A. Two-Phase ‘Modified’ Sinusoidal PWM Technique This form of voltage control is the classical sinusoidal pulse-width-modulation (SPWM) but modified for two-phase system [5]. The pulse-width-modulated wave has much lower order harmonics than the other waveforms. If the desired inverter’s output voltage is sine-wave, two parameters define the control [3]-[4]: Coefficient of the modulation m - equal to the ratio of the modulation and reference frequency. Voltage control coefficient r - equal to the ratio of the desired voltage amplitude and DC supply voltage level.
Generally the synchronous modulation is used. In synchronic modulation a modulation frequency m f is an integer multiple of the reference frequency f .
Output voltage vectors VAS and VBS
are line to line voltages between the middle leg b and another leg a and leg c, Fig. 2. From Fig. 2 we can see that the amplitudes of output voltages are equal, but the complexity of control system SVPWM is much harder like using the control scheme with two-phase sinusoidal SPWM [5]-[6].
Fig. 2. Referenced voltage for each leg of VSI inverter [5].
From the figure also can see that the amplitude of voltage vectors is VDC . Referenced voltages for two-phase SPWM for each leg of VSI inverter are depicted on Fig. 3.
Fig. 3. Referenced voltages of the two-phase sinusoidal PWM technique.
To inverter output voltage analysis the number of math methods can be used [7]-[13], one of them based on Fourier analysis is used. Then, the turn on angle and turn off angle for any modulation interval are calculated, based on the reference sine-wave.
The desired output voltage of the branches has a form:
0
01 02
22
2 2 2 2
ee
e e e e
UU u r sin ;
U U U U u r cos ; u r cos
(1)
To calculate a turn on (
) and turn off (
) angles we compare the DC pulse area with the requested voltage area, as depicted on the Fig. 4 [7].
Fig. 4. Comparison of the voltages area.
For the left and right crosshatched areas of the first output transistors branch the following equations are valid [10]-[13]:
2
01
21 2
21 2
01
2
2
22
2
22
n
m
ee
en
n
m
n
m
ee
en
n
m
UU r sin d U n m
UU r sin d U n m
(2)
After the calculus we obtain for the turn-on and turn-off angles of the first transistors branch the following expressions:
0
0
12 22 1 22 12 22 1 22
n
n
r n cosn cos n m m m r n cosn cos n m m m
(3)
It will be similarly for the second transistor branch.
01
01
12 2 2 1 22 12 2 2 1 22
n
n
rn n sin sin n m m m rn n sin sin n m m m
(4)
So does for third transistor branch.
02
02
12 22 1 22 12 22 1 22
n
n
rn n sin sin n m m m rn n sin sin n m m m
(5) Then we can write the output voltage of the first branch in the form of a complex Fourier series [7], [10]:
Sine referenced voltage for Leg b
Minus cosine referenced voltage for Leg a
Cosine referenced voltage for Leg c
VAS
VBS
0 0.005 0.01 0.015 0.02 0
0.2
0.4
0.6
0.8
1
Leg A Leg B Leg C
732
A. Two-Phase ‘Modified’ Sinusoidal PWM Technique This form of voltage control is the classical sinusoidal pulse-width-modulation (SPWM) but modified for two-phase system [5]. The pulse-width-modulated wave has much lower order harmonics than the other waveforms. If the desired inverter’s output voltage is sine-wave, two parameters define the control [3]-[4]: Coefficient of the modulation m - equal to the ratio of the modulation and reference frequency. Voltage control coefficient r - equal to the ratio of the desired voltage amplitude and DC supply voltage level.
Generally the synchronous modulation is used. In synchronic modulation a modulation frequency m f is an integer multiple of the reference frequency f .
Output voltage vectors VAS and VBS
are line to line voltages between the middle leg b and another leg a and leg c, Fig. 2. From Fig. 2 we can see that the amplitudes of output voltages are equal, but the complexity of control system SVPWM is much harder like using the control scheme with two-phase sinusoidal SPWM [5]-[6].
Fig. 2. Referenced voltage for each leg of VSI inverter [5].
From the figure also can see that the amplitude of voltage vectors is VDC . Referenced voltages for two-phase SPWM for each leg of VSI inverter are depicted on Fig. 3.
Fig. 3. Referenced voltages of the two-phase sinusoidal PWM technique.
To inverter output voltage analysis the number of math methods can be used [7]-[13], one of them based on Fourier analysis is used. Then, the turn on angle and turn off angle for any modulation interval are calculated, based on the reference sine-wave.
The desired output voltage of the branches has a form:
0
01 02
22
2 2 2 2
ee
e e e e
UU u r sin ;
U U U U u r cos ; u r cos
(1)
To calculate a turn on (
) and turn off (
) angles we compare the DC pulse area with the requested voltage area, as depicted on the Fig. 4 [7].
Fig. 4. Comparison of the voltages area.
For the left and right crosshatched areas of the first output transistors branch the following equations are valid [10]-[13]:
2
01
21 2
21 2
01
2
2
22
2
22
n
m
ee
en
n
m
n
m
ee
en
n
m
UU r sin d U n m
UU r sin d U n m
(2)
After the calculus we obtain for the turn-on and turn-off angles of the first transistors branch the following expressions:
0
0
12 22 1 22 12 22 1 22
n
n
r n cosn cos n m m m r n cosn cos n m m m
(3)
It will be similarly for the second transistor branch.
01
01
12 2 2 1 22 12 2 2 1 22
n
n
rn n sin sin n m m m rn n sin sin n m m m
(4)
So does for third transistor branch.
02
02
12 22 1 22 12 22 1 22
n
n
rn n sin sin n m m m rn n sin sin n m m m
(5) Then we can write the output voltage of the first branch in the form of a complex Fourier series [7], [10]:
Sine referenced voltage for Leg b
Minus cosine referenced voltage for Leg a
Cosine referenced voltage for Leg c
VAS
VBS
0 0.005 0.01 0.015 0.02 0
0.2
0.4
0.6
0.8
1
Leg A Leg B Leg C
732
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