Roberto Mrad

Member, IEEE Florent Morel

Gaël Pillonnet

Member, IEEE Christian Vollaire

Philippe Lombard

Member, IEEE Angelo Nagari

Member, IEEE Gael Pillonnet

N-Conductor Passive Circuit Modeling for Power Converter Current Prediction and EMI aspect

Keywords: Conducted EMI, Passive circuit modeling, Impedance matrix, current prediction, Class-D amplifier. Power converter Passive Block (1) Passive Block

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I. INTRODUCTION

Today's electronic applications are mostly dedicated to embedded systems, where autonomy is a major issue. Thus, power management circuits are widely used to increase power efficiency and extend battery life. Nevertheless, those are switching circuits producing high levels of electromagnetic (EM) disturbances, in particular when delivering high supply currents. Moreover, many power converters are often used at a time to fulfill all the power requirements, needed for the multitude of applications in a single device. To increase their electro-magnetic compatibility (EMC), engineers rely on many control methods, such as spread spectrum [START_REF] Nagari | An 8Ω ; 2.5W 1%-THD 104dB(A)-dynamic-range Class-D audio amplifier with an ultra-low EMI system and current sensing for speaker protection[END_REF]- [START_REF] Dousoky | On factors affecting EMIperformance of conducted-noise-mitigating digital controllers in DC-DC converters -An experimental investigation[END_REF], slew rate control [START_REF] Edwards | Efficient low EMI switching output stages and methods[END_REF], [START_REF] Lim | Adaptive voltage slew control used to limit the magnitude of broadband conducted noise emissions for buck derived dc-dc converters[END_REF], pulse position [START_REF] Nakagawa | Pulse position/width collaborative control for reducing EMI noise and ripple voltage in switching DC-DC converter[END_REF], noise shaping [START_REF] Kirlin | Shaping the noise spectrum in power electronic converters[END_REF] and dead time compensation [START_REF] Tallam | Common-Mode voltage reduction PWM algorithm for AC drives[END_REF]. Nevertheless, in most cases these methods are not enough and use of electro-magnetic interference (EMI) filters is unavoidable.

Class-D audio amplifiers are an example of power converters in embedded systems. This type of converter is used in cell phones because of its high efficiency. Moreover, the design of today's Class-D amplifiers provides high audio quality. However, it delivers high power to the speaker (up to 2.3 W [START_REF] Nagari | An 8Ω ; 2.5W 1%-THD 104dB(A)-dynamic-range Class-D audio amplifier with an ultra-low EMI system and current sensing for speaker protection[END_REF]) through switching voltages/currents (around 500 kHz with less 10 ns rise/fall time). With switching, high frequency harmonic, ringing and overshoot, this amplifier is a major source of EM disturbance in cell phones. For this reason, an EMI filter is usually inserted at the output to prevent disturbance through the PCB. Nevertheless, such passive circuits are very bulky and occupy relatively a huge area in a cell phone compared to the amplifier size. For that, a well designed filter is much needed to reduce the cost and filter area.

The choice of EMI filters is generally based on expertise and their performance is tested on prototypes. For that, EMI methods and models are greatly needed before production. However, traditional methods such as time domain simulations become exponentially complicated when they are used for EMI issues. Therefore, designers use frequency models, called also macro-models. Macro-models are frequently used for EMI purposes because quite often system integrators have little information about the components used and their studies are based on measurements. Such models have good accuracy, but in most cases they are dedicated to a specific application or specific problem so their use cannot be generalized. In previous work, methods like [START_REF] Labrousse | Common-mode modeling of the association of switching cells: Application to an electric-vehicle-drive system[END_REF]- [START_REF] Costa | Modeling of conducted common mode perturbations in variable-speed drive systems[END_REF] study only the Common Mode (CM) and their accuracy does not exceed 30 MHz. In [START_REF] Zheng | A simulation platform of switchedmode power supplies for EMI filter design[END_REF]- [START_REF] Grassi | Channel characterization and EMC assessment of a PLC system for spacecraft DC differential power buses[END_REF], the method has two different models for common and Differential Modes (DM), but it is not able to take into account the mode conversion. Methods in [START_REF] Regue | Common and differential mode characterization of EMI power-line filters from s-parameters measurements[END_REF]- [START_REF] Reuter | Impedance analysis of automotive high voltage networks for EMC measurements[END_REF] study the EMI performance of a single part from the system, but do not deal with system EMI performance and EMI emissions. Otherwise, [START_REF] Foissac | System simulation for EMC network analysis[END_REF] takes into account the system emissions, but the model is very compact and cannot be used for system improvement.

In this paper, a frequency model for converter systems with N active conductors and ground reference is presented. The major goal is to improve system design in terms of EMC. Using segmentation, a well-known technique, the present approach allows the designers to separate the different system parts into blocks and detect the EMI defects. Then, a library of blocks can help designers improve their design at the early stages. In addition, common mode, differential model and mode conversion are taken into account. Finally, this approach generates a compact macro-model which can be used in further EMI simulation. Having a short simulation time, this model can be easily used in an optimization process.

After segmentation, passive blocks are modeled by impedance matrices. These are an accurate frequency model for passive and linear circuits. They can take into account the system non-idealities, such as stray elements and impedance of the PCB track in a simple way, without having to search for all the stray elements in order to have the correct high frequency behavior. Moreover, many simulation programs offer the possibility to import network matrices (usually as S-parameters), such as ADS Agilent, Saber and CST, etc. However, Nport impedance matrix theory is not clearly explained in the literature. Thus, this paper develops the required theory as well as the calculation details. In addition, three methods for impedance matrix determination are presented, and then the validity and the limitation of each one are discussed.

Section II explains the theoretical background for the method, describing the case of a converter with N-conductors. Section III discusses the details for the method application on a Class-D audio amplifier system with 2 active conductors. Finally, the conclusion in section IV.

II. PASSIVE CIRCUIT MODELING AND CURRENT PREDICTION FOR N-ACTIVE CONDUCTOR CIRCUITS

In the proposed method, the considered power system is divided into functional sub-circuits which are called blocks. The system is represented by a chain of blocks as shown in Fig. 1. In this chain, on the one hand there are the active blocks, such as switching amplifiers and power converters etc. On the other hand, there are the passive blocks such as filter, load and tracks, etc. Note that passive blocks can also be divided into several blocks if needed. The sub-blocks are modeled in the same way as a normal block. This will help improve the design of a given system part, such as filter and PCB layout. However, the following requirements must be verified before applying this method:

• No feedback control for the converter coming from the following passive blocks. In the opposite case, output voltages are dependent on the passive circuit which makes the model invalid.

• No significant inductive, capacitive or electromagnetic coupling between blocks. In other words, a block impedance matrix is not related to other blocks which means their is no block behavior change after segmentation.

• Block should be sufficiently short to be considered electrically short (block dimensions ≪ wavelength).

• Passive components are used in their linear region (no saturation effects and low or no hysteresis effect, etc.).

A. Active blocks model

Active blocks with N active conductor can be modeled with N electric sources and a 2N × 2N impedance matrix as shown in Fig. 2. Determining the electric sources and the output impedance matrix cannot be generalized. Each particular circuit has its dedicated set of measurements. Many measurement possibilities can be imagined, such as: open circuit (if possible), loaded, open circuit when shorting (if possible) or loading other ports, etc. Some, examples for output impedance extraction can be found in [START_REF] Miyashita | Measurement of the source impedance of conducted emission using mode separable lisn: Conducted emission of a switching power supply[END_REF]- [START_REF] Ibuchi | A study on emi noise source modeling with current source for power conversion circuit[END_REF]. However, in some cases, depending on the converter output impedance, the filter impedance and the considered frequency range, the output impedance matrix can be neglected. In this case, the converter will be modeled only by N electric sources that model the converter switching.

Z OUT (2N 2N) AC 1 AC N V OUT 1 V OUT N AC 2 Switching Output impedance V OUT 2 ͘ ͘ ͘ Fig. 2. Active block model. Passive Block (n) V 1 V N V 2N V N+1 I 1 I N+1 Fig. 3. Type A : 2N +1 ports. Passive Block (n) V 1 V N I 1

B. Passive blocks types and models

Passive blocks are modeled by impedance matrices relating voltages to currents. Two types of blocks can be distinguished, for simplification, they will be called Type A and Type B blocks for the rest of this paper. Type A is intended for the blocks situated in the middle of the chain and can be connected from both sides with 2N + 1 ports (N inputs and N outputs plus a ground) as shown in Fig. 3. Type B is intended for blocks situated at the end of the chain and can be connected only from its inputs with N + 1 ports (Fig. 4). More precisely, Type B blocks are intended for loads because usually they only have an electrical input but with other types of physical output (mechanical, thermal and acoustic, etc.). According to Fig. 3, Type A blocks have 2N voltages and 2N currents, so their impedance matrix dimension is 2N × 2N as shown in equation [START_REF] Nagari | An 8Ω ; 2.5W 1%-THD 104dB(A)-dynamic-range Class-D audio amplifier with an ultra-low EMI system and current sensing for speaker protection[END_REF].

   V 1 . . . V 2 N    =    z 1 1 . . . z 1 2N . . . . . . . . . z 2N 1 . . . z 2N 2N    •    I 1 . . . I 2N    (1)
Similar to Type A blocks, Type B blocks have N × N impedance matrix because they have N voltages and N currents as shown in equation [START_REF] Yeh | An electromagnetic interference (EMI) reduced high-efficiency switching power amplifier[END_REF].

   V 1 . . . V N    =    z 1 1 . . . z 1 N . . . . . . . . . z N 1 . . . z N N    •    I 1 . . . I N    (2)

C. Impedance matrix determination

Impedance matrix determination can be done using several methods and in this paper three methods are studied. The first uses measurements with an Impedance Analyzer (IA), the second uses measurements with a Vector Network Analyzer (VNA) and the third uses simulation with Advanced Design System (ADS) program [START_REF]ADS Advenced Design System[END_REF]. Note that the impedance matrix for passive circuits is symmetric with respect to the diagonal due to the reciprocity theorem (Z ij = Z ji ; ∀i ; ∀j ; i = j).

Passive Block (n) V i I i Fig. 5. Z ii measurements. Passive Block (n) V i I i V j = 0
1) Impedance matrix determination by IA: Determining an impedance matrix using an IA requires 2 steps. First of all, diagonal elements are measured. Equation (3) and Fig. 5 show that z ii is the input impedance of the i port.

z ii = V i I i Ij =0 ; i = j ; j = 1 . . . (2)N (3)
In practical term, the IA is connected directly to port the i th port to measure z ii when the remaining ports are open circuits (I j = 0 A ; j = i).

The second step is to measure the off-diagonal elements (z ij ; i = j). Equation ( 4) shows how to determine the cross elements theoretically.

z ij = V i I j Ii=0 ; i = j ; i = 1 . . . (2)N (4)
Nevertheless, it is not possible to apply this in practice with a single port IA and it is delicate to apply with dual-port IA. According to equation (4), this method needs a current injection on the j th port and exact voltage amplitude and phase sensing on the i th port with I i = 0 A. Instead, equation [START_REF] Edwards | Efficient low EMI switching output stages and methods[END_REF] gives an alternate user-friendly method for the determination of the cross elements.

z ij = z jj (z ii -z ijsc ) ; i = j (5)
Where z ijsc is the impedance measured from the i th port when the j th port is in Short Circuit (SC) (Fig. 6), z ii is the i th diagonal element presented by (3), z ij is the i th j th element presented in (4). However, if the difference between z ii and z ijsc is smaller than the minimum accuracy of the IA, equation ( 5) may cause some errors.

2) Impedance matrix determination by VNA: A VNA can measure the scattering matrix. To determine an impedance matrix of a passive block, the scattering matrix is measured, so (6) can be used to convert it to an impedance matrix.

Z = (I -S) -1 • (I + S) • Z 0 (6)
Z is the impedance matrix, S is the scattering matrix, I is an identity matrix and Z 0 is the characteristic impedance of the VNA (usually 50 Ω).

Note that, a VNA can introduce significant measurement uncertainty. The latter depends on the frequency range, the element type (diagonal or cross element) and the measured impedance level compared to 50 Ω [26]- [START_REF] Ye | De-embedding errors due to inaccurate test fixture characterization[END_REF]. 3) Impedance matrix determination by simulation: The advantage of simulation is determining the impedance matrices before construction, so the designer needs to build fewer prototypes. Moreover, the only frequency band limitation is the modeling details, which means the model validity. Otherwise, the band can be chosen by the user. Also, if needed, it is possible to study the influence of the uncertainty and the disparity of the components due to the fabrication procedure [START_REF] Ferber | Conducted interferences of power converters with parametric uncertainties in the frequency domain[END_REF].

D. Impedance matrix association

Impedance matrices can be associated in order to get one resulting matrix, containing their impedance behavior seen by the converter as shown in Fig. 7. The latter also shows that there are two types of association. On the one hand, there is the Type A + Type A association, and on the other hand, there is the Type A + Type B association.

1) association of 2 Type A blocks: Two Type A blocks are considered to describe this association: The first called k with an impedance matrix Z k and the second called m with an impedance matrix Z m . Z k and Z m are divided into 4 submatrices equal in terms of dimension (N × N ) [START_REF] Nakagawa | Pulse position/width collaborative control for reducing EMI noise and ripple voltage in switching DC-DC converter[END_REF].

Z k = Z k 1 1 Z k 1 2 Z k 2 1 Z k 2 2 ; Z m = Z m 1 1 Z m 1 2 Z m 2 1 Z m 2 2 (7)
Then, equation [START_REF] Kirlin | Shaping the noise spectrum in power electronic converters[END_REF] gives the association method.

Z km = Z km 1 1 Z km 1 2 Z km 2 1 Z km 2 2 (8)
with :

Z km 1 1 = Z k 1 1 -Z k 1 2 (Z k 2 2 + Z m 1 1 ) -1 Z k 2 1 Z km 1 2 = Z k 1 2 (Z k 2 2 + Z m 1 1 ) -1 Z m 1 2 Z km 2 1 = Z m 2 1 (Z k 2 2 + Z m 1 1 ) -1 Z k 2 1 Z km 2 2 = Z m 2 2 -Z m 2 1 (Z k 2 2 + Z m 1 1 ) -1 Z m 1 2
It is also possible to use the transfer matrix (matrix that gives the output voltages and currents in terms of the input voltages and currents [START_REF] Mrad | Differential passive circuit modelling with pentapole impedance matrices -application to an integrated audio switching amplifier for portable devices[END_REF]) for this association. This type of association is not used in section III but has been validated in [START_REF] Mrad | Conducted EMI prediction for integrated Class D audio amplifier[END_REF].

2) Type A and Type B blocks association: To explain how to associate a Type A and a Type B blocks the km block (Type A ⇒ 2N × 2N matrix dimension) presented before and a load block Z L (Type B ⇒ N × N matrix dimension) are considered. If Z kmL is the resulting impedance matrix, it can be calculated with equation [START_REF] Tallam | Common-Mode voltage reduction PWM algorithm for AC drives[END_REF]. Z kmL is a N × N impedance matrix.

Z kmL = Z km 1 1 -Z km 1 2 (Z L + Z km 2 2 ) -1 Z km 2 1 (9)
In the resulting matrix Z kmL , the diagonal elements (Z ii ; i = 1 . . . N ) represent the input impedance of the passive blocks seen by the converter. The cross elements (Z ij ; i = j ; i = 1 . . . N ; j = 1 . . . N ) represent the couplings between the active conductor through the passive system.

Note that the association techniques of two blocks Type A+Type A or Type A+Type B have been experimentally tested and validated in the case of simulations and measurements but the results will not be shown in next sections. [START_REF] Mrad | Conducted EMI prediction for integrated Class D audio amplifier[END_REF] shows the experimental validation in the case of measurements.

E. Current prediction

After associating all the matrices of the passive blocks, one single impedance matrix is obtained called Z R (N × N ). This matrix is a compact model replacing all the passive blocks in terms of impedance seen by the converter. Knowing the output voltages spectra of the converter, the currents spectra can be calculated using equation [START_REF] Labrousse | Common-mode modeling of the association of switching cells: Application to an electric-vehicle-drive system[END_REF].

   I 1 . . . I N    = (Z R ) -1    V 1 . . . V N    (10)
To have a real predictive method, voltages should be known (measured or simulated) without pre-specifying the load. To this aim, it is important that the input voltages are independent from the passive circuits behind. If the voltages are dependent to the load, the output voltages should be measured or simulated with the final passive blocks. In this case, the method is able to produce a compact model of the converter but it is not possible to predict the current spectrum.

III. EXPERIMENTAL APPLICATION ON A TWO ACTIVE

CONDUCTOR CLASS-D AMPLIFIER The method presented in section II was applied on a Class-D audio amplifier used in hands-free cell phone applications [START_REF] Berkhout | Class D audio amplifiers in mobile applications[END_REF]. This amplifier is a switching Pulse Width Modulation (PWM) converter using the audio signal as reference (the modulation technique is shown in Fig. 8). In hands-free applications, the Class-D amplifier uses an H-Bridge power stage which gives a differential output (2 active conductors). In real applications, this amplifier is followed by an EMI filter, then a speaker.

The studied system has 2 active conductors, which means that the filter has a 4 × 4 impedance matrix and the load (speaker) has a 2×2 impedance matrix. The Class-D amplifier used in [START_REF] Pillonnet | A high performance switching audio amplifier using sliding mode control[END_REF] is a test chip mounted on a test board. The filter (Fig. 9), used in [START_REF]MAXIM : MAX9700B Evaluation Kit[END_REF] is made on a specific filter board for easy current sensing. The filter component values are given in TABLE I. Two different loads were studied; the first is a speaker used in cell phone applications [START_REF]MEGATRON[END_REF], and the second is a dummy load (Fig. 10) that emulates the real speaker (note that the dummy load is considered for the rest of this paper). The dummy load component values are given in TABLE II.

L 1 , L 2 15µH C 1 , C 2 0.033µF C 4 , C 5 0.068µF C 3 0.15µF R 1 , R 2 22Ω TABLE II DUMMY LOAD COMPONENTS VALUES. L 1 , L 2 15µH R 8Ω V In V Ramp V CM V Out V Bat V Bat V PWMp V PWM
The different blocks were equipped with SMB connectors for easy association (disassociation) by a simple plug (unplug).

The boards are shown in Fig. 11.

A. Filter and load impedance matrix determination

To determine the impedance matrix of the two blocks (filter and load) the 3 methods described previously are applied. The filter and load impedance matrices are determined with the same methods but in the following sections only the case of the filter is detailed for conciseness.

1) Filter impedance matrix determination using an IA: The IA is used following the procedure presented in section II-C1 to measure Z ii (i = 1 . . . 4) and Z ijsc (i = 1 . . . 4 ; j = 1 . . . 4 ; i = j). Hereafter, equation ( 5) is used to calculate Z ij . Note that the IA used has a frequency limitation of 110 M Hz so the measurements were done in the range of [10 kHz -110 M Hz].

2) Filter impedance matrix determination using a VNA: The 4-port VNA is used to measure the filter scattering matrix. Then, equation ( 6) transforms the measured 4 × 4 scattering matrix to a 4 × 4 impedance matrix. In this case, I is a 4 × 4 eye matrix and Z 0 is 50 Ω. Note that the minimum frequency possible for the VNA used is 300 kHz, so that the measurements are done in the frequency range [300 kHz -1 GHz].

3) Filter impedance matrix determination by simulation: In the simulation program used, PCB tracks are modeled as strip-lines. PCB material characteristics are taken into account. Components are replaced by the measured component impedance itself or by a specific model provided by the constructor. Filter layout is taken into account and in some cases where the tracks are close to each other, coupling between them is also added using specified strip-line blocks. However, the coupling between components themselves and between components and tracks are not modeled but it might be possible to take them into account [START_REF] Zangui | Near-field coupling between EMC filter components[END_REF]. This is the real advantage of the measuring methods shown before (IA and VNA) where these types of couplings are taken into account without any additional effort. The frequency band can be defined by the user and was defined as [10 kHz -1 GHz].

L 1 L 2 C 3 C 2 C 1 R 1 R 2 C 4 C 5 Switches Speaker V 2 V 1 V 4 V 3

4) Result comparison:

To compare the results from the above three methods, the impedance results of z 11 (diagonal elements) are plotted in Fig. 12 and z 14 (cross elements) in Fig. 13. According to Fig. 12, all three methods give similar results for the diagonal elements. However, Fig. 13 shows that in the case of measurements using IA (grey line) or VNA (black line) there may be an accuracy problem with the cross elements caused by the precision of the measuring device.

In fact, in the case of IA, cross elements are calculated using equation ( 5), and z 11 is subtracted from z 14sc . Nevertheless, at high frequencies z 11 is mainly equal to the filter inductor impedance and PCB stray capacitance; the capacitors impedances are very small. Moreover, at high frequencies z 14sc is approximately equal to Z L1 and the PCB stray capacitance ; the capacitor impedances are very small and C 2 , C 4 , R 1 are in short circuit (see Fig. 14). Hence, in Fig. 15, z 11 and z 14sc overlap at high frequencies which makes the difference less than the minimal precision of the measuring device (IA).

Otherwise, in the case of VNA, the Z matrix is calculated from the filter S matrix using equation [START_REF] Lim | Adaptive voltage slew control used to limit the magnitude of broadband conducted noise emissions for buck derived dc-dc converters[END_REF]. Fig. 16 shows the measured and simulated s 14 . This figure shows that s 14 gets lower then the minimal measuring precision of the VNA in the [10 M Hz -100 M Hz] frequency band. This measurement error propagates to Z 14 which can be seen in Fig. 13. When using simulation, the impedances show mostly an over estimated resonance effects. In addition, it is not able to take into account all the inductive and capacitive coupling between tracks and components. However, it shows correct impedance behavior for the entire impedance matrix all along the considered frequency range. Moreover, determining an impedance matrix with simulation requires no prototype which means virtual EMI tests can be performed before production. Therefore, it is considered a good solution for this application, and these results are considered for the rest of this paper.

B. Current prediction for Class-D amplifier

First of all, the output voltages of the Class-D amplifier are measured with different loads (open circuit, speaker or filter and speaker). The frequency spectrum of these voltages is calculated using the Fast Fourier Transform (FFT) and the results are plotted in Fig. 17. According to Fig. 17 it is possible to say that the output voltages of the Class-D amplifier are quite independent of the load up to 110M Hz. This is due to closed loop voltage feedback, as the modulation is independent of the load, and also to the fact that the Class-D output impedance is negligible when compared to the load impedance. For that, the present method can be used for current prediction and the converter output matrix shown in Fig. 2 can be neglected.

The association method shown in section II-D2 is used in order to join both filter and load matrix in a compact model. This model is represented by a 2 × 2 impedance matrix and contains the impedance of all the blocks seen by the converter. The resulting matrix can be used for current prediction.

1 L 2 C 3 C 2 C 1 R 1 R 2 C 4 C 5 Z 11 IA Low impedance at high frequencies L 1 L 2 C 3 C 2 C 1 R 1 R 2 C 4 C 5 Z 14sc
IA Low impedance at high frequencies SC Fig. 14. z 11 and z 14sc impedances seen by the IA A 10 kHz sinusoidal signal is applied on the audio input of the Class-D amplifier. The power stage is supplied by 3.6 V (similar to the cell phone battery voltage). The output voltages for both active conductors were measured in the time domain at the same time. A 12-bit vertical quantification oscilloscope [START_REF] Lecroy | WaveRunner HRO 64 Zi 12-bit oscilloscopes, 400 MHz[END_REF] is used in order to reduce the noise floor. Note that measuring the voltages at the same time makes the relative phase between the outputs correct, and can be used after FFT. Hence, knowing the output voltages and the resulting impedance matrix, equation ( 10) is used for current calculation. Results can be seen in Fig. 18 and only one of the two currents is shown because they are similar. As can be seen from this figure the calculated current spectrum is in good agreement with the measured one. The current is measured using a current probe with a bandwidth of 120 M Hz [START_REF]Tektronix : current probe, 120 MHz[END_REF].

IV. CONCLUSION

A new use of impedance matrices for EMI purposes was presented in this paper. The passive functional circuits of the system are modeled separately with impedance matrices. Those matrices are then associated to rebuild the system in a single one. The resulting matrix and converter output voltages (input voltages for the passive blocks) can be used for current calculation. If the output voltages of the converter are independent of the load, the open circuit output voltages can be used for current prediction. This approach was applied to a cell phone audio system containing a Class-D audio amplifier. The impedance matrices are determined using three different methods. The theory and validity of each is discussed. Finally, the Class-D amplifier voltages and the resulting matrix for both the filter and the load were used for current calculation. Using the simulated filter and load impedance matrices good results were seen up to 110 M Hz. The accuracy of these methods is due to the fact that the CM, DM and mode conversion are included by default in the modeling process.

This method can be used by system integrators when using the ready-made integrated systems in order to search for the best filter design. The present method permits virtual measurements to be made in order to validate a filter or a routing design in terms of EMI performance, before board prototype construction. It is also possible to make a library of matrices corresponding to different filters and loads in order to check the use of an old design in a new application. In addition this method have a short simulation time which tolerates its implementation in filtering optimization algorithm.

In future work, filter optimization using this method will be studied. Moreover, part of the studies will be dedicated to simulating the output voltages of the converter in order to have APPENDIX A Using [START_REF] Miyashita | Measurement of the source impedance of conducted emission using mode separable lisn: Conducted emission of a switching power supply[END_REF] in [START_REF] Hagmann | Determination of mode conversion on differential lines[END_REF] :

I out km = -(Z L + Z km22 ) -1 Z km12 I in km (21)
Finaly using [START_REF] Tarateeraseth | Systematic electromagnetic interference filter design based on information from incircuit impedance measurements[END_REF] in [START_REF] Hagmann | Determination of mode conversion on differential lines[END_REF] :

V in km = [Z km11 -Z km12 (Z L + Z km22 ) -1 Z km12 ]I in km (22)
The resulting block has only V kmL , I kmL on the input. The resulting impedance matrix is the matrix relating V kmL to I kmL . Hence, [START_REF] Tarateeraseth | Systematic power line emi filter design for smps[END_REF] permit us to deduce equation 9.

Fig. 1 .

 1 Fig. 1. System decomposition into blocks.

Fig. 4 .

 4 Fig. 4. Type B : N + 1 ports.

Fig. 6 .

 6 Fig. 6. Z ijsc measurements.

Fig. 7 .

 7 Fig. 7. Block association technique.

Fig. 8 .

 8 Fig. 8. PWM modulation for Class-D amplifier.

Fig. 9 .

 9 Fig. 9. The filter schematic used.

Fig. 10 .

 10 Fig. 10. The dummy load schematic.

Fig. 11 .

 11 Fig. 11. Class-D amplification system.

Fig. 12 .

 12 Fig. 12. Filter : z 11 impedance results.

Fig. 13 .

 13 Fig. 13. Filter : z 14 impedance results.

L

Fig. 15 .Fig. 16 .

 1516 Fig. 15. Filter : Measured z 11 and z 14sc . with IA

Fig. 17 .

 17 Fig. 17. Class-D amplifier output voltages with different loads.

Fig. 18 .

 18 Fig. 18. Calculated current.

PROOF OF EQUATION 8

In order to prove equation 8, the k and m blocks of sectionII-D1 were re-considered (Fig. 19). In Fig. 19 V in k , V out k , V inm , V outm , I in k , I out k , I inm and I outm are vectors with a dimension equal to N. Those vectors are the voltages or the current for the N ports of k or m blocks.

By definition :

Since the block k output is related to the block m input, then : When using [START_REF] Jettanasen | Common-mode emissions measurements and simulation in variable-speed drive systems[END_REF] and ( 12) :

From ( 11) and ( 13) :

Finally, when using ( 14) in ( 11) and ( 13)

The resulting block has V in k , I in k on the input and V outm , I outm on the output. The resulting impedance matrix is the matrix relating V in k , V outm to I in k , I outm . Hence, [START_REF] Grassi | Channel characterization and EMC assessment of a PLC system for spacecraft DC differential power buses[END_REF] and ( 16) permit us to deduce equation 8.

APPENDIX B PROOF OF EQUATION 9

In order to prove equation 9, the km and L blocks of sectionII-D2 were re-considered (Fig. 20). In Fig. 20 V in km , V out km , V L , I in km , I out km and I L are vectors with a dimension equal to N. Those vectors are the voltages or the current for the N ports of km or L blocks.

By definition :

Since block km output is related to block L input, then :

When using ( 18) and ( 19) : In 1998, he joined the AMPERE Laboratory (UMR CNRS 5005). He is now a Professor at the Ecole Centrale de Lyon. Since then, he has carried out research with AMPERE at the Ecole Centrale de Lyon in the field of the numerical modeling applied to the interaction between electromagnetic field and complex systems. He develops, in particular, specific formulations and numerical methods for the computation of electromagnetic fields in complex structures. One of the fields of applications relates to electromagnetic compatibility.

Philippe


Brief Solutions to Collatz Problem, Goldbach Conjecture and Twin Primes

Mesut KAVAK* I published some solutions [START_REF] Kavak | Complement Inferences on Theoretical Physics and Mathematics[END_REF] a time ago to Goldbach Conjecture, Collatz Problem and Twin Primes; but I noticed that there were some serious logic voids to explain the problems. After that I made some corrections in my another article [2]; but still there were some mistakes. Even so, I can say it easily that here I brought exact solutions for them out by new methods back to the drawing board.

Goldbach Conjecture

The main question about Goldbach Conjecture is pretty clear that is each even number sum of two prime numbers?

The solution

Being p is a prime number, for definition of p 4 > p 3 ∧ p 4p 3 = 2n + 2 ∧ n > 2 ∧ n ∈ Z + ; first number group which is created by n pieces non-prime consecutive positive odd whole numbers, the smallest odd whole number it has is p 3 +2 or p 4 -2n and the biggest odd number it has is p 3 +2n or p 4 -2, contains greater numbers than last number group which contains n -1 pieces consecutive non-prime odd numbers and none of the numbers n group contains, for definition of p 2 > p 1 ∧ p 2p 1 = 2(n -1) + 2 is before n group, and the smallest odd number it has is p 1 +2 or p 2 -2(n-1) and the biggest one it has is p 1 + 2(n -1) or p 2 -2. Also n -1 groups which contain n -1 pieces non-prime consecutive odd numbers and have greater numbers than n groups have are possible; but these groups can never exist until n groups emerge because of the above stated reasons and definitions; because n groups also contain 2 pieces (n-1) 1 and (n-1) 2 consecutive groups which n -2 pieces elements of them are common with n groups.

The difference between n -1 and (n -1) m is not to be of prime number there after last number of first group and before first number of last group of emerging two groups; therefore when two groups emerged as (n-1) 1 and (n-1) 2 , it means that n group already had been emerged spontaneously, and is first n group. Being (n -1) 1 is the first and (n -1) 2 is the second group, as last element of (n -1) 2 group is always bigger than all numbers of (n -1) 1 , it means (n -1) 1 had already been emerged before n group. Also as n -1 group must exist at the place before (n -1) 1 group, then numbers of n group are always greater than numbers of n -1 group for the definition of n and n -1 groups. Here all possible nk groups for m, k ∈ Z + definition are unique; but (nk) m groups which have common element with another groups are not unique.

Being k = 0, for nk = n groups, 1 piece of selected n pieces consecutive odd numbers from 3 to numbers of n + 1 group has to be prime number; so if some tables are made like table 1 and table 2 for each n, they will help about the main question.

When similar tables for the other n groups are made, and when an addition is done by order between selected n pieces of the smallest consecutive x number group elements and el- ements of y groups that each group of y contains n pieces of consecutive odd numbers, the result will be like in the tables. Here, the first group of y is the same with x column. The first and the smallest number of the next y group is greater by 2 than the first and the smallest number of the previous y group. The sum of b is always an even number as well.

As the first y group is also x group and as also the biggest number of x group is always 2n + 1, the biggest number of first b group must always become 2(2n + 1) = 4n + 2. This number is the first number which starts to be common with all the other b even numbers which are formed by different x elements for the same n value in the tables; because it is in the last emerging line. Each even number after this number can be formed absolutely as n pieces the number itself is included as well. As 1 piece of selected consecutive n pieces odd numbers must be prime number until n + 1 group, minimum 1 piece of the even numbers which are in each even number group has n pieces of the same even number must be sum of two prime numbers.

Here, if also 4n+4 number after 4n+2 number is included, if all of emerging n pieces of the same 4n + 4 even numbers are not the numbers of n + 1 group in a table for the same n value, as all results of 4n + 4 and 4n + 2 form set of even num-bers greater than 8, then it means each even number which is greater than 8 absolutely must be sum of two prime numbers.

The equation between a values that first a namely a 1 which gives even numbers in the first line for a = [1, ∞) over 2a + 4, and a n which accepts the first even as 4n + 2 and gives the numbers of 2a + 4n is ( 1)

a 1 = a n + 2n -2 (1)
Here, result of a 1 for a n = 1 is also equal to number of used odd numbers in the first y line to form 4n + 2 number or is equal to number of used numbers which are different than each other in the tables for the same n value, outside of using a n to form 2a n + 4n namely 4n + 2 number; because it is also x + y operation number of forming 4n + 2 even number in the first line; therefore a 1 number over (1) for a n = 2 is required to form 4n + 4, and it must be (2)

N = 2n
(2)

The result

As the result for the above stated information, assume that 4n+2 and 4n+4 numbers cannot be sum of two prime numbers for each n value. Last n pieces of y consecutive odd numbers which are required to form 4n + 2 and 4n + 4 numbers for required n value, must be n group numbers; thus only 2n -n = n pieces y consecutive odd numbers can be used it means being 3 is the first usable number on the first y line; but already this means that all numbers except 1 are non-prime consecutive odd numbers for each n that is impossible. Even if it is assumed that, for the worst possibility n and n -1 groups emerge together by the same numbers as non-unique groups, also the information stated above that this assumption is impossible. Already if it is impossible even for n group, when it is assumed that there are another groups, 2n -

n n=2 n
(3) number of used or usable consecutive y odd numbers will decrease by (3), and is impossible for n > 1 definition. As and if we do not know prime separation, for the worst possibility of number of existent primes in 2n pieces usable consecutive odd numbers on the first y line, assume that for n, there are consecutive unique n groups until last 2n number even if actually they cannot be fitted such that as emerging number of the numbers will be bigger than 2n by this way; but if it is right, then amount will not be important that you can assume that there are (2n) 2 pieces n groups if essence of the function provides this that it is provided here; because the number of the non-prime numbers will increase greater than the primes for assumed n pieces unique groups; so as unique groups are between two primes, there must be 2n pieces prime numbers. As this 2n is also equal to number of the used numbers on the first y line that it is not important which numbers of 2n pieces numbers are prime or not prime here, absolutely minimum 1 piece of each n pieces the same even 4n + 2 numbers and minimum 1 piece of each n pieces the same even 4n + 4 numbers which emerge in each table separately are absolutely sum of two prime numbers. Also it means that all even numbers greater than 8 are sum of two prime numbers. This is also proof of infinite number of twin primes.

Twin primes

Select any unique number group which has n pieces of consecutive non-prime odd numbers. This group has to exist between 2 prime numbers according to the definition stated at the beginning between prime numbers and group numbers; because otherwise there will occur a group like n + 1 group instead of n group that actually infinite number of non-prime numbers can be consecutive. For example, let us take consecutive multiples of 3, 5, 7, 9 and 11 for n = 5 group. Being a is an odd number, any multiples of odd numbers become a(2x + 1) for required x; so over (6x 1 + 3) + 2 = 10x 2 + 5, it becomes x 1 = 5x. Over (6x 1 + 3) + 4 = 14x 3 + 7, it becomes x 1 = 7x. Over (6x 1 + 3) + 6 = 18x 4 + 9, it becomes x 1 = 9x. Over (6x 1 + 3) + 8 = 22x 5 + 11, it becomes x 1 = 11x. Results of 6x 1 + 3 which are odd multiples of 3 become 30x + 3, 42x + 3, 54x + 3 and 66x + 3 for the stated x 1 values. If also these are made equal to each other, being (11

• 9 • 7 • 5 • x) = m
and the first number of the group is multiple of 3, consecutive multiples of the group numbers become by order 6m + 3, 6m + 5, 6m + 7, 6m + 9 and 6m + 11. For more consecutive odd multiples, we can increase the number of used numbers in a group forever.

I selected n 1 , n 2 , n 3 and n 4 consecutive odd numbers in n = 4 group like p 1 n 1 n 2 n 3 n 4 p 2 being p is prime number. Minimum one of these n numbers has to be multiple of 3; because separation of odd multiples of 3 is according to 6x + 3, and so there are always 2 consecutive odd numbers between two consecutive odd multiples of 3. Here, if n 2 becomes odd multiple of 3, then p 2 must be the next multiple of 3 that this is only possible for n = 5. If n 3 becomes odd multiple of 3, then p 1 must be the previous multiple of 3 that this is also possible for n = 5. As it was said, it is possible to form groups have infinite number of consecutive non-prime odd number, namely n = 4 must exist anyway.

If n 1 becomes odd multiple of 3, then n 4 must be the next multiple of 3 and also n 5 becomes the next multiple of 3 after n 4 as n 0 became the previous odd multiple of 3 before n 1 over n 0 n x p 1 n 1 n 2 n 3 n 4 p 2 n y n 5 .

If n 4 becomes odd multiple of 3, then n 5 becomes the next multiple, and n 0 and n 1 become the previous multiples of 3; thus n 1 and n 4 are pretty suitable to be odd multiple of 3.

Here, infinite number off odd consecutive n number can take place after n 5 ; so element number of the next group after n = 4 is not important; but n y is always prime or not, this is important. Over n y = n 5 -2 = (6x + 3) -2, it becomes n y = 6x + 1. Hence, n y never can be only prime number where 2 2 Twin primes

x ∈ Z + ∧ x > 0. It is not prime for required x, and otherwise it is prime for emerging odd numbers between two n y and n y+1 numbers which are a result of consecutive two x and x + 1 values; so when it becomes n y = p 3 , it is a twin prime group between n 4 and n 5 ; thus twin primes are in infinite number.

Collatz Problem

The main question about the Collatz Problem is also pretty clear. When a positive whole number is selected, if the number is an even number then it is divided by 2; otherwise it is multiplied by 3, and after that 1 is added to the result. When the same operation with required option of the problem due to the condition of to be odd or even number of the result is repeated for the last results, can each positive integer which is different than 0 and 1 be reduced into 1?

The solution

If the input number is an even number, and if it is not an even number as 2 n as well for definition of n ∈ Z + ∧ n > 0; being pn is process number, when the input number is divided by pn times 2 or directly by 2 pn , each positive even number absolutely turns into a positive odd number as they can be defined as (2x+1)•2 n for definition of x, n ∈ Z + ∧n > 1∧x > 0; thus we should only work over odd numbers.

a n+1 = 3a n + 1 2 (4)
Over (4), it must become a n+1 = {2, 5, 8, 11, ...., 3x -1} where x ∈ Z + ∧ x > 0 for a limited interval. For the numbers which make a n an odd number, it becomes a n+1 = {5, 11, 17, ...., 6x -1} for a limited interval and the same x definition. Also it becomes a n = {3, 7, 11, 15, ...., 4x -1} over a n+1 odd numbers for the same conditions.

The below is a table over a and a n+1 numbers by order over (4) for a limited interval being E is even and O is odd.

18x -10 = 3(12x -7) + 1 2 (5)
The result of (5) is absolutely even number for each x. For 12x-1 numbers from a n+1 numbers with the same x definition, it becomes (6).

18x

-1 = 3(12x -1) + 1 2 (6)
The result of ( 6) is absolutely odd number for each x. Right this point, the question is this that for (7),

a n+2 = 3a n+1 + 1 2 (7)
when a n+2 becomes an even number and divided by 2 pn , does emerging odd numbers as a result always emerge before a n+1 in set of odd numbers due to number order or can it be bigger number than a n+1 ? As the answer, if the result of the operation of (7) becomes even, to realize of to be reduced of the result of 3a n+1 + 1 2 • 2 pn operation into an odd number which is before a n+1 , the condition of (8) always has to be provided.

1 > 3a n+1 + 1 2 • 2 pn a n+1 (8)
If ( 8) is edited then as (9),

1 > 1 2 pn+1 Å 3 + 1 a n+1 ã (9)
the inequality of (9) always provides this for the definition of

pn > 0 ∧ a n+1 > 1 ∧ pn, a n+1 ∈ Z + .
As a result, when a n+2 is reduced into an odd number, the odd number is always before a n+1 odd, and is smaller than it.

It means that the odd number as a result of a m = 3a n+1 + 1 2 • 2 pn is always smaller than a n+1 number. Even if 3a m + 1 2 becomes even number, again it can be reduced into a smaller odd number than both a m and a n+1 , and it is acceptable for the other repeats as well.

Right this point, a second question emerges that is there a number which always gets bigger and does not become an odd number on (4) infinite chain.

As the answer, there is table for the numbers which do not emerge on table 3. If these numbers are included to the numbers on table 3 as well, the sum is set off odd numbers. There will no other odd number which is not included to the calculations. If some groups are made for 4x + 1 numbers in table 4, there will only emerge 3 groups for 12x -3, 2x +1 and 12x -7 numbers for the same x definition. Being a n+1 = 12x -3, (10) always gives even result.

18x -8 = 3(12x -3) + 1 2 (10)

Being a n+1 = 12x + 1, (11) always gives even number result as well.

18x

+ 2 = 3(12x + 1) + 1 2 (11)
As 12x -7 numbers, they are already the same numbers with a n+1 , and at the result of (7) they always give even number being a n+1 = 12x -7.

As even numbers, they can always be reduced into a smaller odd number than the odd number which makes them even number in the operation of (4) as it was proved; thus if each one of a n = 4x -1 numbers do not get greater by turning into an odd number when (4) is repeated for each n number where n ∈ Z + , it means all positive whole numbers different than 0 and 1 can be reduced into 1.

In table 3, 8x -5 numbers from a n numbers turn into a n+1 number which gives even result over (7); thus the only chance is to give odd result always of 8x -1 numbers over (4) infinite chain. To be realized of this, the same numbers with a n must emerge on a n+1 line in table 3. Also always the numbers which give odd result must emerge on a n+1 line between them. For the same x condition, 4x -1 and 6x -1 operations give a n and a n+1 over (4) for the same x number; so the waited loop occurs or does not occur,

x 1 = 6x 2 4 (12)
(12) shows this over the equality of 4x 1 -1 = 6x 2 -1. For (12), it becomes x 1 = 3t and x 2 = 2t over t ∈ Z + ∧ t > 0 condition; so the problem is reduced into the rule of table 5 below. 6, to occur of the infinite loop, when a 4t number is selected, also 6t number which is under it and on 6t line in table 6 must be even number, and also this 6t number must take place on 4t line again. This condition has to take place for one or more than one number to be broken of the Collatz's reducing chain, and then one or more than one number will not be reduced into 1; but this is impossible; because for t n+1 = 6t n 4 where t > 0 ∧ t, n ∈ Z + , for each t whole number,

t n+1,t = lim n-→∞ 6t n,t 4 (13)
(13) has to be provided for the condition of (14),

t n,t , t n+1,t ∈ Z + ( 14
)
where t n,t = 4t, t is the order number of the number which is waited of starting the loop from it, and n is the repeat number of (13) for each t. As this condition of ( 14), it cannot be provided for each t. For example, for t 1,1 = 4, it becomes 6t 1,1 = 4t 2,1 and so becomes t 2,1 = 6. For t 2,1 = 6, it becomes 6t 2,1 = 4t 3,1 and so becomes t 3,1 = 9. For t 3,1 = 9, it becomes 6t 3,1 = 4t 4,1 and so becomes t 4,1 = 27/2. As it can be seen, t 4,1 Z + and so the condition of ( 14) cannot be provided.

The result

(13) cannot continue forever; because wee need a number which has infinite number of common divisors like 4 ∞ or (2x + 1) • 4 ∞ imaginary numbers. As it can be seen, only we can increase the repeat number by using a t number like 4 m where m > 0 ∧ m ∈ Z + that if m gets bigger, then the repeat will increase; but there is no infinite repeat; hence, any whole number absolutely can be reduced into 1 by changing operation numbers of the Collatz's rule due to the used number.

For the repeats of (4), being 2 m is order number of the selected odd number of a n in table 3 where m > 0 ∧ m ∈ Z + , it gives m + 2 pieces odd number, and then the last one gives even number on (4) for 2 m . Table 7 is a demonstration of this. Here for (15), m + 1 pieces odd numbers or repeats on (4) emerges being a = a 0 which is first input number on (4). You can write your own operation as well.

  Also we can use an operation like (15), and we can derive another operation as well. The below is an example. a = 7 +

Table 1 :

 1 The table for n=2

	x			y
	3 3	5	7	9	11 13 ...
	5 5	7	9	11 13 15 ...
	x	+	y	=	b
	6	8	10 12 14 ... 2a+4
	10 12 14 16 18 ... 2a+8

Table 2 :

 2 The table for n=3

	x			y
	3 3	5	7	9	11 13 ...
	5 5	7	9	11 13 15 ...
	7 7	9	11 13 15 17 ...
	x	+	y	=	b
	6	8	10 12 14 16 ... 2a+4
	10 12 14 16 18 20 ... 2a+8
	14 16 18 20 22 24 ... 2a+12

Table 3 :

 3 a n and a n+1 numbers

	a n	3 7	11 15 19 23 ...
	a n+1 5 11 17 23 29 35 ...
	a n+2 E O E	O E	O ...
		On the table of 3, for 12x -7 numbers from a n+1 numbers
	with the same x definition, it becomes (5).

Table 4 :

 4 The numbers which are not in the previous table 5 9 13 17 21 25 29 ... 4x+1 The numbers which are written thick are also in a n+1 line in table 3. The other numbers are the numbers which are not in table 3.

Table 5 :

 5 The numbers for each t Each number on 4x -1 and 6x -1 is also order number of a n and a n+1 in table 5; thus the number has 3t order number on a n or 2t line and the number has 2t order number on a n+1 or 3t line in table 5 are the same numbers.As the odd numbers on 3t line in table 5 are a n+1 numbers which give even result in table 3, they are elected; thus table 5 turns into table 6.

	2t 2 4 6 8	10 12 ...
	3t 3 6 9 12 15 18 ...

Table 6 :

 6 The other numbers for each t

	4t 4 8	12 16 20	24 ...
	6t 6 12 18 24 130 36 ...
	In table

Table 7 :

 7 The other numbers for each t

	m		a n
	1	7-→	11-→ 17
	2	15-→ 23-→ 35-→ 53
	3	31-→ 47-→ 71-→ 107-→ 161
	.
	.

*kavakmesut@outlook.com.tr

Kavak M. 2016, On the Uncertainty Principle, American Journal of Physics and Applications, Vol. 4, No. 4, pp. 90-123. Available online: osf.io/t8zqw/ *kavakmesut@outlook.com.tr