THE COMPARISON REGARDING THD BETWEEN DIFFERENT
MODULATION STRATEGIES IN SINGLE-PHASE FLYING
CAPACITOR MULTILEVEL PWM INVERTER
ADRIAN ŞCHIOP1
Keywords: Modulation strategies, Flying capacitor, Single-phase multilevel PWM
inverter, Simulink.
In this paper the modulation strategies for single-phase multilevel PWM inverter with
flying capacitors are analyzed. The analyzed strategies are: phase disposition PWM
method, phase-shifted PWM method, the saw-tooth rotation PWM method and the
carrier redistribution PWM method. The results are obtained through simulations by
SimulinkTM.
1. INTRODUCTION
Multilevel inverters are generally classified as: diode-clamping inverters,
flying capacitor inverters and cascade inverters [1]. Several modulation techniques
for three phase flying capacitor multilevel PWM inverter are presented in the
literature [2–7]. Some of these techniques obtain natural voltage balancing for the
flying capacitor like: phase-shifted PWM method, saw-tooth rotation PWM method
and carrier redistribution PWM method while for phase disposition PWM method
the redundancy states are chosen for voltage balancing.
In this paper a comparison between the four methods is made for single phase
flying capacitor multilevel PWM inverter regarding Total Harmonic Distortion
factor (THD). The influence on the THD of the disposition of the carrier signals is
also investigated.
2. DESCRIPTION OF FLYING CAPACITOR MULTILEVEL INVERTER
For the three level flying capacitor inverter the switches Sa1 and S’a1, Sa2 and
Sb1 and S’b1, Sb2 and S’b2, from Fig. 1 are driven complementary. An inverter
leg may be represented as in Fig. 2. We will say that transistors Sa2 and S’a2 form
switching cell a2 and transistors Sa1 and S’a1 form switching cell a1. Let us define
S’a2,
1University of Oradea, 1 Universităţii , 410087 Oradea, E-mail: aschiop@uoradea.ro
Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 55, 3, p. 330–340, Bucarest, 2010
2
Modulation strategies in single-phase flying capacitor inverter
331
the connection functions ya1 and ya2 for switching cell. For example, for a2 cell
from Fig. 2, ya2 = 1 when Sa2 is in ON state and ya2 = 0 when Sa2 is in OFF state.
Depending on adjacent switching states of the capacitor, the current through, for
example Ca1, is ia when Sa2 and S′a1 are in ON state, –ia when S′a2 and Sa1 are in ON
state, or zero when Sa1 and Sa2 are in ON state, or when S′a1 and S′a2 are in ON state.
Consequently, adequate driving of the adjacent transistors can modulate the current
through Ca1. In [8] it was shown that natural voltage balancing occurred when ya1
and ya2 have equal durations for each time period TP. Obtaining the equal durations
of all connection functions of commutation cells for an inverter leg is possible by
using two carrier signals with a TP/2 phase shift between them, or by using carrier
redistribution PWM method or the saw tooth rotation PWM method [2–7].
Ue
C2 Ca1
0
Sa2
Sb2
a2 cell
Sa1
Sb1
Sa2
Cb1
Ue
S’a1
S’b1
S’a2
S’b2
A
C2
a1 cell
Sa1
ia
Ca1
A
0
S’a2
S’a1
B
Fig. 1 – Scheme of the three level flying capacitor inverter.
Fig. 2 – Modelling of inverter leg.
In [9] the expression of phase voltage for flying capacitor n level inverters
depending on connection functions is derived. Customizing this expression for the
three level inverter leg with Sa1, S′a1, Sa2 and S′a2 transistors, this will be:
uA0 = ya2uC2+ uCa1(ya2+ ya1)[(1– ya2)+(1– ya1)](–1)(1–ya1),
(1)
where uC2 and uCa1 are the voltages across C2 and Ca1 capacitors.
The interphase voltage uAB will be:
uAB = uA0 – uB0; uAB = ya2uC2+ uCa1(ya2+ ya1)[(1– ya2)+(1– ya1)](–1)(1–ya1)–
– yb2 uC2–uCb1(yb2+ yb1)[(1– yb2)+(1– yb1)](–1)(1–yb1),
(2)
where uCb1 is the voltages across Cb1 capacitor; yb1 and yb2 are connection functions
for the leg with Sb1, S′b1, Sb2 and S′b2 transistors.
The SimulinkTM model for three level capacitor clamped inverter obtained by
(2) is presented in Fig. 3. SimulinkTM model of flying capacitor multilevel inverter
employs the following blocks of the SimulinkTM library: Source, Sum, Relay,
Constant, Gain, Product and Math Function. The Triangle block is a mask
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Adrian Şchiop
3
subsystem that allows set up of the amplitude, frequency, phase and offset of
carrier signals. The Math Function blocks realizes the (–1)(1–ya1) and (–1)(1–yb1)
operations. The Gain blocks multiplies the values of connection functions with
values of voltages on flying capacitors. In Fig. 3, at the output of the Relay blocks,
the connections functions ya1, ya2, yb1 and yb2 for each switching cell are obtained.
The time length of connection functions depends on carrier signal given by
Triangle blocks and on modulator signal given by Sine Wave block. For PWM
modulation strategies the connection functions yield the value 1 when modulating
signal is greater than carrier signal.
Fig. 3 – Simulink model for the three level flying capacitor inverter.
3. MODULATION METHODS
In the PWM method, the connection functions are obtained by comparison of
the carrier signals with the modulating signals. The case when the carrier signals
are overlapping and are covering a continuous range (mode A) and the case when
the carrier signals are phase shifted between them (mode B) are analyzed.
When the carrier signals are overlapped there are two operation sub-modes:
A1 when using 2(n–1) triangular carrier signals overlapping, (n–1) carrier signals
for one leg and the other (n–1) for the other leg, Fig. 4. In A2 mode are using (n–1)
carrier signals for both legs and two modulating signal phase shifted by T/2, where
T is the period of output voltage. The carrier signals can be in phase like in Fig. 5,
or phase opposition like in Fig. 6. For mode B there are three sub-modes: B1 when
carrier signals are triangular and phase shifted between them like Fig. 7, B2 saw-
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Modulation strategies in single-phase flying capacitor inverter
333
tooth rotation PWM method that uses, in fact, saw-tooth carriers signal, Fig. 8 and
B3 carrier redistribution PWM method that is using as carrier signals a sum of
triangular and trapezoidal signals (Fig. 9).
Fig. 4 – A1 mode modulation in single phase three level inverter.
Fig. 5 – A2 mode modulation with in phase carrier signals for three level inverter.
A
B
v*a
C
D
E
0
F
vt1
va* A
TP
vt1
0
ya1
0
0
vt2
B
C
D
E
TP
vt2
ya1
ya2
vt1, vt2, va*
vt1, vt2, va*
Fig. 6 – A2 mode modulation with phase opposition carrier signals for three level inverter.
TP
1
0
ya2
TP
Fig. 7 – B1 mode modulation for three level inverter.
1
0
0
TP
0
TP
Fig. 8 – B2 mode modulation for three level
inverter.
Adrian Şchiop
vt1, vt2, v*a
334
vt1
1
0
5
vt2
v*a
-1
0
Tp/2
TP
ya2 0
Tp/2
TP
0
Tp/2
TP
ya1
Fig. 9 – Carrier signal for a switching cell when carrier redistribution PWM method is used
for three level inverter.
4. SIMULATION RESULTS
The methods previously presented are analyzed by simulation in SimulinkTM
environment regarding the harmonic content and total harmonic distortion factor
(THD). The analyses are performed for single phase three level and four level
flying capacitor PWM inverter. In the next figures the harmonics of line voltage are
presented for different amplitude and frequency modulation indexes ma and mf.
ma = Am/(n–1)AP; mf = fP/fm,
(3)
were Am and Ap are the peak to peak values of modulator and carrier signals. fP and
fm are the frequencies of carrier and modulator signals.
It must be noted that the analysis refers to the voltage at idling for 50 Hz.
The simulation results are presented for two cases. In the first case mf is
1,250 Hz and ma is 0.9. In the second case mf is chosen as the value from which the
harmonics begin to be grouped around the multiply of modulation frequency. In
both cases the THD is calculated for the first 200 harmonics. For low values of mf,
for graphics clarity, only the first 100 harmonics are displayed.
As one can see from Fig. 10 and Fig. 11, for the A1 mode, if mf > 9, the
harmonics are grouped around the frequencies multiple of mf.
fhA1= kmf fm; mf > 9, k ∈ N*.
(4)
If mf < 9 the harmonics are not grouped around the mf and the fundamental
amplitude of the interphase voltage is dependent on mf.
ma=0.9; mf=25
ma=0.9; mf=25
ma=0,6; mf=9
ma=0,6; mf=9
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Modulation strategies in single-phase flying capacitor inverter
Fig. 10 – A1 mode for three level inverter.
335
Fig. 11 – A1 mode for four level inverter.
For the A2 mode, when the carrier signals are in phase, Fig. 12 and Fig. 13, if
mf > 6, the harmonics are grouped around twice of frequencies of mode A1.
fhA2= 2kmf fm; mf > 6, k ∈ N*.
(5)
If mf < 6 the harmonics are not grouped around the 2mf and the fundamental
amplitude of the interphase voltage is dependent on mf.
ma=0.9; mf=25
ma=0.9; mf=25
ma=0.6; mf=6
ma=0.6; mf=6
Fig. 12 – A2 mode for three level inverter.
Fig. 13 – A2 mode for four level inverter.
When the carrier signals are in phase opposition (A2PO), Fig. 14 and Fig. 15,
if mf > 9, the harmonics are grouped like in A1 mode.
In this case the number of line voltage levels decreases in comparison to the
case when carrier signals are in phase. If mf < 9 the harmonics are not grouped
around the mf and the fundamental amplitude of the interphase voltage is dependent
on mf.
fhA2PO = kmf fm; mf > 9, k ∈ N*.
(6)
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ma=0.9; mf=25
ma=0.9; mf=25
ma=0.6; mf=9
ma=0.6; mf=9
Fig. 14 – A2 mode phase opposition for three
level inverter.
Fig. 15 – A2 mode phase opposition for four
level inverter.
For B1 mode, Fig. 16 and Fig. 17, when the inverter has an odd number of
levels, the number of levels of the output voltage decreases in comparison to the
case when the group of carrier signals for inverter legs are 90o phase shifted. For
B1 mode, if mf > 4 for n odd, or mf > 2 for n even, the harmonics of interphase
voltage appear as side band of frequency fhB1:
fhB1 = kmf (n–1)fm; mf > 4, k ∈ N*; for n odd,
fhB1 = 2kmf (n–1)fm; mf > 2, k ∈ N*; for n even.
(7)
For B1 mode, when the carrier groups for the two legs are 90o phase shifted
(B1 90oPS), Fig. 18 and Fig. 19, if mf > 4 for n odd, or mf > 6 for n even, the
harmonics of voltage interphase appear as side bands of frequency fhB1 90PS:
fhB1 90PS = kmf (n–1)fm; mf >6, k ∈ N* with k ≠ 2(2i+1), i ∈ N; for n even,
fhB1 90PS = 2kmf (n–1)fm; mf > 4, k ∈ N*; for n odd,
where n is the number of inverter levels.
(8)
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Modulation strategies in single-phase flying capacitor inverter
ma=0.9; mf=25
ma=0.6; mf=4
Fig. 16 – B1 mode for three level inverter.
ma=0.9; mf=25
337
ma=0.9; mf=25
ma=0.6; mf=2
Fig. 17 – B1 mode for four level inverter.
ma=0.9; mf=25
ma=0.6; mf=4
ma=0.6; mf=6
Fig. 18 – B1 mode with 90o phase shifted
between carrier groups for three level inverter.
Fig. 19 – B1 mode with 90o phase shifted
between carrier groups for four level inverter.
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Adrian Şchiop
9
ma=0.9; mf=25
ma=0.9; mf=25
ma=0.6; mf=9
ma=0.6; mf=9
Fig. 20 – B2 mode for three level inverter.
Fig. 21 – B2 mode for four level inverter.
In B2 mode Fig. 20 and Fig. 21, if mf > 9, the harmonics appears as side
bands of frequency fhB2:
fhB2 = kmf (n–1)fm; mf > 9, k ∈ N*.
(9)
If mf < 9 the harmonics are not grouped around the mf and the fundamental
amplitude of the interphase voltage is dependent on mf.
For B3 mode, Fig. 22 and Fig. 23, if mf > 4 for n odd, or mf > 2 for n even,
the harmonics of voltage interphase appear as side bands of frequency fhB3:
fhB3 = 2kmf (n–1)fm; mf > 4, k ∈ N*; for n odd,
fhB3 = kmf (n–1)fm; mf > 2, k ∈ N*; for n even.
ma=0.9; mf=25
Fig. 22 – B3 mode for three level inverter.
(10)
ma=0.9; mf=25
Fig. 23 – B3 mode for four level inverter.
The THD for analyzed cases are presented in Table 1 and Table 2. From
Table 1 we can observe that B3 mode leads to obtaining of lowest values of THD.
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Modulation strategies in single-phase flying capacitor inverter
339
Table 1
The THD values of line voltage for mf = 25 and ma = 0.9
A1
A2
3 level 0.3225 0.3106
4 level 0.2137 0.2068
A2 PO
0.6196
0.3852
B1
0.5915
0.1805
B1 90oPS
0.2802
0.2412
B2
0.3106
0.1998
B3
0.2802
0.1732
Table 2
The THD values of line voltage for ma = 0.6 and those values of mf from which the harmonics begin
to be grouped around the multiply of modulation frequency
A1
mf =9
3 level 0.3902
mf =9
4 level 0.3125
A2
mf=6
0.4176
mf=6
0.3249
A2PO
mf=9
1.0378
mf =9
0.5614
B1
mf =4
0.9942
mf =2
0.3204
B1 90oPS
mf =4
0.4014
mf =6
0.4172
B2
mf =9
0.4268
mf =9
0.3193
B3
mf =4
0.4014
mf =2
0.3025
By increasing the number of levels in the inverter the THD becomes lower,
for the same mf, but at the expense of increasing complexity in the control circuit.
However, as can be seen from Table 1, even for the three level inverter the
modulation techniques B1 90oPS and B3 lead to a THD below 0.3, while the
harmonics are clustered around the four multiples of mf, which facilitates their
filtering. From Table 2 we can see that mf values from which the harmonics begin
grouped around the multiply of modulation frequency is dependent of modulation
techniques and for B1, B1 90oPS and B3 modes it depends on the number of the
inverter levels
5. CONCLUSIONS
Several modulation techniques for flying capacitor multilevel PWM inverter
are presented in the literature. In this paper a comparison between these modulation
techniques for single phase flying capacitors three/four level PWM inverter are
analyzed regarding THD and harmonics distribution. The characteristics of
discussed PWM methods, obtained by simulations using developed model
presented in [9], are as follows:
• A1 and A2 with phase opposition modes give the same harmonic distribution.
The harmonics are grouped around the frequencies multiple of frequency
modulation index.
• In A2 mode the harmonics are grouped around the twice frequencies multiple
of frequency modulation index.
• In B1, B2 and B3 modes the frequencies around which the side bands are
grouped depend on the number of inverter levels.
• B3 mode allows one to obtain the lowest values of THD.
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Adrian Şchiop
11
For each modulation type, the mf value from which the harmonics begin to be
grouped around the multiply of modulation frequency was determined. This value
is dependent on modulation technique and for B1, B1 900PS and B3 modes it
depends on the number of the inverter levels.
Received on July 14, 2008
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