Download Comparative Evaluation of Space Vector Based Pulse Width

Comparative Evaluation of Space Vector Based
Pulse Width Modulation Techniques in terms of Harmonic
Distortion and Switching Losses
V S S Pavan Kumar Hari
G Narayanan
Department of Electrical Engineering
Indian Institute of Science
Bangalore, India
Department of Electrical Engineering
Indian Institute of Science
Bangalore, India
[email protected]
[email protected]
ABSTRACT
Voltage source inverter (VSI) fed induction motors are widely
used in variable speed applications. The harmonic distortion
in the line currents of the inverter depends on the switching frequency and the pulse width modulation (PWM) technique employed to switch the inverter. The inverter switching loss is also strongly influenced by the PWM technique
used. This paper compares the harmonic distortion and inverter switching losses due to different PWM techniques at
a given average switching frequency. The paper evaluates a
few specially designed PWM techniques based on space vector approach. It is shown that these techniques lead to reduced line current distortion as well as switching losses compared to well-known techniques such as sine-triangle PWM
and conventional space vector PWM (CSVPWM).
Keywords
Induction motor drives, voltage source inverter, pulse width
modulation, harmonic distortion, space vector, switching
losses.
1.
INTRODUCTION
A voltage source inverter (VSI) is used to supply an AC
motor drive with a voltage of variable magnitude and frequency. The schematic of a two level VSI is shown in Fig.1.
With a fixed DC input voltage, the semiconductor switches
in the inverter are required to be switched in an appropriate fashion to obtain the desired fundamental voltage on the
line side. Various PWM techniques have been reported for
generation of gating pulses for the devices. The choice of
PWM technique strongly influences line current distortion
and switching losses in the inverter [1-2].
Sine-triangle PWM (SPWM) is a simple and popular
method of PWM. In this method, three sinusoidal modulating waves, shifted by 120◦ from each other, are compared
with a high frequency triangular carrier to get the switching pattern for the three phases. A sinusoidal modulating
wave is shown in Fig.2(a). Suitable common-mode voltages
can be added to the 3-phase sinusoidal modulating waves
to achieve better harmonic performance and higher line-side
voltage for a given DC bus voltage [1-2]. Examples of such
modulating waves are shown in Fig.2(b) to Fig.2(d). Such
addition of common-mode components could lead to discontinuous modulating waves as shown in Fig.2(c) and Fig.2(d).
With these techniques, a phase remains clamped to a given
+
S1
C1
VDC
S3
D1
S5
D3
D5
O
R
S4
C2
Y
S6
D4
B
S2
D6
D2
–
Figure 1: A two level voltage source inverter using
IGBTs
DC bus in certain region of the fundamental cycle. A phase
is clamped during the middle 60◦ interval in every half cycle
in Fig.2(d), while it is clamped during the middle 30◦ interval in every quarter cycle in Fig.2(c). These techniques are
termed as 60◦ clamp PWM and 30◦ clamp PWM respectively [3-5]. The average switching frequency of the inverter
with such techniques equals two thirds of the carrier frequency.
In addition to the above well known techniques, this paper describes two space vector based PWM techniques [6-8].
The harmonic distortion and switching losses due to various
PWM techniques are evaluated and compared. It is shown
that the two space vector based techniques result in reduced
line current distortion and lower switching losses than other
techniques at high modulation indices for a typical induction
motor drive application.
2.
SPACE VECTOR APPROACH TO PWM
Space vector based generation of PWM waveforms is explained in this section.
2.1
Inverter states and the space vector plane
A three phase quantity can be transformed into a space
vector having components along two mutually perpendicular axes. The three phase input voltages at the stator terminals of an induction machine can be transformed to a space
vector as shown in Eqn.1, where v s (t) is the stator voltage space vector; vRN , vY N and vBN are the line-to-neutral
(3)−+−
++−(2)
II
III
I
VREF
(7)+++
Vp
α
(4)−++
+−−(1)
(0)−−−
0
IV
VI
−Vp
V
(5)−−+
0
60
120
180
ωt(degrees)
240
300
360
(a)
◦
+−+(6)
ωt = 0
Figure 3: Space vectors of a two level voltage source
inverter
Vp
voltages [9].
0
v s (t) = vRN (t) + vY N (t)ej
2π
3
+ vBN (t)ej
4π
3
(1)
3
−Vp
0
60
120
180
ωt(degrees)
240
300
360
(b)
Vp
0
A two level VSI has 2 or 8 combinations of switching
states, which are known as the states of an inverter. Among
these states, the two states 7 and 0 are known as zero states,
as they yield zero output voltage. The remaining six states
(1 to 6) give a non-zero output voltage, and are known as
active states. The voltage vectors corresponding to the eight
states are shown in Fig.3. The space vector plane is divided
into six sectors by the vectors corresponding to active states.
The sectors are marked as I to V I in Fig.3.
2.2
−Vp
0
60
120
180
ωt(degrees)
240
300
360
240
300
360
(c)
Vp
0
−Vp
0
60
120
180
ωt(degrees)
(d)
Figure 2: Modulating waves for (a) sine-triangle
PWM, (b) conventional space vector PWM,
(c) 30◦ clamp PWM and (d) 60◦ clamp PWM. Peak
value of sine wave is Vm . Vp is the peak of triangular
= 0.8.
wave. Modulation index VVm
p
Dwell times of inverter states
In space vector based PWM, the voltage reference is provided using a revolving voltage reference vector V REF instead of three phase modulating waves. At steady state,
the magnitude of this reference vector is proportional to the
desired fundamental voltage and its frequency equals the
fundamental frequency.
The revolving V REF vector is sampled once in every subcycle Ts , which gives the average vector to be applied in the
given sub-cycle. This average vector can be best realized
by the vectors bounding the sector in which it is present
using the principle of volt-second balance. The reference and
applied volt-seconds are balanced over a sub-cycle duration,
i.e.
V REF Ts = V x Tx + V y Ty + V z Tz
(2)
where Ts is the duration of a sub-cycle; V x and V y are
the active vectors bounding the corresponding sector; V z is
the zero vector. The vectors V x and V y correspond to the
active states x and y respectively. The zero vector V z may
correspond to the zero state 0 or the zero state 7. For sector
I, x = 1 and y = 2. Tx ,Ty and Tz denote the dwell times
of the states/vectors. The expressions for Tx , Ty and Tz are
given by
Tx
=
Ty
=
Tz
=
VREF sin(60◦ − α)
·
· Ts
VDC
sin 60◦
VREF
sin α
·
· Ts
VDC sin 60◦
Ts − (Tx + Ty )
(3a)
(3b)
(3c)
3.
Ts =T
0
1
2
7
Tz
2
Tx
Ty
Tz
2
t
(a)
Ts = 2
T
3
1
2
2T
3 z
2T
3 x
2T
3 y
t
(b)
T
Ts = 2
3
7
2
1
2T
3 z
2T
3 y
2T
3 x
t
(c)
Ts =T
0
1
Tz
Tx
2
2
1
Ty
Tx
2
t
(d)
Ts =T
7
2
Tz
Ty
2
1
2
Tx
Ty
2
t
(e)
Ts =T
1
0
1
2
Tz
Tx
2
Ty
t
(f)
Tx
2
Ts =T
2
7
2
1
Ty
2
Tz
Ty
2
Tx
t
(g)
Figure 4: Seven possible sequences in sector I:
(a) 0127, (b) 012, (c) 721, (d) 0121, (e) 7212, (f ) 1012
and (g) 2721.
where α is the angle measured from the starting of a sector
to V REF in anticlockwise direction.
2.3
In the present section, the switching sequences explained
in previous section are compared in terms of RMS current
ripple and switching energy lost over a sub-cycle.
3.1
0
Possible switching sequences
There are numerous possible ways of applying the vectors for corresponding time intervals. The zero vector can
be applied using the zero state + + +(7) or the zero state
− − −(0). An active state can be applied more than once
in a sub-cycle [5-8,10-11]. The various possible sequences in
sector I are 0127,012,721,0121,7212,1012 and 2721. These
seven sequences are illustrated in Fig.4 [7-8]. In all these
sequences, the number of switchings in a sub-cycle is less
than or equal to three. Also, only one phase switches during
a transition from one state to the next. The sub-cycle duration for sequences 012 and 721 is taken to be 23 of that for the
other sequences because there are only two switchings per
sub-cycle. This facilitates comparison of all the sequences
at the same average switching frequency.
ANALYTICAL EVALUATION OF SWITCHING SEQUENCES
RMS current ripple over a sub-cycle
At any given instant, there is an error between the reference voltage vector (V REF ) and the applied voltage vector.
The error voltage vectors corresponding to the vectors V x ,
V y and V z are illustrated in Fig.5(a). The integral of error
voltage vector is the stator flux ripple vector. The trajectory
of the tip of the stator flux ripple vector is parallel to the
instantaneous error voltage vector [10-11].
For harmonic voltages, an induction motor can simply be
modelled by its total leakage inductance. Therefore, the
integral of error voltage, or the stator flux ripple, is proportional to the current ripple (both differ by a factor of
machine leakage inductance). Hence, flux ripple is a measure of the ripple in line current [10-11].
The trajectory of the tip of stator flux ripple vector for
sequence 0127 is shown in Fig.5(a) for VREF = 0.7VDC and
α = 20◦ . The stator flux ripple vector can be resloved along
the reference axes of a synchronously revolving reference
frame, namely q-axis and d-axis. The q-axis is aligned with
V REF , while the d-axis lags behind the q-axis by 90◦ . The
variation of d-axis and q-axis components of stator flux ripple vector for the sequence 0127 over a sub-cycle is shown in
Fig.5(b). The error volt-second quantities Qx , Qy , Qz and
D marked in the plot are defined in Eqn.4.
Qz
Qx
Qy
D
= −VREF · Tz
= (cos α − VREF ) · Tx
= (cos(60◦ − α) − VREF ) · Ty
= sin α · Tx
(4a)
(4b)
(4c)
(4d)
For sequences 012, 721, 0121, 7212, 1012 and 2721, the
locus of the tip of the stator flux ripple vector and the d-axis
and q-axis components of the vector for the same reference
vector (VREF = 0.7VDC and α = 20◦ ) are shown in Fig.6.
The RMS d-axis and q-axis flux ripple over a sub-cycle for a
sequence SEQ can be calculated using Eqn.5a and Eqn.5b.
The total RMS flux ripple over a sub-cycle for a sequence
SEQ is given by Eqn.5c.
s
Z Ts
1
2
2
(5a)
ψedSEQ RM S =
ψedSEQ
dt
Ts 0
s
Z Ts
1
2
2
(5b)
ψeqSEQ RM S =
ψeqSEQ
dt
Ts 0
q
2
e2
ψeSEQ RM S =
ψedSEQ
(5c)
RM S + ψqSEQ RM S
where SEQ = 0127,0121,7212,012,721,1012 or 2721.
It is possible to derive closed-form expressions for the
RMS flux ripple over a sector for different sequences. Expressions for five sequences (0127, 012, 721, 0121 and 7212)
are presented in [11]. The corresponding expressions for
1012 and 2721 are derived here. All these expressions can be
written in a generalized form given in Eqn.6. The coefficients
in this expression C0 , C1 and C2 for different sequences are
given in Table 1. C0 is a function of α for sequences 1012,
2721 and a constant for all other sequences. C1 and C2 are
ed
ψ
Ts
D
Vy
q
ey
V
t
O
eq
ψ
Tz
2
Tz
2
Ty
Tx
e
ψ
Ve
x
ez
V
V REF
−Qz
2
Qz
+Qx
2
α
Vz
Vx
(a)
t
O
d
Qz
2
0127
(b)
Figure 5: (a) Error voltage vectors corresponding to applied vectors V x , V y and V z , and the trajectory of the
tip of stator flux ripple vector for 0127 (b) Variation of stator flux ripple along d and q axes over a sub-cycle
for 0127.
ed
ψ
ed
ψ
2T
3
2T
3
t
O
2D
3
−2D
3
q
q
t
O
e
ψ
eq
ψ
2T
3 z
2T
3 x
2
2T
3 x
t
O
t
2 (Q +Q )
z
x
3
d
2
Tz
T
3 y
eq 3
ψ
O
d
2Q
3 z
721
2Q
3 z
012
e
ψ
2T
3 y
2 (Q +Q )
z
y
3
(b)
(a)
ed
ψ
ed
ψ
T
T
D
2
D
2
q
e
ψ
−D
2
e
ψ
eq
ψ
Tz
Tx
2
Tx
2
Ty
eq
ψ
−D
2
Ty
Tz
2
Ty
2
Tx
O
t
O
t
O
t
O
q
−Qy
2
d
d
0121
Qz +
Qx
2
−Qx
2
7212
Qz +
Qz
Qz
t
Qy
2
(d)
(c)
ed
ψ
ed
ψ
T
T
t
O
D
D
2
−D
2
q
q
−D
t
O
e
ψ
eq
ψ
Tx
2
Tz
Tx
2
e
ψ
Ty
eq
ψ
Qx
2
d
1012
d
t
O
2721
O
Ty
2
Tz
Ty
2
Tx
Qy
2
t
Qz +Qx
(e)
Qx
+Qz
2
(f)
Qy
+Qz
2
Qz +Qy
Figure 6: Trajectory of the tip of stator flux ripple vector and the variation of flux ripple along d and q axes
over a sub-cycle for sequences (a) 012, (b) 721, (c) 0121, (d) 7212, (e) 1012 and (f ) 2721.
ψeSEQ
SEQ
C0(SEQ)
0127
1
12
0121
1
3
7212
1
3
012
1
3
721
1
3
1
3
1012
2721
RM S
=
q
2
3
4
C0(SEQ) VREF
Ts2 + C1(SEQ) VREF
Ts2 + C2(SEQ) VREF
Ts2
Table 1: Coefficients C0 , C1 and C2 for all the sequences
a = sin (60◦ + α) and b = sin (60◦ − α)
C1(SEQ)
C2(SEQ)
»
»
`
´
`
´
1
32
1 4
√
ab4 + a3 b2 − a2 b3
a3 b − 2ab3 − a2 b2
9
9 3
3 3
–
–
`
´
+ 32 −2b3 − 3ab2 − a2 b + b − a
+a2 + 4b2 − 2ab
»
` 2 2
´
ˆ 3
˜
2
2
1
1 2
√
a b − ab3 + 2a3 b
−b
−
ab
−
2a
b
−
12a
+
3b
3 9
9 3
–
´
`
+ 13 4a2 − 2ab + b2
C1(0121) (60◦ − α)
3
1
√
3
ˆ8
3
C2(0121) (60◦ − α)
a3 + 4ab2 − 8a2 b − 4b3 − 6a + 6b
16a5 − 2b5 + 20ab4 − 56a4 b + 74a3 b2
–
−60a2 b3 + 24a2 b − 3ab2 − 9b3 + 18b − 27a
2√
27 3
(1 − ab)
C0(1012) (60◦ − α)
=
=
ER
+ EY SEQ + EB SEQ
|iY 1 |
|iB1 |
|iR1 |
+ nY ·
+ nB ·
(7)
nR ·
Im
Im
Im
SEQ
where
iR1
iY 1
iB1
ω
φ
8
9
`
−15a2 b2 + 11ab3 + 14a3 b − 4a4 − 4b4
–
+4 (a − b)2
1
9
»
2
3
`
= Im sin(ωt + φ)
= Im sin(ωt − 120◦ + φ)
= Im sin(ωt − 240◦ + φ)
= 2π · fundamental frequency
= power factor angle
The values of nR , nY and nB for various sequences in sector
I are given in Table 2 along with the corresponding duration
of sub-cycle. In any sector, ωt can be written in terms of
α (for sector I, ωt = 90◦ + α). Therefore, the normalized
´
−16a4 − 6b4 + 38a3 b + 25ab3 − 41a2 b2
–
+12a2 − 18ab + 7b2
C1(1012) (60◦ − α)
The switching energy loss in a sub-cycle in an inverter leg
is directly proportional to the current in that phase and the
number of switchings of that phase in the given sub-cycle.
This is based on the assumption that VDC and the device
switching times remain constant. The total switching loss
in the inverter in a sub-cycle is the sum of switching losses
in the three phases. If nR , nY and nB are the number of
switchings of phases R, Y and B, respectively, in a given subcycle for a sequence SEQ, the normalized switching energy
loss in the inverter in that sub-cycle is given by
SEQ
1
3
»
C2(012) (60◦ − α)
´
C2(1012) (60◦ − α)
Table 2: Number of switchings of the phases in a
sub-cycle for different sequences in sector I
SEQ
0127
0121
7212
012
721
1012
2721
Switching energy lost in a sub-cycle
ESU B
˜
C1(012) (60◦ − α)
»
functions of α for all the sequences.
As seen from the equations, the RMS flux ripple over a
sub-cycle for a given sequence is a function of VREF , α and
Ts . These expressions can be used to compare the RMS
current ripple over a sub-cycle for different sequences.
3.2
(6)
nR
1
1
0
1
0
2
0
nY
1
2
2
1
1
1
1
nB
1
0
1
0
1
0
2
Ts
T
T
T
2
T
3
2
T
3
T
T
switching energy loss over a sub-cycle is a function of α and
power factor angle φ.
4.
HYBRID PWM TECHNIQUES
The modulating waves corresponding to CSVPWM, 30◦
clamp PWM and 60◦ clamp PWM were shown in Fig.2(b),
2(c) and 2(d). These techniques are illustrated in the space
vector domain in Fig.7. In CSVPWM, sequence 0127-7210 is
applied in sector I. For 60◦ clamp PWM, sequence 721-127
is applied in the first half of the sector and sequence 012-210
is applied in the second half of the sector. The sequences
for 30◦ clamp PWM are applied in a way opposite to that
of 60◦ clamp PWM.
This section describes two hybrid PWM techniques which
employ sequences 0127, 0121 and 7212.
4.1
2
0127
0,7
1
(a)
2
2
012
721
721
012
30◦
30◦
0,7
1
(b)
0,7
1
(c)
Figure 7: PWM techniques considered for evaluation - 1 : (a) CSVPWM (b) 60◦ clamp PWM
(c) 30◦ clamp PWM.
RM S F lux Ripple over a subcycle
16·10−6
14·10−6
12·10−6
10·10−6
8·10−6
6·10−6
4·10−6
0127
2·10−6
0121
7212
0
0
5
10
15
20
25
30
35
40
45
50
55
60
α(degrees)
Figure 8: RMS flux ripple over a sub-cycle Vs α for
sequences 0127, 0121 and 7212 (VREF = 0.866 p.u.).
Comparison of sequences in terms of RMS
current ripple
The RMS current ripple over a sub-cycle corresponding to
sequences 0127, 0121 and 7212 is compared here. Based on
the expressions for RMS flux ripple over a sub-cycle derived
in section 3.1, the sequences can be compared with each
other and their individual zones of superior performance can
be identified.
The RMS flux ripple over a sub-cycle is plotted in Fig.8
for the whole range of α. Here, VREF = 0.866p.u.(1.0p.u. =
VDC ) and Ts = 100µs. It is evident from Fig.8 that for
VREF = 0.866p.u., sequences 0121 and 7212 yield lesser
ripple than 0127 in the range of 5◦ ≤ α ≤ 55◦ (approximately). Similarly, for different values of VREF , the range
of α over which these sequences result in less ripple will be
different. These sets of VREF and α values for which a sequence gives the least ripple indicate the zone of superior
performance of that particular sequence. It can be observed
from Fig.8 that the flux ripple given by 0121 and 7212 are
close to each other, and 0121 results in less ripple in the
first half of the sector while 7212 yields less ripple in the
second half. Based on this discussion, a hybrid PWM technique can be designed to give reduced current ripple, which
is shown in Fig.11(a). This technique is referred to as the
three-zone hybrid PWM technique [6,8]. If the tip of reference voltage vector in sector I falls in the region marked
0127 in Fig.11(a), sequence 0127-7210 is applied. Similarly
sequences 0121-1210 and 7212-2721 are applied in regions
marked 0121 and 7212. Sequences in other sectors are applied accordingly.
4.2
Comparison of sequences in terms of switching losses
In a similar manner, various switching sequences can be
compared in terms of switching losses based on the switching energy lost in each sub-cycle. The normalized switching
energy loss in a sub-cycle ESU B SEQ is a function of α and
power factor angle φ. The normalized switching energy loss
in a sub-cycle is plotted for sequences 0127, 0121 and 7212
over the whole range of α in Fig.9 (VREF = 0.866 p.u., Im =
1.0 and φ = 30◦ lagging) and Fig.10 (VREF = 0.866 p.u.,
Im = 1.0 and φ = 0◦ ).
From Fig.9 and Fig.10, it can
be observed that 7212 produces lower switching losses than
0127 for all values of α, for power factors 0.866 lagging and
unity. Whereas 0121 gives better performance in terms of
switching losses only for some values of α at different power
factors. For lagging power factor operation, which is typical
in induction motor drives, using 7212 instead of 0121 will
be a better option in terms of switching loss performance.
This will not significantly affect the current ripple due to
the PWM technique as 0121 and 7212 have closer characteristics in terms of flux ripple (Fig.8). Therefore, the threezone hybrid PWM technique in Fig.11(a) can be modified
into a simpler technique shown in Fig.11(b) to give better
performance in terms of switching losses. This technique is
referred to as the two-zone hybrid PWM technique [7].
T otal Harmonic Distortion ΨT HD
0.0175
N ormalized Switching Energy loss
ESU B SEQ
2.75
2.5
2.25
2
1.75
1.5
1.25
1
2
3
0.0125
1
0.01
4,5
CSVPWM(1)
0.0075
60◦ Clamp PWM(2)
30◦ Clamp PWM(3)
0.005
Three zone PWM(4)
Two zone PWM(5)
0.0025
0127
0.75
0.015
5
10
15
0121
20
25
30
35
40
45
50
F undamental F requency (Hz)
7212
0.5
0
5
10
15
20
25
30
35
40
45
50
55
60
Figure 12: THD in stator flux ΨT HD for different
PWM Techniques Vs fundamental frequency.
α(degrees)
Figure 9: Normalized switching energy loss over a
sub-cycle Vs α for sequences 0127, 0121 and 7212
(VREF = 0.866 p.u., Im = 1.0 and φ = 30◦ lagging).
1.4
2
1.2
N ormalized ΨT HD
N ormalized Switching Energy loss
ESU B SEQ
3
2.25
2
1.75
1
0.8
CSVPWM(1)
0.6
4,5
60◦ Clamp PWM(2)
30◦ Clamp PWM(3)
0.4
1.5
Three zone PWM(4)
Two zone PWM(5)
1.25
0.2
1
5
10
15
20
25
30
35
40
45
50
F undamental F requency (Hz)
0127
0.75
0121
7212
0.5
0
5
10
15
20
25
30
35
40
45
50
55
Figure 13: Normalized ΨT HD for different PWM
Techniques Vs fundamental frequency (Normalization is w.r.t. CSVPWM).
60
α(degrees)
Figure 10: Normalized switching energy loss over a
sub-cycle Vs α for sequences 0127, 0121 and 7212
(VREF = 0.866 p.u., Im = 1.0 and φ = 0◦ ).
2
5.
2
7212
0127
(a)
0121
5.1
0127
1
0,7
(b)
Figure 11: PWM techniques considered for evaluation - 2: (a) Three-zone hybrid PWM (b) Two-zone
hybrid PWM
EVALUATION OF PWM TECHNIQUES
IN TERMS OF HARMONIC DISTORTION
AND SWITCHING LOSSES
In section III, evaluation of sequences at a sub-cycle level
is presented. This forms the basis for design and evaluation of various PWM techniques. Comparative evaluation
of PWM techniques is presented here.
The techniques considered for evaluation are conventional
space vector PWM (CSVPWM), 60◦ clamp PWM, 30◦ clamp
PWM, three-zone hybrid PWM and two-zone hybrid PWM.
These techniques are summarized in Fig.7 and Fig.11.
7212
0,7
1.0
1
Evaluation of harmonic distortion
The mean square flux ripple over a sub-cycle can be averaged over a sector to give the mean square flux ripple over a
e RM S , for
sector. Thus, the RMS flux ripple over a sector Ψ
a given VREF and Ts is given by
s
Z π
3
3
2
e RM S =
ψeRM
(8)
Ψ
S SEQ (α) dα
π 0
where SEQ is the sequence applied in that sub-cycle, which
5
2
1.2
4
1
1.0
3
0.8
0.6
Two zone PWM(5)
CSVPWM(1)
60◦ Clamp PWM(2)
Three zone PWM(4)
0.4
30◦ Clamp PWM(3)
Lagging p.f.
0.2
−90 −75 −60 −45 −30 −15
Leading p.f.
0
15
30
45
60
75
90
P ower f actor angle (degrees)
N ormalized Switching P ower Loss
N ormalized Switching P ower Loss
1.4
1.4
1.2
3
1.0
4
depends on the PWM technique employed.
The total harmonic distortion (THD) in current IT HD is
defined as the ratio of RMS value of the ripple current to
RMS value of the fundamental component of the total current.
s
2
2
IRM
IeRM S
S − I1RM S
=
(9)
IT HD =
2
I1RM S
I1RM S
As current ripple is proportional to stator flux ripple, the
harmonic distortion in stator flux can be taken as the analytical measure of harmonic distortion in line current. The
THD in stator flux ΨT HD is given by
ΨT HD =
e RM S
Ψ
Ψ1RM S
5
CSVPWM(1)
0.6
60◦ Clamp PWM(2)
30◦ Clamp PWM(3)
0.4
Three zone PWM(4)
Two zone PWM(5)
5
10
15
20
25
30
35
40
45
50
F undamental F requency (Hz)
Figure 15: Normalized switching power loss for
different PWM techniques Vs fundamental frequency at p.f.= 0.866 lagging (Normalization is w.r.t.
CSVPWM).
depends on the PWM technique employed.
For calculating the normalized switching power loss, the
normalized switching energy loss needs to be multiplied by
the sampling frequency. The normalized switching power
loss is calculated for the four PWM techniques considered
and it is once again normalized w.r.t. CSVPWM. The variation of this normalized switching power loss over the full
range of power factor angle is shown in Fig.14 for all the
PWM techniques. The variation of normalized switching
power loss for different PWM techniques with fundamental
frequency at a power factor 0.866 lagging is shown in Fig.15.
(10)
where Ψ1RM S is the RMS value of the fundamental component of stator flux, which is directly related to VREF . Hence,
for a given switching frequency, ΨT HD can be calculated for
various PWM techniques over the range of VREF .
Fig.12 shows ΨT HD for different PWM techniques as a
function of fundamental frequency. As mentioned in section 2, the sampling interval Ts = 23 T for sequences 012
and 721, while Ts = T for other sequences. The duration T
is chosen as 200µs, yielding an average switching frequency
1
fsw = 2T
= 2.5kHz. Fig.13 shows normalized total harmonic distortion for different PWM techniques. The normalization here is with respect to the THD corresponding
to CSVPWM.
5.2
2
0.8
0.2
Figure 14: Normalized switching power loss for different PWM Techniques Vs Power factor angle for
VREF = 0.866p.u. (1.0 p.u. = VDC ). Normalization is
w.r.t. CSVPWM.
1
Evaluation of switching losses
In order to evaluate the switching loss performance of
different PWM techniques, the total switching loss in the
inverter has to be calculated. The average switching loss
per phase over a fundamental cycle is equal to the total
switching loss (three phases together) averaged over a sector.
Presently, the normalized switching loss is used for comparison.
The normalized switching energy loss averaged over a sector for a given power factor angle φ is given by
Z π
3
3
ESU B SEQ (α) dα
(11)
EAV =
π 0
where SEQ is the sequence applied in that sub-cycle, which
5.3
Discussion
It is evident from Fig.12 and Fig.13 that all the techniques
discussed give a better performance in terms of THD compared to CSVPWM when the drive is operating at higher
modulation indices. The frequency range of superior performance is typically 35Hz to 50Hz. The 30◦ clamp PWM
gives the best performance in terms of THD over the range
of 35Hz to 45Hz. The two-zone and three-zone PWM techniques are almost identical in THD performance. These lead
to a reduction of 42% in THD when compared to CSVPWM
at 50Hz.
Referring to the normalized switching loss curve in Fig.14,
the two-zone PWM technique is the best in terms of switching loss for lagging power factor angles ranging from 0◦ to
60◦ . It gives 30% reduction in switching loss over CSVPWM
at a power factor angle close to 10◦ lagging. For leading
power factors ranging from unity to 0.707, 60◦ clamp PWM
is the best technique, giving a maximum reduction of 22%
at unity p.f.
6.
EXPERIMENTAL RESULTS
CSVPWM, two-zone PWM and three zone PWM techniques are implemented using ALTERA CycloneII based
FPGA platform [12]. A 2.2kW, 415V, 50Hz, squirrelcage
induction motor is operated at no load from a 10kVA IGBT
based two level VSI with a DC bus voltage of 600V. The
average switching frequency used for the PWM techniques
is 2.5kHz. The experimental motor current waveforms are
T otal Harmonic Distortion(%)
11
10
1
9
3
2
8
7
CSVPWM(1)
Three Zone PWM(2)
6
5
30
Two Zone PWM(3)
32
34
36
38
40
42
44
46
48
50
F undamental F requency (Hz)
Figure 17: Measured THD in Line Current for different PWM Techniques Vs fundamental frequency
at an average switching frequency of 2.5kHz.
(a) CSVPWM at 50Hz
(b) Three Zone PWM (Using 0127,0121 and 7212) at 50Hz
shown in Fig.16.
The THD measurements of these waveforms show that at
50Hz, the THD in line current for CSVPWM is 10.07%. The
three-zone technique gives a THD of 6.00%, which is 40.4%
less compared to CSVPWM. This is the best reduction in
THD among the techniques discussed. The two-zone technique gives a THD of 6.43%, which is 36.15% less compared
to CSVPWM. These techniques yield superior performance
at high modulation indices. The typical range of superior
performance is 35-50 Hz. The measured THD values over
this range of frequencies is shown in Fig.17. It is observed
that the reduction in harmonic distortion starts at 38Hz,
and increases towards the rated operating point.
7.
CONCLUSION
A method of evaluating different PWM techniques in terms
of harmonic distortion as well as switching losses is presented. The analysis for any technique can be pursued along
similar lines. Four different hybrid PWM techniques are
compared considering CSVPWM as the benchmark. The
two-zone and three-zone techniques are proved to be better
than CSVPWM, both in terms of THD as well as switching
losses. While the distortion due to two-zone and three-zone
PWM are comparable, the former results in a significantly
reduced switching loss at high lagging power factors among
the techniques considered. Hence, two-zone PWM technique
is well-suited for motor drive applications.
(c) Two Zone PWM (Using 0127 and 7212) at 50Hz
Figure 16: Line current waveforms of the motor at
no load. The measured THD values are: (a) 10.07%,
(b) 6.00% and (c) 6.43%.
8.
REFERENCES
[1] J. Holtz, “Pulse width modulation for electronic power
conversion”, Proc. IEEE, vol. 82, no. 8, pp. 1194-1214,
Aug. 1994.
[2] D. G. Holmes and T. A. Lipo, Pulse Width Modulation
for Power Converters: Principles and Practice, IEEE
Press and John Wiley Interscience, New York, 2003.
[3] J. Kolar, F. C. Zach and H. Ertl, “Influence of the
modulation method on conduction and switching
losses of a PWM converter system”, IEEE Trans. Ind.
Appl., vol. 27, no. 6, pp. 1063-1075, Nov. /Dec. 1991.
[4] A. M. Hava, R. J. Kerkman and T. A. Lipo, “Simple
analytical and graphical methods for carrier-based
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
PWM-VSI drives”, IEEE Trans. Power Electron.,
vol. 14, no. 1, pp. 49-61, Jan. 1999.
G. Narayanan, H. K. Krishnamurthy, Di Zhao and
R. Ayyanar, “Advanced bus-clamping PWM
techniques based on space vector approach”, IEEE
Trans. Power Electron., vol. 21, no. 4, pp. 974-984,
Jul. 2006.
H. Krishnamurthy, G. Narayanan, R. Ayyanar and
V. T. Ranganathan, “Design of space-vector based
hybrid PWM techniques for reduced current ripple”,
in Proc. IEEE-APEC ’03, Miami, Florida,
pp. 583-588, Feb. 2003.
Di Zhao, G. Narayanan and R. Ayyanar,“Switching
loss characteristics of sequences involving active state
division in space vector based PWM”, in Proc.
IEEE-APEC ’04, Anaheim, California, pp. 479-485,
Feb. 2004.
G. Narayanan, Di Zhao, H. K. Krishnamurthy,
R. Ayyanar and V. T. Ranganathan, “Space vector
based hybrid PWM techniques for reduced current
ripple”, IEEE Trans. Indl. Electron., vol. 55, no. 4,
pp. 1614-1627, Apr. 2008.
Werner Leonhard, Control of Electric Drives, Springer
International Edition.
G. Narayanan, “Synchronised pulse width modulation
strategies based on space vector approach for
induction motor drives”, Ph. D. Thesis, Dept. of
Electrical Engineering, Indian Institute of Science,
Bangalore, Aug. 1999.
G. Narayanan and V. T. Ranganathan, “Analytical
evaluation of harmonic distortion in PWM AC drives
using the notion of stator flux ripple”, IEEE Trans. on
Power Electron., vol. 20(2), pp. 466-474, Mar. 2005.
S. Venugopal and G. Narayanan, “Design of FPGA
based digital platform for control of power electronics
systems”, Proc. National Power Electronics
Conference, NPEC 2005, Indian Institute of
Technology, Kharagpur, Dec. 2005.