EMC

European Journal of Scientific Research ISSN 1450-216X Vol.63 No.3 (2011), pp. 368-386 © EuroJournals Publishing, Inc. 2011 http://www.europeanjournalofscientificresearch.com

Modelling of DC Motors Conducted Low Frequency EMI/EMC Disturbance for Automotive Applications R. Kahoul IRSEEM, EA 4353, At the Graduate School of Engineering ESIGELEC Avenue Galilée, BP 10024, 76801 Saint-Etienne Du Rouvray Cedex, France E-mail: [email protected] Y. Azzouz IRSEEM, EA 4353, At the Graduate School of Engineering ESIGELEC Avenue Galilée, BP 10024, 76801 Saint-Etienne Du Rouvray Cedex, France E-mail: [email protected] B. Ravelo IRSEEM, EA 4353, At the Graduate School of Engineering ESIGELEC Avenue Galilée, BP 10024, 76801 Saint-Etienne Du Rouvray Cedex, France E-mail: [email protected] Tel.: +33. (0)2.32.91.59.71 / Fax: +33. (0)2.32.91.58.59 Abstract With the increase of the electrical equipment integration complexity in the vehicle systems, the electromagnetic compatibility (EMC) becomes one of key factors to be respected in order to meet the constructor standard conformity. After the introduction of the industrial context, in this paper, a spectrum modelling method of conducted EMC perturbations generated by DC motors in low frequencies is developed. The methodology illustrating the feasibility of the theoretical approach introduced is explained in details. The handling processes of the test bench employed to measure these perturbations are described. Then, the verification results of the apparent approach enabling the achievement of circuit models containing the different elements constituting the motors considered are also offered. We demonstrate that this approach allows us to model mathematically the schematic diagrams which reproduce the mechanism of the brush-collector contact and the EMC perturbation sources. The models developed were validated experimentally by considering the variation of different parameters (speed and power supply) which characterize the DC motor under test. It was shown that good agreements between the motor behaviours and the models proposed were observed. The model investigated is particularly useful for the conducted EMC analysis disturbing the functioning of embedded power electronic systems.

Keywords: Conducted EMC, DC motor, source impedance model, measurement technique, behavioural apparent approach, low-frequency (LF) and highfrequency (HF) perturbations.

Modelling of DC Motors Conducted Low Frequency EMI/EMC Disturbance for Automotive Applications

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1. Introduction Currently, as the automobile industry bounces back from the great recession, manufacturers are designing their next generation of vehicles with an eye on fuel economy and telematics (information and communication technologies) [1-3]. Increasingly, the cars being designed nowadays represent the latest frontier in the telecommunication revolution. The association of classical electric equipments battery-filter-converter-motor-load remains the most expanded electrical configuration for the modern embedded systems as vehicle automobiles. Because of the integration density increase and the assembly of electrical/electronic circuits gathered in a confined space, certain unwanted effects caused by the electromagnetic compatibility (EMC) and electromagnetic interference (EMI) become unavoidable [4-7]. With the increase in the vehicle electronics, EMI problems understandably are on the rise. EMI issues range from on-board radio interference to catastrophic failure of an engine control module due to power transients. EMI has been considered a potential cause of the recent inexplicable acceleration of certain Toyota models [1]. The problems are expected to get worse as system clock speeds and logic edge rates increase, due to increased EMI emissions and decreased EMI immunity [8] Vehicle electrical systems are a rich source of power transients so different international standards are currently required [9-13]. Most severe have been characterized and have become a suite of standard EMI test pulses, as described in SAE J1113, “Electromagnetic Susceptibility Procedures for Vehicle Components [12].’’ These transients include pulses that simulate both normal and abnormal conditions, including inductive load switching, ignition interruption or turnoff, voltage sag during engine starting, and the alternator “load dump’’ transient. Most automotive testing requirements are defined and issued by regional organizations, i.e. European Union (EC) and American (SAE), international (CISPR, IEC and ISO), and manufacturers (GM, Ford, VW, etc.). More detailed illustrations of these EMC test standards and their recent development can be found in articles and symposium digests [14-16]. It is obvious that the electric energy from the DC-DC converter through chopper circuits associated to the DC motors plays an important role to ensure the good functioning of electric systems [17-19]. As reported in [20-23], among the modules using DC motors, we can quote the basic configuration of the electrical network constituted by the chain battery-filter-converter-motor-load which is commonly found in the automobiles. The DC-DC conversion of electrical energy via choppers associated to the DC motors is currently utilized in most features electrified. Otherwise, the AC-DC converter is generally specific to the traction system applications [24-28]. In other word, as stated in [26-28], the DC-AC conversion is rather applied to embedded energy with vehicles electrical traction. For this type of converter, among the modules using DC motors, there are few classic cases of technologies including Groupe Moto-Ventilator (GMV) and starter-alternators, steering column, power window systems and electrical pumps. Compared to the electrical configurations cited previously, the difference of this subject treated in this paper lies on the modelling of another module employing considerable number of DC motors constituting the seat-motor. Basically, this module contains: • A conventional motor to ensure the longitudinal traction; • A motor equipped with screw to secure the vertical displacement; • A motor equipped with gear to ensure the seat back. The DC motor functioning is characterized by the commutation of brush-collector and the fixed bars. Indeed, the addition of equipments operating in fast switching is susceptible to cause, harmful interference, which can spread through the strong EM couplings existing within the onboard network [29]. Due to the possible conducted coupling and/or caused by the radiated coupling, the EMI can be driven by various interconnection cables. These coupling effects may damage the electrical module in which they were integrated. This can be found in digital and/or mixed electronic circuits as the RF devices and microcontrollers integrated in the command devices [30-32]. To predict the various EMI effects generated by the electrical motors, different models of machine windings [33-38], and magnets [39-41] by using, for example, lumped elements [42-43] were proposed notably for high frequencies. More precisely, a modelling method of the DC motor impedance by using the line impedance

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stabilized network (LISN) is investigated in [29]. According to the best of the authors’ knowledge, till now, few works were published in the literature about the simultaneous modelling of conducted and radiated emissions of DC motors in movement due to the complexity of the bar-collector contact characterization. In this paper, we deal with an innovative mechanism of collector-brush contact in rotation and its influences on the EMC model of DC motors. To do this, these latter are considered as an active circuit and vis-à-vis EMI effects, will be considered as an electrical network comprised of the impedance model introduced in [29] and the source of perturbation equivalent to an apparent Norton source denoted INorton. The originality of the present work lies on the modelling of low frequency (LF) and high frequency (HF) models of the perturbations induced by the DC motors.

2. Characteristics of Motors Under Investigation The DC motor with 12 V supplied is usually considered for the investigation of the vehicle seat horizontal adjustment. As reported in [29], this type of motor can consume a power 80 W for a speed 1150 tr.min-1 (120 rad.s-1) and a normalized current 6 A which is equal to the third of the blocking current. As illustrated in Figs. 1, the DC motor armature can be divided in three compartments: • The power supply: As illustrated in Fig. 1.a, this block specific to each supplier contains the interconnect lines and the supplier pins, a circuit breaker protection, a rotation sensor and an RF filter. Mounted downstream of suppliers, the filter is necessary for the cancellation of various perturbations induced in the area of brush-collector contact. Obviously, the devices surrounding the motor as the filter, the circuit breaker and the rotation sensor will not be considered during the test performed. A pair of brush holders in copper perfectly balanced is assembled in this block (see Fig. 1.a and Fig. 1.b). A carbon-brush is set at the end of each brush holder. • The armature or rotor: As explained by Fig. 1.c, it consists of a laminated magnetic core with 8 teeth, 8 concentric coils and placed in the notches constituting the rotor. Here, we can see that each coil has 27 turns and the winding pitch is three slots. The coil is connected to two successive brushes of the collector. It will be fed by direct contact with these two brushes. • Stator: As pictured in Fig. 1.d, it consists of the metallic fixed part or wound cage containing the rotor and two permanent magnets of strontium fabricated with ferrite sheets. Figure 1: Compartments of the DC motor understudy.


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2.1. DC Behaviour and Functional Modelling Generally, the DC motor is modelled by a series circuit Rw-Lac-E as depicted in Fig. 2.a, where Rw and Lac are, respectively, the apparent resistance and inductance of the armature, E is an EMF. Then, IDC and Ω are the mean values of current and speed respectively. Figure 2(a: Simplified electrical model of DC motor considered for functional simulation.

Open-load speed

Figure 2(b: Operating characteristics of DC motor separately excited.

Ω Ω0

UDC=12.5V

Ib

IDC

Blocking current In rotation, the motor behaves as an open loop, the average current IDC can be expressed from the average value of velocity Ω with the characteristic displayed in Fig. 2.b. The change of UDC can vary considerably the output Ω. Such representation remains valid for the functional simulation especially in DC, where the current i(t) and velocity Ω(t) represent, respectively, the electrical power needed for the functioning and the mechanical power produced by the motor. However, for an effective consideration of all phenomena related to the use of this motor (including the presence of arcing due to improper switching), this presentation is not sufficient. 2.2. Measurements in LF (Time-Domain Measurements) Fig. 3 represents the diagram of experimental set up of the bench considered in this study for the LF conducted emission measurements. For the automotive vehicle applications, the test of the conducted emissions in the LF bands is not based on specific standard. This test is particularly essential because it enables us to measure the perturbations caused by the slow and transient switching of electrical equipments which can be detected only below 20 kHz. The measurement of these emissions is based on the use of current probes and a relevant digital oscilloscope.

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R. Kahoul, Y. Azzouz and B. Ravelo Figure 3: Experimental set up of the bench considered for the LF conducted emission measurements.

2.2.1. Time-Domain Measurements The time-domain measured data were converted into frequency data via the FFT. Of course, the current measurement should be as accurate as possible in order to minimize the probable strong difference between the FFT and the realistic perturbation spectrum. To realize operations as correct as possible, the following conditions are required: • The current probes must present a bandwidth exceeding the limits of the considered frequency band (20 kHz steps for LF). • The oscilloscope bandwidth should allow the time-dependent data acquisitions with a sufficient number of samples for the FFT calculation. 2.2.2. FFT Analysis It is well-known that the FFT is a powerful mathematical operation allowing us to represent a timedependent signal into frequency domain data. It must be used with samples regularly time-spaced. Generally, in power electronic area, two algorithms are used to calculate the Cooley-Tukey and the Sande-Tooke FFT algorithms. These algorithms are, in particular, implemented in the computing tools as Matlab and most of electronic/electrical circuit simulators (Orcad/PSPICE [44], PSIM [45], Simplorer [46]) and also in the digital oscilloscopes. However, in this section, before the FFT operation, the following recommendations should be imperatively verified: • The signal to be processed must be periodical and must reache its steady state. • The acquisition must take place over an interval multiple of the signal period (Tacquisition = NT) over the acquisition interval, the greater the resolution of the FFT will be better. • To realize an FFT result covering the entire frequency range studied (LF or HF), the time step should be defined by the relation: 1 ∆tmin = 2f max . (1) The parameters N, T and ∆t define the minimum number of samples denoted Nsample that must be taken from the oscilloscope: N ⋅T N sample = ∆t . (2)


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3. Test Benches Description and Measurements Performed The recommendations outlined above were taken into account to measure the LF perturbations Ip by using classical current probes. The perturbations were measured at different operating points of the DC motor which are characterized by the parameters (IDC, Ω). The representation of the test bench used is represented by the diagram shown in Fig. 4, where: • The voltage across the battery is VBattery = 12.5 V. • A rheostat is placed between the (+) terminal of the battery and the ammeter input in order to vary the voltage at the motor terminals UDC, and thus, the speed Ω. To avoid their interference, the rheostat and ammeter are placed in upstream of the line impedance stabilized network (LISN). • The hysteresis brake is connected to the motor in order to vary the torque Г and therefore, the average current IDC absorbed by the motor. Figure 4: Test bench used to measure the perturbations generated by the DC motor (in LF and in HF).

Let us denote NIdcNΩ the number of points considered during the measurements. As highlighted in Fig. 5, the points (IDC(i), Ω(j)) with i = {1…NIdc} and j = {1…NΩ} for which the measurements were performed, are selected within the area of the DC motor operation shown in Fig. 5. This area will be limited by the following four ‘critical’ lines: • IDCmin represents the current corresponding to the minimum load (empty seat). • IDCmax: represents the maximum current tolerated by the breaker circuit for protecting the motor which corresponds to a maximum load (IDCmax ≥ Inominal). • Ωmax: represents the maximum speed for the minimum load (Ωmax ≤ Ω0). • Ωmin: represents the minimum speed for the maximum load.

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R. Kahoul, Y. Azzouz and B. Ravelo Figure 5: Operating points for which the EMI measurements are performed.

To cover the entire part of the NIdcNΩ operating area, we vary the motor speed from Ωmin to Ωmax and keeping the electric current IDC(i) constant. Then, for each speed value, the current is varied from IDCmin to IDCmax.

4. Perturbations Generated by the DC Motor During the test, according to the targeted automotive applications, we chose 5x4 operating points defined by the current IDC(A) = [2.5, 4, 5.2, 6, 6.8] and speeds Ω(rad.s-1) = [36.6, 120, 210, 350].

4.1. Perturbations Ip Measured in LF As the measurements of Ip were done in time-domain, as a first step, the recommendations of the FFT described in paragraph 2.2.2 have been fulfilled. Then, we considered the following parameters: • In order to cover the frequency band [20 Hz - 20 kHz], a minimum time-step equal to ∆t = 25 µs (see equation (3)) was considered. 1 f max = 20kHz = ⇒ ∆t min = 25µs 2 ∆t min . (3) •

The acquisition interval is multiple of the rotation period (Trot). As explained in (4), the latter was deduced from the speed of rotation Ω.  Ω ( rad .s −1 ) f rot max = max ≈ 56 Hz ⇒ Trot min ≈ 17.9ms   2.π  Ωmin ( rad .s −1 ) f rot min = ≈ 6 Hz ⇒ Trot max ≈ 166ms   2.π . (4) • By choosing an interval equal to 2Trot, we evaluate that the maximum number of measurement samples: 2.Trot max N samples max = = 6650 ∆t min . (5) Once the oscilloscope was calibrated, we measured the DC motor input current Ip(t) for the 54 operating points adopted. Thus, we calculated the corresponding FFT of the time-dependent data.


375

Thus, the Ip spectrums obtained are plotted in Fig. 6 for the case of current nominal value IDC(4) = 6 A and speeds Ω (rad.s-1) = [36.6, 120, 210, 350]. Figure 6(a: Time-dependent perturbations measured in LF versus speeds Ω.

Figure 6(b: Spectrums of the perturbations (Ip) measured for different speeds Ω.

4.2. Comments on the PerturBations Observed in LF In fact, the continuity of the current absorbed by the motor is interrupted by the material between the collector-brushes. These infrequent interruptions commonly known as transient commutations engender harmonics multiple of the frequency 1/Trot in the spectral decomposition of the current through the motor. The harmonics observed in the most significant steps made in the LF spectrum of Ip are in the frequency band [10 Hz - 20 kHz]. According to their origins, they can be divided into three categories:

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R. Kahoul, Y. Azzouz and B. Ravelo •

•

•

The harmonic 1 (relative to frot): This frequency corresponds to one revolution of the rotor; the origin of the harmonic can be attributed to unbalanced dynamics of the motor that affect the stability of its rotation [29]. The harmonic 2 (relative to the frequency 2frot): The DC motor was studied with two bars diametrically opposite with respect to the collector; the origin of this harmonic can be attributed to geometrical defects of the collector. These harmonics correspond to commutations from the side of rotor, or with respect to a rotating reference. Indeed, during the rotation movement, each bar undergoes two commutations multiplied to the number of brushes. A switching turn is 2frot. Harmonics Ncollector (relative to the frequency Ncollectorfrot): These harmonics correspond to the perturbations from the side of stator, or based on the fixed reference. The DC motor understudy is equipped with 8 collectors which generates harmonics 8frot, 16frot, 24frot, 32frot... Generally speaking, one distinguishes to types of multiple harmonics: o The harmonic Ncollector: This harmonic corresponds to the commutation of the pair of Ncollector collector-brushes per revolution (corresponding to the frequency Ncollectorfrot). When the brushes are perfectly diametrically opposed, then, only these harmonic appeared in the current spectrums. o The harmonic 2Ncollector: Actually, the bars are not completely diametrical opposite and a phase shift exists between the previous spontaneous switching. Each bar will have Ncollector commutations and the two bars together undergo 2Ncollector commutations. In this case, a harmonic with frequency 2Ncollectorfrot appears in the current spectrum.

4.3. Analysis of Parameters Influencing the Perturbation Generation and Propagation We can classify the perturbation generations in two families of parameters: • The operating power: The average current IDC, representing the strong mechanical torque applied to the motor determines the amount of charge stored by winding during the short circuit instant, and thus, the current delivered at the end of this phase. As a first consideration, we assume that IDC presents more influence on the perturbations observed in HF. • The speed of rotation Ω: determines the intensity of the friction surfaces of the brushes and commutator bars. As a second hypothesis, we assume that Ω has more influence on the switching transients causing perturbations in LF. The paths of the perturbation propagation were reduced mainly to the impedance Z of the motor and that of the LISN denoted ZLISN.

5. Apparent Approach Modelling For this apparent approach, the motor will be modelled by its Norton equivalent circuit. The apparent model of DC motor associates the impedance Z defining the propagation paths of the DC motor, in parallel to the equivalent current generator (INorton), which represents the source of perturbation as defined in [29].

5.1. Principle of the Modelling Method Considered The knowledge of the propagation paths and measurement circuits represented by the LF perturbation impedance enables us to extract the source INorton. This latter is expressed by a frequency dependent formula whose the constant parameters are quantified and optimized with the analysis of spectrum measurements and simulations. The influence of physical and geometrical characteristics of the brushes and the collectors on the generation of perturbations is reduced to that of the current IDC and speed Ω.


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The successive steps describing the approach of the apparent model proposed are described by the flowchart depicted in Fig. 7. Figure 7: Illustration of the apparent approach proposed.

Start

IDC

Ω

1. LF perturbation measurement

2. Treatment and analysis of measured data

3. Modeling of measured perturbations Ip (f,IDC ,Ω ) Calculation of the function F MD/MC Impedances: motor, cable, LISN

x

4. Realisation of the perturbation source INORTON (f,IDC ,Ω )

End

i.

As initial step, the perturbations were measured for the points inside the critical area of DC motor operation. The objective is to analyze separately the impact of IDC and Ω parameters on the LF spectrums of Ip varied separately during the measurements. ii. In all behavioural approach, we must manipulate the analyzable data, which are not entirely the case of LF and HF measured spectrums. As intermediate step, the measurements must undergo the adequate treatments before the phase of analysis. iii. Afterwards, the spectrums are modelled by mathematical functions with unique three independent variables f, IDC and Ω as illustrated in (6). Due to the large amount of spectrum to be treated in function of the number of points (NIDC NΩ), this step constitutes the largest phase of the apparent approach proposed. VLISN = function(f, IDC, Ω). (6) The first hypothesis is to express the spectrums according to the first variable f, via a single expression. Then, we determine the constants of this expression with respect to the second variable Ω via a second expression, and finally, one can express the constants of the second term by the third variable IDC with the relation: VLISN = function[f(IDC(Ω)]. (7)

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However, such an approach is directly applicable only on condition that the variables are mathematically well-correlated. This is the case of DC motor operating in open-loop, where a variable can be expressed in function of the other ones thanks to the DC motor characteristic presented in Fig 2.b. In the case of functioning independent to the desired operation for which measurements were made, the speed does not depend to the current. Both variables IDC and Ω cannot be considered as a function in direct correlation and expression (7) can only be confirmed after verification of the conditions of compatibility between the constant mathematical expressions. In summary, the modelling of the perturbation spectrums proposed can be summarized in the following steps: • The model based on the first variable f, regardless of IDC and Ω. • The model based on the second variable Ω, regardless of IDC and f. • The model according to IDC third variable IDC, regardless of Ω and f. • Check the first compatibility condition, where the coefficients of VLISN(f) must change with the variable Ω according to a law similar to VLISN(Ω), regardless of f and IDC. • Check the second compatibility condition, where the coefficients of VLISN(Ω) must change with the variable IDC according to a law similar to VLISN(IDC), regardless of f and Ω. • Finally, we realize the model of the perturbation through expression (7). iv. The spectrum model attributed to Ip(f, IDC, Ω) and that of the transfer function DMF/MC(f) associated together with the impedances of propagation paths lead to the expression INorton(f, IDC, Ω). This will only be valid for the motor understudy under the operating conditions considered. We begin by applying the modelling approaches described above to the spectrums of LF perturbations generated by the DC motor.

5.2. Application and Validation Results This subsection presents the results obtained during the analysis and modelling of LF perturbation spectrums induced by the DC motor presented in Figs. 1.

5.2.1. Spectrum Analysis and Processing As shown by the measurements, from the first harmonic visible below 10 kHz, the rest of the spectrums are similar to the noise at high frequencies. In addition, the LF spectrums, naturally with narrow bandwidth, describe the chaotic variation, making them difficult to analyze. Thus, an acquisition algorithm was developed specifically in order to extract the amplitudes of the perturbations Ip(k) and harmonics multiple of the frequency f(k) and constituting a matrix with row and column according to the specific values of IDC and Ω. Figure 8: Variation of measured (meas.) LF harmonic spectrums versus Ω and the plots of corresponding envelops (env.) extracted.


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To facilitate the LF spectrum modelling, we limit this analysis to those of five harmonics that describe the brush-collector commutation shown in Fig. 8, namely: • The harmonic 2frot defining the commutation views from rotor side. • The first 4 harmonics Ncollectorfrot defining the commutation views from brushes side.

5.2.2. Modelling of Spectrums Because of the typically narrow band characteristic of the spectrums, we prefer to model the variation of each harmonic f(k) independently. This later is modelled first, with the variation of the speed Ω, then, according to the variation of the current IDC. We reduce the number of variables to two and one condition in order to check the compatibility of formulations. To do this, the coefficients of Ip(Ω)|f(k) must be expressed in function of IDC variable according to the law similar to Ip(IDC)|f(k), regardless of Ω. i. Modelling Ip|f(k) according to Ω: From the measurement samples provided by the acquisition algorithm, we plotted on the same curve of Ip|f(k), for the same current and following the speed variation. We underline that the results shown in Figs. 9 are limited to the curves for the two harmonics and two Ncollector. Figure 9(a: LF harmonic variation f(1) = 2frot versus speed Ω.

Figure 9(b: LF harmonic variation f(2) = Ncollectorfrot versus speed Ω.

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As we can see, the amplitudes of harmonics 2frot and Ncollectorfrot, increase linearly with the speed in order to model the variation of 54 curves Ip|f(k) by a common expression. The second order polynomial expressed in (8) was well chosen. As shown in the same figure (solid lines), this polynomial approximation provides a good reproduction of the measured data. I p(Ω ) I , f = α (i, k ).Ω 2 + β (i, k ).Ω + δ (i, k ) DC ( i ) ( k ) . (8) ii. Modelling Ip|f(k) according to IDC: From samples of measurements determined with the acquisition algorithm, we plotted on the same curve, changing Ip|f(k), according to the same speed and power. We point out that the results displayed in Figs. 10 are limited to the two harmonics f(2) and f(Ncollector). Figure 10a: LF harmonic variation f(1) = 2frot versus IDC.

Figure 10b: LF harmonic variation f(2) = Ncollectorfrot versus IDC.

Subsequently, we modelled the variation of 54 curves Ip(IDC)|f(k) by a common expression. The third order polynomial expression shown in (9) was considered. As shown in the same figure (solid lines), this polynomial approximation is in good agreement with the measurements. 3 2 I p(I DC ) , f = c3 ( j,k ) ⋅ I DC + c2 ( j,k ) ⋅ I DC + c1 ( j,k ) ⋅ I DC + c0 ( j,k ) Ω( j) (k ) . (9) iii. Analysis of the formulations compatibility: The variations of constants αi(k), βi(k) and δi(k) relatively to the polynomials on Ip(IDC)|f(k) proposed in (8) have been plotted versus the current IDC. By comparing the curves ak(IDC), βk(IDC), δk(IDC) with curves Ip(IDC)|f(k) displayed in Fig. 10, we find many similarities that confirm the compatibility of the selected formulations.


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Figure 11: Comparison of measured and modelled constants α2, β2, δ2 according to the current IDC.

Fig. 11, given as an example, represents the variation of the constants α2(IDC), β2(IDC), δ2(IDC) specific to the model of the harmonic f(2) and Ncollectorfrot. As we can see, the three constants evolve similarly to those curves Ip(IDC)|Ncollectorfrot as plotted in Figs. 10. Therefore, we modelled the variations of the parameters αk, βk and δk according to the current IDC with the 3rd order polynomials expressed in (10). Obviously, the 43 constants that define the three polynomials αk, βk and δk are specific to each harmonic f(k).  α k (I DC ) = ν '1,k ⋅(I DC )3 + υ '1,k ⋅(I DC )2 + γ '1,k ⋅(I DC ) + χ '1,k  3 2  β k (I DC ) = ν ' 2,k ⋅(I DC ) + υ ' 2,k ⋅(I DC ) + γ ' 2,k ⋅(I DC ) + χ ' 2,k  δ (I ) = ν ' ⋅(I )3 + υ ' ⋅(I )2 + γ ' ⋅(I ) + χ ' 3,k DC 3,k DC 3,k DC 3,k  k DC . (10) Finally, each harmonic is modelled by the following expression: I p ( I DC , Ω ) f(k) = α k ( I DC ) ⋅ Ω 2 + β k ( I DC ) ⋅ Ω + δ k ( I DC ) . (11) For k = {1… 5}, the coefficients of the speed Ω identified are indicated in Table 1. Table 1:

Parameters of current harmonics Ip(IDC,Ω)|f(k) for k = {1…5}.

f(k=1) = 2frot f(k=2) = Ncollectorfrot f(k=3)=2Ncollectorfrot f(k=4)=3Ncollectorfrot f(k=5)=4Ncollectorfrot

ν'1,1 = 7.210-8 ν'2,1 = - 1210-5 ν'3,1 = 7310-4 ν'1,2 =-2.910-8 ν'2,2 =6.410-5 ν'3,2 =-210-2 ν'1,3 =4810-8 ν'2,3 =-1510-4 ν'3,3 =2610-4 ν'1,4 =-5.110-8 ν'2,4 =5.910-5 ν'3,4 =-8310-4 ν'1,5 =-8310-8 ν'2,5 = 2.310-5 ν'3,5 = 5.110-4

υ'1,1 = -8.310-7 υ'2,1 = 1710-4 υ'3,1 = -910-2 υ'1,2 =3.910-7 υ'2,2 =-8.810-4 υ'3,2 =2710-2 υ'1,3 =-5210-7 υ'2,3 =210-3 υ'3,3 =-1.910-2 υ'1,4 =-7.410-7 υ'2,4 =7.810-4 υ'3,4 =1.710-2 υ'1,5 = 1710-6 υ'2,5 = 2810-4 υ'3,5 = 6.4

γ'1,1 = 3.110-6 γ'2,1 = -710-3 γ'3,1 = 3610-2 γ'1,2 =-6110-6 γ'2,2 =410-3 γ'3,2 =-1.1 γ'1,3 =1.810-6 γ'2,3 =-8.310-3 γ'3,3 =2810-2 γ'1,4 =-3.210-6 γ'2,4 =3.210-3 γ'3,4 =-0.55 γ'1,5 = 5510-6 γ'2,5 = 1110-3 γ'3,5= 2210-2

χ'1,1 = -3510-5 χ'2,1 = 8410-4 χ'3,1 = -4410-2 χ'1,2 =-5410-4 χ'2,2 =-5110-4 χ'3,2 =1.4 χ'1,3 =-1.510-5 χ'2,3 = 1110-4 χ'3,3 = -8910-2 χ'1,4 = 4310-7 χ'2,4 = 4210-4 χ'3,4 = 0.11 χ'1,5 = 10-3 χ'2,5 = 1410-4 χ'3,5 = 2710-2

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iv.

Determination of Ip(f, IDC, Ω) final expression: Once the models of harmonic Ip(f, IDC, Ω)|f(k) obtained, we were interested to the final model Ip(f, IDC, Ω) which can combine them together as a single function. For this, two forms of representation are used: Discrete representation: In this case, a weighted expression Ip(f, IDC, Ω) is considered. This performance with simple implementation is based on the definition of the Discrete Fourier Transform (DFT). The amplitudes of harmonics Ip(IDC, Ω)|f(k) are similar to the weights of the spectral lines [1, 5, 6] and multiplied by the periodic sequence of Dirac pulses. The rest of the LF range will be assimilated to the measurement noise.

•

N =5

I p ( f ( k ) , I DC , Ω ) =

∑I

p

( I DC , Ω )

f (k)

. Dirac( f − f (k) )

k =1

 I ( I , Ω ) =  p DC 

•

if

f( k )

f(k) = 2 × f rot , N collecteur × f rot ,...., i × N collecteur × f rot noise

ifnot

. (12) Continuous representation: In this case, a continuous expression of Ip(f, IDC, Ω) is considered. This representation is based on the definition of the series of Fourier and Laplace transform. Indeed, like any deterministic periodic signal, the electrical current absorbed by the motor i(t) is decomposed into an infinite number of sinusoids. Each sine wave is defined by its frequency (multiple of friction), amplitude and phase. k = +∞

i(t ) =

∑ C ( f )⋅e

j⋅2π ⋅

n ⋅t Trot

k

. (13) As the current is a real parameter value, we can then express the Fourier coefficients by considering the Dirichlet convergence theorem: N    2π  2π   + bk (f) ⋅ sin n ⋅ t ⋅  i (t ) ≈ I DC + ∑ a k (f) ⋅ cos n ⋅ t ⋅ Trot  Trot  k =1    . (14) Considering the DC motor as causal, the Laplace transform is well suited to frequency representations of the continuous type were applied to expression (14). Thus, the result obtained represents the spectral envelope of the lines Ip(f, IDC, Ω) expressed by the Laplace parameter (p = j2πf) divided by the signal period Trot. N  p + 2πf ( k )  I p ( f , I DC , Ω ) = ∑  I p(I DC ,Ω ) f(k) ⋅ 2 f ( k ) ⋅ 2 2 p + (2πf ( k ) )  k =1   . (15) In Fig. 12, is plotted the simulation of weighted model from equation (14) and that of the continuous model proposed in (15) for the motor operating point (5 A, 350 rad.s-1). We observe that the models are well-correlated to the measurements. However, the continuous representation is more realistic than the discrete representation. Whether the first or second representations to a different operating point of the measuring points (inside the critical zone), the model Ip(f, IDC, Ω) lead to a fairly accurate estimation of the perturbations generated by the DC motor in LF. k = −∞


383

Figure 12: Comparison between the measurement and the discrete and analog representations of LF harmonic models Ip(IDC(3) = 5A, Ω = 150 rad.s-1).

6. Conclusion This paper develops a modelling technique of conducted EMC effects induced by electrical equipments used in the automobiles. The main focus here is based on the analysis of DC motor conducted emission perturbations in LF from DC to 20 kHz. The EMC model provided is useful for investigating the harmonics caused by probable transient commutations undergone by a larger family of DC motors. The different steps constituting the methodology illustrating the principle of apparent approach method investigated is presented. The model provided can be considered as a tool for the design and precharacterization of motors which does not require a detailed description of electrical equipments. The initial data were translated into mathematical functions depending to the motor characteristics as the feeding power and the rotation speed. It was shown that the model proposed permits us to estimate the EMI generated by DC motors susceptible to damage the surrounding electronic circuits. After analytical description of the approach based on the polynomial model, the reproduction of the measurement motor behaviour was verified. Satisfactory results showing a good agreement between the model sand the measurements were confirmed regarding the variation of the DC motor under test parameters as the frequency, control current and the speed. It is worth noting that the higher the order of the harmonics is and the frequency interval is reduced, the function will be more accurate. In the continuation of this work, we plan to apply the model developed in this paper by taking into account the other physical parameters susceptible to disturb the motor functioning as the temperature effect [47-48].

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