IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. PE-2, NO. 4, OCTOBER 1987

Application of Pulse Ratio Modulation to Fixed-

Frequency Sinusoidal Tracking InvertersARMANDO BELLINI, A. DE SANTIS, GENNARO FIGALLI, AND GIOVANNI ULIVI

Abstract-The application of pulse ratio modulation techniques inthe field of fixed-frequency inverters with sinusoidal output waveformsis discussed. In particular, the criteria for choosing the parameters ofthe output filter and of the compensation network are analyzed, and a

comparison is effected between the results obtained by employing eitherthe aforementioned modulation technique or a fixed-frequency pulse-width modulation technique. The comparison, effected by simulation,has also taken into account a nonlinear load, and it has shown that theemployment of the pulse ratio modulation technique allows a remark-able reduction of the inverter supply overvoltage necessary to use

tracking control techniques with high modulation frequencies. Lastly,the results obtained by simulation have been confirmed by a prototype.

INTRODUCTIONGENERALLY, the output voltage of fixed-frequencyJ inverters with sinusoidal waveform is obtained either

by means of the modulation of a sinusoidal reference volt-age with a triangular carrier voltage, or by means of asuitable off-line computation of the commutating instants.In this latter case the commutating instants are determinedto minimize a suitable quality index or to condition theharmonic content of the obtained waveform by taking intoaccount the output filter which is always present [1].

In this way, the shape of the output voltage is open-

loop controlled, while the closed-loop only acts on themean or the rms value of the output voltage. This ap-

proach presents some disadvantages; in fact, the controlof the mean or the rms value of the output voltage cannotbe effected in a time shorter than the output half-period,and the dynamics of the closed-loop system are adverselyinfluenced by the presence of a filter (or of a samplinghold) in the circuit measuring the output voltage. More-over, the waveform of the output voltage depends on theshape of the load current which is very distorted in manycases, as when the load is an electronic apparatus.A different approach to the problem consists of also

controlling the output waveform by a closed-loop system.To this end, the reference and output waveforms are com-

pared, and the obtained error signal, eventually frequencycompensated, is used as the input of the modulator. Also

Manuscript received September 20, 1985; revised November 6, 1986.This paper was presented at the 1985 IEEE Power Electronics SpecialistsConference, Toulouse, France, June 24-28.

A. Bellini is with the Dipartimento di Ingegneria Elettronica-II Uni-versitd di Roma, via Eudossiana 18, 00184 Rome, Italy.

A. De Santis, G. Figalli, and G. Ulivi are with the Dipartimento diInformatica e Sistemistica-Universitd di Roma, via Eudossiana 18, 00184Rome, Italy.

IEEE Log Number 8715898.

in this case the employed modulation techniques are basedon the comparison between the modulator input signal anda carrier waveform (PWM = pulsewidth modulation).The main disadvantage of this latter approach is con-

nected with the necessity of employing an inverter supplyvoltage much higher than that used in the case of the open-loop control, especially if the period of the carrier wave-form is not much higher than the minimum duration ofthe conduction intervals the inverter can perform.

Therefore, to reduce the oversize it is necessary to limitthe maximum modulation frequency. The limitation isstrongly restrictive for thyristor inverters, and it is ratherconsistent also for transistor inverters of medium or highpower.As will be shown later, this constraint can be greatly

reduced by using a different modulation technique. Thistechnique, known as pulse ratio modulation (PRM), was

employed in dc/dc converters, and it acts at the same

time on both the duration and the frequency of the pulsesgiven by the modulator to ensure that the minimum du-ration of the conduction intervals will always be longerthan a prefixed value [2], [3]. Moreover, as shown in [4],[5], its dynamic behavior is also generally better than thatof a fixed-frequency modulator.The present paper analyzes the possibilities offered by

the PRM technique in the field of fixed-frequency invert-ers. To this end, after a short description of the PRMtechnique, some criteria are presented for the choice ofthe parameters both of the power circuit and of the com-pensation network on the basis of the shape of the loadcurrent. Next, the voltage oversizes necessary for the twokinds of modulation are compared by digital simulation.Lastly, the results obtained by simulation have been con-firmed by some tests performed on a prototype.

PULSE RATIO MODULATION TECHNIQUE

As previously mentioned, the main disadvantage whichoccurs in the employment of a pulsewidth modulator op-erating at high frequencies is connected with the mini-mum duration of the conduction intervals that the powercircuit must perform. In fact, taking into consideration abalanced supply ± V, the mean value VO of the outputvoltage in the inner of a commutation interval is

T -T=±+T + (1)

0885-8993/87/1000-0337$01.00 1987 IEEE

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. PE-2, NO. 4, OCTOBER 1987

where r+ is the duration of the time interval during whichthe output voltage is equal to + Vs, r- is the duration ofthe time interval during which the output voltage is equalto - V, and Tm is the modulation period.

Denoting with r,, the minimum duration of the conduc-tion interval, modulation ratio K is given by

K=VOK =0

and it can vary between the following values:

-(1 - 2Tm/ Tm)

(2)

1/f_

+(1 - 2Tm/ Tm).

The variation range of the modulation ratio can be in-creased if the modulation period is also varied when ratioK varies.A modulation technique allowing this variation is the

PRM technique. In it, denoting by fmM the maximum valueof the modulation frequency, the durations of the conduc-tion intervals and the modulation period are

+= 1/2fmm(l - K)

T = 1/2fmM(l + K)

Tm = 1f[MU(l + K)(1 - K)].

-1 -0.5 0 0.5 1 K

Fig. 1. Waveforms of Tm, 7+, and r-.

(3)Fig. 1 shows the shapes of the durations of the conductionintervals and the period when K varies from - 1 to + 1.

It is possible to remark that* when K is equal to zero, T+ and T- are both equal to

I /2fmM;* when the modulus of K tends to one, the duration of

the shortest interval tends to half of the previousvalue, while the other one tends to infinity.

As shown in Fig. 2, the block diagram of the PRM mod-ulator is very simple [4]. In fact, it is essentially com-posed of an integrator, two comparators, a flip-flop, anda level shifter with an output voltage equal to + VM.The shapes of the most significant circuit waveforms

are shown in Fig. 3. From this figure the durations ofintervals r- and r+ can be found to be

2 Vc= KIVM (1 + vi/VM)

Fig. 2. Block diagram of PRM modulator.

Vi+ VM

V.

I1

Vi-VM I

+ 2VeKr=I/Kl VM(l1 - Vil/VM)

in which the meaning of the symbols can be deduced fromFigs. 2 and 3.Comparing these equations with (3), the values of pa-

rameters K and fmM can be easily obtained as

K = Vi/VM fmM = KlVM/4Ve-Moreover, it is convenient to insert, before the modula-tor, a saturation to limit signal vi between - VM and + VM(i.e., Kbetween -1 and + 1).

Vi V

Vl

V.

Tm

Tm

CRITERIA FOR THE SELECTION OF THE PARAMETERS OFTHE POWER CIRCUIT AND THE COMPENSATION

NETWORKThe main utilization of fixed-frequency inverters with

sinusoidal waveform is in the field of uninterrupted power Fig. 3. Waveforms of signals in PRM modulator (vi = V,,/3).

338

t

-T- NO - IT+ b

BELLINI et al.: APPLICATION OF PULSE RATIO MODULATION

Fig. 4. Block diagram

supplies (UPS). In this application, to reduce the har-monic content of the output voltage, an LC filter must beinserted at the output of the inverter. Therefore, the blockdiagram of the whole converter is the one shown in Fig.4.The selection of the parameters requires an estimation

of the kind of load, which often sinks a very distortedcurrent as in the case of electronic equipment. The mainkinds of distorted currents have either an almost rectan-gular waveform or an impulsive waveform.The kind of first current does not cause relevant distor-

tions on the shape of the inverter output voltage becausethe amplitude of this current is much smaller than the am-plitude of the second one, powers being equal. The sec-ond kind of current, on the contrary, can produce a re-markable distortion near the maximum value of the outputvoltage.The shape of the impulsive current depends on the ac-

tual load; an acceptable approximation of its effects canbe obtained by using triangular pulses with the same areaand the same duration. Consequently, power Pd absorbedby this kind of appliance can be approximated by

Le5 c - LOAD

V.

of converter circuit.

The value of the capacitor can be chosen on the basisof the increase of the first harmonic of the current sup-plied by the inverter when the load is not distorted. Sometests have suggested that a value of C corresponding to anovercurrent of the order of ten percent, with respect to therated value, is acceptable. In this condition, value C ofthe capacitor results in

C = 0.15foVp

(6)

where Pr is the rated value of the power. Higher valuesof C cause more than proportional increases of the over-current.When the load is linear, the maximum value of the volt-

age drop on the inductance is

AVL= 474Lfo PrVP

and it results, generally, in a small percent of Vp also se-lecting a value of f, equal to five timesfo. In contrast, thepresence of a load with a triangular pulse current producesa voltage drop A VLd equal to

AVL = 8PdfoLA 7d2vwhere Vp is the peak value of the first harmonic of theoutput voltage, Ip is the amplitude of the triangular pulse,and y is the ratio between its duration and the half-periodof the output voltage. Therefore, the load will be char-acterized by the percent value of the power connected withthe impulsive current with respect to the total power.

Selection of the Parameters of the Output FilterThe parameters of the output filter should be chosen

taking into account that its natural frequency must bemuch lower than the minimum inverter modulation fre-quency to ensure a sufficient attenuation of the output har-monics. In addition, it is necessary that at frequency fo ofthe output voltage the transfer function of the filter havea modulus close to one and a phase displacement aboutequal to zero. Therefore, undumped natural frequency fnof the filter must be at least five times higher than fo; i.e.,

LC > 10- If 2 (5)

Taking into account that the usual values of -y vary from0.2 to 0.3, it is easy to remark that the voltage drop on

the inductance caused by a triangular pulse current is aboutten times higher than the drop due to a linear load absorb-ing the same power. Therefore, the value of the induc-tance must be chosen taking into account the maximumvalue of the voltage drop and the consequent oversize ofthe supply voltage.

Selection of the Compensation NetworkAfter selecting the parameters of the output filter, it is

possible to determine the loop gain factor, taking into ac-

count its influence both on the tracking error and on theopen-loop gain crossover frequency ( fg). In fact, this lat-ter must be rather higher than fo and lower than the min-imum modulation frequency of the inverter.

In addition, as the phase displacement introduced bythe filter at frequency fg is near 180° and, moreover, the

IPVPY2 (4) (8)

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. PE-2, NO. 4, OCTOBER 1987

modulator gives rise to a further delay, it is necessary tointroduce a lead compensator with a maximum lead angleof almost 50-55°, i.e., with a zero/pole ratio greater than10. The position of the pole of the lead compensator mustbe chosen so that the gain crossover frequency of the com-pensated open-loop transfer function coincides with thefrequency of the maximum phase lead of the network.

COMPARISON BETWEEN THE MODULATION TECHNIQUESThe two modulation techniques have been compared

taking into consideration a circuit characterized by thefollowing values:

* output frequency fo equal to 50 or 60 Hz;* mean modulation frequency fmm equal to 400fo;* undumped natural frequency of the filter f, equal to

1 1 .6ko;* zero/pole ratio of the lead compensator equal to 15;* loop gain factor equal to 30.

The capacitor value has been chosen according to (6).Figs. 5 and 6 show the maximum value of tracking er-

ror eT versus the dc supply voltage V,. Both the values arescaled with respect to the peak value ( Vp) of the outputvoltage.The continuous and short-dashed lines refer to the PWM

technique, with a minimum value of the conduction in-tervals respectively equal to five and ten percent of themodulation period. The dotted line refers to the PRMtechnique chosen in such a way that its average commu-tation frequency is never higher than the PWM one.The diagrams of Fig. 5 were obtained with a linear load

sinking ten percent of the rated power and a distorted loadsinking the remaining 90 percent by a triangular pulsecurrent characterized by a value of -y equal to 0.3. It iseasy to remark that if the minimum conduction intervalthe inverter can perform is not negligible with respect tothe modulation period, then the PWM modulator requiresa higher dc supply voltage. In particular, if the maximumacceptable tracking error is ten percent of the peak outputvoltage, then the PRM technique requires an increase ofthe dc supply voltage of two percent over the peak valueof the output. To obtain the same tracking error with aPWM modulator, the increase should be 12 percent if Tm= 0.O5Tm, and it should be 28 percent if T = 0.lTm.

Fig. 6 refers to a linear load sinking the full ratedpower. In this case also, the PRM technique always al-lows a lower dc supply voltage.

Figs. 7 and 8 present the error waveforms obtained withthe operative conditions shown in Fig. 5 when the maxi-mum tracking error is equal to five percent of Vp. Fig. 7refers to the PRM technique with a dc supply voltage equalto 1.13 Vp; Fig. 8 refers to the PWM technique assumingTm = 0.lTm and a dc supply voltage equal to 1.32Vp. Thedrawn bands represent the ripple envelopes.The results obtained by simulation for the PRM tech-

nique were verified by a prototype loaded by a resistorand Graetz bridge with a capacitive filter. Fig. 9 shows

10%eT

5% -

v/Va p

1.2 1.3 1.41.1

Fig. 5. Tracking errors versus supply voltage for distorted load.

10%eT

5% ±

i 01.1 1.2 1.3 V. / VP

Fig. 6. Tracking errors versus supply voltage for linear load.

the waveform of the error (taken at the output of the inputsumming node) and of the distorted load current obtainedon the prototype in the same operative conditions relatedto Fig. 7.The experimental results are in good agreement with

the simulation ones, particularly concerning the value ofthe tracking error versus the dc supply voltage. In con-trast, during the leading edge of the current pulse the rip-ple is rather lower than the expected one because thesmoothing capacitor of the rectifier takes part in the fil-tering action.

CONCLUSIONThe closed-loop control of the output waveform of a

fixed-frequency inverter is a very interesting approach ifa low distortion is desired, even in the case of loads sink-ing a distorted current. On the other hand, this approachrequires a rather high oversize of the dc supply voltage ifusual PWM techniques are used and the minimum con-duction intervals the inverter can perform are not negli-gible with respect to the modulation period.The paper proposes to reduce the oversize by using a

different modulation technique. This technique, known aspulse ratio modulation, was successfully employed indc /dc converters.To obtain a significant comparison between the pro-

posed technique and the classical one, it has been neces-sary to characterize the typical kinds of load and to definethe selecting criteria for the output filter parameters and

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BELLINI et al.: APPLICATION OF PULSE RATIO MODULATION

T (1 - y)/4 T (1+ y)/4 To/2

Fig. 7. Tracking error with PRM.

TI(l - Y)/4

Fig. 8. Tracking error with PWM.

Fig. 9. Upper trace: distorted load current. Lower trace: tracking error(two percent/div.). Horizontal axis: 2 ms:div. (Drawn from photo-graph.)

for the compensation network. The comparison has shownthat in practical operating conditions the PRM techniqueallows a remarkable reduction of the supply voltage,which results in voltages comparable with those used inopen-loop converters.

REFERENCES

[1] G. Buja and G. B. Indri, "Optimal pulsewidth modulation for feedingac motors," IEEE Trans. Ind. Appl., vol. IA-13, no. 1, 1977.

[2] R. A. Shaffer, "A new pulse modulator for accurate dc amplificationwith linear or nonlinear devices," IRE Trans. Ind., vol. 1-11, p. 34,1962.

[3] P. Burger, "Analysis of a class of pulse modulated dc/dc power con-

verters," IEEE Trans. Ind. Electron. Contr. Instrum., vol. IECI-22,p. 104, 1975.

[4] A. Balestrino, A. Eisinberg, and L. Sciavicco, "Pulse ratio modula-tion: An interesting technique to implement dc/dc conversion," pre-sented at the 2nd IFAC Symp. on Control in Power Electronics andElectrical Drives, Dusseldorf, 1977.

[5] A. Balestrino, G. De Maria, and L.. Sciavicco, "Analysis of modula-tion processes in power convertors," Proc. Inst. Elec. Eng., vol. 125,no. 5, 1978.

Armando Bellini was born in Civitella di Ro-magna, Italy, on October 2, 1941. He received thePh.D. degree in electrical engineering from the

Polytechnic Institute of Milan, Milan, Italy, in_|1 ~~~1966.

He joined the Electrotechnical Institute ofMilan in 1966 as a Researcher in the AutomaticControl Laboratory. In 1967, he moved to the Au-

tomatic Control Department, Electrotechnical In-stitute, University of Rome, Rome, Italy. In 1971,he became an Assistant Professor in the Auto-

matic Control Institute of the University of Rome and then also an Asso-ciate Professor of that Institute. In 1980, he became a Full Professor ofElectrical Drives at the University of Calabria, and presently he is a FullProfessor of Industrial Electronics at the Second University of Rome "TorVergata.'" His major fields of interest are power electronics, speed controlof ac motors by means of static converters, and industrial drives.

Alberto De Santis was born in Rome, Italy, onApril 11, 1958. He received the laurea degree inelectronic engineering from the University ofRome in 1984, and a post-graduate degree in sys-

g_;. tems and automatic control theory in 1986.He joined the Computer Science and Systems

Department of the University of Rome in 1985,working on cc/ac inverters for ac motor drives.

_ Now he is a Researcher at the System Analysisand Computer Science Institute of the NationalResearch Council, Italy. His field of interest is

identification and filtering of stochastic infinite-dimensional systems.

eT'

T0 (I + y)/4 Ta12

. . . . . .

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. PE-2, NO. 4, OCTOBER 1987

Gennaro Figalli was born in Naples, Italy, onJune 18, 1949. He received the Ph.D. degree inelectrical engineering, summa cum laude, from theUniversity of Naples, Naples, Italy, in 1974.

Following his graduation, he was with the Au-tomatic Control Institute of the University ofRome, Rome, Italy, as a Researcher. In 1978, hebecame an Assistant Professor with that Institute.He was also with the Automatic and ElectronicDepartment of the University of Ancona, Ancona,Italy, as an Associate Professor lecturing on power

electronics. Since 1982, he has been an Associate Professor of AutomaticControl at the University of Rome. His major fields of interest are power

electronics, speed control of ac motors by means of static converters, andindustrial drives.

Giovanni Ulivi was born in Rome, Italy, on No-vember 20, 1950. He received the Ph.D. degreein electronic engineering from the University ofRome, Rome, Italy, in 1974.

He joined the Automatic Control Institute ofthe University of Rome in 1976 as a Researcher.His major fields of interest are industrial drives,power electronics, and related applications of mi-crocomputers.

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