Michael Bierhoff, Friedrich W. Fuchs
Christian-Albrechts-University of Kiel
Kaiserstr.2, 24143 Kiel, Germany
Tel. 0049-431-880-6104 mib@tf.uni-kiel.de, fwf@tf.uni-kiel.de
«Current Source Inverter (CSI)», «Harmonics», «Pulse Width Modulation (PWM)», «Passive Filter»
To facilitate the design of ac side filters, this work gives analytical expressions for the root mean square value of the total harmonic phase currents in voltage and current source converters depending on the modulation schemes and passive filter components. Novel equations were found for the calculation of total harmonic phase currents regarding the CSC with LC filter. The results are verified by appropriate measurements for selected PWM schemes and are valid for most common applications.
Apart phase displacement of voltage and current fundamentals power quality considerations can usually be reduced to the analysis of the phase current trajectory as generated by the power converter. This quality is commonly measured by the total harmonic distortion factor, THD, or just simply the RMS value of the harmonic content. When operating a complete converter set up including filter circuitry these properties are simple to determine by an FFT analyzer. Before operation, to facilitate filter design, current harmonics to be expected by the intended filter topology or pulse width modulation strategy have to be predicted and compared with the particular requirements. Usually the choice of the right filter is a trade off between costs and weight of the filter and the compliance to power quality requirements.
Much work has already been dedicated to this subject, especially the current harmonics of the most common three phase voltage source converter have been analyzed extensively. There are approaches to express the total harmonic output current as a function of the modulation index which yield simple equations for each modulation scheme [1]-[4]. For most applications a negligible error was reported although these methods only deal with the influence of the inductive load dues on the pulsed converter voltage based on calculations that are conducted in time domain. More sophisticated attempts utilizing
Fourier transformations [5]-[7] achieve accurate but bulky results which are uncomfortable to use but comprehensively applicable to all kinds of filter topology. In this paper an extension of the calculation method in time domain addressing the CSC operated with LC filter is presented. This was realized with the filter being approximated by two separate consecutive integral operations to simplify the calculation.
Results for different modulation schemes are given and an assessment on the error created by this method is pointed out. Furthermore measurements were carried out at a CSC test stand to prove the analytical results.
In the first section of this article the calculation of the pulsed output quantities is concerned and thereby striking similarities between both converter types are found. The second and third sections are dedicated to the calculation of the phase ripple current for VSC and CSC respectively. This is followed by measurement results taken for the CSC with LC filter and a conclusion.

The presented analysis is based on converter and filter circuitry as displayed by figure 1. A fairly high switching frequency is assumed to permit the following approximations. The calculation methods are based on quasi continuous considerations. Regarding the harmonic frequencies that would start at a range around the switching frequency the voltage source converter is considered to be operated in a combination with an active but solely inductive load only. The current source converter usually is assumed to be operated with an LC filter also with resistive parts being neglected for the considered frequency range. In both cases the inductance L f
constitutes both, filter chokes and load dues. Moreover the dc bus voltage V
(VSC) and the dc bus current I d
(CSC) are assumed to be constant. The complex space vector plane is utilized to facilitate the calculation of RMS values of the pulsed output values by confining the d corresponding integral boundaries to sections of π /3 of one fundamental period due to the well known symmetry. Figure 2 reveals an exemplary command space vector residing in such a section. The switching intervals t
V0
+t
V7
, t
V1
and t
V2
are calculated for the VSC as follows in (1). The corresponding formulas for the CSC are given by (2). Time spans t
V0
and t
V7
from (1) resemble the intervals during which the upper or lower half bridge of the VSC is shortened respectively. During time spans t
I7
– t
I9
from (2) the phase legs of the CSC are shortened each. All other time periods constitute the switching times for the first and second active space vector of each sector. At this stage it is important to point out the deviating definitions of the modulation indices of M
V
= 0..2/ √ 3 for the VSC and M
I
= 0..1 for the CSC.
V
VL
V
VConv i
VL
=i
VConv
V
1
V
3
I
V
5 d β
V
2
M
V
2
V
4
V
6
V d
V
2 β t
V2
/T s
.
V
2
V* =M
V
.
V d
/2 .
cos ( ω t' )
L f
ω t' V
1 t
V1
/T s
.
V
1
V
2 α
α
M
V
IL i
IL
L f
V
IC
C f i
IConv
V
1
V
3 i
IC
V
2
V
4
V
L d
I d
V
5
6
V d
I
2 β
= -I
1 β
β I
2 t
I2
/T s
.
I
2 t
I1
/T s
.
I
1
ω t'
I
1 α
= I
2 α i * =M
I
.
I d
.
cos ( ω t' )
α
I
1
Fig. 1: Power circuit of VSC (top) and CSC (bottom) Fig. 2: Command space vector generation for
VSC (top) and CSC (bottom)

t
V 1
=
2
3
⋅ M
V
⋅ sin


π
3
− ω t '


⋅ T
S
, t
V 2
=
2
3
⋅ M
V
⋅ sin
( )
' ⋅ T
S
, t
V 0
+ t
V 7
= T
S
− t
V 1
− t
V 2
(1) t
I 1
= M
I
⋅ sin
π
3
− ω t '


⋅ T
S
, t
I 2
= M
I
⋅ sin ⋅ T
S
, t
I 7
+ t
I 8
+ t
I 9
= T
S
− t
I 1
− t
I 2
(2)
With the following terms the overall RMS value of the output quantities can be calculated for the
VSC (3),[4] and CSC (4).
VConv
I
~
IConv
=
2 π
3
⋅ T s
⋅ ∫
3
0
π
[ (
V
1 α
2 + V
1
2
β
)
⋅ t
V 1
(
M
V
, ω t '
)
+
(
V
2 α
2 + V
2 β
2
)
⋅ t
V 2
(
M
V
, ω t '
) ]
⋅ d ω t ' = V d
M
V
3 π
=
2 π
3
⋅ T s
⋅
π
−
∫
6
π
6
[ (
I
1 α
2 + I 2
1 β
)
⋅ t
I 1
(
M
I
, ω t '
)
+
(
I 2
2 α
+ I 2
2 β
)
⋅ t
I 2
(
M
I
, ω t '
)
⋅
] d ω t ' = I d
2 M
I
π
(3)
(4)
1
VConv
=
M
V
⋅ V d
2 ⋅ 2
, 1 I
~
IConv
=
M
I
⋅ I d
2
(5)
By determining the fundamental components according to (5) and selecting a generalized modulation index M = M
I
= M
V
.
√ 3/2 = 0..1 a generally valid formula for both converter types regarding the total harmonic distortion, THD, of output quantities for both converter types can be found (6). Figure 3 shows the graphical results for the generally valid THD which are independent of the modulation method as long as linear PWM is applied.
THD
V , I
= harm
~
~
VConv
VConv
= harm I
~
IConv
I
~
IConv
= 1 − M
π
4
(6)
1
0,8
0,6
THD(M )
0,4
0 0,2 0,4 0,6 0,8 1
M
Fig. 3: Generally valid representation of the THD of pulsed output quantities for a VSC and a CSC versus the modulation index M

This section contains an overview of what has been published so far by various other authors. For most calculations regarding the harmonic content of the VSC phase current the ac side is assumed to comprise an active load with a purely inductive impedance for the considered frequencies. With this assumption the ac inductance of the VSC only constitutes an integral element for the pulsed output voltages. The resulting total harmonic current already has been calculated in the past by many other sources, see for example [1]-
[4]. The phase current harmonics indeed are depending on the modulation strategy and switching frequency f s
. Thus table I gives the expressions for the RMS value of the total harmonic current on the ac side for a selection of modulation waveforms (figures show a modulation index of M
V
= 1/ √ 3) along with a triangular carrier each (symmetrical appearance of pulses within one PWM period). The well known corresponding graphs are revealed by figure 4.
Modulation function
Mod 1
1
0
M( ω t)
-1
0
ω t
Mod 2
1
π 2 π
0
M( ω t)
-1
0
ω t
Mod 3
1
0
= 0
M( ω t)
-1
0
ω t
π
π
Mod 4
1
0
M( ω t)
-1
0
ω t
Mod 5
1
0
M( ω t)
-1
0
ω t
π
π
2 π
2 π
2 π
2 π
Expressions for the RMS harmonic current content harm I
~
VL
= harm I
~
VConv
=
V d
M
V f s
L f
1
384
⋅


1 −
8
3 π
⋅ M
V
+
3
4
⋅ M
V
2

 harm I
~
VL
= harm I
~
VConv
=
V d
M
V f s
L f
1
384
⋅



1 −
8
3 π
⋅ M
V
+
9
8

1 −
3
4 π
3
⋅ M
V
2

 harm I
~
VL
= harm I
~
VConv
=
V d
M
V f s
L f
1
384
⋅


4 −
8 + 15
3 π
3
⋅ M
V
+
9
8
2 +
2 π
3
⋅ M
V
2

 harm I
~
VL
= harm I
~
VConv
=
V d
M
V f s
L f
1
384
⋅ 4 −
35
3 π
⋅ M
V
+
9
8


2 +
4
9
3 π


⋅ M
V
2 harm I
~
VL
= harm I
~
VConv
=
V d
M
V f s
L f
1
384
⋅ 4 −
35
3 π
⋅ M
V
+
9
8


2 +
4
9
3 π


⋅ M
V
2

0.04
Mod 3
Mod 4 Mod 5
Mod 1
0.02
harm I
~
VL f s
L f
V d
Mod 2
0
0 0.2
0.4
0.6
0.8
1.0
1.15
M
V
Fig. 4: RMS value of the total harmonic current for different modulation waveforms generated by a VSC
The simplified assumptions for the corresponding calculations in time domain can be described as a simple transfer function (7) for considerations in the frequency domain.
i
VL
V
VConv
= Z f
= ω L f
(7)
In this section the current source converter is analyzed in a straight forward manner extending the approximations mentioned above [1]-[4]. The actual difference to the VSC is constituted by the filter with an order higher by one. Here the capacitor and inductor circuits will be decoupled for the calculation to attain rather simple solutions in time domain as a novel approach. In principle the converter’s equivalent circuit diagram is split up into two independent ones. The resistive losses of the filter are neglected once again and the obtained transfer function approximates the original one as the switching frequency to resonant frequency ratio is increased (8). In this case f s
denotes the switching frequency, f res
is the resonant frequency of the filter, i
C f
IL
is the phase current and i
IConv
indicates the converter output current while L f
specify the filter inductance and capacitance respectively. The relative error e rel
and
caused by this method regarding the accuracy of the magnitude of any harmonic would be determined by (9). Where ν denotes the harmonic order and ω is the fundamental angular frequency, 2 π f.
i
IL i
IConv
=
1
1 − ω 2 L f
C f
≈
1
ω 2 L f
C f for
(
2 π ⋅ f s
)
2 L f
C f
=

 f s f res


2
>> 1 (8) e rel
( ) ( )
2
1
L f
C f
=


ν f
⋅ res f


2
(9)

harm i
IL
L f harm V
IC harm i
IC
C f harm i
Equivalent Circuits
IConv
L f harm i
IL harm V
IL
C f harm V
IC harm V
IC
C f harm i
IC
Approximations harm i
IConv
+ L f harm i
IL harm V
IC harm V
IC
Fig. 5: Evolution of equivalent circuit diagrams for the calculation of the current harmonic content
Initially the capacitor voltage waveform is calculated as all components of the ac side apart of the capacitor are neglected yielding a problem dual to that of the VSC of a first order integrator. In a second step of approximation the resulting capacitor voltage is considered ideally impressed when acting upon the inductor as shown by figure 5. For the CSC different PWM methods can be distinguished by their switched space vector sequence as presented by table II. Numbers 1 and 2 represent the first and second active space vector of one sector of 60 degrees of a fundamental period in the complex plane. Zero denotes any switched zero space vector. Unlike for the VSC, the kind of switched zero space vector does not influence the phase current or capacitor voltage ripple. Symmetrical distribution of the space vector portions that are split within one switching period is presumed for Mod 2 - 4, see also table III. Compliant to [8] the PWM methods for the CSC as observed here are chosen because of their different switching loss behavior. Mod 1 generates the least switching losses whereas Mod 2 – 4 are capable to reach these switching loss values for one specific operating point each only. But in turn they attain less capacitor voltage ripple and thus less phase current ripple. Table III reveals the influence of the modulation method for one exemplary switching period on the pulsed output current and the resulting capacitor voltage V
IC α which coincides with the capacitor voltage of phase 1. In this case the first sector of 60 degrees between space vectors I
1
and I
2
is observed. The appearance of the capacitor voltage V
IC
may vary for Mod 2 - 4 depending on the sector that the instantaneous command current vector is located in.
Denotation Space vector sequence
Mod 1
Mod 2
1 - 2 - 0
1 - 2 - 1 - 0
Mod 3
Mod 4
2 - 1 - 2 - 0
1 - 0 - 2 - 0
I
3
1 i
IConv
II
VI
I
2
I
2
1 i
IConv
I
I
1
1 i
IConv
I
1
I
6
Fig. 6: Space vector representation of the harmonics occurring in the pulsed ac current

Space vector representation of the capacitor harmonic voltage
Trajectory of the pulsed output current i
IConv
during an exemplary switching period
Trajectory of the filter capacitor harmonic voltage harm V
IC α during one exemplary switching period
Mod 1 harm V
IC
(t
µ,1
) t
I0
β harm V
IC
(t
µ,3
) i
IConv1 t i
IConv2 t
I1 t
I2 α i
IConv3 t harm V
IC
(t
µ,2
) t
Mod 2 β harm V
IC
(t
µ,4
) harm V
IC
(t
µ,1
) t
I0 t
I1
/2 t
I1
/2 harm V
IC
(t
µ,2
) t
I2 harm V
IC
(t
µ,3
) α i i i
IConv1
IConv2
IConv3 t t t harm V
IC α t
Mod 3 β harm harm V
IC
(t
µ,2
) t
I2
/2
V
IC
(t
µ,1
) t
I1 t
I0 harm V
IC
(t
µ,4
)
α t
I2
/2 harm V
IC
(t
µ,3
) i i i
IConv1
IConv2
IConv3 t t t
Mod 4 β harm V
IC
(t
µ,1 harm V
IC
(t
µ,3
)
) t
I0
/2 harm V
IC
(t
µ,4
) t
I2 t
I1 t
I0
/2 harm V
IC
(t
µ,2
)
α i
IConv1 i
IConv2 i
IConv3 t t t harm V
IC α t
The total harmonic capacitor voltage as shown in table III is basically generated by integration of the instantaneous harmonic output current by the capacitor as depicted in figure 6 for one exemplary switching period. The coordinate plane has been displaced by the fundamental amplitude to disregard the fundamental component. Hence there are only three directions for the illustrated vectors but depending on their switching sequence assigned by the respective modulation method the total outlines of the space vector diagrams appear differently. The heights of the slopes for the capacitor voltage as given by table

III, column 3, can be calculated by (9) for the most simple case which means in total there are two ramps during one switching period only. This applies for Mod 1 over the total of one fundamental period. The other three modulation methods encounter two 60°-sectors each in which the capacitor voltage exhibits a trajectory as displayed for Mod 4 in table III. These are the sectors where the capacitor voltage contains less ripple as for modulation method Mod 1. For these cases (9) has to be adjusted appropriately.
∆ harm V
IC α , Mod 1
=
C f
I d
⋅ f s
⋅ cos
( )
⋅
(
M
I
− M
I
2 ⋅ cos
( ) )
(9)
Integration of the capacitor voltage trajectories for the most simple case that is Mod 1 yields an expression for the RMS value of the ac ripple current harm Ĩ
IL
according to (10). All final equations for the different modulation methods as treated here are compiled in table IV. The derivation for these expressions have to be done distinctively for each sector in which the capacitor voltage trajectory appears differently to that of modulation method Mod 1 depending on the modulation method.
A ( ω t ) =


1
30
⋅
[ (
M
I
⋅ cos( ω t )
) (
1 − M
I
⋅ cos( ω t )
)
3 ⋅
]
[ (
M
I
− M
I
2 ⋅ cos( ω t )
)
⋅
2 cos( ω t )
]


2
B ( ω t ) =


1
6
⋅
[ (
M
I
⋅ cos( ω t )
) (
1 − M
I
⋅ cos( ω t )
)
2 ⋅
]
[ (
M
I
− M
I
2 ⋅ cos( ω t )
)
⋅
2 cos( ω t )
]


2 harm I
~
IL
=
C f
I d
L f
⋅ f s
2
π
−
2
∫
π
2
[
A ( ω t ) − B ( ω t )
] d ω t (10)
Modulation
Mod 1
Expressions for the RMS harmonic current content harm I
~
IL
=
C f
I d
L
⋅ M f
⋅ f
I s
2
⋅
1
288


0 .
2 − 0 .
75 M
I
2 +
2 .
56
π
M
I
3 − 0 .
25 M
I
4


Mod 2
Mod 3
Mod 4 harm I
~
IL
=
C f
I d
L f
⋅ M
⋅
I f s
2
⋅
1
36864




25

+
.
6
π
−
430
96
.
88
M
+
I
3
36
−
π
3
M
I
2
34 +
20 .
25
π
3
M
I
4




 harm I
~
IL
=
C
I d f
L
⋅ M f
⋅
I f s
2
⋅
1
18432





20
+
.
8
67
π
.
−
24
84
π
M
I
3
3
+
+
90
π
20 .
25
π
3
− 72
3
− 17
M
I
2
M
I
4






0,01
0,008
0,006
Mod 4
harm I
~
IL
L f
C f
I d f s
2
0,004
0,002
0
0 0,2 0,4 0,6 0,8 1
M
I
Fig. 7: Measured RMS ripple current content versus modulation index
In figure 7 the calculated and standardized values of the harmonic current content harm Ĩ
IL
is given versus the modulation index M
I
by the solid lines. Mod 5 represents the modulation method with the least ripple current but at the same time executing the most switching operations increasing the number of switching operations by one compared to Mod 2 - 4. Compliant to table II it would constitute a space vector sequence like 1,2,0,2,1,0 and the alignment of the space vector portions again would be symmetrically arranged over one switching period. The resulting graph in figure 7 was calculated numerically.
The provided measurement results do not account for the VSC as such can be found in the literature [4].
Measurements of the total harmonic current where taken from an IGBT 22kW/400V three phase current source converter operated with modulation methods Mod 1, 2, 3 and Mod 5 since the pulse width modulator that was used, yet is not capable of generating Mod 4. The dc current was set to I d all measurements. The three phase ac side only consists of an LC-filter (L f switching (carrier) frequency was set to f s
= 3.5mH, C f
= 10 A for
= 48µF). The
= 3kHz. The converter’s phase current was measured by a current probe at steady state operation. Measurement values were sampled and plotted with an ONO
SOKKI CF-5120 FFT analyzer. The data evaluation including an Fourier analysis was done using
Matlab TM . The experimental results are represented by differently shaped dots in figure 7.
Filter design is a crucial part of power converter application. Expected current distortions have to be determined in advance to assess the performance of any suggested filter or modulation method. The well known expressions of the total harmonic current of a VSC are presented in a review. For the CSC a novel and easy to handle method has been developed in a similar manner. The presented approach yields simple functions of the modulation index M
I
for the RMS value of the total harmonic current on the ac side of an
CSC with LC filter. Thereby four different modulation schemes have been addressed. A high pulse rate
(>20) and a sufficiently high switching frequency to resonant frequency ratio (>>1) of the LC filter provided, the error generated by this method is negligible which holds true for most applications.
Measurements have been carried out to prove the figured analytical results.

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