Analysis and DSP-based Implementation of Modulation Algorithms for AC-AC Matrix Converters

C Watthanasam, L Zhang, D T W Liang Dept. of Electronic and Electrical Engineering,

University of Bradford, Bradford, BD7 lDP, UK

Abstract - The paper presents two control strategies for a matrix converter. Novel and computationally-efficient real- time control algorithms were developed and implemented using a digital signal processor. A 2 kVA prototype IGBT- based matrix converter supplying an R-L load was constructed. The matrix converter was tested and good performances were achieved using both the real-time modulators. Exlperimental converter waveforms and their respective harmonic spectra are compared. Key features of each modulator ,are highlighted.

I. INTRODUCTION It is well known in the field of AC-AC matrix converter that the pulse-width modulation technique is one of the major factors in determining the performance of this type of converters. An itdeal PWM scheme should be able to: 1) provide independent control of the magnitude and

2) give sinusoidal input currents with adjustable phase-shift, 3) achieve the rnaximum possible range of output voltage,

4) satisfy the conflicting requirements of minimal low-order

Thus far two PWM strategies have been proposed; the Venturini methlod and Space Vector Modulation scheme. The former, originally proposed by Venturini and Alesina in 1980 [ l ] and further investigated by them for an extended output-to-input voltage transfer ratio [ 21, uses the transfer matrix analysis. The latter, developed by Huber et a1 [3], selects directly the valid switching states of the converter.

frequency of the generated output voltages,

and

harmonics and minimum switching losses.

In terms of their basic operating principles, both methods are well developed and, it is probably true to say that, they are capable of meeting all the desirable characteristics listed above. However, due to the complexity of a matrix converter circuit and the requirement for good performance, it is important that the converter system designers have a good understanding of both techniques. Moreover, either algorithm can be implemented efficiently, hence allowing on-line computation of the switch timings while operating at a high switching speed.

Many of the applications which have been reported so far concentrate on one of the modulation methods, and few published works have dealt with the practical issues of the microprocessor-based implementation. It is the aim of this paper to give a comprehensive review of both PWM techniques, and highlight their respective features and drawbacks. Furthermore, novel and computationally efficient algorithms for controlling converter switches have been developed and will be discussed. These allow sinusoidal input currents vvith unity input power factor and controllable

output voltages without low-order harmonics to be generated. Both algorithms were implemented using a TMS320C30 digital signal processor and a digital interface unit. They were applied to control a prototype IGBT-based matrix converter, the results obtained will be shown in the paper.

11. MATRIX CONVERTER CIRCUIT The basic circuit of a three-phase matrix converter, as shown in Fig.1, consists of nine switches arranged in three phase groups. Each of these switches can either block or conduct the current in both directions depending on the gate control signals, thus allowing any of the output phases to be connected to any of the input phases. The input side of the converter is a voltage source and the output is a current source, consequently only one of the three switches connected to the same output phase is allowed to be on at any one time. This prevents the input terminal from being short-circuited and the output current path being open- circuited simultaneously.

111. VENTURINI CONTROL METHOD The rationale of this method is simple. For a given set of three-phase input voltages, a desired set of three-phase output voltages may be synthesized by sequential piecewise sampling of the input waveforms. The duration of each sample is derived mathematically to ensure that the average value of the output waveform within each sampling cycle tracks the required output waveform. This principle can also be applied to control the three-phase input current.

For simultaneous control of both output voltage and input current, the Venturini method considers two given sets of variables, the three-phase input voltage with amplitude V, and frequency U,, and three-phase output current with amplitude Z, and frequency O, . They are given by

1 cos(w,r - 410 - 4n / 3 )

(1) respectively, where @(, is an output displacement angle. The general relationship between the input and output voltages is given by equation

[%I = [M(t)I [VI] > (2)

1 mi, ( t ) mi2 ( t ) m13 ( t ) where [M(t)] = m,,(t) m,,(t) m,,(t) , and [V,] represents a

set of three output phase voltages. Likewise, the input and output currents are related by

I m31 ( t ) m32 (l) m33 (l)

[I,] = [M(t)IT [IO] (3)

0-7803-3500-7/96/$ 5.00 0 1996 IEEE 1053

where [Ii] is the three-phase input current.

From a mathematical viewpoint, the problem requires finding the functions for the elements in matrix [M(t)] when both [V,] and [Iij are defined. In Venturini method, the desired set of output phase-voltages is defined as

(4) and the input line-currents is defined as

(5) I : cos(w,t - $ L - 4 n / 3)

cos(o,t - $[) [I ,] = I , cos(0,t - $, - 277 / 3) ,

where @ [ is the input displacement angle.

It can be seen from eqn.(4) that the third-order harmonics of both the input and output frequencies are added to the desired sinusoidal output voltages. This increases the maximum output-to-input line-to-line voltage-ratio of the converter to 0.866 but does not jeopardize the harmonic performance of the output line-to-line voltages.

Neglecting power losses associated with all the circuit switches, the input power equals the output power. Hence,

Substituting [V,] in eqn.(2) by the defined expression in eqn.(4) and [I,] of eqn.(3), and applying eqn.(6), a general formula for one of the nine elements in matrix [M(t)] is

VIr cos = &I0 cos $(, . (6)

derived as follows: m,(t) = (1 + (1 - 0) QCOS(W,~ + ~ , t - ( 2 ( i + j ) - 41% / 3)

+(I + 0) Q C O S ( C O ~ ~ - ~ , t - 2(i - j )rr / 3 )

-(1/ 6) (1 -\@I) QCos(3W0t +Wit - 2 ( j - l ) ~ / 3 )

-(1/6) ( 1 - / O / ) Q C O S ( 3 0 , t - W i t - 2 ( 1 - j ) n : / 3 )

-(1/ 6 h ) 11 - 1011 QcoS(40it - 2 ( j - l)n / 3 )

+(7 / 6 6 ) 11 - 1011 Qc0s(2Oit - 2(1- j )n / 3)) 13, where i, j=1, 2, 3; Q = V, I V, ; and 0 = tan@, I tan$, Note that 0 is an input-to-output phase transfer ratio it can be adjusted to regulate the input power factor.

It is also worth noting that the mli(t) formula define(

(7)

4j and

- above calculates the duty cycle of a switch in the converter, hence

Bearing in mind the restriction on the control of matrix switches stated in Section 11, the switching functions for the switches connected to the same output phase obey the relationship expressed as

0 5 m,( t ) < 1, i , j=1 ,2 ,3 . (8)

m,,(t)+m,,(t)+m,,(t)= 1 , i=1,2,3. (9)

Subsequently, the pulse sequences for these switches are generated as illustrated in Fig.2. Such pulse signals are also applied to control switches connected to the other two output phases, though each phase may be controlled independently. Consequently, the number of switch commutations per switching periods is 9 (=3x3).

IV. SPACE VECTOR MODULATION TECHNIQUE

In this method the three-phase variables are represented as space vectors, hence the three-phase output voltage of a matrix converter can be expressed as

where V,, defines angular velocity of the vector. Likewise the input three-phase current can be represented as

where $, is the input displacement-angle, and Ii and w, represent the magnitude and rotating velocity of the current vector respectively. Similarly, the three-phase input voltage and output current may also be written as

(10) v = v e J ( w J ) 0 01

represents the magnitude of the vector and o,

(1 1) I , = e j (w-%)

L l and 7, = I, e”O“ t-’n) respectively. (12) = v e i ( w , f )

The principle of the SVM method can be summarised as follows: For a sufficiently small time interval, the reference voltage vector can be approximated by a set of stationary vectors generated by a matrix converter. If this time interval is the sample time for converter control then, at the next sample instant when the reference voltage vector rotates to a new angular position, it may correspond to a new set of stationary voltage vectors. Carrying this process onwards by sampling the entire waveform of the desired voltage vector being synthezised in sequence, the average output voltage would closely emulate the reference voltage. Meanwhile, the selected stationary vectors can also give the desirable phase- shift between input voltage and current. The modulation process thus required consists of two main parts: selection of the switching vectors and computation of the vector time intervals. These are briefly discussed in sequel as a detailed discussion of the SVM method was given in the authors’ previous paper [Sj.

A. Criteria for Selection of Switching Vectors As stated above, only one out of three switches connected to a single output phase can be at the ‘on’ state at a given time. This results in 27 valid switching combinations as listed in Tablel [6]. Each of them defines both an output voltage vector and an input current vector which are also given in the table. A common feature can be observed from the first six combinations, namely, they all generate voltage vectors having the same frequency as that of the input voltage, hence cannot be used to synthesize a different voltage waveform. For the next 18 combinations (1P to 9N) in the table, they can be arranged into three groups, each contains vectors giving zero voltage value to the same output phase. A special feature for these vectors is that the phase angle for each of them is a constant, hence they are named stationary vectors. The magnitudes of these vectors, however, vary with the changes of the instantaneous input line-to-line voltages. Projecting all the stationary voltage and current vectors onto a-P plane, two hexagons are obtained as depicted in Fig.3(a) and (b). The final three combinations in Tablel result in three zero line-to-line voltages, and are called zero voltage vectors.

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Having arranged the available switching combinations for matrix converter control, the SVM method is designed to choose 4 out of 18 stationary vectors at any time instant using the following rules: [ 5 ]

Choosing two groups of stationary vectors adjacent to the sector in which the reference voltage vector locates. For example, if is in sector 2, shown in Fig.3(a), the selected vectors are from 1P to 3N and 7P to 9N. Selecting 8 out of the above 12 vectors taking into account the sector location of the input current vector. This can be illlustrated using Fig.3(b), as stays in sector 1 the further selected switch-combinations are lP,IN, 3P,3N, 7P,7N, 9P and 9N. Finally, choosing 4 vectors out of the selected eight for achieving unity input power factor and maximum output- to-input voltage ratio. In this example, vectors 1P,3N,7N and 9P are chosen.

With both current and voltage vectors rotating independently within six sectors, there are altogether 36 input-output sector pairs, giving therefore 36 sets of switch combinations.

B. Computation of Vector Time Intervals This is based on the theory of space vector modulation, namely, the integral value of the reference vector over a short time interval can be approximated by the sum of products of the two adjacent vectors multiplied by their duty ratios. To apply this principle, the above selected vectors are combined into two sets, thus generating two new vectors v,, and 4, which are adjacent to the output voltage vector . Carrying on using the example shown in Fig.3(a), a relationship between the reference and the selected voltage vectors can be defined as

I,

where Toy and Tau represents the time widths for vectors v,, and vo,, respectively, to is the initial time and Ts is the specified samplle period. This gives the following expression:

where tlp, t3N, t7N and tgp are the time-widths of the associated voltage vectors. Note that in general these time-widths are denoted as tl, t,2, t3 and t4.

To control the phase-angle of the input current vector, the four selected vectors are rearranged into another two sets corresponding to the sector location of the input current vector. Applying the SVM theory again will give two equations of the same form and one of them is written as

1 COS 30 COS 30" COS(Wjt - 30") '3, J T , ~ /[sin 3 0 ~ ] + t , P l l t P I[ -sin ,,.I = 11' 3 [ sin(wit -300) '

(15) Solving the above equations, a simultaneous solution for four different vector times is obtained as t,, = M T, cosr[o0t)sin(60"-o,t) , (16)

t,, = M q, cos(w,t)sin(o,t) , t,, = M q, sin(w,t - 30")sin(60"-wit), t,, = M q, sin(o,t - 30")sin(w,t),

(17) (18) (19)

where the modulation index M is 2V,, / &y, . Note that the above equations are only valid when the reference output- voltage vector stays in output sector 2 while the reference input-current vector is in input sector 1. For different output- input sector combinations, the same equations are applied with the arguments of the sinusoidal terms being changed corresponding to their input and output sectors.

Generally the sum of four vector time-widths is less than a switching period T,, and the residual time within Tq is then taken by a zero vector, giving rise to

A

To reduce the switching losses, the transition from one switch-combination to the next should be performed such that the number of switch commutations is minimised. In this example the sequence of combinations is chosen as 3N-+9P+7N+lP-+OB which results in 7 switch commutations per switching cycle, hence meeting this requirement.

V. COMPARISON OF TWO CONTROL TECHNIQUES From the above analytical discussions, the salient features of both methods can be summarised as follows:

Both methods give the theoretical maximum voltage gain of 0.866 though they use different approaches. This is realised, in Venturini method, by adding third harmonic components of both input and output voltages to the desired output waveform whereas in the SVM method, modulation of line-to-line voltage naturally gives an extended output voltage capability. Both methods achieve simultaneous control of output voltage waveform and input current displacement angle. The computational procedure required by SVM method is less complex than that for Venturini method because of the reduced number of sine function computations. The number of switch commutations per switching cycle for SVM method is 20% less than that of Venturini method, having 7 switch commutations as opposed to 9.

VI. DSP-BASED IMPLEMENTATION OF BOTH ALGORITHMS

The general requirements for generating the switch control signals for a matrix converter on-line in real-time are listed as follows; i) Computation of the switch duty cycles must be completed

within one switching period, ii) Accurate timing of the pulse-pattern output should be

achieved, and iii) The computational process must be synchronised with the

input-voltage sinusoidal cycle.

For both modulation methods, a TMS320C30 digital signal processor (C30DSP) was used for on-line calculation of the switch timings. The C3ODSP is mounted on a PC-compatible

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card, the Evaluation Module (EVM), to facilitate the development process. An additional interface card was used for outputting the calculated switch timings. This comprises of 7 programmable timers, 4MHz clock, a 2Kx8-bit EPROM, an 8-bit U 0 port and a few logic gates and is configured in such a way that either Venturini or SVM control method may be selected by the user. Both DSP and interface card are plugged in a PC 80386 which provides a convenient means for data communication and user interface.

The hardware configuration for implementing the Venturini scheme is shown in Fig.4(a). During each sampling interval T,, the C30DSP calculates six switch duty cycles mij(tk), based on eqn.(7), and converts them to integer time-counts using the on-board clock frequency. As the calculation is performed during the process of pulse-timing output and is completed within T, , it has no effect on the real-time pulse signal generation. Once completed the calculated results are stored in the DSP memory and subsequently loaded into the timers. According to the output pulse-pattern in Fig.2, two programmable timers are used for timing three switches. At the start of a switching period, two timers are loaded with T,, and Tl1 +TI2 respectively and begin the count-down process while their outputs are held ‘LOW’. As soon as the first timer reaches a terminal count, its output goes ‘HIGH’, thus initiating the commutation of switches from Sl l to Slz. At the terminal count of the second timer, commutation from SI2 to SI3 takes place. A third timer is required for timing out T,s which also determines the ‘on’ duration of switch SI?. On completion of a switching period, two timers are re-loaded with the new integer counts and the procedure described is repeated. Loading of the timer is performed by a PC interrupt service routine. The outputs of the two timers can be converted to three switch control signals using a few gates, indicated as a 2-to-3 line logic decoder block in Fig.4(a). Since the other two groups of three switches are also controlled using this approach, seven programmable timers are, therefore, required in the Venturini control scheme.

Figure 4(b) shows the block diagram of the SVM switch- timing controller. The DSP computes the vector-time counts using eqs.(l6) to (19) in the same manner as the Venturini method. An EPROM is used to store the selected sets of stationary and zero vectors. Each of the switching vectors, represented by nine control pulses, is coded as a 6-bit binary number to fit into an 8-bit EPROM. In retrieving, 2-to-4-line decoder IC’s are used to recover the nine control signals. The 10-bit address of each memory location is formed by three figures; the sector number of the output voltage vector, the sector number of the input current and output of four timers. The input and output sector numbers are readily available from the DSP and hence can be directed to six address lines of the EPROM using the 6-bit latches. Four programmable timers can be similarly configured to those in the Venturini scheme but their outputs are directed to four address lines of the EPROM. The extra timer for controlling T, is also required. When the sample period begins, four timers are loaded with TI , Tl+T2, TI+T2+T3 and T,+T2+T3+T4 consecutively. The terminal count of each

switch changes the pulse-pattern of the four-timer outputs and in turn updates the pointer of the look-up table. This effectively generates each of the five switching vectors for a specified time interval as required by the SVM technique. On the completion of T,, four integer counts are updated and the process is then repeated.

In comparison, the C30DSP takes 19 ps to complete the computation for the SVM scheme whereas 50ps is needed for Venturini method. As T, is usually set at the level of hundreds microseconds, during the majority of a sample period the DSP is idled and can be used for feedback control or other tasks. Both schemes require one interrupt routine for outputting the calculated results. Since this demands 50 ps to implement, the maximum sampling frequency of the modulator can be extended to 10kHz.

Any control algorithm for a matrix converter must be synchronised with the input voltage cycle so that the calculation of switch timings can be correctly performed. This is achieved by using the software phase-lock-loop which detects the zero-crossing of the input phase voltage. An interrupt pulse generated at each zero-crossing will cause the DSP to initialise appropriate variables for calculation.

VII. EXPERIMENTAL RESULTS Both of the switch-timing generators were tested using an IGBT-based matrix converter. Each of the matrix switches consists of two IGBTs anti-parallelly connected and two fast- recovery diodes as depicted in Fig.1. A staggered commutation strategy [2,3,7] was implemented by means of a simple clock-controlled circuit [ 81. This ensures the safe commutation of the matrix switches. By choosing an appropriate clock frequency, the logic circuit can eliminate both current and voltage spikes which would normally occur in a matrix converter. With the aid of this circuit, both Venturini and SVM schemes were then applied to control the converter.

Figures 5(a) and (b) show the waveforms of the output current and line-to-line voltage obtained from the Venturini and SVM controller respectively. A switching frequency of 8 kHz is chosen. The output-to-input voltage ratio is kept at 0.866. The load consists of a 3 0 a resistor and a 6mH output-filter inductor. In both cases, the output voltages exhibit no excessive spikes whereas the output currents are the smoothed sinusoidal waveforms. The unfiltered input current associated with the input voltage (phase-to-neutral) under the above operating conditions are shown in figures 6(a) and (b). In either case, the input current is observed to be in-phase with the input phase-voltage, thus giving unity input power factor. Fig. 8 shows the harmonic spectra of the output voltages (line-to-line) at 80Hz modulating frequency. The switching frequency is reduced to 2kHz to show the fundamental and switching-frequency harmonics. In both cases the magnitude of the fundamental component is significantly higher than that of the most significant harmonic component. As expected, the high-order harmonics observed in this figure are due to the switching

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frequency component and those of its sidebands. The high- order harmonics can then be removed by the filter of minimum size. The low-order harmonics have negligible effects on both waveform. The above remarks are also valid for the frequency spectra of the input currents shown in Fig.9(a) and 9i(b). The converter performances under both the Venturini and SVM control are comparable.

Figures 7(a) arid (b) show the generated output line-to-line voltages when reference frequency is changed from 30Hz to 8OHz and output-to-input voltage ratio is varied from 0.8 to 0.99. The results from the experimental test demonstrated that smooth waveform transitions can be achieved and that the response of the converter to a command voltage change can be achieved without delay.

VIII. CONCLUSIONS The Venturini and SVM control techniques were analysed and compared. The DSP is an adequate tool for implementing both the developed control algorithms. A prototype matrix converter was constructed and both switch- timing controllers were tested. Experimental results show that similar technical performances were obtained from the two controllers. However, the SVM scheme may be economically iimplemented using a standard microprocessor instead of DSP.

REFERENCES [ 11 M G B Venturini and A. Alesina, “A New Sine Wave In, Sine Wave Out Conversion Technique Eliminates Reactive Elements,” Proceedings of Powercon7, 1980.

[2] M G B Venturini and A. Alesina, “Analysis and Design of Optimum-Amplitude Nine-Switch Direct AC-AC Converters,” IEEE Trans, on Power Electronics, 1989.

[3] L. Huber, D. Borojevic, N Burany, “Analysis, Design and Implementation of the Space-Vector Modulator for Force-Commutated Cycloconverters,” IEE Proc.B, 1992.

[4] M. J. Maytum, D. Colman, “The Implementation and Future Potential of the Venturini Converter,’’ Conf. Proc. on Drives/Motors/’Controls ‘83, York, England, October 1983.

[5] C. Watthanasarn, L. Zhang and F. Hardan, “Real-time control of direct AC-AC matrix converters using a DSP- based space ve’ctor modulator,” UPEC ‘95, September 1995.

[6] D. Casadei, G. Grandi, G. Serra, A. Tani, “Space Vector Control of Matrix Converters with Unity Input Power Factor and Sinusoidal Input/Ootput Waveforms,” EPE ‘93 Conf. Proc., Brighton, UK, Vol. 5, 170-175, Sept 1993.

[7] P. W. Wheeler, D. A. Grant, “A Low Loss Matrix Converter for AC Variable-Speed Drives,” EPE’93 Conf. Proc., UK, Vo1.5,27-32, 1993

[8] L. Zhang, IC. Watthanasarn and W. Shepherd, “A novel switch sequencer circuit for safe commutation of a matrix converter,” Ellectronics Letters, August 1995, 1530- 1532.

3-phase input input L-C filter

Output filter

Figure1 Basic structure of a matrix converter It Ts +

Figure 2 Conversion of modulation functions to on-off durations of the matrix switches

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U'II (a) Venturini switch-timing controller

1-11

(b) Space vector modulator Figure 4 Hardware arrangements for the converter control

hp r t0pp.d

I I

2 ( 0 . 0 , "Y /d l" P O . , 0 .000 Y LO0011 5021 dS

3 too Y / d l " PO,, 200 .0 Y 50.0031 I"* d C

to 0000 n,s r.e,t,",. T r l 0 g . r Mod.,

Edge

0 . 0 0 0 r 2 0.3 m.,dlr

2 10 0 ""/ 3 700 " / 0.00000 Y 2 0 0 . 0 0 0 "

2 t 2 . 5 0 0 nl"

2: Current (2Ndiv) 3 Voltage (100V/div)

(a) Venturini controller (M=0.99, fo=80Hz)

*p Ct0PP.d

I i

3 ,OO Y,dl" P O * , m o 0 " 50 .001 , I"* dC

I O OOOD ms r.a,,im. Trl9g.r nod.,

E d g e

0 . 0 0 0 s 2 00 n.,dlr

2 l O . 0 my, 3 100 " I 0 00000 Y 200 .000 Y

2 t 2 .500 mv

2 Current (ZNdIv) 3: Voltage (lOOV/div)

(b) SVM controller (M=0.99, fo=80Hz) Figure 5 Output currents and output voltages (line-to-line)

I I I 80 0000 II.

r..,,xm. TllSD.' "*d. eeg.

0 000 . 1 0 0 " I . l d , "

-10 OD00 m.

P I O 0 w , 0 50 0 "/ 0 00000 " -300 000 "

3 x 0 000 " 2 lnplt cumnl (I Ndv) 3 lnprt phase m h v (MVldNI

(b) SVM controller (M=0.99, fo=80Hz) Figure 6 Input phase voltages and input currents (unfiltered)

2 Output cunenl (Wd~v) 3 Oulpul vonage VA0 (IWV/dw)

(a) Venturini controller (M=0.8,fo=30Hz to M=.99,fo=80Hz) hp Sl0pp.d

I -50 000 ",* 50 000 mf r.o,,tm. 7rjgg.r nod.,

Edge

0 000 c 80 0 ",dl"

7. 10 0 m", 3 100 V I

2 -L 2 500 m" 0 00000 " 200 000 "

2 Output cun8nt (ZMdIv) 3 Oulpul volage VA0 (IWWdN)

(b) SVM controller (M=0.8, fo=30Hz to M=0.99, fo=80Hz) Figure 7 Transition of output current and voltage waveforms

Vdnlrmil

(a) Venturhi controller (M=0.8,fn=30Hz)

(b) SVM controller (M=0.8,fo=30Hz) Figure 8 Harmonic spectra of the output line-to-line voltages

-00 *000 1-00 11000 2-00 HI

(a) Venturini controller (M=0.8,f0=30Hz) hw. 0")

(b) SVM controller (M=0.8,fo=30Hz) Figure 9 Harmonic spectra of the input currents

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