IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 1, JANUARY 2013
19
Letters
Spectral Analysis of Matrix Converters Based on 3-D Fourier Integral
Bingsen Wang and Emad Sherif
Abstract—This letter proposes an analytical method based on
3-D Fourier integral to obtain accurate spectra of both the switching functions and the synthesized terminal quantities of a matrix
converter. The challenges associated with the spectral analysis of
matrix converter waveforms are twofold. On one hand, the modulation signal contains both the input and output frequencies. Unlike
the third-harmonic injection in the modulation functions, the input frequency and the output frequency are typically independent
from each other and will not form an integer ratio. On the other
hand, it is very common that the switching frequency or the carrier
frequency is not rational multiple of either the input frequency or
the output frequency. These aforementioned challenges make it a
very challenging task to accurately characterize the spectra of matrix converter waveforms through commonly resorted numerical
methods such as a fast Fourier transform (FFT). The contribution of the proposed analytical method lies in providing accurate
solution to spectral analysis of matrix converters when the FFT
approach fails to characterize the spectral performance of matrix
converters under typical operating conditions.
Index Terms—Fourier series, harmonic analysis, matrix converter, pulsewidth modulation (PWM).
I. INTRODUCTION
HE superior spectral performance of matrix converters and
compact realization continue to spur extensive research
activities in matrix converters [1]–[10]. Significant effort has
been directed to better understanding of the modulation process
and improving the performance of the modulators [2].
The high-frequency-synthesis modulation was first proposed
by Alesina and Venturini for nine-switch direct matrix converters (DMC) [11], [12]. The subsequently developed modulation
schemes can be categorized into two types, namely, space vector
modulation [13]–[16] and carrier-based modulation [17]–[21].
The space vector modulation scheme proposed by Huber et al.
functionally divided the nine-switch DMC into a rectification
stage and an inversion stage [13], [22]. The topological implementation of such two-stage formulation led to indirect matrix converter (IMC) topologies and their variations [23], [24].
More general formulation of the space-vector-based approach
proposed in [14] and a more recent paper [16] provides a unified
T
Manuscript received March 7, 2012; revised May 20, 2012; accepted June
12, 2012. Date of current version September 11, 2012. Recommended for publication by Associate Editor M. (Letters AE) Molinas.
The authors are with the Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824 USA (e-mail:
bingsen@egr.msu.edu; Sherifem@egr.msu.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPEL.2012.2206118
view of different modulation strategies. The carrier-based modulation approaches typically require less computational complexity [18], [19], [25].
The performance metrics of different modulation schemes
mainly concern with the switching losses [7], [15], [26], common mode voltage [27], and harmonic distortion of synthesized
voltage and current waveforms [21], [28], [29]. The evaluation
of the waveform quality typically involves spectral analysis.
The challenges associated with the spectral analysis of matrix
converter waveforms are twofold. On one hand, the modulation
signal contains both the input and output frequencies. Unlike the
intentional third-harmonic injection in the modulation functions
of voltage source inverters, the input frequency and the output
frequency in matrix converter are typically independent from
each other and will not form an integer ratio. On the other hand,
it is very common that the switching frequency or the carrier
frequency is not rational multiple of either the input frequency
or the output frequency [30]. These aforementioned challenges
make it a very difficult task to obtain accurate spectra of matrix converter waveforms through commonly resorted numerical
methods such as the fast Fourier transform (FFT), which may
include the harmonics that do not exist in the actual waveforms.
This letter proposes an analytical method based on 3-D
Fourier integral to obtain accurate spectra of switching functions and synthesized terminal quantities of a matrix converter.
The proposed analytical approach will provide accurate results
without the assumption of particular ratio between the switching
frequency and the fundamental frequencies while the FFT typically fails without such assumptions. The proposed method is
an extension of the double Fourier integral analysis of the output
voltage waveforms of voltage source inverters [30]. This letter is
organized as follows. The formulation of triple Fourier analysis
is introduced in Section II followed by its application to the
pulsewidth modulation (PWM) in Section III. The detailed spectral analysis of matrix converters’ input and output waveforms
is presented in Section IV and verified with FFT in Section V.
A summary discussion in Section VI concludes this letter.
II. TRIPLE FOURIER INTEGRAL
The challenge associated with the nonperiodicity of PWM
waveform was solved by the mathematical treatment called
double Fourier analysis [30]. The most well-known analytical method of determining the harmonic components of a
PWM switched waveforms was first developed by Bowes and
Bird [31], who adapted an earlier approach that was originally
proposed for communication systems by Bennet and Black to
modulated converter systems [32], [33].
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As an extension of the double Fourier series, let a triplevariable function f (x, y, z) be periodic in x-, y-, and zdirections. It is further assumed that x, y, and z are angular
variables and the period in all directions is 2π, i.e.
f (x, y, z) = f (x + 2π, y, z) = f (x, y + 2π, z)
= f (x, y, z + 2π).
(1)
With reference to the double Fourier expansion, a triple
Fourier expansion is proposed. If a function f (x, y, z) is periodic in x-, y-, and z-directions with the period of 2π, the triple
Fourier expansion can be obtained by the following:
+∞
+∞
+∞
Fk m n ej (k x+m y +n z )
(2)
f (x, y, z) =
k =−∞ m =−∞ n =−∞
where
Fk m n
1
=
8π 3
2π
f (x, y, z)e−j (k x+m y +n z ) dxdydz
0
Alternatively, the triple Fourier series can be represented with
real coefficients as shown in (3) at the bottom of the page, where
the Fourier coefficients Ak m n and Bk m n are determined by the
following triple-integrals:
2π
1
Ak m n =
f (x, y, z) cos(kx + my + nz)dx dy dz
4π 3
0
2π
1
Bk m n =
f (x, y, z) sin(kx + my + nz)dx dy dz.
4π 3
0
The real coefficients Ak m n and Bk m n are related to the complex coefficients Fk m n by
Fk m n =
Ak m n − jBk m n
2
III. APPLICATION OF TRIPLE FOURIER SERIES TO PULSE
WIDTH MODULATION
For the PWM in matrix converters, there exist two lowfrequency components called modulation functions and one
high-frequency signal named carrier signal.
Let the three variables x, y, z be functions of time t as defined
by
x(t) = ωc t + θc
y(t) = ωm 1 t + θm 1
z(t) = ωm 2 t + θm 2
where θc is the phase angle of the carrier signal while θm 1 and
θm 2 are the phase angles of the two low-frequency modulation
signals. ωc is the angular frequency of the carrier signal while
ωm 1 and ωm 2 are two independent angular frequencies of the
two low-frequency modulation signals. It is worth noting that
no particular ratio between these frequencies has been assumed
for the subsequent derivation.
The comparison between the carrier signal c(x) and modulation function m(y, z) gives rise to the switching function
h(x, y, z) that determines the switching instants of a particular
switch in a power converter, i.e.
h(x, y, z) = Φ (m(y, z) − c(x))
Ak m n = Fk m n + Fk m n
where Φ(·) is the modified signum function defined as
1 if u > 0
Φ(u) ≡
0 if u ≤ 0.
h(t) = h(x(t), y(t), z(t))
Bk m n = j(Fk m n − Fk m n )
(4)
=
where “ ” denotes the conjugate of a complex quantity.
⎢
⎢
⎢
⎢
⎢
+⎢
⎢
⎢
⎢
⎢
⎣
DC Offset
⎤
k =1
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎢
+
[A0m 0 cos(my) + B0m 0 sin(my)] ⎥
⎥+⎢
⎥ ⎢
m =1
⎥ ⎢
⎥ ⎢
+∞
⎦ ⎣
+
[A00n cos(nz) + B00n sin(nz)]
[Ak 00 cos(kx) + Bk 00 sin(kx)]
k =1
+∞
n =1
+∞ +∞
m = −∞,
m = 0
+∞
+∞
Hk m n ej (k ω c t+m ω m 1 t+n ω m 2 t) (7)
k =−∞ m =−∞ n =−∞
Base Bands
+
+∞
A000
2 f (x, y, z) =
+∞
(6)
The spectrum of each switching function can be obtained
through the following Fourier series expansion:
OR
⎡
(5)
⎤
⎥
⎥
⎥
⎥
⎥
+
[A0m n cos(my + nz) + B0m n sin(my + nz)] ⎥
⎥
⎥
m =1 n =1
⎥
⎥
+∞
+∞
⎦
+
[Ak 0n cos(kx + nz) + B00n sin(kx + nz)]
[Ak m 0 cos(kx + my) + Bk m 0 sin(kx + my)]
k =1 m =1
+∞
+∞ k =1 n =1
Interharm onics Between Two Indep endent Frequencies
+∞ Ak m n cos(kx + my + nz)
+Bk m n sin(kx + my + nz)
n = −∞
n = 0
+∞ +∞
Interharm onics Between Three Indep endent Frequencies
(3)
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 1, JANUARY 2013
21
Fig. 2.
vi1
ii1
vi2
ii2
vi3
ii3
T11
T12
T13
vo1
T21
T22
T23
vo2
T31
T32
T33
vo3
io1
io2
io3
Schematic of a conventional matrix converter.
IV. SPECTRAL ANALYSIS OF MATRIX CONVERTER WAVEFORMS
Fig. 1. Illustration of the unit cube where the integration on the right-hand side
of (8) is carried out. x-axis represents the phase angle of the carrier signal while
y- and z-axes represent the phase angles of the two independent low-frequency
components in the modulation function.
where
Hk m n
1
=
8π 3
2π
As an illustrative example, the schematic of a conventional
matrix converter that consists of three single-pole-triple-throw
(SPTT) switches is shown in Fig. 2. Each of the SPTT switches
can be realized with the semiconductor devices that are also
shown in Fig. 2 although the particular realization is insignificant
to the discussion of the spectral analysis.
h(x, y, z)e−j (k x+m y +n z ) dxdydz. (8)
A. Modulation Process
0
For the modulation function with two independent low frequencies and the carrier signal being triangular, the 3-D integral in
(8) would be carried out in a cube with each edge of the length of
2π as shown in Fig. 1, where x-axis represents the phase angle
of the carrier signal and y- and z-axes represent the phase angles
of the two components in the modulation function. Within the
unit cube, the switching function h(x, y, z) is nonzero only in
the space between the two surfaces that are defined as follows:
m(x, y) − c(z) = 0.
(9)
Furthermore, the line defined as follows is also illustrated in
Fig. 1
ωm 1
ωm 1
x + θm 1 −
θc
y=
ωc
ωc
ωm 2
ωm 2
z=
x + θm 2 −
θc .
(10)
ωc
ωc
The intersections of the line defined by (10) and the surfaces
defined by (9) determine the instants when the value of the
switching function h(x, y, z) alternates between “0” and “1.”
Due to the periodicity, the unit cube is repeated in all three
dimensions.
The procedure of carrying out the integration is a relatively
straightforward process once the integration boundaries have
been defined, as will be explained with reference to conventional
matrix converter in the next section.
Although the modulation process is well understood, the inclusion of this section is to set up the notations that will be
utilized in subsequent sections. The input voltages are assumed
a three-phase balanced set as
2π
vi(v ) = Vi cos ωi t − (v − 1)
for v ∈ {1, 2, 3} (11)
3
where ωi is the angular frequency of the input voltages and Vi
is the amplitude of the input voltages. In addition, the output
currents are assumed a three-phase balanced set defined as
2π
io(u ) = Io cos ωo t − (u − 1)
for u ∈ {1, 2, 3} (12)
3
where ωo is the angular frequency of the output currents and Io
is the amplitude of output currents.
For each throw Tu v with u, v ∈ {1, 2, 3}, the corresponding
modulation function is denoted as mu v while the corresponding
switching function is denoted as hu v . The same single carrier
signal is assumed for all nine throws. The modulation function
proposed by Alesina and Venturini [11] is assumed. The modulation function for throw Tu v is defined in the following general
form:
mu v (y, z) =
2π
1+M cos y−(u + v + 1) 2π
3 +M cos z−(u−v) 3
. (13)
3
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 1, JANUARY 2013
1.0
B. Analytical Spectra of Switching Functions
c(x)
For the switching functions given in (16), the corresponding
Fourier series in complex form is
mu1(x,y,z)+mu2(x,y,z)
mu1(x,y,z)
0
t
Tc
hu1(x,y,z)
Tc
hu2(x,y,z)
t
hu v (x, y, z) =
hu3(x,y,z)
Tc
Fig. 3.
Switching function for each single-pole-triple-throw switch.
where M is the modulation index with 0 ≤ M ≤ 0.5; the phase
angles y and z are defined by y = ωm 1 t, z = ωm 2 t with ωm 1
and ωm 2 further being related to the input frequency ωi and
output frequency ωo by
1
8π 3
1
arccos(cos x).
(14)
π
The switching function hu v (x, y, z) for throw Tu v is then determined in (6). The following constraint among the switching
functions for the same SPTT imposed
1
Hk m n (u, 1) =
8π 3
for u ∈ {1, 2, 3} .
(17)
v =1
According to reciprocal rules, the synthesized input currents can
be determined by
ii(v )
3
hu v (x, y, z)io(u )
=
u =1
for v ∈ {1, 2, 3}.
π m
u 1 (y ,z )
e−j (k x+m y +n z ) dxdydz.
−π −π −π m u 1 (y ,z )
π π π m u 1 (y ,z )
1
e−j (m y +n z ) dxdydz
8π 3 −π −π −π m u 1 (y ,z )
⎧1
⎪
for m = n = 0
⎪
⎪
3
⎪
⎪
⎪
⎪
⎨ M e∓j (u +2) 23π for m = ±1, n = 0
=
(22)
6
⎪
⎪
⎪ M ∓j (u −1) 23π
⎪
e
for m = 0, n = ±1
⎪
⎪
⎪
⎩ 6
0
for m, n = 0
(15)
Hence, the three switching functions for each SPTT are generated in the following manner:
⎧
if v = 1
⎪
⎨ Φ [mu 1 (y, z) − c(x)]
hu v (x, y, z) = 1 − Φ [mu 3 (y, z) − 1 + c(x)] if v = 3
⎪
⎩
1 − hu 1 (x, y, z) − hu 3 x, y, z if v = 2
(16)
for u ∈ {1, 2, 3}. The generated switching functions are illustrated in Fig. 3.
The synthesized output voltages with respect to the neutral of
the input voltages are subsequently determined by
π π
H0m n (u, 1) =
v =1
hu v (x, y, z)vi(v )
−π
hu v (x, y, z)e−j (k x+m y +n z ) dxdydz.
If k = 0
c(x) =
3
π
The evaluation of (20) can be proceeded with the consideration of (16), (13), and (9). The spectrum of each of the switching
functions in one SPTT can be determined in three different cases.
1) When v = 1: By letting v = 1, substituting (16) into (20)
leads to
A triangular carrier signal c(x) can be expressed by
vo(u ) =
(21)
ωm 2 = ωo − ωi .
hu v (x, y, z) = 1.
Hk m n (u, v)ej (k x+m y +n z )
(20)
ωm 1 = ωo + ωi
3
+∞
(19)
where the complex coefficients Hk m n (u, v) are determined by
t
t
+∞
k =−∞ m =−∞ n =−∞
Hk m n (u, v) =
Tc
+∞
(18)
If k = 0
k
+ m 2+n π
Hk m n (u, 1) =
kπ
kπ
kπ
−j [n (u −1)+m (u +2)] 23π
×e
M Jm
M (23)
Jn
3
3
sin
3
where Jn (·) and Jm (·) are the nth and mth order Bessel functions of the first kind, respectively.
2) When v = 3: For the case of v = 3, the derivation can
be proceeded in a manner similar to the case of v = 1. The
evaluation of the Fourier coefficients leads to
1
Hk m n (u, 3) = 3
8π
π π
π [1+m
u 3 (y ,z )]
e−j (k x+m y +n z ) dxdydz
−π −π π [1−m u 3 (y ,z )]
(24)
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 1, JANUARY 2013
If k = 0
⎧1
⎪
⎪
⎪
3
⎪
⎪
⎪
⎪
⎨ M e∓j (u +1) 23π
H0m n (u, 3) =
6
⎪
⎪
⎪ M ∓j u 23π
⎪
e
⎪
⎪
⎪ 6
⎩
0
If k = 0
Hk m n (u, 3) =
sin
× e−j [n u +m (u +1)]
Then, the Fourier spectrum F (ω) of f (t) can be determined by
the convolution of the spectra F1 (ω) and F2 (ω), i.e.
∞
F1 (ν)F2 (ω − ν)dν.
(32)
F (ω) =
for m = n = 0
for m = ±1, n = 0
for m, n = 0
4k
3
2π
3
−∞
(25)
for m = 0, n = ±1
+ m 2+n π
kπ
kπ
kπ
M Jm
M
Jn
3
3
23
(26)
where Jn (·) and Jm (·) are the nth and mth order Bessel functions of the first kind, respectively
3) When v = 2: For the case of k = 0
⎧1
⎪
for m = n = 0
⎪
⎪
3
⎪
⎪
⎪
⎪
⎨ M e∓j (u ) 23π
for m = ±1, n = 0
H0m n (u, 2) =
(27)
6
⎪
⎪
2π
M
⎪
∓j
(u
−2)
⎪ e
3
for m = 0, n = ±1
⎪
⎪
⎪ 6
⎩
0
for m, n = 0
This principle can be applied to obtain the spectrum of the
synthesized output voltages as well since they are the summation
of the products of the time-domain switching functions and
the input voltages as described in (17). In a similar manner,
the spectrum of the input currents described in (18) can be
determined.
For the sinusoidal input voltage waveforms that are defined
in (11), the corresponding spectrum can be calculated by
2π
Vi −j (v −1) 2 π
3 δ(ω − ω ) + ej (v −1) 3 δ(ω + ω )
e
Vi(v ) (ω) =
i
i
2
(33)
where
1 if ω = 0
δ(ω) =
0 otherwise.
The spectrum of switching function hu v (x, y, z) can also be
represented as a function of ω.
Hu v (ω) =
+∞
If k = 0,
Hk m n (u, 2) = −Hk m n (u, 1) − Hk m n (u, 3).
(28)
4) The General Form: Equations (22)–(28) can be reformulated to the general results.
If k = 0
⎧1
⎪
for m = n = 0
⎪
⎪
3
⎪
⎪
⎪
⎪
⎨ M e∓j (u +v +1) 23π for m = ±1, n = 0
H0m n (u, v) =
6
⎪
⎪
2π
M
⎪
⎪
e∓j (u −v ) 3
for m = 0, n = ±1
⎪
⎪
⎪
6
⎩
0
for m, n = 0
(29)
If k = 0
⎧
(3v −1)k
m +n
⎪
sin
π
+
⎪
6
2
⎪
⎪
⎪
⎪
⎪
kπ
⎪
⎨ −j [n (u −v )+m (u +v +1)] 2 π
3
×e
Hk m n (u, v) =
⎪
⎪
kπ
kπ
⎪
⎪
M
J
M
for v = 1, 3
×J
m
⎪ n 3
⎪
3
⎪
⎪
⎩
−Hk m n (u, 1) − Hk m n (u, 3) for v = 2.
(30)
C. Analytical Spectra of Output Voltages and Input Currents
Let Fourier spectra of functions f1 (t) and f2 (t) be F1 (ω)
and F2 (ω), respectively. The product of the two time-domain
functions is denoted as
f (t) = f1 (t)f2 (t).
[Hk m n (u, v) δ(ω − (kωc + mωm 1 + nωm 2 ))] (34)
k ,m ,n =−∞
(31)
where Hk m n (u, v) is determined in (29) and (30). Then, the
spectra of the synthesized output voltages are determined to be
∞
Vo(u ) (ω) =
Hu v (ν)Vi(v ) (ω − ν)dν =
−∞
3
2π
2π
Vi Hu v (ω − ωi )e−j (v −1) 3 + Hu v (ω + ωi )e+j (v −1) 3 .
2 v =1
(35)
For the assumed sinusoidal output current waveforms as defined in (12), a similar derivation results in the spectra of the
input currents
Ii(v ) (ω) =
×
3 Io
2
Hu v (ω − ωo )e−j (u −1)
2π
3
+Hu v (ω + ωo )e+j (u −1)
2π
3
.
u =1
(36)
V. NUMERICAL VERIFICATION OF THE ANALYTICAL RESULTS
The analytical results have been verified against the numerical
simulation based on FFT. To ensure the accuracy of the FFT
results, the interval over which the FFT is conducted has to be
the integer times of the exact period of the underlying waveform.
A time interval of 0.1 s has been chosen for carrying out the
numerical FFT given the rest parameters being listed in Table I.
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 1, JANUARY 2013
TABLE 1
LIST OF PARAMETERS USED IN SIMULATION
fi
Vi
fo
Io
fc
M
60
200
70
50
1200
0.4
0.1
Spectrum of ii1 (A)
Input frequency
Input voltage amplitude
Output frequency
Output current amplitude
Switching frequency
Modulation index
Time interval for FFT
20
Hz
V
Hz
A
Hz
10
5
0
sec
FFT Spectrum
Analytical Spectrum
15
0
0.6
1.2
1.8
Spectrum of vo1 (V)
80
ii1 (A)
50
0
2.4
3
FFT Spectrum
Analytical Spectrum
60
40
20
0
0
0.6
1.2
1.8
2.4
3
f (kHz)
-50
0
5
10
15
Fig. 6. Analytical and numerical spectra of synthesized input current ii 1 , and
synthesized output voltage v o 1 .
100
20
0
Spectrum of ii1 (A)
vo1 (V)
200
-100
-200
0
5
t (ms)
10
0.4
0
0.6
1.2
1.8
2.4
3
FFT Spectrum
Analytical Spectrum
0.3
0.2
0.1
0
0
0.6
1.2
1.8
0.6
2.4
3
FFT Spectrum
Analytical Spectrum
0.4
0.2
0
0
0.6
1.2
1.8
2.4
3
f (kHz)
Fig. 5.
h1 3 .
0
0.6
1.2
1.8
2.4
3
FFT Spectrum
Analytical Spectrum
80
60
40
20
0
0.6
1.2
1.8
2.4
3
f (kHz)
0.4
Spectrum of h12
5
0
0.2
0
Spectrum of h13
10
100
Spectrum of vo1 (V)
Spectrum of h11
FFT Spectrum
Analytical Spectrum
15
0
Fig. 4. Time-domain waveforms of the synthesized input current ii 1 in the
upper panel and the synthesized output phase-to-neutral voltage v o 1 in the lower
panel.
0.6
FFT Spectrum
Analytical Spectrum
Analytical and numerical spectra of switching functions h 1 1 , h 1 2 , and
Fig. 7. Analytical and numerical spectra of synthesized input current ii 1 , and
synthesized output voltage v o 1 with noninteger ratio between the fundamental
frequency and the switching frequency.
Under these conditions, the time-domain waveforms of the
synthesized input current ii1 and the synthesized output phaseto-neutral voltage vo1 are shown in Fig 4.
The analytical spectra of the switching functions h11 , h12 , h13
are compared with the numerical spectra obtained from FFT in
Fig. 5, which clearly demonstrates agreement between the numerical and analytical results for each switching function. Fig. 6
shows the analytical and numerical spectra of synthesized input
line current ii1 and the phase-to-neutral voltage vo1 . Again, very
good agreement between the analytical and numerical spectra
has been observed for both the synthesized output voltages and
input currents.
However, the agreement between the analytical spectra and
the corresponding spectra obtained from FFT cannot be observed if the ratio between the carrier signals and the modulation functions is not kept to particular value. For instance, if the
input fundamental frequency is changed from 60 to 61 Hz, the
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 1, JANUARY 2013
spectrum that is obtained from FFT may contain nonexistent harmonic contents. Moreover, the amplitude of harmonics in FFT
spectrum may be inaccurate as well. Such errors caused by FFT
are evident from Fig. 7. The FFT spectrum of the switching input current ii1 and the output voltage vo1 in Fig. 7 clearly shows
the nonexistent harmonics around the fundamental frequency
and the switching frequency. Similar discrepancy between the
FFT and analytical spectra of the switching function has also
been observed.
VI. CONCLUSION
This letter has presented an analytical approach to characterizing the spectra of the synthesized output voltages and input
currents of matrix converters. The analysis is carried out in two
steps: first, the spectra of switching functions are derived based
on 3-D Fourier integral. Second, the spectra of output voltages
and input currents are obtained according to the frequencydomain convolution.
The analytical approach of the 3-D Fourier integral is demonstrated through the carrier-based PWM of a conventional matrix
converter. It is worth noting that the approach is equally applicable to space vector-based modulation once the equivalent
modulation functions are determined. In addition, the modulation process of IMC can also be analytically characterized using
the proposed approach in this letter.
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