575Bull Pol Ac Tech 64(3) 2016 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol 64, No 3, 2016 DOI 10 1515/bpasts 2016 0064 *e mail r smolenski@iee uz zgora pl Abstract The assurance[.]
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol 64, No 3, 2016 DOI: 10.1515/bpasts-2016-0064 Diophantine equation based model of data transmission errors caused by interference generated by DC-DC converters with deterministic modulation J BOJARSKI1, R SMOLENSKI2*, P LEZYNSKI2 and Z SADOWSKI2 Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora, ul Licealna 9, 65-417 Zielona Gora, Poland Institute of Electrical Engineering, University of Zielona Gora, ul Licealna 9, 65-417 Zielona Gora, Poland Abstract The assurance of the electromagnetic compatibility of sensitive smart metering systems and power electronic converters, which introduce high-level electromagnetic interference is important factor conditioning reliable operation of up to date power systems Presented experimental results have shown that currently binding, frequency domain tests are ineffective for the evaluation of data transmission error hazards The proposed in this paper mathematical, time-domain model, based on Diophantine equation, enables evaluation of data transmission errors caused by interference introduced by DC-DC power electronic interfaces with deterministic modulation In the paper there have been presented possible application areas for the proposed model Key words: electromagnetic compatibility, electromagnetic interference, power electronic converters, Diophantine equation Introduction Power electronic converters are increasingly being used as interfaces in Smart Grid systems [1–3], however due to pulse mode of operation and high dv/dt and di/dt rates they produce a substantial level of electromagnetic interference (EMI) [4–7], which is why power electronic converters are often treated by their designers as the main causes of industrial system malfunctions Conducted EMI can be associated with unwanted currents and voltages that might be coupled, by means of parasitic couplings [8], to data transmission circuits Superimposition of the EMI voltages onto voltages representing binary data signals may lead to data transmission errors Electro-Magnetic Compatibility (EMC), according to the European EMC Directive 2004/108/EC, “means the ability of equipment to function satisfactorily in its electromagnetic environment without introducing intolerable electromagnetic disturbances to other equipment in that environment’’ This general description requires detailed rules on how to evaluate EMC assurance The evaluation is usually based on harmonization standards which provide strict measuring procedures Fulfillment of the requirements of harmonization standards is taken as presumption of EMC compliance However, the increasing number of novel power electronic interfaces generating a high level of electromagnetic interference (EMI), connected with susceptible control systems operating in time domain, has made currently binding frequency domain EMC standards obsolete *e-mail: r.smolenski@iee.uz.zgora.pl Thus, evaluation of the probability of the appearance of data transmission errors, caused by interference generated by power electronic converters, is important for both cognitive and technical reasons Elaboration of the presented mathematical model has enabled explanation of significant differences in experimentally obtained distributions of awaiting times for data transmission errors caused by interference generated by a DC-DC converter with random [9–11] and deterministic modulation The evaluated probability of the appearance of data transmission error may become a useful factor supporting the development of data transmission systems The DC-DC converter was selected as an interference source for mathematical analyses of data transmission errors caused by interference generated by converters with deterministic modulation A detailed description of the interference introduced by a DC-DC converter [12–14] has been presented in our previous paper [15] Deterministic vs random modulation of power electronic interfaces Random modulation of power electronic interfaces is often recommended as an EMI reduction technique [16, 17] In fact the utilization of random modulation does provide ostensible reduction [18, 19] of the interference levels measured, in accordance with EMC standards, in the frequency domain Fig. 1 shows the experimental results for conducted emission generated by a DC-DC converter with deterministic and random modulation, measured using average detector, in accordance with the EN 55022 standard, with the upper limit of the frequency range reduced from 30 MHz to MHz for better clarity of the results 575 Bull Pol Ac.: Tech 64(3) 2016 Brought to you by | New Mexico State University Authenticated Download Date | 1/22/17 2:29 PM 80 Deterministic Random 70 60 50 0.2 1.0 2.0 3.0 4.0 Frequency [MHz] Fig Conducted electromagnetic interference, generated by deterministic and random modulated converters, measured according to EN 55022 standard, using average detector In fact, a decrease in the observed interference levels has exceeded 6 dB This means that the conducted emission, measured according to standards in the frequency domain, has been decreased more than twofold, without investment in converter hardware However, commonly applied data transmission standards use time domain signals Thus, the lowering of maximum levels of interference, caused by even distribution of interference over frequency range in the case of random modulation, does not imply lowering of interference waveforms in the time domain The phenomenon of the appearance of interference caused by power electronic converters is precisely described in the literature [4, 20, 8] The shape of the time-domain, damped oscillatory mode, interference pulse depends on the high frequency impedance of interference paths as well as rising and falling slopes of the converter voltages, exciting interference currents [4] Thus, the waveform of the individual damped oscillatory mode pulse is identical in both the deterministic and the random modulation cases, only the moment of the its appearance is different The typical experimental waveforms of the interference currents generated by a DC-DC converter with deterministic modulation are shown in Fig tered in real situations A comparison of distributions of awaiting times for data transmission errors caused by interference, generated by a DC-DC converter with random and deterministic modulation, has indicated significant dependence of the obtained results on the selected converter switching frequency In order to evaluate this dependence, the time-consuming measurement of transmission errors for various converter switching frequencies have been performed Figure shows probabilities of data transmission error appearance caused by electromagnetic interference introduced by the same power electronic interface with both deterministic and random modulation for various converter switching frequencies (depicted by dots and squares) For each measuring point more than 1000 error events was collected for the probability evaluation The curves presented in Fig differ significantly, indicating the different natures of the phenomena of error appearing in the same system but with different modulation Based on the presented data it is hard to say which modulation is better In the case of the deterministic modulation small changes in switching frequency of the converter bring about significant differences in observed data transmission error probabilities The obtained results encouraged the authors to make an indepth investigation concerning descriptions of the phenomenon of data transmission error occurrence caused by DC-DC converters with deterministic modulation The Diophantine equation-based model of data transmission errors caused by interference generated by DC-DC converters with deterministic modulation, as presented in this paper, constitutes a significant complement for the time-domain model of probability of data transmission error appearance in a system consisting of DCDC converters with random modulation, as presented in our previous work [18] 0.35 Deterministic experimental Random experimental 0.30 Error probability EMI Average Detector Level [dBµV] J Bojarski, R Smolenski, P Lezynski and Z Sadowski 0.25 0.20 0.15 Probability of the appearance of error caused by interference generated by DC-DC converters with deterministic and random modulation In spite of the significant ostensible reduction in level of interference measured using normalized equipment our preliminary, experimental investigations have revealed that random modulation has no clear advantage over deterministic modulation in the context of data transmission errors induced by converter operation Measurements have been made in a system consisting of a DC-DC power electronic converter with both deterministic and random modulation and a series data transmission system, representing distorted microprocessor control circuits, encoun- 576 0.10 15 20 25 30 35 40 45 50 Time between interferences [µs] Fig Dependence between transmission error probability and time between interference pulses generated by deterministic and random modulated DC-DC converters The aim of using the time-domain based model of data transmission to assess the probability of appearance of error in a system consisting of DC-DC converters with random modulation was to formulate a mathematical model based on the Bull Pol Ac.: Tech 64(3) 2016 Brought to you by | New Mexico State University Authenticated Download Date | 1/22/17 2:29 PM Voltage [V] the transmission standard series transmisrorRS in a232 given transmission standard represents In the investigated case ror in a given transmission standard In the investigated caseon sion that232 isthe commonly used in microprocessor systems [18].uatedapp the RS transmission standard represents series transmisRS 232 transmission standard represents series transmislimi Insion the that experimental the voltage levels -3 Vapplicabi ission commonly used in microprocessor systemsfrom [18] that isarrangement commonly used in microprocessor systems [18] limitation tion In thetoexperimental arrangement the voltage levels fromlevels -3 V from down -5 In V the correspond to the binary "1"the and levels from V -3 V experimental arrangement voltage tion is de Diophantine equation based model of data transmission errors down to -5 V correspond to the binary "1" and levels from of V from up to V down correspond the binary "0", thus"1" addition the V to -5 V to correspond to the binary and levels +to the binary "0", thus addition of the up to V up correspond , higher than V, to the "1" andof the interference signals to V pcorrespond to the2 binary "0",binary thus addition +i Diop + − , higher than V, to the binary "1" and interference signals p i signals than about V, to the binary "1" and interference p , higher , lower than V, to the binary "0" brings data transp “1” – ∆S1 ∆S2 ∆S3 ∆S4PWM carrier ip− and pthan 2 V, to ithe"0" binary “0” brings about data tran , lower V, than to the binary brings about data transi−, ,lower i lower than V, to the binary "0" brings about data transp A mission error i dt transmission error mission error mission error.bits of information are sent in equal time gene When the consecutive the consecutive of information sent intime equal simplified WhenWhen the consecutive bits ofbits information are sentare in are equal mini When the consecutive of information sent in equal time (const) and bits ∆t interference signal intervals ii = time intervals (const) and i = ¢t and = ∆t ∆t¢t(const) intervals ∆tintervals αA ∆ti = ∆t (const) and Fig 6Fig sh 1 stitu stituting t {∆t, ∆S λ )} (1 − < dt < Time [s] ∆S0¢S )}− λ(1)}− λ )} (10 − 0< < dt < < 02min min{∆t, {¢t, (λ1∆S p+ p+ p+ p+ p− p− p− < dt < {∆t, data data tran bit checking then the probability ofofthe the appearance of error inerror N-bit frame then the probability of appearance of of error inof N-bit frame ∆t DCc then the probability of the appearance inframe N-bitDC-DC frame then the probability the appearance error in N-bit can be expressed by [18]: can be expressed by [18]: was elab was can be expressed can be expressed by [18]: by [18]: Time [s] t0 t1 t2 t3 t4 t5 t6 the pheno the NN N k verters w Fig Simplified scheme describing influence of the interference k k vert P(X = k), (1) − (1 − p = ) P e error = k), (1)errors (1) pe ) = P(X = (1 Perror1 − Perror = ∑ − 1p− P(X k),(1) ∑ e )(1 − ∑ ind caused by DC-DC converter with random modulation on the transk=0 erro k=0 k=0 mission signal where: where: where: where: 4.1 Tim 4.1 period of the carrier function, ∆S0 average equation, average period thecarrier carrierfunction, function, ∆S0period ¢S0 – average period ofofthe average of the carrier function, ∆S equ = 1, 2, (i.)2, independent random random variables of distri-of ofterminist =are 1, 2, ) are independent random variables distri∆S knowledge of the occurrence of the phenomenon of data trans- ∆Si (i¢S (i = 1, …) – are independent variables i (i = i1,U2,((1 )−are independent random variables of distriterm ∆Si butions )∆S (1 λ )∆S λ λ∈)∆S 1), 01, − , ,0(1 + ), ), λ∈ (0, 1),1), period, th butionsλ U ((1 mission error caused by interference introduced by PEIs The distributions (( − λλ+))∆S ¢S (),1 + λ )(0, ¢S λ 2 (0, 00 0 Error probability λ )∆S0 , (1 + λrepresenting )∆S0 ), λ ∈ the (0, 1), U ((1 model should be as simple as possible, yet should enable as- butions dt – is the − established value duration of Bull Pol Ac.: Tech XX(Y) 2016 sessment of the probability of a transmission error occurrence theBull interference signal that2016 can cause transmission error, Pol Ac.: Tech XX(Y) Ac.: Tech.by XX(Y) 2016 during the transmission of a frame containing N-bits in the pres- Bull Pol depicted a gray stripe, ence of interference with given parameters pi+ , pi– – points for i = 1, 2, … determine moments of “+” Figure shows schematically the carrier function repreand “ −” signal appearance, respectively, senting both deterministic (¢Si = const) and random modulan – number of bits in the transmitted frame, tion (¢Si = var) of PEI The ti depicts transmitted signal value ti – moment of i-th bit sending checking moments while pi+ and pi– depict simplified interference signals introduced by operation of the converter 0.35 Real interference signals are usually more complex than Random experimental Random model data signals, however measured interference signals can be sim0.30 plified for the sake of the model [18] In practice, the required parameters of the simplified interference signal can be deter0.25 mined on the basis of the time domain measurements of the interference voltage at the terminals of the transmission data 0.20 receiver The parameters of the simplified interference signals are depicted on the interference voltage waveform presented 0.15 in Fig. 4 The width dt of the simplified interference signals pi+ and 0.10 – pi should be measured at a level that causes the appearance of 15 20 25 30 35 40 45 50 error in a given transmission standard In the investigated case the RS 232 transmission standard represents series transmisTime between interferences [µs] sion that is commonly used in microprocessor systems [18] In the experimental arrangement the voltage levels from −3 V Fig Experimental and simulation dependencies between transmisdown to −5 V correspond to the binary “1” and levels from sion error probability and time interval between interference pulses generated by random modulated DC-DC converters −3 V up to 5 V correspond to the binary “0”, thus addition + of the interference signals pi , higher than 2 V, to the binary dt Voltage [V] Time [µs] -2 Fig Experimental waveform of interference voltage with depicted dt parameter of simplified interference signal Figure shows the probabilities of the appearance of a data transmission error observed in the experimental arrangement with random modulation and corresponding probabilities evaluated on the basis of the probability of the appearance of a time-domain based model of data transmission error in a system consisting of DC-DC converters with random modulation, as presented in our previous work [18] Both curves the experimentally obtained as well as the evaluated on the basis of the model fit well, however the model applicability is limited to random modulated converters The limitation concerning minimum demanded level of ¢Si variation is described in the paper [18] 577 Bull Pol Ac.: Tech 64(3) 2016 Brought to you by | New Mexico State University Authenticated Download Date | 1/22/17 2:29 PM peri J Bojarski, R Smolenski, P Lezynski and Z Sadowski Voltage [V] A p+ D αA p+ p− p− p+ p− interference number M Generally, it can be written that d-th discrete time interval of i-th interference signal appears in moment: p+ Time [s] N M((1 − ® )(i mod 2) + (i div 2)) + t + d,(3) i = 0,1,…,t = 1 − M, − M, …,0,d = 0,1,…,D − 1, J Bojarski, R Smolenski, P Lezynski and Z Sadowski where t is+ the moment of the appearance of the zero interfer4.5 Estimation of the probabili p2 p+ p− p− ence1 signal data transmission error caused Fig Simplified scheme describing influence of the interference J Bojarski, R.J.Smolenski, Bojarski, R P.by Smolenski, Lezynski and P.converters Lezynski Z Sadowski andwith Z Sadow DC-DC dete αAmodulation on the interference caused by DC-DC converter with deterministic 4.4 Superimposition of data and interference signals The M number ing Diophantine equation-based transmission signal A + A− the bit-checking moEstimation 4.5 Estimation of the probab of p0 p+ p+ p+ p+ p0 p− p− p− interference signal 3 on 4.5 is superimposed Time [s] an applicability of the proposed D D D ment, when the equation, with earlier assumed conditions, is data transmission data transmission error cause N model the calculations correspon bit satisfied: by DC-DC by converters DC-DC with conver de number αA αA interference interference tained results presented in Fig M M number number ing Diophantine ing Diophantine equation-base eq Diophantine equation basedFig model of data describing 3of the interference 6 Simplified scheme influence estimation of the probability of the Time Time [s] Nj − M((1 − ®)(i mod 2) + [s](i div an 2)) − t − d = 0, (4) applicability an applicability of the proposed of transmission errors caused by interference caused by DC-DC converter with Ndeterministic modulation on the mission error caused by interfere N model the calculations model the calculati correspo bit bit generated by DC-DC converters transmission signal0 5 converters with deterministic modu number number tained resultstained presented resultsinprese Fig with conditions with deterministic modulation tions presented below Fig Simplified Fig 6.scheme Simplified describing scheme influence describing of the influence interference of the interference estimation ofestimation the probability of theofpro t J Bojarski, R Smolenski, P Lezynski and Z Sadowski 2 {0,1,…, 1}, could be by assumed that for some M, N, Di with ∈ N, j there is a rela-the nT−HEOREM DC-DC caused converter by DC-DC withconverter deterministic deterministic modulation onmodulation on(5) theerror mission The quadruples (j mission caused error by interfe cause Figure shows the scheme of data andcaused interference signals tionship A + transmission signal transmission signal equation (4) givendete par converters deterministic mo constituting the basis equation 4.5 t Estimation probability 2ofphantine the appearance of are a with p+ p+ p+ p− p− p− {1 − M, of −the M,…,0}, d {0,1, …, Dwith −converters 1}, based model forp1the Diophantine D τ , ∆t = N τ , dt = (D − 1) τ , α M ∈ N = M ∆S tions presented tions below presented belo of data transmission errors caused by interference generated data transmission error caused by interference generated J Bojarski,byR Smolenski, P Lezynski and Z Sadowski t = (N j mod (−M) − d − (1 − J Bojarski, R.beSmolenski, P assumed Lezynski and Z Sadowski DC-DC The scheme and known N M ®(® M ) could assumed could be that for some that M, for N, some D ∈ N M, there N, D is ∈ a N relathere is a relaby DC-DC converters with deterministic modulation us, , αA converters with deterministic modulation interference the interval between the interference Then M determines HEOREM HEOREM The quadruples The T T M A N j − t − d − (1 − α )M k number tionship tionship was elaborated on the basis of the knowledge about the equation 4equation-based is known of as Diophantine [21–24] ing The Diophantine order to validate 4.5 Estimation themodel probability the appearance of(4 p2+ p4+ p6+ p+ p3− p5− p− iIn =equation of physical pulses, equation phantine (4)of equation area given pa+ A + + + + N the time between bit value checking moments, D the − − M D p− Estimation of theproposed probability ofphantine the by appearance p p p p p p Time [s] interference 2generation 3DC-DC convert- 4.5 0of the phenomenon by an applicability of the Diophantine equation-based data transmission error caused interference generated distretized time the interference as , ∆S ∆t0 simplified N τ , dt ∆t = = (D N τ− , 1)dt τpulse, , = (D αM −∈shown 1)N τ, α M ∈ N M τof =M ∆S0 = D N where jappearance =∈ 1,t modulation = ob.(N ,ofn − 1}, t= (N j{0, mod (−M) j−mod d −d(− (1∈ caused by interference generated ers with deterministic modulation and3 the occurrence ofbiterrors data 4.5.transmission Estimation oferror thecorresponding probability of the model the calculations to deterministic experimentally by DC-DC converters with us interference in Fig αA M number M the {0, 1} Then M determines Then determines interval the between interval the between interference the interference number by DC-DC converters with deterministic modulation tained results presented in Fig have been performed of given parameters [15, 19] a data transmission error caused by interference ing Diophantine equation-based model toj − validate N jgenerated −In t −order dAn −Nus(1 t− α )Md αA induced by interference interference M i =the a data transi=2 Time [s]value pulses, NDiophantine the pulses, time between N the time bit between checking bitprobability value moments, checking D moments, the D Fig.Time Simplified scheme describing influence of the interference estimation of the of the appearance of number 4.1 discretization In order to apply the by DC-DC converters with deterministic modulation using an applicability of the proposed Diophantine equation-based ing Diophantine equation-based model In order to validate M signal model assumptions The n bits of informa- Proof By substitution of (6) into ( 54.2 Data N4 distretized time ofdethe an simplified time of theinterference simplified interference pulse, as shown pulse, as shown caused bythe DC-DC converter deterministic modulation on the Time [s] mission error caused by interference introduced by DC-DC equation, time has been0 with discretized with the step τ.distretized For Diophantine equation-based In order validate model calculations corresponding to experimentally obd bit applicability of the the proposed where jtoequation-based =∈ where {0, 1, an j .=∈ , n −{0, 1},1, tion are transmitted in N5 discrete time intervals There ismodel noDiophantine number transmission signal N in Fig in Fig converters with deterministic modulation requires the assumptained results presented in Fig have been performed An terministic modulation the triangle carrier function has a fixed applicability of the proposed Diophantine equation-based model In the further investigation, there w {0, {0, 1} oblack of generality if we assume bit of information is model that the zero calculations corresponding to 1} experimentally bit numberof the Simplified scheme describing influence interference tions presented below estimation of the probability of the appearance of a data trans of information bits disturbed by i period, 0thus Fig ¢S1 i6 = ¢S for i = 1, 2, …, moreover, it the calculations corresponding to experimentally obtained results 0 = const, sent in the moment Therefore, the moment of the j-th tained results presented in Fig.nbitbits have been performed An signal 4.2 Data model signal assumptions assumptions The nhave bits been of The informaof informaProof By substitution Proof By of substituti (6) into caused that by converter with modulation on model the inmission error caused by interference introduced by DC-DC assumed condition that the 0-bit of could for M, D there presented Fig. 2 performed An estimation of the could be be assumed assumed thatDC-DC for some some M, N, N,appearance D4.2 ∈deterministic NData is a relais expressed by: HEOREMof1.theThe quadruples j,no d,t, i) satisfying DioFig Simplified scheme describing influence tion of the estimation probability of is(the appearance of athe data transareinterference transmitted tion are in transmitted NT discrete intime N discrete intervals time There intervals There is no converters with deterministic modulation requires moment t0 error tionship transmission signal tionship probability of the appearance of a data transmission causedthe assump phantine equation (4) are given parametrically by caused by DC-DC converter with deterministic lack modulation on the In the further In investigation, the further invest there mission error caused by interference introduced by DC-DC Nif j, jby = interference 0, if.that we ,n− introduced (2) of generality lack of generality we assume assume zero bit that of information zero bitDC-DC of information is converters is with detertions presented below by EFINITION Let D ∆t = N τ , dt = (D −sent 1) τ ,in the α M0∈ N transmission ∆S signal of information of information bits disturbed = M τ, with deterministic modulation requires thebelow assump- bitsby αof)M k))j-th mod = (Nmodulation jTherefore, mod (−M) − the dmoment −the (1 − moment theconverters Therefore, 0isministic moment the moment the of j-th bit the bit(M), ÂS0=M,Ât=N,dt=(D 1) ,đM there requires assumptions presented could be assumed that for some M, N, Dsent ∈ Nin a trelaHEOREM The quadruples ( j, d,t, i) satisfying the0-bit Dio T assumed condition assumed the 4.3 Interference signaltions model assumptions In(1practice presented appearance isappearance expressed is by:expressed by: )that =condition #{( j, d,t,t L(t0(6) Then M determines the interference tionship the interval between N jphantine − tbelow − d −equation − α )M kbit (4) are given parametrically by moment t moment t + k, i = 0 time much shorter than the1.duration of the interfertimefor between checking moments, D the could bepulses, assumed that some M,value N, Dbetween ∈checking N there is aisrelaThenNMthedetermines the bit interval the interference Theorem ( j,d,t,i) Diophantine NM j,T∈HEOREM j = 0,N j, ,1 nThe −jThe =quadruples 0,M quadruples , n − where ( (2) j,the d,t, i) satisfying ((2) j, satisfying d,t, i) satisfying the Dio-(4) , dtsignal, = (D − 1) τ , canα N ∆S0 = M τ , ∆t = N τ ence which cause transmission error The bit checkα )M k)) mod (−M), t = (N j mod (−M) − d − (1 − distretized time of the simplified interference pulse, as shown EFINITION EFINITION Let Let D D tionshippulses, N the time between bit value checking moments, D the phantine equationjequation (4) are where =∈ {0, given 1,(4) of , nparametrically −given 1}, dparametrically ∈ {0,by 1, , D −by 1} except and k ∈specific are ing time is the time required for determination the voltage Unfortunately, case (6 in Fig Then M determines the interval between the interference distretized time of the simplified interference pulse, as shown jIn−practice t − d − (1 α )M k bit 4.3 Interference signal{0, model signal assumptions model N assumptions bit In−practice 1}.or j,L(t d,0 L(t0is) = = NNτ ,the dt = between (D − 1)level τ ,4.3 αInterference Mchecking N M τ , ∆t ∆S0in=Fig. 6 + k, (−M), = − d − (1Inthat is∈connected with t“0” “1” biti representation prime numbers, there no#{( general pulses, time bit value moments, D the = (N j mod (−M) − α )M k)) mod checking time checking is muchtime shorter is much than the shorter duration than the of the duration interferM of the interferterference signals appear in time intervals M and the width of Thus, a numerical evaluation must where ( j, d,t,where i) satisfying ( j, (6) d,t, i) (4) sati Data signal model assumptions The bits of which informadistretized time of the simplified pulse, aswhich shown Proof By substitution of− (6) into ence signal, ence signal, can cause transmission can error The bit checkbit checkThen M4.2.determines the interval between theninterference interference where {0, k,(4) nThe − According 1}, d ∈ {0,to1,the ,above D (6) − 1} and k ∈ Ntojcause − t −transmission d j−=∈ (1 α1,error )M the single, simplified signal is equal D of discrete time intermentioned tion are transmitted in N discrete time intervals There is no in Fig i = + k, Data signal modelbitassumptions Theing nmoments, bits ofisinformation time the ing timerequired is the time for required determination for determination voltage of be theUnfortunately, voltage theUnfortunately, except specific excep ca pulses, 4.2 N the time between value checking D time the {0,investigation, 1} In the further there will M of the vals Without the lackof it can be assumed that the andanalyzed interferencenumber signals are determ lacktransmitted of generality ifdiscrete we assume that zero bitThere ofthat information isofisgenerality, are in N time intervals is no lack level is connected level that with connected “0” or “1” with bit “0” representation or “1” bit representation InInprime numbers, prime there numbers, is no gene ther distretized time of the simplified interference pulse, as shown of information bits by interference signals, forand theinterference interference appears not later than zero signal theinto of where j =∈ {0, 1,M.1, disturbed intervals ,substitution ndata ∈ {0, 1,appearance (4) .…, ,aof Dnumerical k ∈ mu sent in theif04.2 moment Therefore, the moment of the j-th bit Data signal model assumptions The nsignal bits informaProof By of (6) generality we assume that zero bit ofzero information is sent in of where j = 2 {0, …, n− −1}, 1}, d 2 {0, 1, D− −1} 1}theand terference signals terference appear signals in time appear intervals in time and the width Mdof and of the width Thus, Thus, evaluation aatnumerical ev in Fig assumed condition that the 0-bit information will emerge Generally, it can be written that d-th discrete time interval of sume that t is discrete uniform di appearance is expressed by: tion are transmitted in N discrete time intervals There is no {0, 1} the moment Therefore, the moment of the bit appearance k 2 {0,1} thej-th single, simplified the single, signal simplified is equal signal to D is of equal discrete to D time of discrete intertime interAccording to According the above to mention the ab moment t0 In the further investigation,[1there be analyzed the numbe i-th interference signal appears inismoment: − M,will 0], then the probability of t if we assume that zero vals bit information is expressed lack by: of generality vals Without theoflack Without the lack ofitofgenerality, can be assumed it can that bedisturbed assumed the and thatinterference the and interference signals are sig N j, j = 0, The , n −n1 (2) of generality, information bits by interference signals, fordete the 4.2 Data signal model assumptions bits of informamission error (using the Theorem Proof By substitution of (6) into (4) sent in the moment Therefore, the moment of the j-th bit EFINITION Let D mod interference 2) + (iappears div 2))signal +not t +substitution d, thannot M ((1interference − α )(i zero Proof By of (6)than into (4) data zero signal later appears zero later data signal zero signal the appearance the of appearance the interferenc of th assumed condition that the 0-bit of information will emerge a (3) caused by interference signal intr tion are transmittedappearance in N discrete time intervals There is no(2) is expressed by: j, j = 0, − i1. Generally, can bebit written it can that be d-th written discrete that d-th time discrete interval time of interval of sume that t sume is discrete that t uniform is discr 4.3 InterferenceNsignal model…, n assumptions = 0, 1,In practice ,ittGenerally, = 1− M, − M, , 0, d = 0, 1, , D − 1, ) = #{( j, d,t, i) : t = t }, (7) L(t In the further investigation, there will be analyzed the number 0 moment t with deterministic modulatio lack of generality if we assume thatthan zero bitduration of information is In the further investigation, there will verter be[1analyzed the interference signal appears signal in moment: appears in moment: − M, 0], then [1 number −the M, probability 0],the then theop checking time is much shorter the1.i-th interferN j,the jwhere =i-th 0, interference t is ,of nthe − (2) of information bits disturbed by interference signals, for L(t0 ) moment of the appearance of the zero interferwhere ( j, d,t, i) satisfying (4) sent in 4.3 the moment Therefore, the moment of the j-th bit EFINITION Let D model assumptions In practice bit of information bits disturbed by interference signals, for theerror mission error mission (using the Theore enceInterference signal, whichsignal can cause transmission error The bit check1(using α )(i mod 2) α + )(i (i div mod 2)) 2) + + t + (i div d, 2)) + t + d, M ((1 − M ((1 − ence signal assumed condition that the of information will emerge 1by − at P(error) = ∑ appearance is expressed by:required checking time istime much shorter for than themodel duration of assumed that the0-bit 0-bit of information will emerge (3) (3) caused by interference caused interfere signal ing time is the determination of the theinterfervoltage 4.3 Interference signal assumptions In practice bit condition Unfortunately, except specific cases in which N and M are co2 (7in i) : t M = tt00=1−M }, L(t )1,= .#{( t.0 M, i representation = 0,the 1, duration bit , it check= 0, 1In− 1,the M, moment prime 2interfer, t−moment =M, 1numbers, − , 0, M,0, 1, is, 0, no ,dDgeneral = − 0, 1, , Dj,verter −d,t, 1, ence which cantime cause transmission error The at t02 d−=there with deterministic verter with determin modula levelsignal, that ischecking connected with “0” or “1” bit is much shorter than of ) formula for function L(t N j, j = 0, , n − 4.4 Superimposition (2) of data and interference signals The (4) where (evaluation j,the d,t, i) satisfying ing time is signals the time required for the voltage L( where tofisthe the where moment tofis D the ofEFINITION the moment appearance of the of appearance zero interferof thebezero interferterference appear in time intervals M and width ence signal, which candetermination cause transmission error bit checkThus, a numerical must Note that the probability of1 the interference signal isThe superimposed on1.theLet bit-checking mo- performed 0ap level that issimplified connected with or “1” representation InDefinition Let above mentioned ence signal.time enceintersignal the single, signal is“0” equal to Dbit of discrete ing time is the time required for determination of the voltage According to the assumptions both the data Unfortunately, except specific cases in which N and M are co − P(error) = P(error) = ∑ (7)M mean ∑ is M the arithmetic ment,Inwhen the equation, conditions, is i) :mitted 4.3 Interference signalappear model practice bit of with earlier assumed 20 =1− #{(there j, d,t, t =but t0 data },formula L(t ) = are terference signals in assumptions time intervals Morand the width tfor =1−M tL(t vals Without thethat lackisofconnected generality, it can be assumed that the and interference level with “0” “1” bit representation Insignals deterministic, the moment of prime numbers, is no general function 0) occurrence for a specified period satisfied: checking time is much shorter than thenotduration ofintervals the interferthe single, simplified signal is equal to of discrete time interL(t ) = #{( j,d,t,i): t = t }, (7) 4.4 Superimposition 4.4 Superimposition ofwidth data and of interference data and interference signals The signals The zero interference signal appears later than zero data terference signals appear inDtime Msignal and thewhere of the appearance of the interference signal is random Let’s asThus, a numerical evaluation must be performed 0 ( j, d,t, i) satisfying (4) ference signal The numerical ana ence signal, which can cause transmission bit checkNotewith that theNote probability that the of proba the )(isuperimposed mod 2) + div − tbit-checking − dtouniform = 0,the (4) Nassumed jto− Minterval ((1 −α vals Without thesingle, lack of generality, can beThe that the interference signal is signal is(i superimposed on the on bit-checking mo-mentioned moGenerally, itthe can be written that signal d-thiterror discrete time of simplified isinterference equal D of discrete time intersume that t According is2)) discrete distribution the interval the above assumptions both data R software environment forthe statisti ing timezero is the time required for determination of the voltage mitted data is mitted the arithmetic data is the mea Unfortunately, except specific cases in deterministic, which is Nofand Mtransare interference signal appears notmoment: than zero data where (earlier j,d,t,i) satisfying (4) ment, when ment, the equation, when the with with earlier conditions, assumed i-th interference appears in vals signal Without the lack oflater generality, it can besignal assumed that the [1 −equation, M, 0], then the probability of theisconditions, appearance databut andassumed interference signals are the comoment ar o with conditions [26] occurrence for occurrence a specified for perio a sp level that is connected with “0” orsignal “1” bit representation Insatisfied: satisfied: ) prime numbers, there is no general formula for function L(t zero interference appears not later than zero data signal mission error (using the Theorem of Total Probability [25]), the appearance of the interference signal is random Let’s as Fig shows the probabilities of th M ((1 − α )(i mod 2) + (i div 2)) + t + d, i ∈ N,of j ∈interval {0, 1, a .of , by n −interference 1}, that ference signal ference The numerical signal Thean terference signals appear in time andd-th the width Generally, it canintervals be writtenMthat discrete time (3) caused signal introduced a(4) DC-DC consume t(iis discrete uniform the interva Thus, numerical evaluation must be performed (5) mission error for with different conve α )(i mod 2) α + )(i (i div mod 2)) 2) − + t − div d = 2)) 0, − t (4) − d = 0,bydistribution N j − M ((1 − N j − M ((1 − i = 0, 1, , t = − M, − M, , 0, d = 0, 1, , D − 1, R software environment R software environm for stati t in ∈ moment: {1time − M,inter2 − M, According , 0}, ∈ {0, above ,D 1}, 578simplified Bull Pol Ac.: 64(3)the 2016 i-thsignal interference signal verterdwith modulation, isThe equal to:Tech [11, − .M, 0],−then the probability of the appearance ofdata data trans the single, is equal to Dappears of discrete todeterministic the mentioned assumptions both solid curve shows probabilitie with conditions with conditions [26] [26] moment Probability (using L(tMexico where the t is lack theMmoment of mod the appearance of thet + zero interfermission the Theorem of Total [25]) you by | New State University L(t0 ) vals Without of generality, it2)can that theN, α and are but of toerror the 0) andassumed known N, M ∈ (α Minterference ∈ N) Brought the while the + (ibe div 2)) + d, ((1 − α )(i 1experimental 1theresults, deterministic, signals Fig showsFig 7probabilities shows thecon pro of Authenticated ence signal (3) caused by interference signal introduced by a DC-DC − = − P(error) = i ∈ N, j ∈ i {0, ∈ N, 1, , j n ∈ − {0, 1}, 1, , n − 1}, zero interference signal appears not later than zero data signal The equation is known as Diophantine equation [21, 22, 23, abilities evaluated using the propos ∑ ∑ the appearance of the interference signal is random Let’s asi = 0, 1, , t = − M, − M, , 0, d = 0, 1, , D − 1, M 2 M (5) (5) Download Date | 1/22/17 2:29 PM mission error mission for different error for d t0 =1−M verter is equal =1−M 24].time ences tbetween the to: presented curve t ∈ {1 − M, 2tof − ∈ M, {1 − M, , 0}, 2− M, d ∈t .{0, , 0}, 1, with d, D∈deterministic −{0, 1},1, , D −modulation, 1}, Generally, it can be written that d-th discrete interval sume that is discrete uniform distribution with the interval The solid curve The shows solid curve probabili sho (8) 4.4 Superimposition of data and interference signals The 4A p+ D bit p+ p− number Voltage [V] Voltage [V] Voltage [V] Voltage [V] Voltage [V] Voltage [V] ther ther investigation, investigation, there there will will be analyzed analyzed the the number number moment t0 be mation ation bits bits (2) disturbed disturbed by by interference interference signals, signals, for for the the EFINITION Let condition condition that that the theD0-bit 0-bit of of information information will will emerge emerge at at n practice bit (7) L(t0 ) = #{( j, d,t, i) : t = t0 }, 00 Diophantine equation based model of data transmission errors f the interferON ON 1 Let Let The bit check- where ( j, d,t, i) satisfying (4) = #{( #{( j,j,d,t, d,t,i)i) :: tt = =except tt00}, }, specific cases (7) (7) in which N and M are coL(t L(t00)) = of the voltage Unfortunately, Unfortunately, except specific cases in which N and M are error probabilities for small changes of converter switching sentation Inprime numbers, there is no general formula for function L(t0 ) d,t, d,t,i)i) satisfying satisfying (4) co-prime(4) numbers, there is no general formula for function frequency, presented in Fig. 2 In has been revealed that these the widthL(tof) Thus, Thus, a numerical evaluation be performed a numerical evaluation mustmust be performed unexpected differences result directly from the divisibility of ately, ately, except except specific cases cases in in which which NN and and M M are are coco-assumptions both the data ete time inter-specific According the above mentioned According to thetoabove mentioned assumptions both the numbers The higher the smallest common divisor of time be).but the moment mbers, mbers, there there is is no no general general formula formula for for function function L(t L(t 00) umed thatdata the andand interference signals are deterministic, but the moment of checking moments (N) and the time between interinterference signals are deterministic, tween bit umerical umerical evaluation evaluation must must be be performed performed o data signal the appearance of the interference signal is random Let’s asof the appearance of the interference signal is random Let’s ferences (M), the smaller the error probability is gg to tointerval the the above above mentioned assumptions assumptions both the the data data me of mentioned sume t is discrete uniform distribution with the interval assume that t isthat discrete uniformboth distribution with the interval In-depth understanding of the side-effect phenomena, acference erence signals signals are are deterministic, deterministic, but but the the moment moment of of appearance [1 − M,0], then of the appearance of data trans-of data companying [1 − M,the 0], probability then the probability of the trans- the application of new technologies, usually enrance rance of of the the interference interference signal signal isis random random Let’s Let’s asmission error (using the(using Theorem Total asProbability [25]), ables development and application of low cost mitigating techmission error the of Theorem of Total Probability [25]), caused by interference signal introduced by a DC-DC converter niques The proposed model for the appearance of error caused t tt isis discrete discrete uniform uniform distribution distribution with with the the interval interval (3) caused by interference signal introduced by a DC-DC con,then D −the 1, with deterministic modulation, is equal to: by interference, generated by a converter with deterministic ,,.then the probability probability of of the the appearance appearance of of data data transtransverter with deterministic modulation, is equal to: modulation, might be directly used by practitioners In the case error rror (using (using the the Theorem Theorem of of Total Total Probability Probability [25]), [25]), L(t0 ) L(t0 ) zero interfer0 of the observed high probability of the appearance of error, 1 1 yy interference interference signal signal introduced introduced by by aa1DC-DC DC-DC concon-= − − in significant decreasing of the data transfer rate, the P(error) = ∑ ∑ resulting M t0 =1−M Mequal hh deterministic deterministic modulation, modulation, isis equal to: to: t0 =1−M (8) switching frequency of the locally installed converters with de (8) signals L(t00)) L(t00)) terministic modulation should be slightly changed Producers 11 00 11 L(t 11 L(t 11 00The that= of error ofinconverters trans=the 11− −probability = = hecking∑ ∑mo-11−− Note ∑ ∑ of the appearance with deterministic modulation might offer such M Marithmetic 22 M M tt00=1−M =1−Mis tt00=1−M =1−M 22mean of the probabilitiesfixture mitted data is the of error conditions, without investments in hardware (8) occurrence for a specified period(8) of the appearance of interthe of the appearance of error in transthe probability probability of the appearance of error in transTheofnumerical analyses were performed using Noteference that the signal probability the appearance of error in trans−ta =the 0, arithmetic (4) mean of tad isis the arithmetic mean of the the probabilities probabilities of error error computing mitted data is the arithmetic mean offor the of probabilities of error oc-and5 R software environment statistical graphics Conclusion e for period of appearance of ce for aa specified specified period of the theperiod appearance of interinter- of interference currence[26] for a specified of the appearance The were performed using R softwareof data Thetransproposed Diophantine equation-based model of the probgnal The numerical analyses were performed using ignal Thesignal numerical were performed using Fig.numerical 7analyses showsanalyses the probabilities of the appearance environment for statistical computing and graphics [26] ability of the appearance of transmission error caused by ine environment statistical computing and graphics re environment for statistical computing and graphics (5) for mission error for different converter switching frequencies − 1}, Figure shows the probabilities of the appearance of data terference generated by DC-DC converters constitutes a sigThe solid curve shows probabilities calculated on the basis of transmissionof error different converter frequencies nificant complement for a time-domain based model of the ws the the appearance of transows the probabilities probabilities of thefor appearance of data dataswitching transthe experimental results, while the dashed curve depicts probThe solid curve shows probabilities calculated on the basis of probability of the appearance of data transmission error in converter switching frequencies error for different converter switching frequencies nrror [21,for 22,different 23,experimental abilities evaluated using proposed model.probObserved differthe results, while the the dashed curve depicts system consisting of DC-DC converters with random moducurve calculated on of curve shows shows probabilities probabilities calculated on the the basis basis of result from inaccuracy of ences between the presented curves abilities evaluated using the proposed model Observed differ- lation, as presented in our previous work The mathematical mental results, while curve depicts probimental results, while the the dashed curve depicts probences between thedashed presented curves result from inaccuracy of model explains significant differences between probabilities of valuated using the proposed model Observed differevaluated using the proposed model Observed differthe parameters of the DC-DC converter and data data transmission errors caused by interferences generated by Bull.transmission Pol Ac.: Tech XX(Y) 2016 ween presented curves result from ween the the system presented curvesdespite resultinherent from inaccuracy inaccuracy of the experimen- deterministic and random modulated converters, observed in However, inaccuracy of of tally determined parameters, measured in a real arrangement, the Diophantine equation-based model still capable Bull Bull Pol Pol Ac.: Ac.: Tech Tech.isXX(Y) XX(Y) 2016 2016 of predicting the probability of error to a satisfactory level of accuracy Moreover, the performed analyses, using the proposed model, have enabled the authors to understand the physical and mathematical reasons for significant differences observed in 0.35 Deterministic experimental Deterministic model Error probability 0.30 0.25 0.20 0.15 experimentally obtained results It has been revealed that the probability of the appearance of transmission error in systems with deterministic modulated converters depends on the smallest common divisor of time between bit-checking moments and the time between interferences Besides cognitive value, the presented model, based on the Diophantine equation and physical knowledge about interference induced transmission errors, can be used in engineering practice The model can be applied in order to: ● determine probability of transmission error appearance in real situation, based on experimental waveforms, ● make a decision about investments in EMI reduction techniques, assuring reliable data transmission, ● increase the data transfer rate by small changes of the converter switching frequency, ● elaborate new, time-domain based, EMC standards, etc References 0.10 15 20 25 30 35 40 45 50 Time between interferences [µs] Fig Experimental and simulation dependencies between transmission error probability and time between interference pulses generated by DC-DC converters with deterministic modulation [1] G Benysek, M Kazmierkowski, J Popczyk, and R Strzelecki, “Power electronic systems as a crucial part of Smart Grid infrastructure – a survey’’, Bull Pol Ac.: Tech 59, 455–473 (2011) [2] R Strzelecki, G Benysek, and M Jarnut, “Power quality conditioners with minimum number of current sensors requirement’’, Przegląd Elektrotechniczny (11), 295–298 (2008) 579 Bull Pol Ac.: Tech 64(3) 2016 Brought to you by | New Mexico State University Authenticated Download Date | 1/22/17 2:29 PM J Bojarski, R Smolenski, P Lezynski and Z Sadowski [3] R Strzelecki, G Benysek, Z Fedyczak, and J Bojarski, “Interline power flow controller-probabilistic approach’’, Power Electronics Specialists Conference 2, 1037–1042 (2002) [4] J Luszcz, “Motor cable influence on the converter fed AC motor drive conducted EMI emission’’, Compatibility and Power Electronics, 386–389 (2009) [5] F Diouf, F Leferink, F Duval, and M Bensetti, “Wideband impedance measurements and modeling of DC motors for EMI predictions’’, IEEE Trans Electromagn Compat 57 (2), 180–187 (2015) [6] R Smolenski, M Jarnut, G Benysek, and A Kempski, “AC/ DC/ DC interfaces for V2G applications – EMC Issues’’, IEEE Trans Ind Electron 60 (3), 930–935 (2013) [7] R Smolenski, M Jarnut, G Benysek, and A Kempski, “CM voltage compensation in AC/DC/AC interfaces for smart grids’’, Bull Pol Ac.: Tech 59 (4), 513–523 (2011) [8] S Pasko, M Kazimierczuk, and B Grzesik, “Self-capacitance of coupled toroidal inductors for EMI filters’’, IEEE Trans Electromagn Compat 57 (2), 216–223 (2015) [9] A Trzynadlowski, F Blaabjerg, J Pedersen, R Kirlin, and S Legowski, “Random pulse width modulation techniques for converter- fed drive systems-a review’’, IEEE Trans Ind Appl 30 (5), 1166–1175 (1994) [10] J Salt and A Sala, “A new algorithm for dual-rate systems frequency response computation in discrete control systems’’, Appl Math Model 38 (23), 5692 – 5704 (2014) [11] H Khan, E Miliani, and K Drissi, “Discontinuous random space vector modulation for electric drives: A digital approach’’, IEEE Trans Power Electron 27 (12), 4944–4951 (2012) [12] S Jabrzykowski and T Citko, “A bidirectional DC-DC converter for renewable energy systems’’, Bull Pol Ac.: Tech 57, 363 – 368 (2009) [13] A Tomaszuk and A Krupa, “High efficiency high step-up DC/ DC converters – a review’’, Bull Pol Ac.: Tech 59, 475–483 (2011) [14] A Ales, J.-L Schanen, D Moussaoui, and J Roudet, “Impedances identification of DC/DC converters for network EMC analysis’’, IEEE Trans Power Electron 29 (12), 6445–6457 (2014) 580 [15] J Bojarski, R Smolenski, A Kempski, and P Lezynski, “Pearson’s random walk approach to evaluating interference generated by a group of converters’’, Appl Math Comput 219 (12), 6437–6444 (2013) [16] A Elrayyah, K Namburi, Y Sozer, and I Husain, “An effective dithering method for electromagnetic interference (EMI) reduction in single-phase DC/AC inverters’’, IEEE Trans Power Electron 29 (6), 2798–2806 (2014) [17] Y.-S Lai, Y.-T Chang, and B.-Y Chen, “Novel random-switching PWM technique with constant sampling frequency and constant inductor average current for digitally controlled converter’’, IEEE Trans Ind Electron 60 (8), 3126–3135 (2013) [18] R Smolenski, J Bojarski, A Kempski, and P Lezynski, “Timedomain-based assessment of data transmission error probability in smart grids with electromagnetic interference’’, IEEE Trans Ind Electron 61 (4), 1882–1890 (2014) [19] R Smolenski, Conducted Electromagnetic Interference (EMI) in Smart Grids, Springer, 2012 [20] A Kempski, R Strzelecki, R Smolenski, and Z Fedyczak, “Bearing current path and pulse rate in PWM-inverter-fed induction motor drive’’, Power Electronics Specialists Conference, 2001 PESC 2001 IEEE 32nd Annual, 4, 2025–2030 (2001) [21] T Andreescu, D Andrica, and I Cucurezeanu, An Introduction to Diophantine Equations: A Problem-Based Approach, Birkhäuser, 2010 [22] R D Carmichael, Diophantine Analysis, CreateSpace Independent Publishing Platform, 2013 [23] J Salt and A Sala, “A new algorithm for dual-rate systems frequency response computation in discrete control systems’’, Appl Math Model 38 (23), 5692 – 5704 (2014) [24] R D Carmichael, The Theory of Numbers and Diophantine Analysis, Dover Publications, 2004 [25] A Papoulis and S U Pilla, Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York, 2002 [26] R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, 2008 Bull Pol Ac.: Tech 64(3) 2016 Brought to you by | New Mexico State University Authenticated Download Date | 1/22/17 2:29 PM ... descriptions of the phenomenon of data transmission error occurrence caused by DC- DC converters with deterministic modulation The Diophantine equation- based model of data transmission errors caused by interference. .. applicability of the proposed of transmission errors caused by interference caused by DC- DC converter with Ndeterministic modulation on the mission error caused by interfere N model the calculations model. .. Probability of the appearance of error caused by interference generated by DC- DC converters with deterministic and random modulation In spite of the significant ostensible reduction in level of interference