A Boyer

S Ben Dhia

A Durier
email: andre.durier@irt-saintexupery.com

A new Voltage Measurement Probe for investigating Radiated Immunity Test

Keywords: Radiated immunity, voltage probe, anechoic chamber, validation

Diagnosis of failures induced during radiated immunity tests in anechoic chamber and validation of field-toline coupling models require measurements of the induced voltages at the input of electronic equipments. Although such measurements seem obvious at first sight, they can become complicated and erroneous due to measurement conditions. This paper describes the design a miniature high-frequency voltage probe to solve this issue. This probe is validated experimentally by a comparison with simulation.

I. INTRODUCTION

Radiated immunity (RI) tests in (semi)-anechoic chamber, as those defined by standards IEC61000-4-3, ISO11452-2 or DO160 are necessary to guarantee the safe operation of electronic equipments illuminated by electromagnetic disturbance. These tests require a precise control of all measurement parameters to ensure a sufficient repeatability of the results. Moreover, anticipating risks of failures needs an analysis of the field to line or equipment coupling mechanisms, making modeling and simulation unavoidable [START_REF] Durier | Comparison of voltages induced in an electronic equipment during far field and near field normative radiated immunity tests[END_REF] [2] [START_REF] Leone | On the coupling of an external electromagnetic field to a printed circuit board trace[END_REF]. However, reliable measurement tools are also required to validate simulation results.

Field-to-line coupling models have been developed these last decades, such as [START_REF] Taylor | The response of terminated two-wire transmission line excited by a non uniform electromagnetic field[END_REF] or [START_REF] Paul | Frequency response of multiconductor transmission lines illuminated by an electromagnetic field[END_REF]. They were usually validated by measurements based on TEM cell or capacitive coupling clamp, but hardly on anechoic chamber measurements. When it happened, it was current measurement with inductive coupling clamp and not voltages induced on line terminals or equipment input. Several practical difficulties arise when such a voltage measurement is done:

• the frequency range of these measurements (up to 1 or 6 GHz), imposing constraints on the measurement receiver.

• the presence of the measurement receiver may alter electromagnetic field locally. It may be placed outside the shielded room and connected to the device under test through coaxial cables

• the presence of cables contributes to a significant parasitic coupling of the incoming radiated disturbance, which compromises the repeatability of the RI test

• the measurement receiver must be as compact as possible and embed its own energy source to limit direct coupling of the incoming radiated disturbance.

The analysis of existing and usual measurement equipments used in EMC laboratory shows that no commercial solution exists to cover this need, certainly because it is not associated to any EMC compliance requirement. This paper aims at presenting a high frequency voltage measurement probe, miniature enough to limit parasitic coupling of the incoming disturbance, batterypowered, and designed only with off-the-shelf components. After a description of its principle and a presentation of its characteristics, an experimental validation is provided. The relevance of the measurement results of the voltage induced on a line under test during RI test is evaluated by a comparison with simulation.

II. PRESENTATION AND CHARACTERIZATION OF THE VOLTAGE PROBE

A. Description of the design

The first main constraint is the limitation of radiated disturbance coupling on the probe, which prohibits the use of coaxial cable and requires electro-optical conversion. However, the second main constraint is the frequency range, which must be at least 1 GHz to cover usual RI test range. It reduces the number of affordable measurement devices. As only the amplitude of sine waveform is under concern, the easiest solution consists in using a RF power detector, which converts the amplitude of a RF signal into a continuous voltage. Commercial miniature RF detection module exists, such as Mini Circuit ZX 47-60-LN-S), but they are not battery powered and they have no optical output. Battery and output cables may contribute to radiated disturbance coupling. Thus, a home-made solution based on off-the-shelf components seems to be the only available solution at an acceptable cost.

Numerous commercial RF detector integrated circuits are available, with typical sensitivity ranging between -60 and +20 dBm. On a 50 Ω input, the voltage ranges between 220 µV and 700 mV. Their typical bandwidth starts at some MHz and stops at several GHz (sometimes at more than 10 GHz). These characteristics are compatible with requirements of RI tests. Moreover, these circuits are low-cost, low power, have a small temperature drift and are mounted in miniature packages.

Package

The RF detector LT5534 developed by Analog Devices [START_REF]LT5534 -50MHz to 3GHz RF Power Detector with 60dB Dynamic Range, Datasheet, Linear Technology[END_REF] has been selected because of its small size and its simplicity. Its characteristics are given in Table I. This logarithmic-based detector has a bandwidth and a 60 dB dynamic range compatible with RI requirements.

The DC voltage provided by the RF detector has to be digitalized before its optical transmission. This task is ensured by the optical probe Langer A-100. With an input voltage range 0 -10 V and a 25 kHz bandwidth, it is compatible with the signal provided by the RF detector. The choice of the optical probe is justified by its compactness, its high resolution (12 bits), its low consumption (3 mA) and its high robustness to electromagnetic disturbance (> 200 V/m). The detector also needs a input impedance matching network with a RC filter to stabilize and filter the output signal. The probe is powered by a 4.8 V miniature rechargeable microbattery, with a capacity of 150 mA.h. The autonomy of the probe is estimated to 10 hours. The battery is followed by a low dropout regulator to provide a fixed 3.3 V voltage to the RF power detector and optical probe. The electrical schematic is presented in Fig. 1. A prototype has been developed on a 40 x 20 mm four-layer FR4 board (Fig. 2).

Preliminary illumination experience on the RF detector showed that the RF detector is sensitive to direct coupling of the incoming radiation, which may corrupt the measurement result. In order to preserve the integrity of the measurement results, the RF detector must be shielded. The sensitive active circuits, i.e. the RF detector and the LDO regulator, are placed within a shielded cabinet (Wurth WE-SHC) in order to minimize the direct coupling of the incident radiation on the voltage probe. The shielded cabinet is connected to an inner ground plane of the four-layer board. Except the RF input signal, all the signals which enter or leave the shielded cabinet are filtered by ferrite beads to limit the conducted coupling of the incident radiated disturbance to the LDO regulator and the RF detector.

B. Characterization

The sensitivity of the RF power detector varies according to the input signal power and frequency. A calibration is required prior to measurements. The characterization results of the probe sensitivity according to power and frequency are shown in Figs. 3 and4. The calibration of the probe is based on the results of Fig. 3. The sensitivity is nearly constant up to 1 GHz and loses 25 % at 3 GHz. The probe can be used under the minimum frequency given by the detector manufacturer, at least at 10 MHz, as shown in Fig. 4. The probe input is 50 Ω matched in order to limit return loss. The input reflection coefficient is shown in Fig. 5. It remains almost constant with a magnitude lower than -25 dB up to 1.3 GHz. Above this frequency, it increases but remains less than -8 dB at 3 GHz.

III. COMPARISON OF MEASUREMENT METHODS

A. Case study description

The relevance of voltage measurement provided by the proposed probe during RI test is tested. It consists in measuring the voltage induced at the terminal of a device under test (DUT) when it is illuminated by a radiated disturbance in semi-anechoic chamber. The measurement results with the proposed probe will not only be compared with simulation (see part IV), but also to direct measurement done with a spectrum analyzer connected to the DUT by a coaxial cable and placed outside the chamber. The DUT is a 75 mm long 2.5 mm wide 50 Ω microstrip line designed on a 100x100 mm FR4 printed circuit board (PCB). The experimental set-up is described in Fig. 6.

B. Characterization of the sensitivity of the probe to incoming radiated disturbance

Initially, the voltage probe is placed in the anechoic chamber and is not connected to any trace under test. This measurement aims at characterizing the error introduced by a direct coupling of the incoming radiation on the probe, which may disturb its operation. The probe is placed at 1.3 m to the injection antenna. It is placed horizontally; the RF connector is oriented to the antenna. The power injected to the antenna is set to produce a constant electric field. The incoming wave is vertically polarized. Fig. 7 shows the evolution of the power measured by the probe in this configuration.

The power measured by the probe is not affected by the incoming radiation up to 1.2 GHz. The measured power is less than -65 dBm, i.e. the sensitivity threshold of the RF detector. Above 1.2 GHz, the effect of a direct coupling to the probe becomes visible. A peak appears at 1.7 GHz, where the measured power is equal to -39 dBm, i.e. a voltage of 2.5 mV on a 50 Ω load. This power remains small in comparison to the voltage that may be induced by radiated coupling on large structures. This peak is certainly due to the resonance of the cavity formed by the inner ground plane of the PCB and the shielded cabinet placed around the RF detector and the LDO regulator. The input-output filtering and the shielding of the probe PCB should be improved to reduce the sensitivity of the probe, especially the RF detector, to the direct coupling of radiated disturbance.

C. Measurement results and analysis

Measurements have been done for two orientations of the line under test: the PCB is placed horizontally and the illuminated trace is oriented perpendicularly to the antenna. Three measurements have been performed and compared: with the proposed voltage probe (after calibration) and two others with a spectrum analyzer and two types of 2 m long 50 Ω coaxial cables. They differ from their shielding: the first cable has only one shielding layer while the second has two shielding layers. The position of both cables in the chamber is nearly identical. Measurement results are presented in Fig. 8. Although the measurement types present similar trends and orders, significant differences are visible, especially under 200 MHz. In this frequency range, the far-field hypothesis is not valid and the chamber resonates. These differences are due to the coaxial cables that contribute to radiated disturbance coupling as parasitic antenna. The coupled field is not only related to the characteristics of the trace under test, but also to those of the coaxial cables which connects the trace under test to the spectrum analyzer.

Sensitivity floor LT5534

The observed frequency-dependent variations with both measurements made with spectrum analyzer are explained by the length of cables which introduces multiple resonances. Measurements made with our miniature probe present less variations because of its small size and its low contribution to incoming field coupling. The measured variations are due to non-constant field over the frequency range.

It must be underlined that the measurements done with both types of cable show differences that can reach 10 to 20 dB, especially below 200 MHz. Field coupling depends on the common-mode impedance of cables which is related to the nature of the shielding. Any change in the internal construction of the cable and its position induces a large measurement variability, unlike the proposed measurement probe.

IV. SIMULATION AND VALIDATION

Experimental results show that the proposed probe improves the repeatability of the measurement of the coupled voltage on an illuminated line. However, the validity of the measured voltage must be tested, so that it is compared to prediction based on a far-field coupling model.

A. Description of the model

The trace under test was initially characterized with a vector network analyzer to extract its per-unit-length (pul) parameters. These values were also compared with theoretical pul parameters, computed with the freeware IC-EMC [START_REF]IC-EMC v2.9[END_REF], which provides a good agreement. Then a Taylor model was built to determine the voltage coupled on its terminals when it is illuminated [START_REF] Taylor | The response of terminated two-wire transmission line excited by a non uniform electromagnetic field[END_REF]. The field recorded by the electric-field probe is used as excitation values in the simulation. Fig. 9 illustrates the geometrical and the electrical equivalent models. Vertical electric EZ and tangential magnetic Hx fields contribute to generate distributed electromotive force and current along the line, as given by ( 1) and [START_REF] Caniggia | SPICE-like models for the analysis of the conducted and radiated immunity of shielded cables[END_REF]. They contribute to the voltages induced across each terminal of the line, whose expressions are given by ( 3) and ( 4).

( ) ( ) ( ) ( )  - + Γ + Γ + = - - - - L E eq H eq eq eq y L k y L k dy dI Z dV Z Z Z e e V 0 1 2 1 2 2 2 2 2 1 1 ) (ω (3)
where Zeq1 and Zeq2 are the terminal load impedances seen from a point y of the line (( 5) and ( 6)). Γ1 and Γ2 are the reflection coefficients of termination load ZL1 and ZL2, k is the wavenumber and Zc the characteristic impedance of the microstrip line. However, Taylor model is not sufficient to take into account the presence of the ground plane and the PCB substrate. They modify the distribution of the E and H fields around the line, due to the superposition of the incoming excitation wave and the reflected waves at the substrate-air interface and ground plane. A correction is provided to compute the distribution of E and H fields within the substrate, based on the model described in [START_REF] Scott | Practical Simulation in Printed Circuit Boards and Related Structures -First Edition[END_REF]. The considered geometrical model is described in Fig. 10. The substrate is illuminated by an incident wave, with an electric field value Einc and an angle of arrival θ2. Here, the model is only described when the E field is in the xz plane, but equivalent formulations can be found when the E field is perpendicular to the xy plane. Within the substrate (medium 1), the E and H field complex amplitudes are given by ( 7) to (9). C1 + is the complex amplitude of the incident wave in the substrate and is given by (10), where k1 and k2 are the wavenumbers in media 1 and 2 ((12) and ( 13)). The model has been implemented in the freeware IC-EMC [START_REF]IC-EMC v2.9[END_REF].

( ) ( ) ( )

ky th Z Z ky th Z Z Z y Z L C C L C eq 1 1 1 + + = (5) ( ) ( ) ( ) ( ) ( ) y L k th Z Z y L k th Z Z Z y Z L C C L C eq - + - + =

B. Comparison measurement -simulation

A comparison between measurements made with the proposed probe and simulations is shown in Fig. 11, for two orientations of the trace under test. An acceptable agreement is found between measurement and simulation. The maximum gap is observed between 100 and 200 MHz. In this frequency range, the far-field condition is not met perfectly and chamber resonance enhances measurement variability. This result confirms the relevance of the measurement done with our proposed probe for the characterization of the voltage induced on a line illuminated in a (semi-)anechoic chamber.

Sensitivity floor LT5534

Fig. 1 .Fig. 2 .

 12 Fig. 1. Electrical schematic of the proposed voltage probe

Fig. 3 .

 3 Fig. 3. Evolution of the probe output DC voltage according to the input signal power and frequency (in MHz)

Fig. 6 .

 6 Fig. 6. Experimenal set-up: characterization of the induced voltage across the terminal of a line under test during radiated immunity test The trace under test is placed on a non-conductive fixture at 1 m above the chamber ground. It is illuminated by a RF disturbance radiated by a log-periodic antenna placed at 1.3 m. The antenna is excited by a sine waveform signal produced by a RF signal synthesizer and amplified by a 50 W power amplifier. The test frequencies cover the range 30 -1000 MHz. Tests have been done with vertically polarized electric field, whose magnitude is about 20 V/m. The E-field intensity is controlled by an electric field probe (model EP600 from Narda). Preliminary tests have proved that the field around the line under test could be considered as homogeneous. The uncertainty on the incoming field intensity is about 10 %, except between 80 and 200 MHz where it reaches 25 % due to chamber resonances.

Fig. 7 .

 7 Fig. 7. Characterization of the sensitivity of the probe to the incoming radiated disturbance

Fig. 8 .

 8 Fig. 8. Comparison of the power coupled on the line under test measured with the three different measurement methods

Fig. 9 .

 9 Fig. 9. Equivalent model of the illuminated trace (Taylor model)

Fig. 10 .

 10 Fig. 10. Incident and reflected waves on a two-layer planar structure

Fig. 11 .

 11 Fig. 11. Comparison between measurement and simulation of the voltage coupled on the trace under test: orientation 1 (top), orientation 2 (bottom)

TABLE I

 I

	.	MAIN CHARACTERISTICS OF RF DETECTOR LT5534 [6]
		Bandwidth	50 -3000 MHz
		Dynamic range	-60 -0 dBm
		Typical sensitivity	41 mV/dB
		Temperature drift	0.025 dB/°c
		Power supply voltage	2.7 -5.25 V
		Current consumption	7 mA


Saliha Chetouani
email: saliha.chetouani@irt-saintexupery.com

Sébastien Serpaud

Alexandre Boyer
email: alexandre.boyer@insa-toulouse.fr

Sonia Ben Dhia

Fast and efficient approach to predict EMC immunity of complex equipment after a component change

Keywords: Obsolescence, Conducted immunity, Multiport, S-Parameters, Indirect measurement, De-embedding. I

This paper describes a fast methodology for managing the obsolescence issues of components in industrial equipment (aeronautical and/or automotive). The objective is to predict Electromagnetic Compatibility (EMC) non-compliance risk or guarantee a non-regression of EMC performances, in conducted immunity, after a component change. This could be achieved through an equivalent test at the component level if the residual voltage at its pins could be estimated. Accordingly, this approach aims to decline conducted immunity requirements, from the connector at the input of an equipment to an obsolete component placed on one of its boards. Which must constitute a faster and cheaper solution than systematic EMC qualification test of the equipment each time a component is replaced.

INTRODUCTION

A simple comparison of the commercial lifetime of electronic components (approximately 3 to 5 years) and that of electronic equipment and systems (15 to 40 years for an aircraft, for example) shows the inevitable problem of rapid obsolescence of components. To remain compliant with standards, any change of an obsolete component must lead the Original Equipment Manufacturers (OEM) to requalify their products to prove the non-regression of their EMC performances. The requalification usually requires timeconsuming tests, significant costs and delays. Currently, the standard EMC measurement method is the only authorized method to validate EMC of an equipment. This paper propose an approach based on the EMC measurement at component level to estimate the non-regression of equipment in the context of the component obsolescence.

In context of the immunity test, the proposed approach propose to decline the disturbance in the boundary of component. Since, the EMC risk assessment could be done by a conducted immunity test done on the component to replace. The type of disturbances and the applied levels should be representative of the component environment at the equipment level and how it is mounted. The Direct Power Injection (DPI) method or the Near-Field Scanning in Immunity (NFSi) measurement method could be used to test this constraint at the component level. This paper presents a methodology to evaluate the risk level of immunity regression due to a change of a component in a complex electronic equipment. Fig. 1. presents an example of typical aeronautic equipment considered in this study: e.g. four (or more) daughter boards mounted on one or two backplane boards, inside a metal enclosure and interconnected to harness through a wide connector. Different approaches have been deployed for managing electronic component obsolescence. A common approach proposes a complete and exact modeling of the equipment according to a bottom-up vision. It consists in the development of models at the component level, then at the electronic board level and finally at the equipment level, which are interconnected together [START_REF] Dhia | Modélisation l'immunité Électromagnétique des composants en vue de la gestion de l'obsolescence des systèmes et modules électroniques[END_REF] [2] [3] [START_REF] Ayed | Développement de méthodologies pour l'extraction et la construction des macromodèles d'immunité électromagnétique appliqués aux circuits intégrés[END_REF]. They are usually based on electromagnetic modeling 3D and Full-Wave simulations [START_REF] Durier | Using the EM simulation tools to predict EMC immunity behavior of an automotive electronic board after a component change[END_REF]. However, these approaches quickly find their limits because of the diversification, the complexity of the equipment and their EMC requirements. The long calculations time required to build models is also a main limitation. In addition, the exact information regarding each part of the equipment is not necessarily known because of confidentiality reasons. These drawbacks severely limit the use of these methods and could be, in some cases, not envisaged due to the cost and delays.

In this paper, an experimental approach to deal with this issue is presented. It consists in declining the external perturbations from the equipment level (a common mode disturbance injected into one or more equipment inputs) which arrive to the obsolete component (placed on a daughterboard). Then the residual level of disturbance applied to the component can be estimated, which will serve as a reference level for an immunity test on the new component. The purpose of this study is to predict, from some partial measurements at the input of the equipment, the level of risk due to a component change.

If non-linear effect could be neglected, an S-parameter based black box can efficiently model the filtering effect of the equipment. It build from measurements done between the input connector of the equipment and the pad of the component to be replaced. However, putting in practice such a measurement is not straightforward as the measurement ports at component side are not accessible easily. The originality of the proposed approach is to use an indirect Sparameter measurement method based on [START_REF] Maeda | Mathematics of 2r-Port S-Parameter Estimation by the r-port Measurements[END_REF] to build the black box model of equipment.

From this equipment model, it is possible to decline the normative constraint declined at the input of equipment to the input of component such as the common or differential voltage applied to the pins of the component to change.

Note the modeling of the test setup is not covered by the study. Here we assume a declined constraint at the input of equipment (connector) as a common mode generator.

After a general description of the proposed approach, an experimental case study dedicated to the validation of the method is presented.

II. GENERAL ASSUMPTIONS OF THE PROPOSED APPROACH

As described in the introduction, this study concerns the obsolescence of components of industrial equipment (e.g. aeronautical, automotive). However, to simplify the study, the following assumptions and choices are made: • The studied systems are assumed reciprocal and linear.

The non-linear behavior of the components (e.g. active circuits, protection structures, etc.) is neglected.

• The general ground of the system, which is usual in electronic equipment mounted in shielded enclosure, could be different from the local component ground (usually a local ground plane at PCB level).

• The conducted injection on the harness is of the common mode type.

• The disturbance induced by the injection system and driving on the harness is modeled as an equivalent source placed at the input connector of the equipment.

• The component terminals are not accessible to place an S-parameter measurement port, but the component can be removed and replaced by some passive loads.

The purpose of the proposed approach is to determine the influence of the equipment on the propagation of a conducted disturbance to an obsolete component, only from measurements made on the input of the equipment. It aims to estimate either the single-ended or the differential voltages applied to the different pins of the component to be replaced.

III. DESCRIPTION OF THE INDIRECT MEASUREMENT METHOD

S-parameters are a powerful formalism to characterize the electromagnetic behavior of passive electronic devices, especially in the case of very complex equipment and components. The state of art on S-parameters estimation of a 2-port device without directly measuring the two ports is quite rich. Reference [START_REF] Ou | Determine Two-Port S_Parameters from One-Port Measurements Using Calibration Substrate Standards[END_REF] presents a synthesis on some methods and proposes a simple and an interesting technique. It consists in determining the four S-parameters from three measurements done at the only accessible port, by connecting three loads on the second inaccessible port. The previous method can be extended to the case of an N-port device, formed by any number of ports. We provide in the following part a generalized technique, which was developed by [START_REF] Maeda | Mathematics of 2r-Port S-Parameter Estimation by the r-port Measurements[END_REF]. It is based on the same principle and allows estimating analytically the S-parameters of multiport device with hard-to-access ports, which models the filtering effect applied by the equipment on the conducted disturbance from the input connector to the obsolete component.

A. General description of the method

We consider a system with N ports (N = 2r), as illustrated in Fig. 2. The r ports on the left, denoted by the index a, are considered as the direct-measured ports (in the case of an equipment, the pins of the input connector). The other r ports on the right represent the indirect-measured ports, inaccessible by the measurement probe, denoted by the index u.

The matrix [Stot] of the multiport system could be written in terms of three submatrices Saa, Sau and Suu, defined in (1). As the system is supposed, the submatrix Sua is equal to the transpose of Sau (Sau T ). The matrix a = [aa T au T ] T contains the incident power waves and b = [ba T bu T ] T the reflected power waves.

=

(1)

where Saa, Sau, Suu ∈ C × and aa, ba, au, bu ∈ C × .

In order to estimate all the S-parameters, we connect a set of known loads (1-port) to the u ports. Their reflection coefficients constitute the diagonal of the load matrix noted [SL]. The matrix designates the measured S-parameters from the direct-measured ports. The equivalent source at the input of the multiport inject the incident waves aa, it follows that:

= (2)
Contrary to the waves aa, the incident waves au result from the reflection of bu on the loads, therefore:

= (3)
Substituting the terms in ( 2) and ( 3) to (1) yields to (4).

= + ( - ) (4)
where I is the identity matrix.

Equation ( 4) is the main formula connecting the matrix of the measured S-parameters to the three corresponding submatrices of the multiport. The method to extract the three submatrices are described in the following parts.

B. Estimation of Saa

There are two ways to estimate the Saa matrix. If perfectly matched loads can be placed at inaccessible ports (SL = 0), Saa is directly equal to the matrix of the measured S-parameters using this load-set according to (4). Equipment IC

1 Sl ( ) × St St St St St St St St St St St St St St St ⋯ ⋯ ⋯ St St St St St St ⋮ ⋮ ⋮ ⋱ ⋮ St St St St St St ⋯ St (11)
Equation ( 11) allows calculating the square of the different terms of the 1 st row of the matrix S

. By repeating the operation for each of the other ports (r+2, …., 2r), we obtain all the coefficients of S and then the matrix S as the inverse of the matrix S .

It should be noted that the square root operation implies two solutions of S (+ S and -S ). The choice of the sign does not affect the magnitude of the calculated parameters (Sau, Suu) and the estimated single-ended voltages. However, it involves a phase inversion. In addition, it influences the magnitude and the phase of the differential voltage. The ambiguity of the sign could be resolved by a solution proposed in [START_REF] Maeda | Indirect Measurement Method for S-Parameters with a Recursive Procedure[END_REF]. Which is not envisaged in our study, since it requires adding an extra known connection between the direct and the indirect ports. We have developed a new correction algorithm, explained in the next part, to solve this inverse problem of the sign of Sau matrix.

D. Sign ambiguity correction algorithm

In order to choose the right solution between the two mathematical solutions given by the previous calculation, we propose an analysis based on the physical characteristics of the studied systems. We supposed that the systems are mainly composed of transmission lines with some EMC filters. The transmission S-parameters between input and output ports have a negative phase. This is explained by the causal nature of the systems. For example, for a two-port network, the transmission matrix (chain matrix) can be expressed according to the characteristics of the transmission line (the characteristic impedance Zc, the propagation constant G, the length L, the propagation speed and the angular frequency ) as follows:

= ℎ( . ) . ℎ( . ) ℎ( . )/ ℎ( . ) (12)
The transmission parameter S12 can be expressed using the conversion relations between the ABCD matrix and the S matrix [START_REF] Reveyrand | Multiport conversions between S, Z, Y, h, ABCD and T parameters[END_REF] .

= 1 exp ( . ) = exp (-. )

where = + . (α is the attenuation constant ≥ 0 and = / is the phase constant ≥ 0). Therefore:

ℎ ( ) = -β. = -ωL ν (14)
Hence, the phase of the transmission parameter S12 is negative whatever the frequency. This demonstration could be generalized to an N-port network.

The correction algorithm contains therefore two stages:

-First, in low frequencies, we calculate the phases of the two Sau values and we choose the value whose phase is negative. The signs of the other parameters (extra diagonal parameters) are determined from the signs of these transmission parameters (diagonal parameters).

-Then, we start with a frequency point, corrected in the first step, and we calculate the difference (in absolute value) between the phases of two successive points. The value of this difference must be between two thresholds (low threshold 60° and high threshold 340°). Otherwise, we should reverse the sign of the second point.

E. Estimation of Suu

The submatrix Suu can be calculated directly be replacing Saa and Sau obtained previously in (15). A reflection coefficient (SL (i) , for example) can be used with its corresponding measurement (S ( ) ).

= - ( - ) (15)
F. Estimation of S-parameters of the component to replace Finally, the estimation of these three sub-matrices allows creating the matrix [Stot], which represents the influence of the equipment on the propagation of the conducted disturbance to an obsolete component. Once determined, the S-parameters of the component, noted as SIC, can be easily estimated. In (15) SL is replaced by SIC, therefore leading to (16).

= ( + ( - ) ) ( 16
)
One measurement is conducted with connecting the obsolete component to obtain , the measurement Sparameters when it is connected. The calculated matrix is a diagonal matrix corresponding to the S-parameters associated with each port of the component. The matrix multiplication in (16) removes the sign ambiguities of .

IV. ESTIMATION OF THE APPLIED VOLTAGE ON THE TERMINALS OF THE FUTURE COMPONENT

This section explains how to calculate the residual voltage at the components pins from the S-parameter matrix [Stot] estimated previously. A differential voltage, expressed as the difference of the residual voltages applied on two pins, must be computed.

Let j be a port of a multiport system, as presented in Fig. 2. The port voltage Vj is the sum of the incident and the reflected voltage noted by Vji and Vjr respectively. The incident and reflected power (aj and bj) related to the port j can be expressed by the incident and the reflected voltage as well:

= √ and = √ ( 17
)
where Z0 is the port impedance.

The single-ended port voltage Vj can be expressed according to (18).

= + = ( + ) (18)
Without any loss of generality, a 4-port device is considered, with two direct ports and two indirect ports. Let Vp be the internal voltage source of a simultaneous injection of a common source noise on the direct ports, and Sp the internal reflection coefficient as shown in Fig. 3. The incident and reflected powers applied on the different ports are related by (19) [START_REF] Maeda | An Indirect Measurement Method for Multiport S-Parameters with Reduced Number of Measurements[END_REF]. The proposed case study, shown in Fig. 4, is a 4-port board containing two coupled microstrip lines routed on a FR4 substrate 4-layers PCB. This case study represents a simple equipment to illustrate easily the proposed approach. The studied component is represented by two passive loads (560 Ω and 100 Ω) soldered on SMA connectors and placed on ports 3 and 4 respectively (Fig. 5).

Fig. 1 .

 1 Fig. 1. Propagation of a conducted disturbance to an obsolete component; Example of an aeronautical equipment and its boards (top)

Fig. 2 .

 2 Fig. 2. Schematic illustration of the studied system with N = 2r ports

  rewritten in a more compact way:= +Thus, the quantities a and b can be expressed in terms of the estimated S-matrix [Stot] as follows[START_REF] Maeda | An Indirect Measurement Method for Multiport S-Parameters with Reduced Number of Measurements[END_REF]: combining (20) and (21) with (18), the residual differential voltage between the indirect ports can be calculated according to (22).

Fig. 3 .

 3 Fig. 3. Example of a common source disturbance injected on the direct ports V. CASE STUDY

Fig. 4 .

 4 Fig. 4. Case study description (PCB board)

ACKNOWLEDGMENT

This study is supported by the IRT Saint-Exupery foundation (a French Institute of Technology) under the FELINE project. The authors would like to thank IRT, LAAS and all the project partners (Safran, Airbus, Continental, Liebherr, Actia, Thales, etc.) for funding and supporting this work. Also, special thanks to André Durier for providing the printed circuit board used in the experimental validation.

If it is complicated to obtain suitable matched loads (especially for very high frequencies), Saa can be estimated using three load-sets of arbitrary value as follow: we conduct two measurements (i, j) with two different known-load sets. We note by SL (i) and SL (j) the reflection coefficients of loads used in the measurements ( ) and ( ) respectively. By reversing the both sides of (4) and taking the differences of equations for the two measurements, (5) can be derived.

Expanding [START_REF] Durier | Using the EM simulation tools to predict EMC immunity behavior of an automotive electronic board after a component change[END_REF] and reversing both sides of the equation leads to [START_REF] Maeda | Mathematics of 2r-Port S-Parameter Estimation by the r-port Measurements[END_REF].

A third measurement ( ) is performed. By replacing j in ( 6) by k, ( 7) can be written.

If the three loads are selected such as, their reflections coefficients are related to a non-zero constant k1 defined by ( 8), ( 6) and ( 7) can be combined to extract the submatrix Saa according to [START_REF] Maeda | An Indirect Measurement Method for S-Parameters which is based on Reduction to Eigenvalue Problem[END_REF].

where:

C. Estimation of Sau

Estimating the Sau matrix is the most complex part. The authors of this technique have proposed different methods [START_REF] Maeda | An Indirect Measurement Method for Multiport S-Parameters with Reduced Number of Measurements[END_REF] [9] [START_REF] Maeda | Indirect Measurement Method for S-Parameters with a Recursive Procedure[END_REF]. We present in this part, the easiest one to use and we mention the references of the others, which are more mathematically complicated.

Replacing Saa expression in [START_REF] Maeda | Mathematics of 2r-Port S-Parameter Estimation by the r-port Measurements[END_REF] with its formula defined by (9) leads to a quadratic equation in terms of Sau. We defined, in the previous part, the loads matrix in terms of the reflection coefficients, such as:

-SL (i) = diag [sl (i) , …...., sl (i) ], for the 1 st measurement; -SL (j) = diag [sl (j) , …..., sl (j) ], for the 2 nd measurement; -SL (k) = diag [sl (k) , ….., sl (k) ], for the 3 rd measurement;

To estimate Sau we conduct additional measurements noted by ( ) , using a set of loads consisting of the first and the second loads as follows:

-SL (f) = diag [sl (j) , sl (i) , ..., sl (i) ], for the 4 rd measurement;

The indirect-measured ports, from r+2 to 2r, must be terminated by the load used in the first measurement. Only the port r+1 must be terminated by the load used in the second measurement.

Replacing the terms SL (k) in [START_REF] Ou | Determine Two-Port S_Parameters from One-Port Measurements Using Calibration Substrate Standards[END_REF] by SL (f) gives:

Denoting the left hand side of [START_REF] Maeda | Indirect Measurement Method for S-Parameters with a Recursive Procedure[END_REF] as M and the elements of the matrix S as , the above equation becomes: The objective is to estimate, only from the S-parameter measured at the direct ports (port 1 [a1] and port 2 [a2]) the residual voltage at the component pins (port 3 [u1] and port 4 [u2]). The S-parameter matrix of the equipment can be developed as shown in Fig. 6. The three submatrices can be estimated from III. The results will be compared to a direct 4port measurement at the Vector Network Analyzer (VNA) from 100 kHz to 3.5 GHz. We present in Fig. 7. and Fig. 8. an example of some parameters measured directly and those estimated indirectly (using the indirect method). Next, a simultaneous injection of a 1 V common source noise is injected on the direct ports (1 and 2). The residual singleended voltage on each pins of the component is calculated, as presented in Fig. 9. Then, the differential voltage induced on the component is computed, as shown in Fig. 10.

The direct voltages are calculated, by (18) to (21), using the S-parameters of the DUT measured directly using the 4port direct measurement. The estimated single-ended voltages present good agreement up to 3 GHz. The difference observed in high frequencies is due to the variations in the characteristics of the charges used on the indirect ports. In addition, the differential voltage is well estimated with good precision up to 3 GHz.

VI. CONCLUSION

This paper proposes to use an indirect measurement method for S-parameters to deal with the obsolescence of components in complex electronic equipment. The method consists, first, in elaborating the equivalent "black box" model of the transfer function between the input connector of an equipment and an obsolete component mounted in one of its boards. Then, this multiport S-parameter model is used to estimate the residual voltages induced on pins of the component to replace.

We have developed a new sign correction algorithm which corrects the phase inversion of the all the parameters. Therefore, it allows giving the exact value of the magnitude and the phase of the estimated differential voltage.

An application of this method on a PCB board was presented, compared to the results of a full measurement using the VNA. The results are good from 100 kHz to 3 GHz. Other studies are in progress in order to apply this approach on a real integrated circuits in complex equipment. Finally, this method could quickly bring satisfactory results when faced with the problem of the obsolescence of components in complex electronic equipment.

Direct Method Indirect Method (voltages)

Direct Method Indirect Method (Diff. voltage)