Fig. 2. Block diagram of VCCS
The general structure of the VCCS is based on two modules, the control module (CM) and
the execution module (EM). The control module includes the following internal blocks:
– adder (A), producing the error signal fbreferr uuu −= ,
– regulator of the output current with the gain factor of k0,
– M-order multi-dimensional sample-and-hold system (MSHS) consisting of M connected in
parallel sample-and-hold amplifiers.
The execution module consists of:
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– M-connected in parallel half-bridge type inverters,
– output filter (LEM,i: i = 0.1,..., M-1),
– current transducer (CT), producing the feedback voltage ufb, being proportional to the
VCCS output current iL.
The EM is loaded by the ZL impedance and LLL RLjZ += ω. The VCCS is controlled by
the reference voltage uref at the input of the CM.
The sampling moments and PWM carrier signals in individual channels of the converter
are shifted with each other by Ts/M, where Ts is a master sampling period. As a result, from
the point of view of system stability, the converter “transfer function” is preferably modified
[12]. Thanks to this, the regulator gain in the VCCS control system can be increased compared
to the one-channel converter. It has a very positive impact on the performance of the VCCS
because a control algorithm allows for more accurate match of the VCCS output current in
a reference one.
In this particular case of operation of the VCCS in a rectifier system its structure has been
slightly modified. It has been necessary due to the way of including the pulse transformer in
this system based on a virtual ground. As a result, the 2 x 2-channel modified inverter in the
EM of the VCCS has been used. The block scheme of the VCCS is presented in Fig. 3.
Fig. 3. Block diagram of VCCS for driving pulse transformer
The modulation used in the pulse modulators is two-sided and asymmetric. The load of the
VCCS – the pulse transformer – is included in the circuit in a differential manner. That is to
say, the voltage at the output of the ,x0
INV inverters pair has to be in anti-phase with respect
to the voltage at the output of the x,1
INV pair. Because of M = 2, sampling moments in the
1
SHA and 0
SHA should be shifted with each other by a Ts/2 time.
3. Stability issue of a controlled current source
One of the most important aspects of the current modulator work is its stability. This anal-
ysis will include an important factor that occurs in the operation of a real system, which usual-
ly is not respected. This is an aliasing phenomenon – characteristic for sampled-data systems.
The mathematical model of a control system of a multi-channel converter for the stability
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analysis has been proposed e.g. in [12]. This description is based on the extension of the Whit-
taker-Kotelnikov-Shannon (WKS) sampling theory – the Generalized Sampling Expansion
(GSE) – being formulated by Papoulis [14].
Assume signal )()( 2ℜ∈ Ltx , the space of the square-integrable functions, where ℜ is
a real number domain, and its Fourier transform )ω(jX exists. Assume also that sampling of
x(t) is uniform and ideal (i.e. with utilization of Dirac series:
()
max
ω
π
:δ≤−
∑s
n
sTnTt ,
where Ts is the sampling period, and y(t) is the signal at output of a sample-and-hold amplifier
(SHA) as a 0-order extrapolator of a sampled signal. The relationship between ω)( jX and
ω)( jY apart a static (time invariant) component, also contains a dynamic (time variant) com-
ponent, i.e.
∑
∞
−∞=⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
ns
T
njX π2
ω,
so the formal transfer function of the SHA does not exist. The dynamic component is also
related to the aliasing effects, by which the high frequency poles are folded back into lower
frequencies. Although respecting the static part only of equivalent transfer function of SHA
gives, in most cases of a system stability analysis, satisfying results, the crucial knowledge is
that the aliasing mechanism can cause loss of stabilization at critical frequencies [12, 15, 16].
From the point of view of the GSE, general setting is that a signal x(t) is processed by a linear
multi-dimensional sampling system (MSS). Suppose now that x(t) is a common input to M
sampling systems and each individual sampling system is a sub-system of the MSS – Fig. 4.
Fig. 4. MSS with single ZOH block as signal extrapolator (E)
Each sub-system samples at a rate of 1/M times of the Nyquist rate. Assuming individual
delay
10,1,...,τ−== M:i
M
T
is
i,
the x(t) can be retrieved from its samples x*(t) by using the inverse Fourier transform formula:
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()
⎪
⎭
⎪
⎬
⎫
⎩
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛+= ∫
∑∑
−
⎥
⎦
⎤
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛+−
∞
−∞=
−
=
dω
ω2
1max
max
ω
ω
ω
1
0
max
s
T
M
i
ntj
n
M
i
seT
M
i
nxtx . (4)
The overall sampling rate still satisfies the Nyquist criterion [14]. The MSS, being the con-
sequence of GSE, makes it possible to reduce the required sampling frequency to 1/M, com-
paring to one-dimensional sampling system (WKS sampling theorem), working at the same
sampling rate. Considering the system from the other side, when sampling it at a rate of 1/Ts
its effective sampling rate is equal to M/Ts. A Nyquist band is now M-times extended. In the
case of a control system of a real multi-channel converter an essential difference in relation to
GSE assumption appears. It consists in the fact that each individual converter channel includes
a single SHA and individual channels output signals are summed at the output of the convert-
er. This modification of the original GSE concept is necessary due to the different position in
the system of a signals summing node. The second reason results from the limitation of dy-
namic parameters of real power electronics devices used in inverters. Thus, the VCCS control
system includes now the multidimensional sample-and-hold system (MSHS) instead of the
MSS [13]. The proposed small-signal (linear) model of the VCCS with the 2nd order MSHS,
being the object of further considerations, is shown in Fig. 5.
Fig. 5. Small-signal model of VCCS based on 2-channel converter
The proposed model is an IOM, SISO and LTI one and can be completely described in
terms of a continuous-time system with a transfer function
() ()
()
()
()
ω1
ω
ω
ω
ω
ref jK
jK
jU
jI
jG
al
L
+
== .
The system stability analysis takes advantage of the Nyquist criterion, e.g. [17]. Hence, the
characteristic equation 0ω)(1ω)( =+= jKjR al of the model, respecting (4), takes the form:
()
()
[]
()
()
0
ωω
ωω
1ω
max
max
MHS
0=
++−
−
+= ∑
−=
N
Nn LLEMs
s
CT RLLnj
njK
krjR , (5)
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where: ω)(
MHS jK is the static component in a relationship of the MSHS input and output
signals, rCT is the gain factor of CT, k0 is the gain of the regulator, s
ω is the sampling fre-
quency and Nmax is the number of respected aliases.
Research of the VCCS small-signal model have defined the static component in the rela-
tionship of input and output signals of the MSHS as follows:
()
()
⎪
⎪
⎩
⎪
⎪
⎨
⎧
−∈∧
−
==
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
−
−
4
1
4
1
ττ
2
1
2eωcos
2
ωSa
1e
2
ωSa
ω
10
2
ω
2
ω
MHS
, a
T
, a: MaT
T
: M
T
jK
s
T
j
s
s
T
j
s
s
s
. (6)
It is crucial for further utilization of a multi-channel inverter concept to determine the max-
imal regulator gain ( 0
k) to make the system stable. In order to determine this value it is neces-
sary to determine magnitude of ω)( jKal for the critical phase value
{}
πω)(arg ±=jKal . It
occurs at a frequency of
()
,...,, : n
T
n
s
n210
π
12ωπ,±±=+= .
Having solved (5) the maximal regulator gain can be expressed by the following formula:
⎪
⎪
⎩
⎪
⎪
⎨
⎧
±=∧=
+
=∧==
+
<
4122
021
max0
/ a : M
Tr
LL
a , M :M
Tr
LL
k
sCT
LEM
sCT
LEM
,. (7)
The maximal theoretical value of the regulator gain is 2-times over the gain of the regula-
tor in the case of a one-channel converter. It gives higher quality mapping of the VCCS output
current in the reference signal.
4. Simulation model of rectifier system
The fundamental block of a current modulator is the VCCS based on the 2-channel con-
verter that powers a pulse transformer [10]. Selected waveforms in the simulation model for
the target shape (i.e. triangular) of the reference signal are shown in the following figure are
shown in Fig. 6. These are also related to the two cases of an order of the MSHS i.e. M = 1
and M = 2. The PWM carrier frequency c
T=100 μs. The nominal magnitude value of the ref-
erence voltage ,n
Aref = 10 A, and ,BEM,,AEM,,BEM,,AEM, LLLL 1100 === = 1.2 mH. Parameters of the
simulation model are consistent with parameters of the real system, being described in Sec-
tion 5.
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a)
b)
Fig. 6. Waveforms of reference voltage uref, modulator current iCM, and error signal uerr for the case of
triangular shape of a reference signal and: a) M = 1, k0 = 35; b) M = 2, a = ¼; k0 = 40. Also currents in
individual channels of VCCS (iL, 0, A and iL, 0, B) are shown in Fig. b). Magnitude of the reference signal is
equal to nominal one
In the case of the two-channel VCCS shapes of the modulator current and reference signal
almost coincide, in contrast to the standard VCCS, where these signals clearly differ from
each other. For the VCCS in a two-channel version the first pulse modulation component in
the modulator current is at a frequency of 4/Tc, instead of 2/Tc and its magnitude is over
3-times lower compared to a one-channel converter.
A very adequate and reliable criterion of the quality of a converter output signal can be the
converter control error given by the following equation:
%
u
uu
%
u
u100100ε2
ref
2
fbref
2
ref
2
err
CTR
−
== . (8)
Assuming nominal conditions of the simulation model work, the value is as follows:
CTR
ε= 5.88%: M = 1, CTR
ε= 3.63%: M = 2. Thus, in sense of this criterion a two-channel
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VCCS makes it possible to improve the quality of the output current about 62% compared to
a one-channel VCCS solution.
The simulation model has been “powered” by 3 H230 V, 50 Hz grid. The nominal DC out-
put power of the rectifier system ,n
P
DC has been set at 6 kW. While the current modulator is
disabled the THD value for the power grid current (iA, iB and iC) is approximately equal to
12.5% in a 5 kHz band. This is the typical value for a standard 12-pulse rectifier. If the current
modulator is enabled, the deformations of power grid currents (Fig. 7) are now of an over
order less, i.e. THD = 1.35% while the output power is equal to the nominal value and
THD = 2.93% while the output power is equal to the 10% of this one.
a)
b)
Fig. 7. Waveforms of power grid currents iA, iB and iC in rectifier simulation model while current modu-
lator is enabled and: a) output power is equal to the nominal value, b) output power is equal to 10% of
nominal value
The impact of the number of converter channels on the quality of power grid currents can
be evaluated on the basis of the i
ε error. This one is related to the difference of a iA (iB, iC)
current and its 1st harmonics – similarly to the converter control error (8). A value of this error
is a function of both, the rectifier output power DC
P and M:
21:ε
DC,
DC , M
P
P
f
n
,M
i=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=.
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The graph of this function is shown in Fig. 8. Also, the relationship of 2
εi, and 1
εi, is pre-
sented.
The quality of the power grid current in the case of utilization of the two-channel converter
is increased about 10.8 ÷ 28.3% compared to the standard one, while its average value is equal
to 11.8%. In consequence, power loss in the rectifier system (mainly in transformers) can be
reduced by means of a two-channel converter solution.
Fig. 8. Curves of i
ε and reciprocal relationship of error functions (gi, 2/ gi, 1) while M = 1 and M = 2
5. Laboratory model of current modulator
The laboratory model of the part of a rectifier system has been investigated. The aim of the
study has been the validation of theoretical assumptions and research results of a rectifier
simulation model – mainly in the relation to the quality of the modulator current. The labora-
tory model has consisted of these blocks of the full rectifier circuitry which are on the grey
background in Fig. 1. Its diagram has been consistent with Fig. 3. The secondary winding of
the pulse transformer has been short circuited.
Basic technical parameters of the laboratory prototype are given below:
– DC link voltage in the multi-channel converter: 60 V,
– nominal magnitude of the modulator current: 10 A,
– inductance of the single coil in the VCCS: 1.15 ÷ 1.27 mH,
– signal sampling frequency in the control module: 20 kHz,
– PWM carrier frequency: 10 kHz.
The control module in the laboratory prototype has been based on the ALS-G3-1369 [18]
DSP evaluation board with Analog Devices ADSP-21369 SHARC® DSP and two
P3-5-550MFE LABINVERTERs [18] in the execution block.
Laboratory model tests have been carried out for the current modulator magnitude being in
the range of 10% ÷ 100% of the nominal one. Also, two cases of converter configuration have
been tested i.e. M = 1 and M = 2. In Fig. 9 and Fig. 10 exemplary waveforms in the laboratory
prototype are shown while the current modulator magnitude is equal to the nominal one. Assuming
nominal conditions of the laboratory model work, the value of the control error is as follows:
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CTR
ε≅ 9.8% while M = 1 and CTR
ε≅ 5.9% while M = 2. Thus, in sense of the control error
criterion the two-channel VCCS makes it possible to improve the quality of the output current
about 66% compared to a one-channel VCCS solution.
Fig. 9. Selected waveforms in laboratory prototype of current modulator: reference voltage,
modulator current, and error signal while M = 1
Fig. 10. Selected waveforms in laboratory prototype of current modulator: reference voltage, modulator
current, and error signal while M = 2; the current modulator magnitude is equal to the nominal one
6. Conclusions
In this study one of the ways leading to increasing the quality of power grid current of tra-
ditional diode rectifiers is presented. In order to obtain this a current modulator in the common
DC circuit of two 6-pulses rectifiers has been utilized. The modulator is based on a wide-band
power electronics controlled current source with a multi-channel converter. The control algo-
rithm respects aliasing effects, having the influence on the system stability, taking place in
sampled-data systems. They make it possible to maximize crucially effective gain of the regu-
lator and, in consequence, better matching of the modulator output current in the reference
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signal. As a result the total THD factor of the power grid current is lowered approximately
11% ÷ 28%, compared to the standard converter. It gives opportunity of decreasing power
losses in the rectifier power grid transformers. The solution of a rectifier is especially attrac-
tive in the case of a higher power of the load, since power of the current modulator is equal to
2.5% of the rectifier DC output power. Thus, higher complexity of the proposed converter
solution has only a minimal effect on the cost of the whole rectifier system.
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