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Worst-Case Induced Disturbances in Digital and Analog Interchip Interconnects by an External Electromagnetic Plane Wave—Part I: Modeling and Algorithm Jorge L. Lagos, Student Member, IEEE, and Franco Fiori, Member, IEEE

Abstract—This paper deals with the susceptibility of electronic units to radiated electromagnetic (EM) interference and specifically, it focuses on the parasitic coupling of EM fields with printed circuit board interconnects like microstrip lines. To this purpose, a uniform lossless microstrip line illuminated by an EM plane wave is considered and the voltages at the line terminations are evaluated referring to the Baum–Liu–Tesche equations. Based on this, a new algorithm to identify, frequency-by-frequency, the incidence angles and the polarization of the impinging field that gives rise to the maximum induced voltages at the line terminations is presented. Index Terms—Electromagnetic interference (EMI), interconnects, microstrip line, printed circuit board (PCB), susceptibility.

I. INTRODUCTION N the past decade, the widespread use of wired and wireless electronic systems in almost all fields of human life has brought about even higher levels of electromagnetic (EM) pollution. As a consequence, modern electronic systems in automotive, avionic, industrial applications, and, in general, any electronic system related to safety, should operate properly even if its nominal signals are corrupted by disturbances that wiring interconnects collect from the surrounding environment. Cable harnesses are effective antennas and the interference they collect could propagate into electronic modules if proper electromagnetic interference (EMI) filters at the connector level are not included. For this reason, the susceptibility of wiring harnesses to EMI has been widely investigated [1]–[3] and several EMI filters for both common-mode and differential-mode interference have been developed over the past decades [4]. However, EM fields can also couple at the module level with wiring interconnects, like the metal traces of printed circuit boards (PCBs), connector leadframes, and flexible buses. Such unwanted coupling becomes effective at higher frequencies and is of particular concern whenever metal shields are not effec-

I

Manuscript received October 7, 2009; revised May 15, 2010; accepted September 22, 2010. Date of publication November 11, 2010; date of current version February 16, 2011. The authors are with the Electronics Department, Politecnico di Torino, 10129 Torino, Italy (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2010.2085005

tive or not employed. In these cases, nominal signals of such electronic modules and, in particular, those with low swing can be corrupted by radiated EMI. As a result, the nominal operation of high-sensitivity circuits like analog-to-digital converters, high-gain amplifiers, and filters (including switched-capacitor circuits), as well as that of low-voltage digital circuits, can be affected by such disturbances, leading to transient or permanent operation failures. Furthermore, radiated EM fields also couple with the wiring interconnects (leadframe and bonding wires) of integrated circuits, but induced voltages are usually negligible as long as package interconnects are significantly shorter than the interference minimum wavelength. Given that the maximum size of mass-production IC packages is usually smaller than a few centimeters, the disturbance collected by the package interconnects can be neglected up to several gigahertz, hence, in what follows, the direct coupling of external EM fields to IC interconnects (at package and chip level) will be neglected. In this paper, the magnitude of disturbances affecting the nominal signals of analog and digital interconnects is evaluated referring to a uniform, lossless microstrip line, which is illuminated by an impinging EM plane wave. The EM field to microstrip line coupling has been evaluated referring to the Baum–Liu–Tesche (BLT) equations [1], [5] and, more precisely, to the model proposed by Leone and Singer [6]. Referring to this last model, it immediately follows that the magnitude of EMI-induced disturbances depends on several parameters like the field incidence, polarization angles, and the magnitude and phase of the impedances loading the microstrip terminations. Based on these, and aiming to quickly evaluate the maximum voltages at the terminations of a given microstrip line illuminated by an EM plane wave of a given magnitude and frequency, this paper proposes a new algorithm, which allows for identifying the angles of maximum coupling (incidence and polarization), hence the maximum induced voltages at the interconnect terminations, for a set of frequencies falling in a given range. The paper is organized as follows. Section II summarizes the model that has been considered to evaluate the voltages at the microstrip line terminations by an impinging EM plane wave of given incidence angles and polarization. Section III presents a new algorithm to identify the angles, which maximize the voltages at the line terminations. In Section IV, the accuracy and computation time of the proposed algorithm are evaluated and compared with those of the direct calculation of the voltages

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Fig. 1.

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Microstrip cross section.

induced at the microstrip line terminations. Finally, Section V draws some concluding remarks. Fig. 2.

II. COUPLING OF AN EM PLANE WAVE WITH A MICROSTRIP LINE Common electronic modules are made of PCBs that include ICs like microprocessors as well as passive components connected among them by means of metal traces of various length and shape. In general, the prediction of the voltages induced at the terminations of module-level interconnects by an external EM field can be obtained by means of cumbersome EM simulations, which do not provide direct insight on the impact of design parameters, e.g., trace length, trace width, and termination impedances on the circuit susceptibility to radiated EMI. Furthermore, the prediction of common EM simulators is obtained referring to a specific EM excitation and is usually very time consuming. On the basis of these considerations, and aiming at the evaluation of the maximum magnitude of the disturbances affecting the nominal signals of analog and digital interconnects, a microstrip line illuminated by an EM plane wave is considered and the voltages at the microstrip line terminations are evaluated referring to the model described in [6].

EM plane wave impinging on a microstrip line.

B. EM Plane Wave to Microstrip Line Coupling The voltages VNE and VFE that are induced by an incident EM field at the near-end (NE) and far-end (FE) terminations of a microstrip line can be expressed in the frequency domain referring to the BLT equations [1], i.e., VNE =

(1 + ρNE )(ρFE S1 + ej β L S2 ) ej 2β L − ρNE ρFE

(2)

VFE =

(1 + ρFE )(ej β L S1 + ρNE S2 ) ej 2β L − ρNE ρFE

(3)

where ρNE and ρFE are the reflection coefficients of the impedances loading the NE and FE ports, and L is the trace length. Such coefficients are defined as follows: ρNE,FE =

ZNE,FE − ZC . ZNE,FE + ZC

(4)

(1)

The microstrip line propagation constant is given by √ (5) β = k0 r,eff √ where k0 = 2πf 0 μ0 is the free-space wavenumber, while f is the frequency. In these expressions, the factors S1 and S2 model the external EM excitation of the microstrip line and referring to the Agrawal method [1], they can be expressed as follows:  1 (6) v0 + vx+ − vL ej β L S1 = 2   1  S2 = − v0 + vx− ej β L + vL (7) 2 where  0 v0, L = Ez (xNE,FE , z)dz, xNE,FE = {0, L} (8) −h and  L Ex (x)e±j β x dx (9) vx± =

higher order modes start being supported and the conventional TL model is not valid anymore. For common PCB traces, fg ,stat ranges from 1 to 10 GHz [7]. In the remainder of the paper, a uniform lossless microstrip line is considered and it is assumed that the impinging EM plane wave illuminating the microstrip line has a frequency lower fg ,stat .

in which the components Ex and Ez of the EM field close to the microstrip line derive from the specific excitation considered. In the remainder of the paper, the case of an arbitrary linearly polarized plane wave propagating toward the microstrip line from the upper half-space with angles of incidence θ and φ is considered (see Fig. 2). The angle γ defines the field polarization [6].

A. Microstrip Line In this paper, a uniform microstrip line of width w, trace thickness t, dielectric thickness h, relative dielectric constant r , with a cross section like that sketched in Fig. 1 is considered. In the lower frequency range, this structure supports only a quasi-TEM mode [7], so that signal propagation is well predicted by solving the telegraphers’ equations. As a consequence, it is common to refer to per-unit-length parameters or to the characteristic impedance ZC and to the effective relative permittivity r,eff . These transmission-line (TL) parameters can be evaluated using more-or-less accurate closed-form expressions that depend on the relative permittivity r , and the w/h and t/h ratios. However, for frequencies greater than fg ,stat ≈

2.13 × 107 √ (w + 2h) r + 1

0

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Based on the aforementioned, (8) and (9) can be written as follows: 1 − e−j k 2 z 2h −j k x x N E , F E v0, L = E0 fz (θ, γ) , e jk2 z xNE,FE = {0, L} vx± = E0 fx (θ, φ, γ)

(10)

ej (±β −k x )L − 1 j(±β − kx )

(11)

Vp ort,m ax = −fmincon(−Vp ort (f, X, C), DX , X0).

where E0 is the incident field magnitude, while kx = k0 sin θ cos φ  k2 z = k0 r − sin2 θ fz (θ, γ)  and

cos γ sin θ r

(12) (13) (14)

2 k2, z cos φ cos γ + sin φ sin γ cos θ . fx (θ, φ, γ)  j2hk0 k02 r (15) The expressions (14) and (15) are valid under the hypothesis that the microstrip line has a substrate thickness smaller than the substrate wavelength, so that 2hk2, z  1. This model, which has been validated in [8], is now employed to calculate the maximum voltages at the microstrip line terminations. 

III. ALGORITHM FOR THE EVALUATION OF WORST-CASE EM-INDUCED VOLTAGES The model shown in the previous section has highlighted that the EM-induced voltages at the NE and FE ports of a microstrip line depend on several parameters like the angles of incidence of the impinging field (θ, φ), the angle of field polarization (γ), and the reflection coefficients of the NE and FE loading impedances (ρNE , ρFE ). Based on these and aiming at the evaluation of the maximum EM-induced voltages at the microstrip NE and FE ports, also when frequency-dependent termination impedances are considered, this section shows an algorithm, which allows for identifying the worst-case illumination, i.e., the set of angles, which maximizes, for a given frequency range, the coupling of the EM plane wave with a given microstrip line. A. Optimization Problem To the aforementioned purposes, (2) and (3) can be expressed as follows: Vp ort = Vp ort (f, X, C),

port = {NE, FE}

(16)



where X = (γ, φ, θ) is a vector grouping the illumination angles and C is a vector grouping all the remaining conditions of the problem, i.e., termination impedances, TL parameters, and incident field amplitude C = C(ZNE , ZFE , L, h, w, r , E0 ).

can be obtained by solving the optimization problem of maximizing the nonlinear function Vp ort with respect to the vector X. To this purpose, a commercial optimization tool based on the sequential quadratic programming method is used. Specifically, the MATLAB function fmincon() is employed, which allows the constrained minimization of the objective function −Vp ort (f, X, C) given a starting point X0 and the domain DX in which X is bounded

(17)

Thus, given the characteristics C of the problem, the worst-case induced voltages at the ports of the TL at a given frequency

(18)

Notice that since fmincon() performs constrained minimization, the negative of the induced voltage magnitude is passed to it as the objective function for achieving maximization. A description of the optimization algorithm employed by this function is found in [9]. It must be noted that since, in general, the function Vp ort (f, X, C) has several local maxima, the direct application of (18) does not guarantee the convergence to the global maximum. In fact, the application of (18) for the same conditions C and restrictions DX might converge to different local maxima for different values of the starting point X0 passed to fmincon(). Therefore, a special procedure is followed for applying (18) in order to determine the worst-case induced voltages at microstrip line terminations. B. Algorithm In order to find the overall maxima of Vp ort in a given frequency range, the iterative procedure illustrated in Fig. 3 is used (recall port = {NE, FE} denotes either ports of the TL). The process comprises three phases, namely: a local optimization loop, a merging phase, and a verification loop, which are executed in that order at every iteration. The final output of the procedure is the vector V p ort,worst with the absolute worst-case induced voltages in the desired frequency range. The operation of the procedure is as follows: 1) The process begins by reading the problem conditions C, a completely arbitrary initial optimization point X01 , and the vector f covering the desired frequency range. It then enters the local optimization loop. 2) In the local optimization loop, at each iteration, the fmincon() function is applied for each frequency in f according to (18), using the current starting point X0i for the optimization. This procedure results in the output vector V p ort,worst, i , and the procedure continues to the merging phase. 3) As the name implies, in the merging phase, the vector V p ort,worst, i is merged with all the previously obtained vectors V p ort,worst,m (1 ≤ m < i) to form an updated version of the absolute worst-case induced voltage vector V p ort,worst . At each frequency, the entries in V p ort,worst are updated by taking the maximum of all the V p ort,worst,m vectors obtained so far. 4) In the verification loop, the procedure checks whether the updated vector V p ort,worst comprises the global maxima. Specifically, the objective function Vp ort is evaluated at arbitrary illumination points X arb,k swept from a subset {X init , . . . , X final } of the entire illumination domain DX .

LAGOS AND FIORI: WORST-CASE INDUCED DISTURBANCES IN DIGITAL AND ANALOG INTERCHIP INTERCONNECTS

Fig. 3.

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Flowchart of the procedure employed for determining the worst-case induced voltages.

For each arbitrary illumination point X arb,k , the difference Vdiff between the resulting induced voltage vector V p ort,arb,k and the current V p ort,worst is calculated at every frequency f . If for all the frequencies in f , this difference is negative, the loop continues to process the next X arb,k . If, however, the difference is positive at any frequency (indicating that the merging process has not yet provided the global maximum at these frequencies), the loop determines the maximum value of this difference Vdiff ,m ax,k and the frequency fdiff ,m ax,k at which it occurs, and then, continues to process the next X arb,k . After having processed all the X arb,k , the values of the differences Vdiff ,m ax,k are analyzed. In case they are all null, meaning that for all frequencies in f , the points in V p ort,worst yielded values greater than those of the arbitrary illumination cases, the V p ort,worst vector is considered to carry the global maxima of Vp ort and the iterative process is finished. On the contrary, if this condition is not met, the arbitrary illumination case X arb,k for which the difference

Vdiff ,m ax,k was maximum is chosen as the new initial point for optimization X0i , and the process returns to the local optimization loop for a subsequent iteration. It should be noticed that a critical factor for the correct operation of the described procedure is the selection of the verification subset {X init , . . . , X final }, which should include a number of points sufficiently large and appropriately spaced in order to be a representative sampling of the entire illumination space DX . This, in turn, might have to be traded off with CPU computation time. For the case of the microstrip structure with infinite ground plane, the illumination domain is as follows: DX = Dγ × Dφ × Dθ = [0, 360◦ ] × [0, 360◦ ] × [0, 90◦ ]. (19) In order to illustrate the aforementioned procedure, a microstrip line with the parameters listed in Table I, which has a characteristic impedance ZC ≈ 120Ω, is considered. The worst-case coupling at the FE port of this structure under a unitary-magnitude incident field (E0 = 1V/m) is analyzed, for the case of the

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TABLE 1 MICROSTRIP PARAMETERS FOR ALL THE NUMERICAL EXAMPLES PRESENTED

resistive terminations RNE = 40 Ω and RFE = 360Ω. An implementation of the flowchart in Fig. 3 that covers the [10 MHz, 6 GHz] frequency range at 10 MHz steps is employed. In this implementation, the points X arb,k of the verification subset are formed by sampling the illumination space DX at 11 × 13 × 13 = 1859 uniformly spaced points. The evolution of the iterative process as applied to this example is further detailed in Table II, where the output of the optimization loop is reported at two particular frequencies f1 = 3.51GHz and f2 = 4.0 GHz, along with the results of the verification loop. The magnitude of the FE-coupled voltage function VFE (X, f ) at these two frequencies is also plotted in Fig. 4(b) and (c), respectively, where the voltages at the starting illumination points X0i used for the optimization process and those at the resulting worst-case illumination directions X worst,i are also shown. As reported in Table II, for this particular case, the process performs six iterations before all the global maxima of the VFE (X, f ) function are determined at all the computation frequencies. Notice also that the starting illumination point X0i used for the optimization loop at each iteration (after the first one) is taken as the arbitrary illumination point X arb,k for which the difference VFE (X arb,k , f ) − VFE,worst (f ) was maximum in the preceding iteration. With respect to the optimization process, it can be noted that the worst-case coupled voltages at f1 and f2 are attained in the fifth and third iterations, respectively, for all the other X0i , the optimization loop converges to points not corresponding to the global maxima of the VFE (X, f ) function, as confirmed by Fig. 4(a) and (b). With respect to the verification process, it is observed in Table II that the value of the maximum difference VFE (X arb,k , f ) − VFE,worst (f ) decreases after each iteration, until it finally reaches zero, meaning that the iterative process has succeeded in determining all the worstcase coupling directions and their associated voltages at all the computation frequencies. IV. VALIDATION In order to validate the efficacy of the proposed algorithm, a comparative analysis has been carried out against bruteforce calculations, i.e., results obtained by direct calculation of the coupled voltage VFE by sampling the illumination domain DX at large number of points and taking VFE,m ax as the maximum of the values found. To this purpose, the same TL structure analyzed in the previous section has been con-

sidered (L = 10 cm, w = 0.22mm, h = 0.8 mm,  = 4.2, and ZC ≈ 120 Ω) with the NE port loaded by RNE = 40 Ω and the FE port by a RFE = 360 Ω. Analyses have been carried out for frequencies in the range [10 MHz, 6 GHz] and frequency step Fstep = 10 MHz. The performance of both the brute-force method and the proposed algorithm for estimating the worst-case coupled voltage can be analyzed in terms of the number N of points X i used to sample the illumination domain DX . For the brute-force method, these points are directly used to estimate the worst-case coupled voltage as VFE,m ax = max(VFE (X i )), while in the proposed algorithm, they constitute the set {X init , . . . , X final } of points used in the verification loop. Considering a uniform sampling of DX , N will be a direct function of the angular step δ used to perform the sampling. Thus, a comparative analysis of the brute-force method and the proposed algorithm can be carried out by embedding each of these procedures in an iterative scheme, where the procedure for estimating VFE is run, starting from a rough angular step δ = δ0 , and then, the value for δ is decreased at each iteration (effectively increasing the number of points in DX ) until the relative improvement in VFE over the previous iteration is below a certain threshold. Since for the illumination problem DX = Dγ × Dφ × Dθ = [0, 360◦ ] × [0, 180◦ ] × [0, 90◦ ] (note the original Dφ = [0, 360◦ ] can be halved due to symmetry), the value for the initial rough angular step is set to δ0 = 90◦ . Also, with the purpose of simplifying the analysis and speedingup the convergence of the iterative schemes, the angular step is decreased by halving its value at each iteration (δi = δi−1 /2). Finally, in order to provide a measure of the relative error incurred in the estimation of VFE at each iteration, the output of both procedures is compared against a reference set of worstcase values, denoted VF E ,r ef , which are obtained by sampling DX at a small angular step of δ = 1◦ and applying the bruteforce method. The results to be presented in the following show that this choice for δ produces a sampling of DX with N large enough (N = 5 946 031 points), such that the direct calculation of the maximum coupled voltage values over the sampled set of points provides a good approximation for the asymptotic limit of VFE at each frequency. The processing time needed for calculating these reference values corresponds to execution on a 2.40 GHz Intel Core 2 Duo processor with 2 GB of RAM. In order to quantify the evolution of the brute-force and proposed algorithm iterative schemes, three metrics have been taken into consideration, namely the maximum relative increment over the last iteration (RILI), the maximum relative error against the reference values (REAR), and the processing time. At the ith iteration, the first two are defined as follows:

 VFE,i (fj ) − VFE,i−1 (fj )  RILIi = max j VFE,i−1 (fj )

 VFE,i (fj ) − VFE,ref (fj )  REARi = max . j VFE,ref (fj ) Table III shows the performance of the proposed method and the brute-force method in terms of computation time RILI and REAR, which have been obtained referring to the microstrip

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Fig. 4. 3-D plots illustrating the voltage that is induced at the FE termination of a resistively loaded microstrip line by an impinging EM plane wave versus the illumination direction (θ, φ) at (a) f1 = 850 MHz and (b) f2 = 4.1 GHz (R N E = 40Ω, R F E = 360 Ω, and microstrip parameters as in Table I). The radial and azimuthal angles vary within θ ∈ [0, 90]◦ and φ ∈ [0, 360]◦ , respectively, while the coupled voltage is reported on the vertical axis in millivolt (vertical polarization is considered in both cases). The voltage at the starting illumination points X 0i used for the optimization process (black squares) and at the resulting worst-case illumination points X w o rst, i (white circles) are also indicated (see Table II).

and terminations mentioned as earlier. These results have been obtained by iteratively executing both the brute-force method and the proposed algorithm until the maximum relative improvement between iterations was less than 1%. Also reported in this table are the partial and total processing times employed by each method, along with the relative time saving of the proposed algorithm with respect to the brute-force method. As can be seen from Table III, the iterative execution of both approaches for diminishing δ results in correspondingly diminishing RILI and REAR values, meaning that the progressive estimations ob-

tained for VFE converge toward the reference values VF E ,ref at each iteration. However, the two algorithms attain convergence for different δ values. The brute-force method stops after seven iterations (δ ≈ 1.41◦ and N = 2 154 945 points), while the proposed algorithm only after three (δ ≈ 22.5◦ and N = 765 points). For the RILI threshold taken into consideration (1%), the proposed algorithm achieves a time saving of 94.67% with respect to the brute-force method. The same analysis has been repeated with different termination obtaining similar results.

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TABLE II EVOLUTION OF THE PROCEDURE IN FIG. 3 APPLIED TO A RESISTIVELY LOADED MICROSTRIP LINE (R N E = 40 Ω AND R F E = 360 Ω) AT f1 = 850 MHz AND f2 = 4.1 GHz (PARAMETERS AS IN TABLE I; VOLTAGES IN mV, FREQUENCIES IN GHz, AND ANGLES IN DEGREE)

TABLE III ITERATIVE ALGORITHM EVOLUTION FOR THE MICROSTRIP LINE OF LENGTH L = 10 CM, Z C = 120 Ω, R N E = 40 Ω, R F E = 360 Ω, AND RILI THRESHOLD: 1.0%

V. CONCLUSION In this paper, the BLT equations, which allows calculating the voltage induced at the terminations of a microstrip line by an impinging EM plane wave, have been used to find the field incident angles and polarization angles that maximize the voltage induced at the line terminations. The effectiveness of the proposed algorithm in terms of computation time and accuracy has been proved performing several computer analysis. The algorithm proposed in this paper is used in the companion paper [10] to evaluate the influence of the interconnects design parameters and termination impedances of the worst-case induced voltages.

[8] M. Leone, “Radiated susceptibility on the printed-circuit-board level: simulation and measurement,” IEEE Trans. Electromagn. Compat., vol. 47, no. 3, pp. 471–478, Nov. 2005. [9] Optimization Toolbox User’s Guide, The MathWorks Inc., Massachusetts, 8th. ed., 2007. [10] J. L. Lagos and F. Fiori, “Worst-case induced disturbances in microstrip inter-chip interconnects by an external electromagnetic plane wave—Part II: Analysis and validation,” IEEE Trans. Electromagn. Compat., DOI: 10.1109/TEMC.2010.2089566, to be published..

Jorge L. Lagos, photograph and biography not available at the time of publication.

REFERENCES [1] F. Tesche, M. Ianoz, and K. Torbjorn, EMC Analysis Methods and Computational Models, New York: Springer-Verlag, 1997. [2] C. R. Paul, “Frequency response of multiconductor transmission lines illuminated by an electromagnetic field,” IEEE Trans. Electromagn. Compat., vol. EMC-18, no. 4, pp. 183–190, Nov. 1976. [3] F. Rachidi, “Formulation of field to transmission line coupling equations in terms of magnetic excitation field,” IEEE Trans. Electromagn. Compat., vol. 35, no. 3, pp. 404–407, Aug. 1993. [4] H. W. Ott, Noise Reduction Techniques in Electronic Systems, 2nd ed. New York: Wiley, 1988. [5] A. K. Agrawal, H. J. Prince, and S. H. Gurbaxani, “Transient responce of multiconductor transmission lines excited by a nonuniform electromagnetic field,” in Proc. IEEE Trans. Electromagn. Compat., Feb., 1980, vol. 22, pp. 119–129. [6] M. Leone and H. L. Singer, “On the coupling of an external electromagnetic field to a printed circuit board trace,” IEEE Trans. Electromagn. Compat., vol. 41, no. 4, pp. 418–424, Nov. 1999. [7] R. K. Hoffmann, Handbook of Microwave Integrated Circuits. Norwood, MA: Artec House, 1987.

Franco Fiori (M’02) received the Laurea and Ph.D. degrees in electronic engineering from the Polytechnic University of Torino, Turin, Italy, in 1993 and 1997, respectively. From 1997 to 1998, he was a Leader of the Electromagnetic Compatibility (EMC) Group, R&D Division, STMicroelectronics. In 1999, he joined the Electronics Department, Polytechnic University of Torino, where he is currently an Associate Professor of electronics. Since 2006, he has been a Scientific Leader and a Researcher of the Istituto Superiore Mario Boella, an advanced research institute on telecommunications based in Torino, Italy. He has authored or coauthored more than 90 papers published in international journals and conference proceedings. His research interests include analog circuit analysis and design, nonlinear circuits, device modeling, and electromagnetic compatibility.