A Carrier Based PWM Algorithm for Indirect Matrix Converters

Bingsen Wang Giri Venkataramanan Department of Electrical and Computer Engineering

University of Wisconsin-Madison 1415 Engineering Drive

Madison, WI 53706 USA bingsen@cae.wisc.edu; giri@engr.wisc.edu

Abstract— In this paper, the indirect matrix converter is systematically studied with the single-pole-multiple-pole representation. A carrier based PWM algorithm is developed in two steps. First, the continuous modulation functions for all the throws are derived based on the desired sinusoidal input currents. Then the switching functions are derived from the modulation functions with focus on the zero current commutation. The proposed PWM algorithm is verified by numerical simulation and hardware experimentation on a laboratory prototype matrix converter.

I. INTRODUCTION Various attractive features of the matrix converter, such as

a potential for high power density through elimination of bulky passive components, high quality input and output current/voltage waveforms, are leading to continuous research efforts to enable their adoption widely [1-6]. While the matrix converter was first studied as a class of frequency changers realized using controlled switches by Gyugyi [7], significant progress in high frequency synthesis was made by Venturini in 1980 [8-10], with voltage transfer ratio limited to one half. An improved modulation strategy with a voltage transfer ratio of 0.866 was published in 1989 [11, 12]. In addition to Venturini’s modulation methods, indirect modulation methods based on a fictitious dc link concept [13-18] and space vector techniques have been developed further [17, 19-23]. Irrespective of the modulation methods used, robust commutation is critical, as has been extensively investigated and examined [24-30]. As an alternative to the standard 9-switch topology, the rectifier-inverter-without-dc-link structure and its variations have been carefully examined [16, 31-37], leading to a simple and robust commutation on the basis of its topological properties [34].

A space vector modulation scheme for the indirect matrix converter (IMC) was developed with focus on the desired output voltage in [34]. In this paper, a carrier based PWM algorithm is proposed and derived based on the desired sinusoidal input currents and output voltages. The derivation proceeds with the calculation of modulation functions followed by the generation of switching functions. Compared

to the space vector modulation scheme, the proposed carrier based PWM algorithm demands less computational complexity, while retaining the features of simple and robust commutation. Thus, resources of the digital signal processor commonly employed in the practical realization will be available to assume complex computational tasks for the controller rather than the modulator. Furthermore, with more increasingly available field programmable gate array (FPGA), the firmware implementation of carrier based PWM algorithm may feature the reliability of hardware and the flexibility of software.

The paper is organized as follows: In Section II the IMC is represented using single-pole-multiple-throw (SPMT) switches. The modulation functions are derived in Section III followed by switching functions presented in Section IV. The numerical simulation and the experimental results are presented in Section V and Section VI, respectively. A summary of the paper is presented in the concluding section.

II. IMC REPRESENTED BY SPMTS A family of various IMC realizations has been developed

from the topology shown in Fig. 1 by imposing additional operating constraints such as unidirectional power flow and/or limited power factor range. Some of the realizations that introduce additional operating constraints feature reduced number of controlled-switch count [35]. The focus of this paper is on the IMC topology without reduced switch count, since bidirectional-power capability and the wide power factor range of this converter are of primary interests to the authors. However, the PWM algorithm developed here can be readily applied to the reduced-switch topologies with little or minor modifications.

The IMC topology shown in Fig. 1 consists of a cascaded connection of two bridge converters. One bridge is connected to a set of voltage-stiff sources va, vb and vc, which would be formed using capacitors in practical implementations. The other bridge is connected to a set of current-stiff sources iu, iv, iw, which would be realized using inductors in practical implementations. The bridge connected to the voltage-stiff sources acts like a current source converter in terms the voltage constraints at the three-terminal port and current

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constraints at the two-terminal port, i.e. the voltage-stiff sources may never be short-circuited and the link current may never be open-circuited. Hence, we may call this bridge as current source bridge or CSB. In a dual manner, the bridge connected the current-stiff sources acts like a voltage source converter in terms of the voltage constraints at the two-terminal port and the current constraints at the three-terminal port, i.e. the stiff currents may not be open-circuited and the link voltage should never be short-circuited. Thus, we may call this bridge as voltage source bridge or VSB.

These inherent operational constraints to the IMC topology can be accommodated by the graphic representation as shown in Fig. 2, where the CSB is represented by two single-pole-triple-throw (SPTT) switches Sp and Sn and the VSB is represented by three single-pole-double-throw (SPDT) switches Su, Sv, Sw. With this arrangement, the throws in each switch can be modulated independently without violating the terminal constraints.

III. MODULATION FUNCTIONS For three-phase ac power conversion, the stiff voltages and

stiff currents at the ac ports shown in Fig. 2 may be assumed to be balanced three-phase quantities given by

va +-

+-

+-

ip

vpn

-

+

vb

vc

iu

iv

iw

CSB VSB

a

b

c

u

v

w

Fig. 1 Schematic of the IMC topology under study.

vc

+-

vb +-

iava

-

iu

iv

iw

ip

+

-

vpn

Sp

Tap Tbp Tcp

Sn

Tan Tbn Tcn

Sw Su SvTwp

Twn

Tup

Tun

Tvp

Tvn+

ib

ic

vu

vv

vw

n

Fig. 2 Schematic of the IMC represented using SMPTs.

( )( )( )( )( )( )

( )( )( )( )( )( )

cos cos

2 2cos ; cos3 32 2cos cos3 3

a i i u o o

b i i v o o

c i i w o o

v V t i I t

v V t i I t

v V t i I t

α β

π πα β

π πα β

= =

= − = −

= + = +

(1)

where Vi is source voltage amplitude; Io is the current source amplitude; αi(t) is the phase angle of the voltage source, given by ( ) 0i i it tα ω α= + ;

βo(t) is the phase angle of the current source, given by ( ) 0o o ot tβ ω β= +

The desired fundamental components of the currents at the ac terminals of the CSB may be expressed as

( )( )( )

_ _

_ _

_ _

cos ( )

cos ( ) 2 / 3

cos ( ) 2 / 3

β

β π

β π

=

= −

= +

a ref i ref i

b ref i ref i

b ref i ref i

i I t

i I t

i I t

(2)

where ( ) 0β ω β= +i i it t .

Consequently, the power factor angle on the CSB becomes

0 0φ α β= −i i i (3)

A. CSB Modulation Functions Each fundamental period of the ac quantities at the CSB

port may be divided into six sectors as shown in Fig. 3. If we assume the moving average of the dc link current to be constant, the modulation of the CSB would be identical to that of a PWM current source ac-dc converter. If the desired reference input phase current is maximum during one sector, then the corresponding throw of Sp is closed during the entire sector. The dc link current returns through one of the three throws of Sn, through appropriate commutation. Similarly, if the reference phase current is minimum during one sector, the corresponding throw of Sn is closed during the entire sector. The dc link current returns through one of the throws of Sp through appropriate commutation. Notice, if the throws corresponding to the same phase of Sp and Sn are closed simultaneously (termed a zero state of the CSB), each of the ac line currents are zero. Thus by varying the interval of the zero state, the current source converter may be modulated to regulate the amplitude of the ac fundamental component of the line currents, with a constant dc link current.

ia_ref ib_refic_ref

Sector 2Sector 1 Sector 4Sector 3 Sector 6Sector 5

Fig. 3 Waveforms of typical sinusoidal reference currents at the CSB ac

port, divided into six intervals.

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Alternatively, in the proposed modulation strategy for the IMC, the amplitude of the ac fundamental component of the line currents are regulated by modulating the VSB, without using zero states for the CSB and not compromising any waveform quality. For example, throw Tap is closed Tan is open during the entire interval of Sector 1, and the corresponding modulation functions map is 1 and man is 0. The dc link current flows through Tap, and returns through Tbn or Tcn. In order for this modulation approach to be successful a key modulation assumption needs to be satisfied, namely, the averaged link current may be regulated (by appropriately modulating the VSB, as will be illustrated in the following subsection) to follow the reference current ia_ref during Sector 1. Then, it follows that the modulation functions mbn and mcn can be determined to be

_

_

_

_

b refbn

a ref

c refcn

a ref

im i

im i

= −

= −

(4)

It may be noticed that the amplitude of the reference currents Ii_ref does not appear in (4), and thus the modulation functions of the CSB only determine the phase angle of the synthesized current relative to the stiff voltage source. In other words, while the power factor angle of CSB currents is set by the modulation functions, amplitude of the synthesized currents are determined indirectly by the VSB (naturally, through power balance considerations). Extending this strategy to the other sectors of the CSB ac current references, the modulation functions for each throw of the CSB of can be calculated as tabulated in TABLE I. The corresponding waveforms of modulation functions are also plotted in Fig. 4.

B. VSB Modulation Functions Having determined the modulation functions for the

throws of CSB during the entire period, we proceed to calculate the modulation functions of the VSB, to synthesize appropriate output voltage, while validating the key modulation assumption from the previous subsection. The modulation functions for each of the three phase-legs for the VSB can be expressed as

( )( )( )

cos ( )

cos ( ) 2 / 3

cos ( ) 2 / 3

u o o

v o o

w o o

m M t

m M t

m M t

α

α π

α π

=

= −

= +

(5)

where Mo is the modulation index and 0 < Mo < 1 and the phase angle ( ) 0α ω α= +o o ot t . If injection of triplen

harmonics are adopted, Mo can reach 2/√3 without over-modulation. The power factor angle φo at the VSB terminals is defined by

0 0φ α β= −o o o

(6)

Validation of the key modulation assumption is made through making the modulation index Mo to be a time varying function, as described further. Furthermore, in the conventional sine-triangle scheme, the modulation functions

TABLE I. MODULATION FUNCTIONS FOR THE THROWS OF CSB.

Sector 1 Sector 2 Sector 3 Sector 4 Sector 5 Sector 6

map 1 _

_

a ref

c ref

ii

− 0 0 0 _

_

a ref

b ref

ii

−

mbp 0 _

_

b ref

c ref

ii

− 1 _

_

b ref

a ref

ii

− 0 0

mcp 0 0 0 _

_

c ref

a ref

ii

− 1 _

_

c ref

b ref

ii

−

man 0 0 _

_

a ref

b ref

ii

− 1 _

_

a ref

c ref

ii

− 0

mbn_

_

b ref

a ref

ii

− 0 0 0 _

_

b ref

c ref

ii

− 1

mcn_

_

c ref

a ref

ii

− 1 _

_

c ref

b ref

ii

− 0 0 0

map mbp mcp

man mbnmcn

Sector 2Sector 1 Sector 4Sector 3 Sector 6Sector 5

1

0

1

0 Fig. 4 Typical waveforms of modulation functions for the various throws of

CSB.

for each phase leg is shifted and scaled to obtain the modulation functions for throws of the switches. For instance, the modulation functions for the throws Tup, Tvp and Twp are

( )

( )

( )

12

12

12

uup

vvp

wwp

mm

mm

mm

+=

+=

+=

(7)

Now, the average of the link current may be determined to be p up u vp v wp wi m i m i m i= + + (8)

Substituting the expressions for the output currents iu, iv and iw from (1) into (8), the averaged link current may be determined to be

3 cos4p o o oi M I φ= (9)

The key modulation assumption requires that <ip> should follow ia_ref during Sector 1, and ic ref during Sector 2, etc.

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corresponding to the unity modulation index rows of TABLE I. , under each sector. Thus during Sector 1,

_

3 cos4a ref o o oi M I φ= (10).

In order to validate (10), the amplitude Mo of the VSB modulation function, being the only free variable may be determined to be

( ) ( )_4 cos ( )3 cos

i refo i

o o

IM t t

Iβ

φ= (11)

using (2) during Sector 1. Through reciprocity, Mo(t) during Sector 1 may also be expressed as

( ) ( )_4 cos ( )3 cos

o refo i

i i

VM t t

Vβ

φ= (12)

The variation of Mo(t) during other sectors of the CSB current reference waveforms can be determined in a similar manner, as tabulated in TABLE II. The resultant modulation function amplitude Mo(t) together with the modulation functions for each phase-leg of the VSB are plotted in Fig. 5.

TABLE II. VSB MODULATION FUNCTION AMPLITUDE IN DIFFERENT SECTIONS.

Mo(t)

Sector 1 ( )_4 cos ( )3 cos

o refi

i i

Vt

Vβ

φ

Sector 2 ( )_4 2cos ( ) 33 coso ref

ii i

Vt

Vπβ

φ− +

Sector 3 ( )_4 2cos ( ) 33 coso ref

ii i

Vt

Vπβ

φ−

Sector 4 ( )_4 cos ( )3 cos

o refi

i i

Vt

Vβ

φ−

Sector 5 ( )_4 2cos ( ) 33 coso ref

ii i

Vt

Vπβ

φ+

Sector 6 ( )_4 2cos ( ) 33 coso ref

ii i

Vt

Vπβ

φ− −

mu(t) mv(t) mw(t)

Mo(t)

t

t

πωi

2πωi

3πωi

2πωo

πωo

Fig. 5 Waveforms of VSB modulation amplitude Mo(t) and phase leg

modulation functions mu(t), mv(t), mw(t).

IV. SWITCHING FUNCTIONS Once the modulation functions are derived, the switching

function hxy for each throw Sxy can be determined with special care of the switching sequence.

A. CSB Switching Functions The CSB switching functions are generated by comparing the modulation functions shown in Fig. 4 with a linear carrier at the switching frequency. However, the alignment of the PWM pulses with respect to the carrier has to be carefully chosen to eliminate abrupt discontinuities at sector boundaries. It may be observed from Fig. 4, that during each sector there are only two unclamped (or active) modulation functions, one of them decreasing and the other increasing and all the other modulation functions are clamped (or inactive) at either zero or unity. At each sector boundary one of the active modulation functions becomes inactive (clamped at zero or unity), and an inactive modulation function becomes active. In order to preclude any discontinuities in PWM pulses at sector boundaries, the following simple conditions needs to be satisfied: For positive throws, increasing pulse widths have to be left aligned and decreasing pulse widths have to be right aligned; for negative throws, increasing pulse widths have to be right aligned and decreasing pulse widths have to be left aligned. Such a strategy using a saw-tooth carrier to generate the switching function hbn and the hcn is obtained by inverting hbn as illustrated in Fig. 6. It can be observed that all PWM pulses are continuous during all sector boundaries.

B. VSB Switching Functions Switching events of VSB switching events are coordinated

with the CSB switching events, such that CSB commutation takes place when the link current is zero [34]. To illustrate the coordination between the VSB and CSB, a single switching cycle of during “Sector 2” from Fig. 6 is zoomed in as illustrated in Fig. 7. It may be noted that the carrier waveform of the VSB carrier triangle is not symmetrical as would be typical. Instead, the rising interval d1Ts and the falling interval d2Ts of VSB carrier are determined by the CSB switching functions. In this manner, the two CSB commutation events in each switching cycle always take place during the zero states of VSB, i.e. when either all the three top throws or all the three bottom throws of the VSB are connected to the link bus. During the zero states, the link current is zero. Therefore CSB can commutate with zero link current, which eliminates the switching losses, and the need for overlap time, eliminating any possible short circuit between stiff voltages feeding the CSB.

V. NUMERICAL SIMULATION VERIFICATION In order to verify the modulation scheme described in

Section III and Section IV, a numerical simulation has been carried out on a detailed model built using Matlab-Simulink®.

Fundamental period of the CSB ac quantities divided into 6 sectors

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d1Ts

mup

mvp

mwp

d2Ts

hup

hvp

hwp

mbp

hap

hbp

VSB modulationfunctions

VSB switchingfunctions

CSB switchingfunctions

CSB modulationfunctions

CSB carrier

VSB carrier

VSB zero states

Fig. 7 Waveforms illustrating of the coordination between the VSB and the CSB commutation processes

Schematic of the power circuit used in this simulation is shown in Fig. 8 with the parameters listed in TABLE III. Selected waveforms from the simulation results are plotted in Fig. 9 - Fig. 11. The currents drawn from the source and load currents are shown in Fig. 9. Fig. 10 illustrates the source voltage of ‘a’ phase va, the filter capacitor voltage of ‘a’ phase vCf-a, the link voltage vpn and the source current of ‘a’ phase ia. In this simulation, unity power factor on the input side is selected. Fig. 11 shows the output line-line voltage vuv, the line-neutral voltage vu, the link voltage vpn and the load current iu.

va +-

vb +-

vc +-

iu

vpn

-

+

Lf

Cf

ia

ib

ic

+

-

vCf-a

iv

iw

+

-

vuv

vu+ -

Lload Rload

Fig. 8 Schematic of the power circuitry used for simulation and

experimental test.

TABLE III. LIST OF PARAMETERS FOR SIMULATION

va,b,c 100 Vpk_l_n fswitching 10 kHz Lf 200 μH Rload 8 Ω Cf 30 μH Lload 10 mH finput 60 Hz foutput 40 Hz

Sector 1 Sector 2 Sector 3 Sector 4 Sector 5 Sector 6

hap

hcn

hbp

han

hcp

hbn

CSBcarrier

Left aligned

Left aligned

Right aligned

Right aligned

mbn mbp mcn mcp man map

Fig. 6 Waveforms illustrating the generation of CSB switching functions.

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0 0.01 0.02 0.03 0.04 0.05-5

0

5

i a,b,

c (A)

0 0.01 0.02 0.03 0.04 0.05-10

-5

0

5

10

i u, v

, w (

A)

t (s) Fig. 9 Waveforms of source currents (top panel) and load currents (bottom

panel) obtained using simulations.

0 0.01 0.02 0.03 0.04 0.05-100

0

100

v a

0 0.01 0.02 0.03 0.04 0.05-100

0

100

v Cf-a

0 0.01 0.02 0.03 0.04 0.050

100

200

v pn

0 0.01 0.02 0.03 0.04 0.05-5

0

5

i a

t (s) Fig. 10 Waveforms of source voltage, filter capacitor voltage, link voltage

and source current, obtained using simulations, from the top to bottom, respectively.

0 0.01 0.02 0.03 0.04 0.05-200

0

200

v AB

0 0.01 0.02 0.03 0.04 0.05-200

0

200

v pn

0 0.01 0.02 0.03 0.04 0.050

100

200

v pn

0 0.01 0.02 0.03 0.04 0.05-10

0

10

i A

t (s) Fig. 11 Waveforms of load l-l voltage, load l-n voltage, link voltage and

load current obtained using simulations, from the top to bottom respectively.

VI. EXEPRIMENTAL RESULTS In order to further validate the proposed modulation

algorithms, a prototype of the converter has been constructed in the laboratory. A photograph of the prototype converter is shown in Fig. 12. The DSP board is used as the controller platform. The modulation scheme described in proceeding sections is implemented in the FPGA on the DSP board.

The measured gating signals for the six throws in CSB and VSB are shown in Fig. 13. It can be observed at no instant the top and bottom throws in the same phase leg are closed simultaneously, i.e. no zero states in the CSB modulation. Fig. 14 illustrates that the CSB only commutates when the link current is zero. The top three traces in Fig. 14 are switching signals for the three top throws in VSB. The switching events, as shown in the bottom trace, only happen when the top throws in VSB are all closed or all open, which corresponds to zero link current for either case. Selected waveforms (corresponding to the simulation waveforms from Fig. 9 - Fig. 11) are shown in Fig. 15 - Fig. 16, illustrating excellent conformity.

Fig. 12 A photograph of the prototype of the converter used for obtaining

experimental tests.

0 0.005 0.01 0.015 0.02-10

01020

T ap (V)

0 0.005 0.01 0.015 0.02-10

01020

T bp (V)

0 0.005 0.01 0.015 0.02-10

01020

T cp (V)

0 0.005 0.01 0.015 0.02-10

01020

T an (V)

0 0.005 0.01 0.015 0.02-10

01020

T bn (V)

0 0.005 0.01 0.015 0.02-10

01020

T cn (V)

t (s) Fig. 13 Waveforms of measured of gating signals for the CSB throws: from

top to bottom Tap, Tbp, Tcp, Tan, Tbn, Tcn, obtained using the laboratory prototype.

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0.0365 0.0366 0.0367

-10

0

1020

T up (

V)

0.0365 0.0366 0.0367

-10

0

1020

T vp (

V)

0.0365 0.0366 0.0367

-10

0

1020

T wp (

V)

0.0365 0.0366 0.0367

-10

0

1020

T ap (

V)

t (s) CSB Switching Events

Fig. 14 Illustration of the zero current commutation of the CSB, from top to bottom are measured gating signals for throws Tup, Tvp and Twp of the VSB

and Tap of the CSB obtained using the laboratory prototype..

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045-5

0

5

i a, b

, c (

A)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

-5

0

5

i u, v

, w (

A)

t (s) Fig. 15 Waveforms of source currents (top panel) and load currents

(bottom panel) obtained using the laboratory prototype.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045-100

0

100

v a (V)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045-100

0

100

v a-C

f (V)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450

100

200

v pn (

V)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045-4-2024

i a (A)

t (s) Fig. 16 Waveforms of source voltage, filter capacitor voltage, link voltage and source current, obtained using the laboratory prototype, from the top to

bottom respectively.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045-200-100

0100

v uv (

V)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

-1000

100

v u (V)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450

100

200

v pn (

V)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045-5

0

5

i u (A)

t (s) Fig. 17 Waveforms of load l-l voltage, load l-n voltage, link voltage and

load current obtained using the laboratory prototype, from the top to bottom respectively.

VII. CONCLUSIONS A carrier based modulation scheme has been presented for

the IMC, by identifying them to be a cascade connection of a CSB and a VSB. The modulation functions for the two bridges are developed from the modulation functions derived from the desired CSB currents and VSB voltages. Then the switching functions are generated with particular focus on the coordination between the CSB and VSB switching events to realize a robust commutation. The proposed algorithm has been verified by both numerical simulation and hardware experiments, validating the effectiveness of the modulation scheme. The modulation strategy is simple to implement using general purpose digital signal processors in conjunction with FPGAs as well as using application specific digital signal processors with built in PWM operational modules.

Although the 18-active-switch topology is used in the analytical development and laboratory tests, the proposed modulation algorithm can be applied the reduced-switch-count topologies with no or minimal modifications. Further the modulation functions can be extended to the conventional matrix converter (CMC) by the mapping relationship between CMC and IMC, which will be explored in the future publications.

ACKNOWLEDGMENT The authors would like to acknowledge support from the

Wisconsin Electric Machine and Power Electronics Consortium (WEMPEC) at the University of Wisconsin-Madison. The work made use of ERC shared facilities supported by the National Science Foundation (NSF) under AWARD EEC-9731677.

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