Pulse modulation by switches of high frequency of alternating current
circuits and application of this method
MARCIS PRIEDITIS, IVARS RANKIS, AGRIS TREIMANIS
Institute of IEEI
Riga Technical University
Liepaja, Liedaga iela 3
LATVIJA
Abstract: - This paper is devoted to analysis of AC pulse modulation systems and calculation method explanation
for such systems. Analysis of load filter parameters are performed by regressive estimation of filtering system
parameters. The utility of such system has been reviewed in the context of the application of 3-phase induction
motor.
Key-Words: - pulse modulation; AC; motor control; filtering; regressive estimation; unipolar modulation.
Received: May 27, 2021. Revised: March 18, 2022. Accepted: April 16, 2022. Published: May 23, 2022.
1 Introduction
Evaluating the available publications on the
regulation of processes in DC and AC circuits [1],
one can observe that in the main research, there are
devices with elements and circuits of DC current. In
this case, the regulation of processes is performed by
the method of pulse regulation using pulse-width
modulation PWM and switching device frequencies
up to the several hundred kilohertz [1-5]. Devices
with alternating current circuits are either supplied
with intermediate circuits and DC nodes in which
high-frequency pulse regulation is carried out, or
controlled by electronic devices operating at the
frequency of the supplying alternating current, i.e.,
the method of changing the duration of current flow
through the circuit during each half-period of the
supplying alternating current is used. In this case,
single-position controlled semiconductor elements -
thyristors or triacs, controlled only by the moment of
switching, are used. Controllability is achieved by
changing the unlocking interval in the half-cycle of
the supply AC voltage, i.e., the pulse-phase method
of AC regulation is used [6, 7]. Many theoretical
studies have been devoted to the development and
study of such control systems for processes in
alternating current circuits [1]. As a result, many
practically applied systems and devices have been
created [1, 8].
The use of the pulse-phase method of AC regulation
leads to a significant deterioration in the quality of
power supply as well as in load. Firstly, the
waveform of the currents in both nodes is far from
sinusoidal, which is reflected in the increased values
of the harmonic current distortion indicators of the
circuits under current conductivity and the current
distortion indicator Iv=I(1)/I. Ш. Secondly, the
method itself is based on the introduction of a forced
phase shift between fundamental waves of currents
and supply voltages, which is reflected in low values
of the cosϕ(1) of the shift angle ϕ(1) of the
fundamental harmonic current relative to the supply
alternating current wave. As a result, the power factor
- the ratio of the real power P and the apparent power
S, defined as the product of the multiplication of Iv
and cosϕ(1) which is evaluated to obtain small
values, i.e., the power quality is poor. The indicated
disadvantages - strong distortion of the current shape
and forced shift of the fundamental current wave are
not easy surmountable, which means that there are no
great opportunities to improve the state in the
direction of improving the power quality without
extra hardware.
Considering the successful and multifunctional
application of pulse regulation of DC circuits [9], it
could be assumed that effective use of this method of
current regulation is also possible to be applied in AC
circuits, either. Of course, all electronic switches
used in AC circuits must be of alternating current,
i.e., allow carrying out regulation regardless of the
direction of the current at the moment of switching
the AC circuit. This requirement of course
complicates the realization of AC switches.
Basically, technical solutions are reduced either to
the creation of circuits with two series-connected
transistors of opposite polarity of conductivity [10] or
to the inclusion in the diagonal of a single-phase
diode bridge of one commutable transistor [11]. The
use of transistor alternating current switches allows
high-frequency modulation of processes in the AC
circuits, which in turn allows synthesizing both new
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fundamental waves of alternating voltages of system
elements and currents of the same type. The use of
transistor alternating current switches allows high-
frequency modulation of processes in the AC circuits,
which in turn allows, moreover, these fundamental
waves of electrical signals can be both with an
adjustable amplitude and an adjustable phase shift
with respect to the voltage (current) wave.
At present, AC systems with high-frequency
modulated switches in the scientific and technical
literature aspects, have not been concerned enough.
Therefore, there is no great clarity both with the use
of methods of analysis and calculation of processes,
and with multifunctional application. Taking into
account the possibility of generating new
fundamental waves of voltages and currents, the
analysis can be built on the basis of generating the
effective (RMS) values of modulated currents and
voltages considering their interphase shifts, i.e., use
the method of vector analysis, as shown in [12].
However, for a reasonable application of the method,
it is necessary to accumulate many verified analyzed
solutions, which seems to be the purpose of this
research work. Offering filter parameter assessment
methods could give great benefit in other applications
[13, 14] as well as they could be used on smart grid
development technologies [15].
2 Algorithm of unipolar modulation of
alternating current
The load with this method during one period of
modulation Tm (electronic state on-off) is connected
periodically during the half-period of the supplying
alternating current u1 to the same pole of the supply
voltage, but during the off-interval of the switch the
load voltage is at the zero level (i.e. load is
shortened). To the load during the half-period of the
supply voltage, there are two values u1 and 0 across
the load. For this, in each half-cycle, two states
should be used, one connects the load to the source,
and the other shortens the voltage of the load (see the
schematic in Fig. 1).
In the circuit Fig. 1, it is possible to obtain a
modulated fundamental harmonic of the load voltage
in a directly proportional (concurrent) form with
respect to the supply voltage wave, and in an inverted
form. For the first, one of the switches of the direct
connecting circuit S1/S2 must always be turned on
over half-cycle of supply voltage. To implement the
second option, one switch must always be turned on
in the reverse circuit S4, S3. For example, to
implement the first option, that is the switch S2, but
switch S1 is providing on-duty operation Dcon; to
implement the second – the switch S3 could stay
constantly turned-on, but switch S4 provides on-duty
operation DR for the second option. Zeroing of the
load voltage is performed through a permanently
switched on switch from one circuit and a
complementary (1-D) modulated switch of another
switch control circuit. For example, in the
coordinated control mode, when S1 is locked, and S2
remains permanently on, (1-D)con interval is carried
out through S3, which complements the S1. Thus, for
coordinated control mode Dcon corresponds to S1 and
S2, and (1-D)con corresponds to S2 and S3. In the
reverse case of control DR corresponds to S4 and S3,
and (1-D)R – to the mode S4 and S2. The latter is
complementary to the S4 switch.
Fig. 1. Scheme for direct proportional (concurrent con) and inverse I
proportional modulation of the load voltage.
To balance the loads on the switches, when the
polarity of the supply voltage is reversed, the
connection can be made to the other pole of the
supply voltage. Thus, two more switches are required
to implement the second half cycle. Later, as appear
in Figure 1, switches S1 and S3 carry out control at
the one polarity of the supply voltage, switches S2
and S4 – performing control in opposite polarity of
the voltage of one-phase alternating supply voltage.
If the supply voltage is also negative, switch S2 is
performing the modulation (D interval), whilst S1
staying constantly on, and the load is short-circuited
with the switches S4 (see the switching table).
Table I Switching table.
nl
>0
<0
uld(1)
>0
0
<0
>0
0
<0
S1
on
off
off
on
on
on
S2
on
on
on
off
off
on
S3
off
on
on
on
off
Off
S4
off
off
on
on
on
Off
D
1-D
1-D
D
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The fundamental harmonic of the modulated load
voltage could be described as:
where D٠U1m is the amplitude of the fundamental,
and dividing this indicator by √2 is the RMS value of
the fundamental.
Examples of load voltage diagrams for various
options are shown in Fig. 2. In the first case, where
both voltages coincide in polarity at the conditionally
positive polarity of supply voltage, the S1 switch
performs the direct modulation function Dcon(duty)
by periodically connecting the load to the power
source, while the S3 switch constantly turned-on and
switch S2 implements the zero off-duty pause (1-
D)con. If the polarity of the supply voltage has
switched to be negative, the function Dcon is
performed by the S2 switch and (1-D)con – the S4
switch.
In the case when both voltages – supply and load
– have conditionally opposite signs whereas the
supply voltage has a positive polarity, the DR and (1-
D)R functions are performed by the S4 switch and S2
switches (but at the same time S3 remains constantly
on), respectively. In the fourth version of the
algorithms, when, with different polarities, the
supply voltage has a conditionally negative sign the
interval D is carried out with the switch S3, but 1-D
with S1 (but with constantly on S4 switch). As a
result, for the implementation of all four algorithms,
a special block for control of the polarity of the
supply and load voltages is needed and a logical node
is required to select one of the four algorithms.
From the obtained diagrams, it is possible to
determine the effective and amplitude values of the
voltages of the circuit elements (see the data table II).
As it can be observed from calculations and
experiments, amplitude meanings are well
complying, but RMS meanings measured with
voltmeter are different as it was correctly found from
the results of Fourier transformation. Due to lack of
voltmeters, measuring the RMS values of
fundamental waves it should be difficult to use
measured vectors but only calculated could be
admissible with certain accuracy. The problem
initiates at operation with modulated signals, while
RMS meanings are far from estimated by diagrams
of voltages.
Fig. 2. Simulated diagrams of modulated load voltage Dcon=DR=0.5,
U1m=325 V, 50 Hz; fM=500 Hz . a – an concurrent waves of supply and
fundamental harmonic of load voltage; b – reverse waves of supply and
load fundamental waves.
Table II Comparison of experimentally obtained and calculated
results.
Not changed
polarities
Reverse polarities
Amplitude voltage of
the fundamental
wave, V
Amplitude voltage
of the
fundamental,
RMs values of
fundamental for
concurrent
voltages
for reverse waves
162.7 V ; experimental
162.1 V
-162.7V;
experimental
– 162.1 V
Calcul. 114.0 V,
experimental
114.5 V
Voltmeter
– 162.367
The modulated voltage has a poor indicator of the
quality – the THD (total harmonic distortion) factor,
i.e., it contains many higher harmonics (see Fig. 3).
Fundamental harmonic of current developed from
such voltages currents can be described as i1 = D٠ild
٠sin ωt , the effective RMS value of such a current is
defined as I1rms = D.Ild/√2 , and the apparent power of
the power supply is defined as
٠ ٠
Real power is defined as Pld= Ild2٠ R and if there are
no other devices with real power in the circuit, then
the power factor will be defined as:
a
b
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Fig. 3. Diagram of amplitudes of higher harmonics in content of
modulated load voltage.
3 Filtering modulated load voltage
To obtain a sinusoidal alternating current under
specific load corresponding to the fundamental
harmonic voltage, it is necessary to suppress all other
harmonic orders (see Fig. 3). In parallel to the load,
capacitor C is connected (Fig. 4), but in series with
the circuit the inductor L is introduced. If the both
elements are selected correctly, the load voltage is
slightly different from the sinusoid with the mains
frequency ω=2πf, where f is the frequency of AC
voltage supply source. THD indicator for the load
voltage becomes closer to the zero value. Modulated
voltage uD, RMS value of which is DU1 is not
sinusoidal at all, but in the vector diagram it can be
replaced by the voltage vector of the fundamental
harmonic, and as consequently the load voltage UC
can be determined from the vector expression
which is assumed to be sinusoidal, and
calculated from the specified equation as:
The inductor L current can be calculated by
considering the capacitor and load currents data (see
vector diagram for voltage vectors and currents in the
scheme Fig.5).
٠
In this case, the inductor current vector leads the load
voltage vector by the angle φ, which can be
calculated by:
Fig. 4. Schematic for realization of harmonic filter for modulated load
voltage.
Fig. 5. Vector diagram for voltages and currents in the scheme of filter.
The inductor voltage will be defined as:
٠
where
factor represents the relation
between RMS current of an inductor and RMS
voltage of load circuit, and the vector of this voltage
is ahead of the current IL vector by 90 degrees. The
total sum of both vectors is equal to the RMS value
of the modulated
.
By substituting the values of the expressions, it is
obtained:
or
.
Denoting the expression under the root as b, yields:
.
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4 Regressive estimation of filtering
system parameters on quality of the
system
Considering that, the angle it
is possible to analyze the influence of each parameter
on the value of the load voltage UC. The best
approach is to use the regressive expression method
– for example getting an algebraic expression
where X is the present value of parameter mean
value of the entire family of experimental results,
,… - normalized values of influence
parameters 1,2,3…; a, b, c…,indicators of influence
of each parameter .
If, for example, the capacitance of a capacitor is
considered in the range between 50 μF to 200 μF and
the wired load resistance R value is between 1 ohm
to 100 ohm, then at an angular frequency of ω=314
1/s (50 Hz), the following table of angle φ values
calculated by the formula should be considered.
Table III φ values calculated by the formula.
R*
C*
φexp. Gr
φform. Gr
-1
-1
0,9
-4,3
-1
+1
3,6
8,8
+1
+1
81
75,8
+1
-1
57,5
62,7
If the ratios are described as
where
– the average value of the
calculation results, the indicator A takes into account
the influence of the resistor parameter on the angle
value
and the indicator B of the influence of the C
parameter on the angle.
As a result of indicator calculations
φ=35.75+35.5R*+6.55C*, where
, diagram
of factors influencing the angle φ dependence in
accordance with the regression formula obtained is
presented in Fig.6.
Fig. 6. Diagram of factors influencing the angle φ in accordance with
the regression formula obtained
Having carried out calculations using the obtained
regression formula, we obtain the values with index
form, which are also given in table IV. As you can
see, the values calculated by the obtained formula for
the reference points coincide quite well with those
calculated by the exact expression.
The dependence of the inductor current IL on the load
voltage at different parameters of the elements R
and C conducting research on the indicator a:
Carrying out similar actions as in the first case, we
create a table of indicators a found about the
expression as:
Table IV Table of indicators a
R*
C*
aexp.
aform.
-1
-1
1
0.988
-1
+1
1
1.01
+1
+1
0.064
0.0513
+1
-1
0.0186
0.0287
Hence it is expressed as
As a matter of fact, in this case the coincidence of the
results in terms of the analytical expression and the
statistical one is quite close, too. Knowing the
indicator a, the inductor current is calculated as:
As it can be observed, if, with other parameters
unchanged, a increases, then the inductor current also
grows. In this case, the more R*, the smaller becomes
a (see Fig.7), and at R* max becomes zero.
Accordingly, the inductor current magnitude is also
small.
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Fig. 7. Diagram of dependence an indicator a on parameters of filter
load circuit according the regression formula developed.
Similarly, it is possible to investigate changes in
the load voltage, but already depending on the three
parameters of the elements R, L, C (see the table to
establish the values of b for various parameters), The
inductance of the inductor is considered in the zone
from 1 mH to 20 mH.
Table V Values of b for various parameters.
R*
L*
C*
b
bform
-1
-1
-1
1.043
8.276
-1
-1
+1
1.06
8.364
-1
+1
-1
31.4
23.536
-1
+1
+1
31.35
23.624
+1
-1
-1
0.995
-6.524
+1
+1
-1
0.597
8.736
+1
+1
+1
1.024
8.824
+1
-1
+1
0.98
-6.436
From the data table we can get the regression
expression
which is illustrated in the Fig.8.
Fig. 8. Dependence diagram of load voltage level indicator b on
parameters of filter elements by a developed regression formula.
We remind that the effective value of the load
voltage is defined as the division of the effective
value of the fundamental wave of the modulated
voltage DU1 by the indicator b, which depends on the
parameters of the three elements
.
Considering the graph of the dependence of b on
the parameters, it is evident that there is a strong
dependence of the magnitude of the inductance of the
inductor L. The larger the L, the larger the indicator
b also. It should be noted that the indicators of the
influence of R and L have opposite signs with equal
absolute values. This means that with an increase in
R, in order to stabilize the value of b and the level of
load voltage, the value of L should be increased, i.e.
the inductor should be characterized by the
nonlinearity of the inductance depending on the
current – at lower currents through the inductor the
inductance of the winding should be higher than in
higher currents.
The values of the angles between different
currents and voltages of the node are very important.
One of the most important is the angle between the
current IL and the modulated voltage vector UD ,
which can be called the input shift angle of the node
vectors. Using the similar method as previous) it is
possible from table of calculated data (see the table
by exact expression) express angle between
vector of modulated supply voltage and load voltage
Table VI Calculated data (see the table by exact expression)
express angle φ2 between vector of modulated supply voltage
and load voltage.
R*
L*
C*
-1
-1
-1
17.74
8.47
-1
-1
+1
17.73
-0.776
-1
+1
-1
88.17
108.47
-1
+1
+1
91.78
90.236
+1
-1
-1
0.186
18.71
+1
-1
+1
0.182
9.46
+1
+1
-1
148.3
118.7
+1
+1
+1
107.7
109
Fig. 9. Phasor diagram of load node.
The entry angle is usually leading (i.e., the current
vector leads the modulated voltage vector, see fig. 9).
Again, with increasing inductance, the angle
increases in magnitude.
For some parameters of the elements, the angle is
close to zero, i.e. the node works as an active one,
however, with others, the lead can reach 90 and more
degrees in the leading direction. The
expression for the also can be easily visualized. If
both angles ϕ and ϕ1 are known, then the angle
between the vector of the modulated voltage and the
vector of the load voltage is determined as:
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Fig. 10. Diagram of parameters influence on the angle between vector of
modulated supply voltage and load voltage in accordance with the
developed regression formula.
By examining the considered relations, one can
get a general idea of the nature of the influence of the
parameters of the elements. Figure 10 shows some of
the obtained diagrams of currents and voltages of the
elements of the filtering system at C = 100 µF, R =
10 Ω, L = 5 mH. Normalized parameters of elements
in the case are
and
.
To check the correctness of the expressions obtained
for the modulated voltage filter, a calculation was
carried out by expressions obtained and a comparison
of the calculation results with the experimentally
obtained data (based on a computer model). The
following values of the parameters of the voltage
modulation system were used: RMS value of supply
AC voltage U1=230 V, duty ratio for modulation
D=0.5, reduced resistance of the load R=10Ω,
inductance of the filter coil L=5 mH, capacitance for
load resistance C=100 μF.
The results of the comparison are summarized in
a table, which shows the calculated and
experimentally obtained values of the RMS of the
modulated voltage UD(1), the RMS voltage on the
plates of the load capacitor Uс, the effective value of
the inductor current IL , the values of the ratio factors
b and a, as well as the values of the angles of the
phase shift ϕ between the IL and UC , the angle ϕ2 of
the shift between the voltage vectors UD(1) and the
load one UC, as well as the phase shift ϕ1 between the
vectors IL and the vector of the modulated voltage
UD(1) .
Table VII. Comparison between experimental results and
calculated results
UD(1)
, V
UC V
IL , A
φ,
deg
φ2
deg
φ1
deg
Exp
calcul
Exp calcul
Exp calcul
Exper
calcul
Exper
calcul
Exper
calcul
114.5
115
119.16
119.34
b=0.96
bcalc=0.9636
12.53 12.51
aexp=0.105
acalc=0.1048
16.63
17.43
9.03
9.38
7.6
8.05
As you can see, the coincidence of the calculated and
experimentally obtained results is even very good,
which confirms the possibility of calculating and
analyzing complex modulated alternating current
systems using vector methods, representing vectors
with effective values of currents and voltages based
on the generated by modulations new fundamental
waves of currents and voltages. Evaluation of the
accuracy of the obtained regression expressions
shows that since they were obtained for a wide range
of changes in the input parameters of the system
elements, the accuracy is often low, especially when
using the obtained regression expressions for
calculation of the ratios of parameters, i.e., a, b, ϕ, ϕ2
. So, for example, using the data adopted in the
construction of the table ϕ2= 9.03 deg, but according
to the regression expression this parameter is 47.8
deg .. However, the regression expressions give a
very good idea of the influence of individual
parameters on the overall result.
5 Technical example of using unipolar
switch modulation
As a technical example, consider the use of three
unipolar modulated switch AC regulators for smooth
start-up of a three-phase induction electric motor
with a cage rotor (three-phase asynchronous squirrel-
cage motor) from a network of three phase (phase-
neutral) voltages A1-N1, B1-N1, C1-N1 with a smooth
increase the RMS value of these voltages in respect
to the phases of an electric motor during starting the
motor. The induction motor in this system also has a
neutral point NM (fig. 11).
In each of the three phases of the supply network
and the motor, two oppositely switched electronic
switches of alternating current conductivity are
switched on: when S1 conducts alternating current,
then S2 does not conduct current at all (locked) and
vice versa, when S1 is locked, the switch S2 conducts
alternating current, thereby creating a zero voltage
interval for the phase of motor. If both switches
switch at a relatively high frequency (about 100 times
higher than the frequency of the mains supply
voltage) and the S1-on state in the switching period
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is active with a relative duration D (duty ratio), then
the S2-on interval in the switching period TM has a
relative duration of 1-D. In this case, the fundamental
harmonic of the phase (neutral-phase) voltage of the
motor is represented by a wave
Fig. 11. The induction motor connection.
where is the amplitude of the voltage supply
phase-neutral, ω= 2π is the angular frequency of
the supplying alternating voltage. The effective
(RMS) voltage of the fundamental wave of the phase
voltage of the motor is defined as
If for a soft start it is supposed to linearly increase
this effective voltage, then the starting process is
characterized by a linear increase in time in the
indicator D from the minimum value D = 0
corresponding to the minimum necessary RMS
voltage value to the D = 1 (if the nominal values of
the network and the motor voltages are equal).
Change of D can be carried out by fixing the
interception points of a unipolar saw-shaped voltage
ust with a modulation frequency = 1/TM with slowly
variable control voltage UC over the modulation
period. Signal for unlocking the switches S1 in all
three phases is supplied when the control voltage
exceeds the instantaneous values of the saw-tooth
voltage with an amplitude Ustm , i.e. the relative
duration of the switched-on S1 (Duty ratio) is
expressed as
and the relative duration of switching on the switches
S2 is as
Modeling of this system is carried out at a switch
modulation frequency of 5 kHz with an asynchronous
electric motor of the resistance of the stator and rotor
windings (reduced to the stator), respectively, 0.294
ohm and 0.156 ohm with the inductance of these
windings 1.39 mH and 0.74 mH, magnetizing
inductance 41 mH, number of poles 6 and moment of
inertia 0.4 kg.m2. The motor parameters correspond
to approximately 15-20 kW rated power. Diagrams
of the instantaneous current of the motor, power
supply, corrected control voltage and speed of the
motor shaft are shown in Fig. 12.
Fig. 12. Diagrams of the instantaneous current of the motor, power
supply, corrected control voltage and speed of the motor shaft.
The control voltage was corrected with the
deduction of the component approximately
proportional to the current amplitude of the motor
from the linearly increasing control voltage, which
allows better linearization of the character of the
motor speed rise at starting. To ensure a smooth and
linear increase in the speed of the motor during start-
up, for any peculiarities of the influence of the motor
load, it is necessary to use a significantly advanced
start-up control with the control of the magnetic field
of the motor during start-up.
Figure 13 shows the experimentally taken
diagrams of currents of all three phases of an
asynchronous electric motor in the middle of a
smooth start of an electric motor in the proposed soft
start system. As you can see, the waves of currents
are close to sinusoidal, which is a consequence of the
generation of fundamental waves of voltage of the
phases of the motor by modulated switches.
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Fig. 13. Experimentally taken diagrams of currents of all three phases of
an asynchronous electric motor.
Fig. 14. Output voltage.
To obtain sinusoidal diagrams of the power supply
currents in phases of the supplying alternating
current, it is necessary to turn on the L-C filter
modulated by the current. In the case here L=1
mH/phase but capacitance is 50 μF/ phase. The filter
turns out to be quite light, since the modulation
frequency during the start-up process is taken quite
high at the level of 5 kHz.
6 Conclusion
Using the mentioned approach, a vector diagram
for a complex circuit has been created, on the basis
of which the main parameters of such a circuit -
voltage values on the load capacitor, inductor current
RMS values, shift angles between different voltages
and currents, etc. calculation.
The results obtained in terms of expressions
coincide very well with those obtained in practical
models, which prove the validity of the approach.
The value of the filter voltage is sensitive to an
increase in the load on the circuit connected to the
capacitor, but as the statistical processing of the
calculations for a sufficiently wide range of element
parameters shows, this decrease can be compensated
by a decrease in inductance with a nonlinear
inductance.
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