Adrien Guérin

Olivier Devauchelle
email: devauchelle@ipgp.fr

Vincent Robert

Thierry Kitou

Céline Dessert

Amélie Quiquerez

Pascal Allemand

Éric Lajeunesse

Amélie Quiquerez

Eric Lajeunesse

Stream-Discharge Surges Generated by Groundwater Flow

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The typical hydrograph of a river draining a small catchment (i.e. the time series of its discharge) increases steeply during rainfall, and declines slowly afterward, as groundwater reservoirs empty into the drainage system [START_REF] Sefton | Dynamic response characteristics of the plynlimon catchments and preliminary analysis of relationships to physical descriptors[END_REF]. Catchments thus shape their response to rainfall by storing water, and then releasing it into the network of streams that drains them [START_REF] Harman | A similarity framework to assess controls on shallow subsurface flow dynamics in hillslopes[END_REF][START_REF] Kirchner | Catchments as simple dynamical systems: Catchment characterization, rainfall-runoff modeling, and doing hydrology backward[END_REF]. Understanding this process is a formidable task: before it reaches a stream, rainwater infiltrates into the vadose zone [START_REF] Maher | Vadose zone infiltration rate at hanford, washington, inferred from sr isotope measurements[END_REF], is absorbed by the roots of trees [START_REF] Mares | Examining diel patterns of soil and xylem moisture using electrical resistivity imaging[END_REF], and eventually joins the groundwater zone, where it flows through heterogeneous and fractured rocks [START_REF] Berkowitz | Characterizing flow and transport in fractured geological media: A review[END_REF][START_REF] De Marsily | Dealing with spatial heterogeneity[END_REF][START_REF] Goderniaux | Partitioning a regional groundwater flow system into shallow local and deep regional flow compartments[END_REF]. There is little doubt that it is groundwater that sustains the recession limb of a hydrograph after a rainfall event [START_REF] Brutsaert | Regionalized drought flow hydrographs from a mature glaciated plateau[END_REF]. During the event itself, however, the significance of its contribution to the rising limb of the hydrograph remains debated [START_REF] Kirchner | A double paradox in catchment hydrology and geochemistry[END_REF]. Most streams remain chemically close to groundwater even at their peak discharge, indicating that groundwater can respond quickly to rainfall [START_REF] Jasechko | Substantial proportion of global streamflow less than three months old[END_REF][START_REF] Mcdonnell | Where does water go when it rains?: Moving beyond the variable source area concept of rainfall-runoff response[END_REF][START_REF] Neal | Chemical studies of chloride and stable oxygen isotopes in two conifer afforested and moorland sites in the british uplands[END_REF]; M. [START_REF] Sklash | Environmental isotope studies of storm and snowmelt runoff generation[END_REF]. For this to happen, the vadose zone must promptly transmit the rainfall signal to the water table, but the mechanisms by which it does so remain controversial [START_REF] Mcdonnell | How old is streamwater? open questions in catchment transit time conceptualization, modelling and analysis[END_REF]. Likely candidates include water flowing through non-capillary cracks [START_REF] Beven | Macropores and water flow in soils[END_REF][START_REF] Mcdonnell | A rationale for old water discharge through macropores in a steep, humid catchment[END_REF][START_REF] Mcglynn | A review of the evolving perceptual model of hillslope flowpaths at the maimai catchments, new zealand[END_REF][START_REF] Tromp-Van Meerveld | Threshold relations in subsurface stormflow: 2. the fill and spill hypothesis[END_REF], and ridging in the capillary fringe [START_REF] Abdul | Laboratory studies of the effects of the capillary fringe on streamflow generation[END_REF][START_REF] Cloke | Using numerical modelling to evaluate the capillary fringe groundwater ridging hypothesis of streamflow generation[END_REF][START_REF] Fiori | Numerical experiments of streamflow generation in steep catchments[END_REF] M. G. [START_REF] Sklash | The role of groundwater in storm runoff[END_REF].

Once it reaches the water table, rainwater raises the pressure in the groundwater zone.

How the water table responds to this change in the pressure field, and ultimately pushes water into the drainage network, depends on the aquifer's geometry. In steep and shallow aquifers, the rainfall signal travels as a kinematic wave driven by gravity [START_REF] Beven | Kinematic subsurface stormflow[END_REF][START_REF] Tani | Runoff generation processes estimated from hydrological observations on a steep forested hillslope with a thin soil layer[END_REF]. In more typical aquifers, however, it is the pressure gradient that drives the flow through the porous rock. In general, one then needs to consider both the horizontal and vertical components of the groundwater flow which, in steady state, determine the flow pattern and the distribution of the transit time through the aquifer [START_REF] Cardenas | Potential contribution of topography-driven regional groundwater flow to fractal stream chemistry: Residence time distribution analysis of tóth flow[END_REF][START_REF] Toth | A theoretical analysis of groundwater flow in small drainage basins[END_REF]. Dynamical simulations in unconfined aquifers prove more challenging.

When permeability decreases steeply with depth, or when an impervious horizon bounds the groundwater flow, the latter is mostly horizontal, and one can combine Darcy's law to the shallow-water approximation to derive the Boussinesq equation [START_REF] Boussinesq | Recherches théoriques sur l'écoulement des nappes d'eau infiltrées dans le sol et sur le débit des sources[END_REF][START_REF] Dupuit | Etudes theoriques et pratiques sur le mouvement des eaux courantes[END_REF]. [START_REF] Brutsaert | Regionalized drought flow hydrographs from a mature glaciated plateau[END_REF] used this equation, in its original non-linear form, to estimate the hydraulic properties of an unconfined aquifer based on the recession limb of its hydrograph. This method, and the associated power-law of the recession limb, has since become a staple of groundwater hydrology (Troch et al., 2013, and references therein).

Only recently, however, has the Boussinesq equation been invoked to represent the rising limb of a hydrograph [START_REF] Pauwels | Estimation of aquifer lower layer hydraulic conductivity values through base flow hydrograph rising limb analysis[END_REF]. Based on numerical solutions of the linearized Boussinesq equation, [START_REF] Fiori | Old water contribution to streamflow: Insight from a linear boussinesq model[END_REF] showed that it could simulate a complete hydrograph, and explain the chemical composition of the groundwater that feeds a stream during a rainfall event. [START_REF] Pauwels | Confirmation of a short-time expression for the hydrograph rising limb of an initially dry aquifer using laboratory hillslope outflow experiments[END_REF] confirmed the validity of this equation in gently sloping laboratory aquifers submitted to a sudden rainfall event, showing that unconfined aquifers react virtually instantly to a rainfall input. These results, however, must be reconciled with the power-law recession of the hydrograph, which can result only from the non-linear Boussinesq equation. Laboratory experiments achieve this reconciliation when the aquifer's outlet, which represents a stream, coincides whith the impermeable base of the aquifer [START_REF] Guérin | Response of a laboratory aquifer to rainfall[END_REF]. In this configuration, the flow can enter a new asymptotic regime (the "storm-flow regime"), during which the aquifer's discharge increases in proportion to R 3/2 s , where R s is the recharge rate.

If it occurs in nature, this non-linear regime would especially amplify the most intense rainfall events. To our knowledge, however, it has never been identified in the field. Here, we combine high-frequency field measurements with the classical Boussinesq approximation to seek out the storm-flow regime in a small catchment.

Field Setup

To identify the storm-flow regime in a natural setting, we instrumented an eighthectare catchment in the volcanic island of Basse-Terre, in the Guadeloupe archipelago, French West Indies (Figure 1a). There, the pristine forest of the Guadeloupe National Park covers an at least 10 m-thick layer of unconsolidated clay [START_REF] Buss | Depth profiles in a tropical volcanic critical zone observatory: Basse-terre, guadeloupe[END_REF][START_REF] Clergue | Influence of atmospheric deposits and secondary minerals on li isotopes budget in a highly weathered catchment, guadeloupe (lesser antilles)[END_REF] (Figure 1b), the hydraulic conductivity of which typically ranges from about 10 -6 to 10 -5 m s -1 depending on compaction and composition (Colmet-Daage & Lagache, 1965, Supporting Information Text S1). An electrical resistivity tomography survey revealed a homogeneous aquifer, with no visible horizon in the clay layer (Supporting Information Text S1 and Figure S1). This shallow catchment (its surface slope is 14 • on average) is drained by the Quiock creek, a stream less than 2 m wide. Outside this stream and its tributaries, the catchment shows no indication of surface runoff-the ground is permanently littered with decaying leaves.

Modern pressure transducers can record high-frequency measurements over months without human intervention, making it possible to monitor accurately the response of groundwater to a long series of rainfall events. We installed eight such transducers in the Quiock catchment, seven of which in piezometric wells arranged in a linear array which extends, perpendicularly to the stream, over 30 m (Figure 1c). This disposition allows us to reconstruct the shape of the groundwater surface every minute. We placed the eighth transducer in a stream gauge to record the discharge of the Quiock creek at the same frequency.

In addition, a tipping-bucket rain gauge measures precipitations less than 10 m away from the wells, below the canopy (Supporting Information Text S2).

Observations

We find that the Quiock catchment, like most catchments of its size, distorts the rain signal (Figure 1d). Right after the beginning of a rain event, the runoff increases quickly, sometimes tenfold within a few minutes. After the rain has stopped, the stream's discharge begins a recession that lasts until the next event, often days later.

Remarkably, although the groundwater surface lies a few meters below ground, it rises virtually instantly (Figure 1e), so much so that our measurements cannot tell which, of the groundwater or the stream, reacts first (Supporting Information Text S4 and Figure S2).

This observation indicates that the pressure jump induced by fresh rainwater propagates through the vadose zone at a velocity of a few millimeters per second at least, before the water table responds to it. This fast transfer of the rainfall signal, common to many catchments [START_REF] Abdul | Field studies of the effects of the capillary fringe on streamflow generation[END_REF][START_REF] Sidle | Stormflow generation in steep forested headwaters: a linked hydrogeomorphic paradigm[END_REF][START_REF] Tromp-Van Meerveld | Threshold relations in subsurface stormflow: 2. the fill and spill hypothesis[END_REF], remains the subject of active research [START_REF] Cloke | Using numerical modelling to evaluate the capillary fringe groundwater ridging hypothesis of streamflow generation[END_REF].

To interpret our observations, we first need to estimate the geometry and hydrological properties of the aquifer. We represent the former as simply as possible, by assuming that the groundwater flow is mostly orthogonal to the river. Accordingly, we estimate its horizontal extension L a as half the average distance between two rivers, namely L a = A/(2L r ), where

A is the area of the catchment, and L r the total length of its drainage network. Based on the Lidar map of Figure 1a, we find L a ≈ 40 m.

We further assume (i) that there exists, at the catchment scale, a representative hydraulic conductivity K (Sanchez-Vila, [START_REF] Sanchez-Vila | Representative hydraulic conductivities in saturated groundwater flow[END_REF], and (ii) that most of the groundwater flows horizontally in an active layer, above the stream elevation. If correct, these assumptions open the toolbox of the Boussinesq approximation: the pressure head in the aquifer is hydrostatic, and the Darcy flow it drives is induced by the local elevation of the water table [START_REF] Boussinesq | Recherches théoriques sur l'écoulement des nappes d'eau infiltrées dans le sol et sur le débit des sources[END_REF][START_REF] Brutsaert | Regionalized drought flow hydrographs from a mature glaciated plateau[END_REF][START_REF] Dupuit | Etudes theoriques et pratiques sur le mouvement des eaux courantes[END_REF]. Strictly speaking, approximation (ii) (often referred to as the "fully-penetrating stream") holds only when an impervious horizon joins the stream, and confines the groundwater flow above itself. We have no indication that such is the case in the Quiock catchment. One often finds, however, that the hydraulic conductivity of unconsolidated aquifers quickly decreases with depth, possibly due to compaction [START_REF] Mckay | Hydrogeology and pedology of saprolite formed from sedimentary rock, eastern tennessee, usa[END_REF][START_REF] Montgomery | Hydrologic response of a steep, unchanneled valley to natural and applied rainfall[END_REF][START_REF] Schoeneberger | Directional saturated hydraulic conductivity and macropore morphology of a soil-saprolite sequence[END_REF]. We expect that such a vertical gradient of conductivity would confine most of the flow to the layers lying above the stream. We cannot assess the validity of this point a priori; instead, we will deem it plausible as long as the catchment's behavior accords with it [START_REF] Harman | A similarity framework to assess controls on shallow subsurface flow dynamics in hillslopes[END_REF]. Asymptotic regimes, in particular, are sensitive to the physical mechanism that drives them [START_REF] Barenblatt | Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics[END_REF], and those of the Boussinesq equation would break down if the shallow-flow approximation were grossly inadequate. In the following, we interpret two of them as the signature of a shallow flow.

Averaging the Boussinesq equation over a sufficiently long period yields the textbook expression for the shape of a steady water table (Supporting Information Text S5):

h RMS = R K x (2 L a -x) (1)
where x is the distance to the stream, R is the average recharge rate and h RMS = h 2 is the root mean square of the water table elevation. That h RMS appears in the above equation, as opposed to the average elevation of the water table h , results from the non-linearity of the Boussinesq equation (Eq. ( 2), Supporting Information Text S5). Here, we define h RMS with respect to the stream's elevation, in accord with approximation (ii). Fitting the ratio R /K to our observations (Figure 1c), and estimating the recharge rate as the ratio of the average stream discharge to the area of the catchment, we find a catchment-scale hydraulic conductivity of K = 5.6×10 -6 m s -1 , within the range of expected values for unconsolidated clay (Supporting Information Text S5).

Integrating h RMS over the entire catchment yields an estimate of the aquifer volume that, on average, holds water above the stream's level :

V r = πAL a 4 R K ≈ 1.9 × 10 5 m 3 . (2)
This volume occupies a significant part of the available space in the aquifer, between the stream's level and the ground surface (Figure 1c). When rainfall reaches the water table, the aquifer's matrix gets refilled with water, in proportion to its fillable porosity φ f -the ratio of the pore volume available to the rising water table to the total volume [START_REF] Acharya | Analytical expressions for drainable and fillable porosity of phreatic aquifers under vertical fluxes from evapotranspiration and recharge[END_REF][START_REF] Park | A simple model for water table fluctuations in response to precipitation[END_REF][START_REF] Sophocleous | Combining the soilwater balance and water-level fluctuation methods to estimate natural groundwater recharge: practical aspects[END_REF]. Multiplying V r with the fillable porosity thus yields an estimate of the volume of water, V a = φ f V r , that the aquifer stores above the stream's level, on average. In the framework of the Boussinesq approximation, this is the total volume of groundwater that, on average, would be released into the stream during a prolonged drought; we name it "available volume" on that account. Red dots indicate the transition between the equilibrium front and the rising plateau, at x = L f .

Dividing this volume by the average discharge of the catchment yields a characteristic time,

T c = V a / Q which, at this point, is merely a mathematical definition. We will see, however, that it will prove a convenient parameter to represent the reaction of the groundwater flow to a rainfall event.

We now use the recession of the water table after a rain to evaluate the validity of the Boussinesq approximation in the Quiock catchment, and to characterize its shallow aquifer [START_REF] Rupp | Analytical assessment and parameter estimation of a low-dimensional groundwater model[END_REF][START_REF] Troch | The importance of hydraulic groundwater theory in catchment hydrology: The legacy of wilfried brutsaert and jean-yves parlange[END_REF]. Once rainfall has stopped, the water table decreases in all piezometric wells, although at different rates, as the aquifer drains slowly into the stream. Since we monitor simultaneously the stream discharge and the water table elevation, we may compare their evolution to the predictions of the Boussinesq approximation, and adjust the aquifer's hydraulic properties to fit the theory to our observations. During a dry period, the water table should decrease as the inverse of time, while the discharge of the stream decreases as the squared inverse of time [START_REF] Brutsaert | Regionalized drought flow hydrographs from a mature glaciated plateau[END_REF]. The Quiock aquifer accords with this theory (Supporting Information Text S6 and Figure S3), and we find a drainable porosity of φ d ≈ 5.3 × 10 -2 , an ordinary value for clay [START_REF] Batu | Aquifer hydraulics: a comprehensive guide to hydrogeologic data analysis[END_REF]. This analysis also yields another estimate for the hydraulic conductivity, K ≈ 4.1 × 10 -6 m s -1 , which is consistent with the estimate based on equation (1). These findings supports the assumption, so far unsubstantiated, that the permeability of the aquifer decreases quickly below the stream's level, and encourage us to use the non-linear Boussinesq equation during the early stage of a rainfall event as well. In the next section, we analyze our observations in search of the storm regime.

Storm-flow Regime

After a drought, the water table is low and the stream discharge recedes. The next rainfall event abruptly increases the groundwater pressure throughout the shallow aquifer, which responds by expelling more water into the neighboring stream. Following [START_REF] Guérin | Response of a laboratory aquifer to rainfall[END_REF], we now idealize this scenario by considering an initially empty aquifer suddenly recharged at rate R s . Under these assumptions, the groundwater flow enters the storm-flow regime of the Boussinesq equation, the mathematical expression of which was derived by

Guérin et al. (2014) (Supporting Information Text S7

). Here we propose a simpler derivation which better illuminates the mechanics of this peculiar regime, at the cost of mathematical rigour.

Figure 2 shows the theoretical shape of the water table as it swells to accommodate the rainfall input [START_REF] Guérin | Response of a laboratory aquifer to rainfall[END_REF]. We now approximate this mathematical solution by splitting it into two connected regions. Far from the outlet, the water table rises at velocity R s /φ f , unaffected by the groundwater lost to the stream. Meanwhile, a smooth front of length L f connects the outlet to this rising plateau. Along this front, the water table is virtually at equilibrium with the recharge. Since, in this simple reasoning, the rising plateau delivers no water to the front, the groundwater discharge per unit length of stream is R s L f -collected entirely by the front. The front's shape is that of a steady water table in an aquifer of length L f , that is, equation ( 1) in which we substitute R with R s , and L a with L f . However, the front is not exactly at equilibrium, since it needs to match the height of the rising plateau; this requires that the front's thickness increase linearly with time, according to L f = √ KR s t/φ f , where t is the time elapsed since the beginning of recharge.

This matching yields the approximate shape of the water table in the storm-flow regime (red lines in Figure 2). Although not rigorous, this procedure yields the same expression as [START_REF] Guérin | Response of a laboratory aquifer to rainfall[END_REF], but for a prefactor of order one.

Unfortunately, in the Quiock catchment, the spatial resolution of our water-table measurements does not allow for a direct comparison between the actual water table and the solution derived above. Indeed, after one hour of sustained and intense rainfall, we expect the equilibrium front to extend over a few meters only (Figure 2). Instead, we may calculate the amount of groundwater the aquifer delivers to the stream during a rainfall event, and compare it to the associated discharge surge we measure in the stream, ∆Q s . To do so, we first derive the groundwater flow that exits the aquifer in the storm regime at time T s (Supporting Information Text S7):

∆Q s ≈ 2L r √ K φ f R 3/2 s T s . ( 3
)
This expression is similar to the equation derived by [START_REF] Guérin | Response of a laboratory aquifer to rainfall[END_REF], but for a numerical prefactor of about 0.73, which is accessible only to a thorough mathematical derivation of the storm-flow regime. As expected, we recover the exponent of 3/2 that distinguishes the storm-flow regime. The above equation also shows that the storm flow, like most asymptotic regimes in dissipative systems, does not depend on initial conditions (i.e. the shape of the water table before the rainfall event). How decent an approximation this mathematical feature will prove in practice is, at this point, open to question.

Equation (3) involves parameters that are sometimes difficult to measure (L r , K and φ f ), and the recharge rate raised to a non-integer exponent-a quantity whose dimensions are hardly meaningful. To produce a more presentable version of Eq. ( 3), we divide it by the average water balance for the catchment, namely Q = A R . We thus get

∆Q s Q ≈ 2 L r K R A φ f R s R 3/2 T s . (4)
Finally, using Eq. ( 2) to express the characteristic time T c of the aquifer in terms of the hydraulic properties of the latter, we rewrite Eq. ( 4) as:

∆Q s Q = C T s T c R s R 3/2 , ( 5
)
where C ≈ 0.57 is the numerical prefactor derived by [START_REF] Guérin | Response of a laboratory aquifer to rainfall[END_REF] (Supporting Information Text S7). The above equation is equivalent to Eq. ( 3), but perhaps more telling. Its advantages are that (i) all quantities are made non-dimensional using their average value, and (ii) the hydrological properties of the aquifer are all lumped into a single free parameter, the characteristic time T c (or, equivalently, the available volume V a ).

These advantages, however, come at a cost: the presence of average quantities ( Q , T c and R ) in Eq. ( 5) might suggest that the prefactor of this power-law depends on the average hydrological conditions. In fact, however, a change in the average rainfall rate would affect Q and T c R 3/2 in the same proportion, and Eq. ( 5) would thus remain unaffected-so long as the groundwater flow has entered the storm-flow regime. We now compare the storm-flow regime with our field measurements. Let us consider a rainfall event such as the one of Figure 3a, which we idealize with a constant recharge rate R s over a time T s (Supporting Information Text S3). The aquifer responds to this input by delivering more water to the stream, the discharge of which thus surges (Figure 3b).

Assuming that this surge is due mostly to the groundwater input, we can measure ∆Q s on the hydrograph of the Quiock stream (Supporting Information Text S3), and normalize it with Q , the average stream discharge. We now wish to compare this relative discharge surge to the storm-flow regime, embodied by Eq. (3). In doing so, we assume that the aquifer, or at least its shallower, most reactive part is virtually empty before the rainfall event. This, strictly speaking, is not true unless the stream dries out entirely. Still, this approximation can hold when the water table has enough time to recede between two rainfall events, but it is unlikely to hold when rains quickly follow each other.

Between January 2015 and October 2017, we have identified 137 isolated rainfall events, and measured the associated discharge surge ∆Q s , rainfall intensity R s , and total volume of water delivered to the catchment V s (Supporting Information Text S3). Figure 3 shows these measurements in the coordinates suggested by equation ( 5), namely the normalized discharge surge ∆Q s / Q , and the product of the rainfall duration, T s , with the power 3/2 of the normalized rainfall intensity, R s / R , measured as shown on Figs. 3a andb. Both quantities spread over almost three orders of magnitude, revealing a positive correlation (the Pearson coefficient is 0.55 in logarithmic space), despite a significant scatter (about one order of magnitude). Fitting a power law to our data by orthogonal distance regression yields an exponent of 0.98 ± 0.09, close to the exponent of one predicted by equation ( 5).

Assuming this exponent is indeed one, we find that a characteristic time of T c ≈ 10.4 days best fits our data, which corresponds to an available volume of V a ≈ 2 400 m 3 . We thus find that the fillable porosity is about φ f ≈ 1.3 × 10 -2 , a value less than a fourth of the drainable porosity φ d , as measured based on the recession flow-not a surprizing observation [START_REF] Acharya | Analytical expressions for drainable and fillable porosity of phreatic aquifers under vertical fluxes from evapotranspiration and recharge[END_REF]. These values however, should be treated with caution, as they inherit the uncertainty associated to the dispersion of the data in Figure 3 (at least a factor of 5).

Among the many hypotheses that allow one to derive equation ( 5), the existence (and location) of a horizontal impervious layer below the stream's level is arguably the least substantiated. If, for comparison, we assume that such a layer lies a few meters below the stream's level, the groundwater flow would not enter the storm-flow regime.

Instead, we would expect it to enter the linear counterpart of this assymptotic behavior (provided the Boussinesq approximation still holds), and the stream's discharge would then increase like R s √ T s [START_REF] Pauwels | Estimation of aquifer lower layer hydraulic conductivity values through base flow hydrograph rising limb analysis[END_REF][START_REF] Pauwels | Confirmation of a short-time expression for the hydrograph rising limb of an initially dry aquifer using laboratory hillslope outflow experiments[END_REF]. Despite the scatter of Figure 3, there is little doubt that this linear regime does not fit our observations, thus supporting, in retrospect, the assumption of a fully-penetrating stream. Still, this simplifying hypotheses can only be a crude model of the Quiock aquifer, the actual geometry of which probably contributes to the dispersion of the data around the storm regime.

Often, the volume of groundwater that a catchment contains affects its response to the rainfall signal. This sensitivity to initial conditions could also explain part of the dispersion in Figure 3 [START_REF] Biswal | Study of dynamic behaviour of recession curves[END_REF][START_REF] Botter | Nonlinear storagedischarge relations and catchment streamflow regimes[END_REF][START_REF] Kirchner | Catchments as simple dynamical systems: Catchment characterization, rainfall-runoff modeling, and doing hydrology backward[END_REF]. Indeed, we may only expect the storm-flow regime to be a decent representation of the groundwater flow if the aquifer is essentially empty before the rainfall event. (Even in the framework of the Boussinesq approximation, the initial shape of the water table influences the response of the groundwater flow to recharge.) To evaluate the state of the groundwater flow before a rainfall event, we measure the discharge Q i of the stream just before it rises, for the 137 events of Figure 3 (Supporting Information Text S8).

Surprisingly, once detrended according to equation ( 5), the response of the stream's discharge appears uncorrelated with the ratio Q i /∆Q s , even when the latter becomes larger than one-that is, when we would expect the storm-flow regime to break down (Supporting Information Figure S4a). A different picture emerges if we normalize the initial discharge with the average discharge of the stream (Q i / Q then becomes our proxy for the groundwater's state, Supporting Information Figure S4b). The prefactor of equation ( 5) increases significantly with this ratio, showing the influence of initial conditions on the groundwater's response to rainfall events, but this correlation disappears for the most isolated events (Q i / Q less than about 0.1). Repeating this analysis with the elevation of the water table in piezometer G confirms these observations, although with a lesser statistical significance (Supporting Information Figure S4c andS4d).

It is remarkable that the power-law relation associated to the storm-flow regime appears to hold even for rainfall events that are not, strictly speaking, isolated from the previous ones-although such events probably cause some of the scatter visible in Figure 3.

If we restrict the analysis to the most isolated rainfall events, thus reducing our data set to only ten points, we find that a critical time of T c ≈ 49 days best fits our observations (Supporting Information Text S8). Although less significant statistically, this value might be more relevant physically. If so, the available volume of water in the catchment would be closer to V a ≈ 11 000 m 3 , and the associated fillable porosity would be φ f ≈ 6.0×10 -2 -very close to the drainable porosity.

Like most field observations, our measurements are highly variable, and the Boussinesq approximation can only provide a rudimentary model of the groundwater dynamics during a rainfall event. Even where this approximation is appropriate, we can only expect the stormflow regime to occur during rainfall events that are isolated from previous ones. Nonetheless, this regime expresses itself unambiguously in the hydrograph of the Quiock Creek, thus displaying the typical robustness of asymptotic regimes [START_REF] Barenblatt | Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics[END_REF]. Among the rainfall events we have identified over 3 years, only a few qualify as floods-none of them catastrophic. The trend shown on Fig. 3c, however, shows no sign of abating, and the scaling law of the storm-flow regime might fit more severe events than those of our data set.

We suggest that the storm-flow regime takes place in many catchments where groundwater flows through a shallow unconfined aquifer. It would thus contribute to the widespread non-linearity of small catchments [START_REF] Botter | Nonlinear storagedischarge relations and catchment streamflow regimes[END_REF][START_REF] Buttle | Hydrologic coupling of slopes, riparian zones and streams: an example from the canadian shield[END_REF][START_REF] Kirchner | Catchments as simple dynamical systems: Catchment characterization, rainfall-runoff modeling, and doing hydrology backward[END_REF][START_REF] Tromp-Van Meerveld | Threshold relations in subsurface stormflow: 2. the fill and spill hypothesis[END_REF]Uchida, Tromp-van Meerveld, & Mc-Donnell, 2005). Of course, we would need more high-frequency rainfall and discharge measurements to support this hypothesis. Provided such time series, the plot of Figure 3 makes it straightforward to calibrate equation ( 5), which in turn can be implemented either as a source term in watershed models [START_REF] Thompson | Application of the coupled mike she/mike 11 modelling system to a lowland wet grassland in southeast england[END_REF], or included in low-dimensional, physically-based models of the groundwater dynamics [START_REF] Basso | A physically based analytical model of flood frequency curves[END_REF][START_REF] Kirchner | Catchments as simple dynamical systems: Catchment characterization, rainfall-runoff modeling, and doing hydrology backward[END_REF]. We trust this could improve flood forecasting in catchments dominated by shallow groundwater. In addition, the calibration of equation ( 5) yields the average volume of groundwater, V a , that a catchment stores above the stream level-an estimate of the amount of water that will be released during a prolonged drought.

Finally, as a solution of the non-linear Boussinesq equation, the storm-flow regime reconciles the classical drought flow with the quick response of groundwater to rainfall. This encouraging finding bolsters the renewed interest in Boussinesq's theory that high-frequency measurement devices have fostered [START_REF] Troch | The importance of hydraulic groundwater theory in catchment hydrology: The legacy of wilfried brutsaert and jean-yves parlange[END_REF].

Figure 1 .

 1 Figure 1. Instrumented catchment of the Quiock creek, Guadeloupe, French West Indies. (a) Map of the catchment (16 • 10 36 N, 61 • 41 44 W). Dots indicate piezometric wells. (b) Quiock creek about 10 m upstream of measurements. (c) Cross section of the shallow aquifer. Blue line shows the root mean square of the water table profile, hRMS, from January 2015 to October 2017. Red dashed line corresponds to equation (1). Brown shaded area, grey lines and black dots indicate ground surface, boreholes and pressure sensors respectively. (d) Time series of the stream discharge and rainfall rate (April 2015). (e) Time series of the water-table elevation, with respect to stream level, at 2.4, 21 and 29 m from the river.

Figure 2 .

 2 Figure 2. Theoretical shape of the water table during the storm-flow regime. For illustration, Rs = 10 mm h -1 , φ f = 1.3 × 10 -2 and K = 5.6 × 10 -6 m s -1 . Blue lines: self-similar solution of the Boussinesq equation[START_REF] Guérin | Response of a laboratory aquifer to rainfall[END_REF]. Red dashed lines: approximate self-similar solution.

Figure 3 .

 3 Figure 3. Relationship between rainfall intensity and associated surge in the stream discharge. (a) Rainfall event. Blue line shows actual time series superimposed over its mathematical representation (red dashed rectangle). The rectangle's area, Rs Ts, and duration, Ts, are the integral and variance of the actual signal, respectively. (b) Surge of stream discharge caused by the rainfall event of panel A, with amplitude ∆Qs. (c) Normalized discharge surge, ∆Qs/ Q , as a function of the product of the duration of the rainfall event, Ts, with the normalized rainfall intensity, Rs/ R , to the power 3/2 for 137 isolated rainfall events between January 2015 and October 2017. Error bars indicate measurement error (Supporting Information Text S3). Only a few error bars, selected for representativity, are shown. Rimmed marker indicates event of panels A and B. Solid red line shows proportionality (equation (5) with Tc = 10.4 days). Dashed red lines represent the standard deviation of Tc in logarithmic space.

Acknowledgments

All the time series used in this paper were collected by the Observatoire de l'eau et de l'érosion aux Antilles (ObsERA, INSU-CNRS). They are available at http://webobsera.ipgp.fr/.

We thank the members of the Observatoire Volcanologique et Sismologique de Guadeloupe, and specially O. Crispi, for technical support. We are grateful to the Parc National de la Guadeloupe, and specially to S. La Pierre de Melinville and M. Gombauld, for granting us access to the field site. We thank S. Basso for suggesting, during the review process, to introduce the characteristic time of equation ( 5).


Daniel Cortes-Borda

Ksenia V Kutonova

Corentin Jamet

Marina E Trusova

Francoise Zammattio

Charlotte Truchet

Mireia Rodriguez-Zubiri

Francois-Xavier Felpin

Optimizing the Heck-Matsuda Reaction in Flow with a Constraint-Adapted Direct Search Algorithm

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Beatrice Dal

Francesca Frontini

Giancarlo Luxardo

Agnès Steuckardt

Indexing and Linking Text in a Large Body of Family Writings

THE PROJECT

The Corpus 14 project aims at building a digital edition of correspondences of French soldiers in WW1. Writers are soldiers and their families, with little formal education. The main objectives are: -the study of the Great War and its legacy -the evolution of the linguistic uses in wartime

RESEARCH OUTCOMES

  Steuckardt, A. (ed) (2015). Entre villages et tranchées: l'écriture de poilus ordinaires. Uzès: Inclinaison. Roynette, O., Siouffi, G. and Steuckardt, A. (eds) (2017). La langue sous le feu : mots, textes, discours de la Grande Guerre. Rennes: Presses universitaires de Rennes. La Grande Guerre des gens "ordinaires". Conference, Montpellier, 2018. L'écriture des Poilus ordinaires. Travelling exhibition since 2014. RELATED STUDIES • Most notable projects (Mapping the Republic of Letters; Early Modern Letters Online; Clavius on the Web) work on the correspondences of intellectuals. An exception is The Migrant Letter Digitised (O'Leary and Moreton, 2017). • A TEI Correspondence Element was introduced in the 2.8.0 version of TEI P5. • Project Testament des poilus -different genre, same writerswas an inspiration for the indexes. THE CORPUS • The corpus is comprised of almost 1800 correspondences written by 37 writers in 11 areas for a total of almost 500,000 tokens. • The transcripts are encoded in conformance with the TEI guidelines, which allowed to keep the alignment between the text and the facsimile. • Sources are browsable and downloadable either from the ORTOLANG repository or from a TXM portal (with CQP queries). ANNOTATION TEIheader -<correspDesc> for the identification of sender(s), addressee(s), relationship, date and place of sending Ongoing development: -Annotation of places, people, organisations. <placeName> <persName> <orgName> -Internal indexes for references + link to external resources (DBpedia, Geonames) THE VISUALISATION 1) Identify places and persons evoked in the correspondences between soldiers and their families 2) Trace the level of mobility of soldiers and family members The visualisation interface was developed in collaboration with Pietro Barbieri, Chiara Capone and Luca Ciccone, MSc students in computer science, supervised by Marina Ribaudo, associate professor at DIBRIS, Università degli Studi di Genova.


Brasil Junior
email: brasiljr@unb.br

Rafael C F Mendes

Julian Lacroix

Ricardo Noguera

Taygoara F Oliveira

Keywords: Hydrokinetic turbines, Propeller horizontal axis turbines, BEM methods, Glauert theory

A design study of propeller hydrokinetic turbines is presented. The main objective is to evaluate the optimized geometry for horizontal axis hydrokinetic turbines, take into account the runner design with a different number of blades. The optimized blade geometry is determined by the Glauert theory for axial free flow turbines (hydrokinetic and wind turbines). The performance of the different blades is evaluated in the entire range of operating conditions, using Blade Element Momentum Theory (BEMT). Computational Fluid Dynamics (CFD) computations were also carried out to evaluate the detailed features of the fluid flow.

INTRODUCTION

The hydrokinetic turbines are placed in a context of an emergent renewable technology that had an important and fast innovation development during the last decade. Low environmental impact conversion systems can be implemented using this technology for sites with hydrokinetic potential in rivers or in marine environment [1][2][3][4][5]. The kinetic energy of the water streams can be converted to electricity by means of different models of free flow hydro turbines and the modern design methodologies have been employed to obtain optimized machines, augmenting the performance of the turbines and reducing the cost of the technology. It is an important alternative for the sustainability of small and medium energy systems, with applicability in many worldwide regions.

One considers, in the present paper, a horizontal axis propeller turbine with a diameter D, with N b blades, with a rotation speed of ω (see Figure 1). Those machines are aligned to the stream direction, converting the kinetic energy of the water flow, with velocity V 0 , to mechanical power. The turbines can be installed in the bottom of a river or in a marine site (Figure 2). The machines can be placed also near the free surface using floaters or pylons. From the turbomachinery technology point of view, those turbines belong to the same family of wind turbines or to the propeller hydraulic machines. The design approaches employed for those conventional turbines can be adapted for this specifically situation [6][7][8][9].

The main problem to be discussed here is the influence of the number of the blades for given operating conditions and the desired performance level. To answer this question, few hydrodynamical design approaches can be employed. The first possibility is to use the design approach to optimize runner geometries of conventional axial hydraulic turbines (Kaplan or propeller). Another possibility is to employ the design approach of wind turbines. Both methodologies, used to define the best shape of the blades, are very close one to each other, but some differences have to be pointed out.

The first approach inherits the aspects for the classical dimensioning of the axial runners in conventional hydraulic machines. It is based on the integral equations of fluid mechanics (Euler equation) and in the determination of the hydrodynamical efforts using the velocity triangles on the runner inflow and outflow cross sections. This methodology has been historically employed to the "confined" runners and many classical developments have evolved to obtain geometries with high hydraulic efficiency (see Rodrigues et al. [10],Cruz et al. [11] for instance). The main problem with this approach is that it does not take into account the condition of the free flow and the reduction of the axial velocity in the front of the runner. To be precise, an optimum runner designed to a confined flow rate (close to that one related to the velocity V 0 ) can have the worst performance in free flow condition. It is justified by the fact that the flow blockage effect, dependent on the geometry of the runner and its rotation speed, reduces the flow velocity in the inflow plane in front of the blades. Consequently, it decreases substantially the converted hydrodynamics power, which is scaled to the cube of the effective inflow velocity.

Considering the arguments of the last paragraph, the wind turbines methodologies [12][13][14], which considers as its hypothesis the free flow condition and its variation through the axial direction, seems more adapted to the design of free flow runners. Taken it into account, the methodology for the design of free flow turbines (wind and hydrokinetic) will be employed in the present study.

In general, the free flow runners design approach is based on the Glauert theory [15], with several improvements for applications in wind turbines [15][16][17][18], propellers [19] and hydrokinetic devices [9]. Some ad hoc corrections associated with complex physical flow features (wake hydrodynamics, tip blade effects, etc.) have been proposed and the cavitation limits for the water flow have been also considered, constraining the rotation speed of the runner in particular for large hydrokinetic machines [14].

The key point of the Glauert formulation lies in the actuator disc approach and the Boundary Element Momentum Theory (BEMT), which considers the balance of linear and angular momentum in the runner, as well as an estimate of the hydrodynamic forces in each section of the blade. In the present paper, this methodology will be employed for the blade shape design and to the performance analysis, evaluating the influence of the number of blades for optimized geometries to runners with 2 to 4 blades in the performance of the turbine. The BEMT methodology is validated by comparisons of the performance estimates to the CFD computations.

MATERIAL AND METHODS

Rotor kinematics

The performance of a hydrokinetic turbine is expressed by the power coefficient, which is parameterized by its dependence to the Tip Speed Ratio, denoted respectively by C p and λ. Those variables are classically defined as

(1) (2)
In Equations ( 1) and (2) P, A and R denote respectively the power, area and radius of the rotor, and ρ is the water density.

In other to define the flow kinematics of the machine in the runner, firstly a parameterization of the axial velocity on the rotor inflow plane is written as (see Figure 3):

(3) Additionally, on the outflow plane, the tangential velocity in the rotating wake -in opposite to the rotor rotation, can be expressed by a model of free vortex written as (4) Equations ( 3) and (4) introduce the axial and tangential induction factors denoted by a and a', respectively. Those variables are dependent to the radial position r.

In the rotor middle-plane (between A and B sections in Figure 3) and at a radial position r, the velocity triangle can be defined as illustrated in Figure 4, such that the inflow angle φ, can be determined by From Equations ( 5) and ( 6) a kinematic relationship between a and a' can be obtained in the form (7)

Using this equation, for a specific value of axial induction factor a and for a local velocity ratio λ r , the tangential induction factor can be calculated from the Equation ( 7) by ( 8)

Axial and angular momentum balance

The axial and angular momentum conservation principles, applied to the fluid flow through the runner (from sections A to B), give us expressions for the thrust and torque differentials as resulting from an integral 1D balance in an anular section of radius dr (for more details see [12] or [13] for instance) as being In those equations, dT and dM denote the parcel of the thrust and torque in the runner related to the section.

The hydrodynamic power in dr can be computed from Equation ( 10) as ( 11)

Or, in a dimensionless form, (12) Therefore, the power coefficient for entire runner can be computed by integrating Equation ( 12) in the radius range given

(1 3)
The observant reader should notice that λ r could be interpreted as a dimensionless radius. Equations ( 12) and ( 13) establishes a direct dependence of the runner performance to the radial distribution of the pair {a,a'}. Some important remarks have to be pointed out:

Remark 1: The optimization of the performance lies in the maximization of the C p , that is directed related to the maximum value of the term f(a,a')=a'(1-a).

Remark 2: Classically, the Betz theorem establishes an optimum of C p equal to 16/27, which is equivalent to a=1/3 and a'=0 (see [12] for instance). The equations ( 12) and (13) show that this limit is not valid in the case of a rotating wake, and a new optimum point has to be considered in this more realistic situation, taking into account the changing in the angular momentum in the fluid flow through the rotor.

Remark 3: Considering the relationship between a' and a, given by the Equation ( 8), the optimization problem related the search of the maximum condition for f(a,a') can be simplified to a one-dimensional problem, only dependent on the variable a, considering that a'=a'(a). It is the aim of the Glauert theory [14].

Optimum flow conditions for runner design

Given the turbine nominal operation variables {ω,V 0 }, which defines the values of λ, and for each radial position λ r . The best value of dC p , related to the maximum condition of f(a,a'), is simply obtained by the derivative of the function f to a, i.e.:

(14) From Equations ( 7) and ( 8), substituting the term da'/da in Equation ( 14), the following conditions for {a,a'} verifies the maximum power coefficient increment in Equation ( 12), i.e.:

(15) and ( 16) Therefore, for a given value of λ r , Equations ( 15) and ( 16) can be solved for the values of {a,a'}. After all, Equation ( 5) can be also employed to obtain the value of the inflow angle φ.

The mathematical problem to determine the optimum axial induction factor, given by the Equation ( 15), can be solved numerically using any non-linear solver (e.g. fsolve function in Matlab). Some analytical approaches can also be used to obtain the third order polynomials roots (see [21], for instance).

Considering the optimum conditions for the fluid flow through the rotor, the following algorithm can be employed to determine the first group of variables for the blade design:

Algorithm A.1: Optimum flow conditions

Step 1: Given the design data {ω,V 0 ,D}

Step 2: Compute λ and {λ r,i ; i=1:Nsec}

Step 3: Solve non-linear problem (15) obtaining {a i ; i=1:Nsec}

Step 4: Compute {a' i ; i=1:Nsec}, using the Equation ( 16).

Step 5: Compute {φ i ; i=1:Nsec}, using the Equation ( 5).

Step 6: Compute an estimate of the maximum C p using the numerical integral (13).

In this algorithm the blade is divided in Nsec radial sections.

Blade Element Momentum Theory

The BEMT method is a useful methodology to blade design of free flow turbines. It can be also employed to compute the machine performance for any value of λ. This method is presented in many textbooks and some additional ad hoc corrections have been formulated and reported in the literature, allowing good realistic results for the turbine performance modeling [16][17][18].

In the BEMT, Equations ( 9) and ( 10) are complemented by the computation of the hydrodynamic forces in a blade element dr by the hydrofoil theory. For a rotor with N b blades the thrust and the moment in a section of width dr can be computed by: From a design point of view, Algorithm A.1 give us the values of the induction factors (a and a') and the inflow angle φ, which establishes the optimized flow conditions for each radial position. The rotor design is complemented with the definition of the blade shape, expressed by the chord c and twist angle θ for different sections distributed through its length, i.e., {c i ,θ i ; i=1:Nsec}.

Using Equations ( 9) and ( 17), and estimating the tip and hub losses from ( 21), an optimal relation for the chord radial distribution can be explicitly obtained in the form (24)

The blade-twist angle is obtained by taken into account an optimum attack angle of the foil. In the present paper, the optimum attack angle α op is defined as the angle related to the maximum value of C L /C D . Therefore, for a given hydrofoil characteristics, with optimum attack angle of α op , the pitch angle can be computed from the inflow angle (Equation ( 5)) using ( 25)

For rotors with a number of blades greater than four and high solidity (σ ≡ N b c / (2pr), the hydrodynamic interference of the flow through the neighbor's blades has to be considered. A correction interference factor related the cascade flow, which is dependent on the solidity and the twist angle, can be taken into account, changing the Equation ( 22) to ( 26)

The classical correction factor F casc proposed by Weinig [30] was updated in the present work to consider the variation of chord lengths along the radial position, providing a corrected version of C n .The original plot proposed by Weinig, showed in Figure 5, was digitalized and a spline fitting is applied to obtain a Matlab function F casc (σ -1 ,θ). This correction is currently assumed for the situations with σ -1 <3.

[Figure 5: Cascade flow correction [30] -Original plot and fitted surface.]

The mathematical formulation present on the last sections allows the proposition of the following algorithm to obtain the blade geometry:

Algorithm A.2: Blade geometry

Step 1: Given the design data Nb and variables obtained in the algorithm A.1 {a i φ i ; i=1:Nsec}

Step 2: Run XFOIL to obtain hydrofoil polars and {α op , C L (α op ),C D (α op )}

Step 3: Compute the twist angle { θ i =φ i -α op ; i=1:Nsec}

Step 4: Compute chord {c i ; i=1:Nsec}, using the Equations ( 21), ( 22) and (24).

Step 4: Compute σ -1 and correct the values of chord dimension if it is necessary.

Influence of the number of blades in runner design

The optimized blade geometry, issued from the methodological approach presented in the last sections, is explicitly related to the number of blades by means of the Equation 24. It is clear that for a greater number of blades thinner shapes are obtained (smaller chord dimensions).

For a design condition of a given tip speed ratio λ, the optimum inflow conditions {a.a',φ} give only one optimum design value of C p (from algorithm A.1 and Equation 13). It is independent of the number of blades. On the other hand, the shape of the blade, given by the Equation ( 22) for the chord length, is dependent on the number of blades.

The volume of the entire rotor is related to the amount of material involved in the manufacturing the rotor. From the Equation ( 22), the dimensionless blade volume can be calculated by the following relation: (26) In this equation, Af is the dimensionless area of the standard airfoil section.

The volume depends on to the number of blades through the term [F(r,N b )/N b )], which is a decaying function of Nb. This is an indication that the amount of material necessary to rotors manufacturing is also a decaying function of the number of blades. This methodology preserves the estimation of the power coefficient in optimum conditions for any number of blades. Some works on wind turbine design report that rotors with a larger number of blades have slightly better power coefficients values (Duquette et al. [31]). It is a key point that will be investigated in the next section, using numerical methodologies to evaluate the performance of the different runners configurations. It is not clear, just at this point, that it is verified in the framework of BEMT-Glauert mathematical formulation for blade optimum blade design.

RESULTS AND DISCUSSIONS

Optimized blade geometry

The design methodology, presented in section 2 and summarized in algorithms A.1 and A.2, was implemented in a Matlab code. It allows us to compute the blade shape for a given set of input parameters{N b ,U o ,ω}. The blade radius -R, is defined considering the target machine power. An iterative approach can be used, computing successively the power coefficient (from Equation ( 13)) and the nominal Tip Speed Ratio, up to the convergence to the desired power.

In the present paper, the blade design methodology uses the foil NACA4415 geometry, with polars curves obtained from the software XFOIL with a typical Reynolds number of 1.0e+06. Only the value of α op and the evaluation of C L,op and C D,op , are needed for the algorithm A.2.

Figure ( 6) presents the twist angle for a four blades rotor, for different values of λ. The associated C p (which is independent of the number of blades) is obtained for whole range of tip speed ratio. It is a classical result that can be used to define preliminarily some design parameters (rotation speed for instance).

In Figure ( 7), different blade geometries are presented for rotors with four blades. This result shows that the thinner shapes are obtained for faster machines. Unfortunately, the tip cavitation is a constrain that limit the machine rotation speed and consequently the design tip speed ratio. It can be circumvented by means of an alternative chord design that consider the cavitation limit [4], which in a practical point of view augments the dimension of the chord near to the tip. All of those prelimina results are direct consequence of the design algorithm, with all classical ad hoc corrections (Tip, Hub, and Cascade). A better performance estimate will be computed in the next section

BEM and CFD computations -Performance evaluation

Numerical simulations using BEM and CFD methods were carried out to evaluate the performance and the fluid flow patterns for typical rotors with 2, 3 and 4 blades (Figure 9).

The BEM simulations were obtained using QBLADE [32]. In that code, the BEMT equations are implemented in a robust and friendly simulation platform, equipped with a XFOIL interface for the polars curve computation. This numerical tool permits the performance analysis in the entire range of TSR, considering the corrections for tip and hub losses, as well as the Buhl-Glauert correction for high values of axial induction factor. All designed blades geometries were considered for the BEM simulations, using properly QBLADE setting parameters.

The CFD computations were performed ANSYS-CFX 15, with simulation domain shown in Figure (11). The k-ω/SST turbulence model was used in two different sub-domains: The first one is a parallelepiped that defines the large fluid region influenced by the turbine. The second one is an immersed cylindrical region that contains the runner. In this volume, the flow is described in a rotating reference frame with the same rotation speed of the turbine runner, in such a way the rotor rotation is taken into account by the action of Coriolis and Centrifugal accelerations. The simulations were performed always in a time-dependent flow regime.

The mesh generation was performed using the ANSYS-CFX/ MESH software. Tetrahedral elements were used in an unstructured mesh, with local refinement in the wake and near wall regions, as illustrated by Figure ( 12). This refinement guarantees small values of the non-dimensional distance from the wall, y+, an important condition to assure good results for k-ω turbulence models. Table 1 presents three different mesh densities used for simulations of 4-blade rotors in the framework of a mesh convergence previous study. The convergence of Cp is achieved for meshes around 5.5 million nodes. The same mesh density was used for different runners, keeping the same refinement used for the reference condition (4-blade runner). The results obtained for the reference conditions realistically describe all the flow patterns (mainly the wake hydrodynamics) and had allowed good previsions of the machine performance, as also discussed in [33]. In Table (2), numerical results for nominal conditions (V 0 =2.5 m/s and λ=1.6) are summarized. The values of the power coefficient, obtained from CFD and BEM computations, are consistent with any number of blades.

Results for a wide range of TSR are presented in Figure ( 13). It could be observed that BEM and CFD simulations agree in the neighborhood of the nominal operation condition (λ=1.6). For higher values of TSR, the BEM results overestimate the performance of the turbine and it is more accentuated for the 2-blade rotor. For the 3-blade and 4-blade rotors, the performance levels are predicted coherently by the two approaches.

All results have shown an increase in the hydrodynamic performance for rotors with more blades, confirming some experimental considerations of a similar study of Dunquette et al [31] for wind turbines.

A slight difference in the power coefficient curve for 3-Blade or 4-Blade is observed. In a practical point of view, for hydrokinetic turbines, the small handicap of 3-blade rotor in comparison to the 4-blade rotor with less volume (Figure 11), induces the use of the last design choice considering the present value of nominal TSR.

[Table 2]: Performance of different rotors obtained by CFD and BEM with V 0 =2.5 m/s and λ=1.6.

Flow visualizations

In order to evaluate the hydrodynamic aspects of the fluid flow for differently designed runners, the flow patterns of the wake flow are presented and discussed in this section. The flow visualizations are obtained from the post-processing of CFD computations. All flow visualizations are processed for the nominal condition for TSR.

In

CONCLUSIONS

In this work the hydrodynamic influence of the number of blades in the propeller hydrokinetic turbines is discussed.

The results have pointed out that better power coefficients are related to rotors with 3 or 4 blades. The classical Glauert-BEMT methodology produces thinner blades (with reduced volume) for runners with larger number of blades.

The difference in the performance of 3 and 4 blades runners is very small, indicating that from the hydrodynamic point of view these two runners are quite equivalent.

The flow visualization of the unsteady wake by means of the CFD results has shown the influence of the blade design in the wake region. Rotors with small number of blades trend to

[Figure 1 :

 1 Propeller horizontal axis hydrokinetic turbine.] [Figure 2: Example of commercial and experimental models of horizontal axis hydrokinetic turbines [22-29].]

[Figure 3 :

 3 Flow through an axial runner.] [Figure 4: Machine kinematics -Velocity triangles in a blade section at r and radial cross-section dr.]

  (17) and (18) c is the chord length and F(r,Nb) is a correction function used to take into account complex flow phenomena in the rotating blades (e.g tip and hub losses). In the present work, Prandtl's tip and hub losses correction are considered, using:(19)

  17) and (18) the variables Cn and Ct are calculated using the lift and drag hydrofoil coefficients (CL and The NACA four digits foils are used in the present work, with parameters extracted from international databases for its geometry. The polar curves (C L (α) and CD(α)) are obtained from the open source code XFOIL [20].

[ 7 :

 7 Figure 6: (a) Blade twist and (b) power coefficient for optimized conditions w/hub and tip corrections.] Optimized blades for rotors with different tip speed ratios.] [Figure 8: Optimized blades for rotors with different tip speed ratios.] On Figure (8), blade geometries for rotors with a different number of blades are illustrated. The 3D view is also shown in Figure (9). Rotors with many blades are related to the thinner shapes. The estimated power coefficients for all rotors are the same (considering the present the design methodology). The volume of the blades is smaller for rotors with a greater number of blades as shown in Figure (10). It is also observed in Equation (24).

[Figure 9 :

 9 Rotors with different number of blades.] [Figure 10: Dimensionless blades volume for rotors with different number of blades -λ= 2.5.]

[Figure 11 :

 11 CFD simulation -domain dimensions.][Figure12: CFD simulation -mesh details for 5.5 millionnodes.]

  Figures (14), a general 3D flow visualization using streamlines and Q-criterion levels[34], illustrate the formation of the rotating wake for the different rotors. Tip and hub vortices can be identified. The tip vortex structure can also be observed in the 2D visualization of the axial velocity levels in the longitudinal plane (Figure (16)).The extension of the wake influence region can be observed in Figure (17). The axial velocity levels and turbulence kinetic energy show that wake is larger for the rotors with more blades and the shadowing effects of the rotating blades are more pronounced in extension and in turbulence intensity. Two-dimensional flow visualizations on circular cross sections are presented in Figures (18)-(19). A tomography of the levels of axial velocity and the kinetic energy of turbulence illustrate the influence of the wake in the rear region of the machines. The wake influence region is larger for 3 and 4-blade runners, but the levels of turbulence intensity are quite equivalent. The tip vortices were properly identified. For 2-blade rotors, the transient variations in the wake are more intense. It can be verified in the thickness of vortices filaments shown in Figure (14). [Figure 14: 3D flow visualizations.] [Figure 15: Visualization of dimensionless levels of axial velocity (u x /U 0 ) and kinetic energy of turbulence (k/U 0 2 .] [Figure 16: 2D cross sections visualizations of dimensionless axial velocity-u x /V 0 .] [Figure 17: 2D cross sections visualizations of dimensionless kinetic energy of turbulence-k/

[Table 1]: Different mesh sizes for 4-blade rotor with V 0 =2.5 m/s and λ=1

  .6.

		Mesh Nodes	y+	C p
	Mesh 1	426 216	323.6	0.381
	Mesh 2	1 285 254	117.8	0.386
	Mesh 3	5 572 162	1.6	0.389


Denis Labrousse

Mickaël Petit

François Costa

Keywords: bypass, DC-link capacitor, decoupling, GaN transistor, high speed switching, overvoltage, switching cell, switching transient

Mechanisms responsible for ringing and oscillations in power converters at transistors turn-offs are, first, studied using small signal modelling. It is explained why a 50 % derating must be applied to high-speed transistors. Experimental measurements validate the analytical predictions. The influence of the distance between a switching cell and its decoupling capacitors is then studied; it appears that using a low-inductancethough simplelayout results in a stray inductance as low as 11 nH when the capacitor is 30 cm away from the switching cell, enabling degrees of freedom for thermal management.

Study of the Impedance of the Bypassing Network of a Switching Cell -Influence of the Positioning of the Decoupling Capacitors I. INTRODUCTION Wide BandGap (WBG) devices are much faster than their silicon counterparts. Parasitic elements due to layout, connections and to components themselves result in voltage overshoots, oscillations and losses that drive designers to oversize devicesa costly measure. One way to limiting stray inductances consists in placing the devices as close as possible to one another. This, however, may result in poor thermal performances whilst complicating system integration. There is therefore a trade-off between thermal performances, electrical stress, and Electro-Magnetic Compatibility (EMC).

Reaching optimal designs requires a comprehensive and quantitative understanding of this trade-off.

The aim of this paper is two-fold. First, in section II, a simple and linear model of a switching cell is proposed so as to predict turn-off oscillations frequencies and amplitudes. Then, in section III, given a specific layout, the relationship between the loop inductance and the distance between the switching devices and its decoupling capacitors is experimentally studied. Section IV concludes the paper.

II. OSCILLATION MECHANISMS IN A SWITCHING CELL

A. Introduction

Many research papers have developed models aiming at predicting oscillations frequencies and amplitudes, either at turn-off [START_REF] Zhang | Characterization and Modeling of a SiC MOSFET's Turn-Off Overvoltage[END_REF]- [START_REF] Teulings | MOSFET switching behaviour under influence of PCB stray inductance[END_REF] or turn-on [START_REF] Williams | Diode Turn-On Time Induced Failures in Switching Regulators[END_REF], [START_REF] Zhang | Characterization and Modeling of a SiC MOSFET's Turn-On Overvoltage[END_REF]. These studies usually model switching loops with laddered linear RLC circuits and current-or voltage-sources. Oscillations are studied either in time- [START_REF] Zhang | Understanding the limitations and impact factors of wide bandgap devices' high switching-speed capability in a voltage source converter[END_REF], [START_REF] Teulings | MOSFET switching behaviour under influence of PCB stray inductance[END_REF], [START_REF] Li | A study on the effect of DC-link decoupling capacitors[END_REF]- [START_REF] Witcher | Methodology for switching characterization of power devices and modules[END_REF] or frequency- [START_REF] Li | A study on the effect of DC-link decoupling capacitors[END_REF], [START_REF] Chen | A frequency-domain study on the effect of DC-link decoupling capacitors[END_REF] domain.

These models, however, usually require an extensive knowledge of the circuit parasitic elements. Even if this can partly be inferred from simulations, accurate valuesand especially damping coefficients in semiconductorscan be difficult to obtain, in particular due to their dependences with frequency, bias and temperature. Similar comments can be made regarding switching times. So as to avoid the cumbersome and unreliable task of estimating switching times and resistances, the model developed in this paper assumes zero-switching times, i.e. that edges are infinitely steep. The circuit is then studied in a worst-case scenario, resulting in an over-estimation of the oscillations amplitudes. This assumption will nonetheless prove to be quite reasonable when dealing with fast, WBG, devices.

B. Modelling 1) Introduction

Let us consider (Fig. 1) a switching cell bypassed by two capacitors: a small one, 𝐶 𝑋7𝑅 , a priori ceramic-based, for local bypassing, and a bulkier DC-link capacitor 𝐶 𝑏𝑢𝑙𝑘 . The circuit is powered by a voltage source 𝑉 𝑏𝑢𝑠 and supplies a load modelled as a current source. Some lumped parasitic inductances are shown in Fig. 1. This figure also highlights three loops that may oscillate: -a High Frequency (HF) loop, including the switching cell itself and the local bypass capacitor; -a Low Frequency (LF) loop, including both capacitors; -a Very Low Frequency (VLF) loop, including the power source and the DC-link capacitor.

So as to simplify the analysis without losing the relevance of the study for field-use, it is assumed that the parasitic inductances fully decouple the loops: the oscillations can then be studied independently in each loop, which have distinct ringing frequencies and do not interfere. Papers have studied this matter, e.g. [START_REF] Li | A study on the effect of DC-link decoupling capacitors[END_REF], [START_REF] Zhang | Methodology for switching characterization evaluation of wide band-gap devices in a phase-leg configuration[END_REF]. The circuit of Fig. 1 can then be simplified (Fig. 2), were:

{ 𝐿 1 = 𝐿 𝑟1 + 𝐿 𝑋7𝑅 𝐿 2 = 𝐿 𝑟2 + 𝐿 𝑏𝑢𝑙𝑘 + 𝐿 𝑋7𝑅 ≈ 𝐿 𝑟2 + 𝐿 𝑏𝑢𝑙𝑘 𝐿 3 = 𝐿 𝑃𝑆𝑈 + 𝐿 𝑏𝑢𝑙𝑘 (1)
are the HF, LF, and VLF loop inductances.

Furthermore, the equivalent circuit when 𝑄 𝐿𝑆 is off and the load current flows through 𝑄 𝐻𝑆 is given in Fig. 3. 𝐶 𝑜𝑢𝑡 models the parasitic capacitance of the switching node -𝐶 𝑜𝑢𝑡 is a priori mainly due to the output capacitance of 𝑄 𝐿𝑆 . The asymptotic impedance diagram of the network, seen from the load is given in Fig. 4.

Loop capacitances can be defined as:

{ 𝐶 1 = (𝐶 𝑜𝑢𝑡 -1 + 𝐶 𝑋7𝑅 -1 ) -1 ≈ 𝐶 𝑜𝑢𝑡 𝐶 2 = (𝐶 𝑋7𝑅 -1 + 𝐶 𝑏𝑢𝑙𝑘 -1 ) -1 ≈ 𝐶 𝑋7𝑅 𝐶 3 = 𝐶 𝑏𝑢𝑙𝑘 (2)
The series-resonant frequencies of the HF and LF loops are:

𝑓 𝑟1 = (2𝜋√𝐿 1 𝐶 2 ) -1 , 𝑓 𝑟2 = (2𝜋√𝐿 2 𝐶 3 ) -1 (3)
The parallel-resonant frequencies are:

𝑓 𝑎,𝑖 = (2𝜋√𝐿 𝑖 𝐶 𝑖 ) -1 , 𝑖 ∈ ⟦1, 3⟧ (4)
Assuming that the loops are decoupled was justified since, a priori and due to component sizes:

𝐶 𝑜𝑢𝑡 ≪ 𝐶 𝑋7𝑅 ≪ 𝐶 𝑏𝑢𝑙𝑘 , 𝐿 1 ≪ 𝐿 2 ≪ 𝐿 3 (5)
This leads to time-domain decoupling of transient phenomenaas will be detailed hereunder.

At 𝑄 𝐿𝑆 turn-off (and, more generally, in a hard-switching converter supplying an inductive load, at turn-off) the voltage is switched prior to current. Two transients, a voltage step and then a current one, are therefore generated, leading to oscillations due to distinct mechanisms. The next two subsections describe those mechanisms.

2) Voltage switching

During the first phase of the switching, the switching node voltage goes from 0 to 𝑉 𝑏𝑢𝑠 . During this phase, the Low-Side (LS) transistor can be modelled as its output capacitance whereas the High-Side (HS) switch can be modelled as a voltage step going from 𝑉 𝑏𝑢𝑠 to 0. This step will excite the loops one after the other, propagating to the power supply.

During the first instant following the step, before notable oscillations can be observed in the LF loop (i.e. such that 𝑡 ≪ 𝑓 𝑎2 -1 ) and since 𝐶 𝑜𝑢𝑡 ≪ 𝐶 𝑋7𝑅 , the voltage across 𝐶 𝑋7𝑅 remains quite constant. The circuit can therefore be simplified as shown in Fig. 5. To put it another way: after the step and before the voltage across 𝐶 𝑋7𝑅 has had time to notably change, since 𝐶 𝑜𝑢𝑡 ≪ 𝐶 𝑋7𝑅 , the circuit can be simplified as shown in Fig. 5. The HF loop therefore oscillates at frequency 𝑓 𝑎,1 and with amplitude 𝑉 𝑏𝑢𝑠 , the voltage 𝑉 𝑠𝑤 being expressed as:

𝑉 𝑠𝑤 = 𝑉 𝑏𝑢𝑠 ⋅ (1 -cos(2𝜋𝑓 𝑎1 𝑡)) ( 6
)
The overvoltage is then equal to 𝑉 𝑏𝑢𝑠 . In other word: the peak voltage across 𝑄 𝐿𝑆 is twice the input voltage.

The step then propagates to the other loops but with decreasing amplitude, yielding oscillations of negligible amplitudes.

If the quality factor 𝑄 1 of the HF loop was not infinite, one could show that the overvoltage would be given by:

𝑉 ̂𝑉,1 = 𝑉 𝑏𝑢𝑠 ⋅ 𝑒 -𝜋 Λ ⁄ , Λ = √4𝑄 1 2 -1 (

3) Current switching

After the voltage is switched, the current switching occurs. During this transition, the High Side (HS) transistor can then be modelled by a short, the LS one by its output capacitance and the load by a current step -Fig. 6.

The current step will excite resonances, leading to oscillations, in the HF, LF and VLF loops one after the other, propagating from the switching node to the input power supply. Under the assumption that the loops are decoupled, each one can be modelled as a simple circuit with inductance 𝐿 𝑖 (𝑖 ∈ ⟦1, 3⟧, cf. ( 1)), capacitor 𝐶 𝑖 (𝑖 ∈ ⟦1, 3⟧, cf. ( 2)), and a current step of amplitude 𝐼 𝑙𝑑 in parallel. The oscillations have frequencies 𝑓 𝑎,𝑖 (4) and amplitude:

Δ𝑉 𝐼,𝑖 = 𝐼 𝑙𝑑 ⋅ 𝑍 𝑐,𝑖 , 𝑍 𝑐,𝑖 = √ 𝐿 𝑖 𝐶 𝑖 , 𝑖 ∈ ⟦1, 3⟧ (8)
Where 𝑍 𝑐 is the characteristic impedance of loop 𝑖.

C. Experimental validation

Voltage overshoots were measured on a prototype and compared to predictions from our model. An integrated GaN switching cell (LMG5200, 80 V -10 A -15 mΩ 𝑡𝑦𝑝 , from TI [START_REF]LMG5200 Datasheet[END_REF]) was used. It was bypassed by a MKP DC-link capacitor ( 𝐶 𝑏𝑢𝑙𝑘 = 15 µF) together with a X7R-ceramic bypass capacitor ( 𝐶 𝑋7𝑅 = 1 µF). The supply voltage was set to 𝑉 𝑏𝑢𝑠 = 30 V . Typical switching waveforms are given in Fig. 7. Three ringing periods can be seen, corresponding to the three loops: HF loop between 2 ns and 20 ns; LF loop between 100 ns and 2 µs; VLF loop between 2 µs and 20 µs.

The amplitudes of the HF and LF oscillations where measured for various load current values. As shown in Fig. 8, the amplitude of the measured HF oscillations (red squares) do not depend upon the load current 𝐼 𝑙𝑑 and are very close to the ones predicted by our model for the voltage switching (red line assuming 𝑄 1 = ∞, yellow line using the measured 𝑄 1 -value). On the other hand, the amplitude of the measure LF oscillations (blue squares) increase linearly with 𝐼 𝑙𝑑 (fit in blue), with a slope 1. It has also been observed that the amplitude and the frequency of LF oscillations have, as expected, a dependence in 𝐶 𝑋7𝑅 -0.5 whereas the HF oscillations do not depend on 𝐶 𝑋7𝑅 .

As a result, these measurements tend to validate the model developed in this section.

D. Discussion: on the steep-edge assumption

It has been assumed that edges have zero-rise time, which implies that the edges are fast enough to excite oscillations in the loops. Formally, this means that the bandwidth of the edges, with rise time 𝑡 𝑟 , is greater than the loop oscillation frequency 𝑓 𝑟𝑒𝑠 . For the HF loop, this can be written:

1 𝜋𝑡 𝑟 > 𝑓 𝑟𝑒𝑠 ⇔ 𝑡 𝑟 < 2√𝐶 1 ⋅ 𝐿 1 (9)
For various 𝐶 𝑜𝑢𝑡 and loop inductance values, we give in TABLE 1 the maximum rise time so that the assumption is valid. The first three lines correspond to cells using GaN transistors, lines number 2 and 3 are more related to SiC and Si MOSFETs whereas the last one is given for reference, corresponding to a traditional power module. It appears that the switching times calculated are similar to those that can be measured in prototypes, validating the assumption in the case of switching cells using fast, WBG, transistors.

E. Conclusion

The bypass network of a switching cell is composed of several loops that can oscillate when excited by switching events. A simple model that does not require extensive knowledge on the circuit under study whilst considering fast switching devices was developed. It shows that two mechanisms are involved in the oscillations: voltage and current switching. Both mechanisms yield to voltage overshoots in the switching loops, with various amplitudes and phase-shifts. Current switching yields oscillations with amplitude proportional to the switched current, whereas voltage switching yields oscillations with amplitude close to the switched voltagei.e. close to the supply voltage. This last overshoot is highly detrimental since it requires a 50 % derating of the switch voltage rating. Assuming a minimal loop inductance, little can be done to avoid this overshoot but slowing down the switch (e.g. by artificially increasing the gate resistance) or increasing the cell output capacitance.

III. ON THE INFLUENCE OF THE DISTANCE BETWEEN BYPASS CAPACITORS AND SWITCHING CELLS

A. Introduction

Being able to put bypass capacitors at some distance from the switching cell would simplify circuit design and reduce cost. As a matter of fact, it would enable several cells to share decoupling capacitors and heatsinks while loosening thermal constrains. It would also simplify mechanical designs and allow manufacturing circuits with unconventional form-factors.

This study aims at understanding to what extent the distance between a switching cell and its decoupling capacitors can be increased without excessively degrading EMC or increasing either thermal or electrical stress. To this end, the relationship between loop inductance and distance between a switching cell and its decoupling capacitors is studied.

To this end, a switching cell was built its output voltage measured given various decoupling network arrangements.

B. Circuit under study

An integrated GaN half bridge (LMG5200, 80 V -10 A -15 mΩ 𝑡𝑦𝑝 , from TI [START_REF]LMG5200 Datasheet[END_REF]) was used; dead-times were set to 5 ns. Its power supplied was decoupled by a DC-link capacitor (875 V -15 µF, MKP, ref. B32678G8156K000 [START_REF][END_REF]) and a ceramic local bypass capacitor (X7R, 470 nF -1 kV, ref. 2220Y1K00474KXTWS2). A 70 mm-wide, not terminated, Edge Coupled Microstrip transmission line was used between the capacitor and the switching cell. The distance between the decoupling capacitors and the cell are 𝑑 𝑏𝑢𝑙𝑘 and 𝑑 𝑋7𝑅 (Fig. 9). The circuit was manufactured using the top-copper layer of a 35/370/35 µm-thick FR4-based PCB, the bottom copper layer being used as a ground plane. The circuit was powered by a voltage source 𝑉 𝑏𝑢𝑠 = 30 V. It was used in a double pulse test, supplying a broadband inductive load with a current 𝐼 𝑙𝑑 = 5 A.

C. Measurement method

The switching node voltage was measured using a Teledyne-Lecroy HDO8108 oscilloscope (12 bit, 1 GHz, 2.5 GSa/s) and a PP008 10:1 passive voltage probe (500 MHz, 10 MΩ//9.5 pF), connected to the circuit with an ad-hoc, low-inductance setup [START_REF] Zhang | Methodology for switching characterization evaluation of wide band-gap devices in a phase-leg configuration[END_REF].

The capacitances 𝐶 𝑜𝑢𝑡 , 𝐶 𝑏𝑢𝑙𝑘 and 𝐶 𝑋7𝑅 were measured using an E4990a impedance analyser, at 100 kHz and at 30 V biasing voltage:

𝐶 𝑋7𝑅,30V 𝑑𝑐 =
was then fitted on the measurements, the vector 𝛽 being used as fitting parameter. The 𝜔 𝑘 and capacitances values were then used to estimate the loop inductances using Thomson's formula [START_REF] Thomson | On Transient Electric Currents[END_REF], (3)-(4).

Model [START_REF] Zhang | Methodology for switching characterization evaluation of wide band-gap devices in a phase-leg configuration[END_REF] appeared to be a good fit for the measurements (Fig. 10), the capacitances nonlinearities having limited impact on the result.

D. Loop inductance 1) LF Loop

The effect of the distance 𝑑 𝑏𝑢𝑙𝑘 between the switching cell and the DC-link capacitor on the LF loop inductance was studied at 𝑑 𝑋7𝑅 = 0 (Fig. 11). As expected, 𝐿 2 increases with 𝑑 𝑏𝑢𝑙𝑘 , due to the increase of the loop area. The rise is however very slow since it is limited to 18 nH (+64 %) when 𝑑 𝑏𝑢𝑙𝑘 goes from 0 to 260 mm. Furthermore, the capacitor self-inductance (ESL: 𝐿 𝑏𝑢𝑙𝑘 = 18.4 nH) accounts for up to 66 % of the overall loop inductance.

2) HF Loop

The DC-link capacitor having been removed from the circuit (i.e. 𝑑 𝑏𝑢𝑙𝑘 = ∞), the impact of 𝑑 𝑋7𝑅 on the HF Loop inductance was studied (Fig. 11). These measures show that the smallest achievable inductance is 1.2 nH and that the increase of 𝐿 1 vs. 𝑑 𝑋7𝑅 is very slow: 0.3 nH/cm. The HF loop inductance is then only 11 nH when the switching cell is 30 cm away from its bypass capacitor.

E. Small signal measurements

Another prototype was manufactured, with layout similar to the first one but incorporating a factice switching cell, with 𝑄 𝐿𝑆 replaced by its output capacitance and 𝑄 𝐻𝑆 shorted. The impedance seen from the load was then measured using an E4990a impedance analyser (stimuli amplitude: 500 mV, no bias). The linear model from Fig. 3 was then fitted on the measurements.

The measured inductances were in good agreement with those obtained with the first prototype. In particular the same slope of 0.3 nH/cm was found. Though, an offset due to the use of a factice switching cell, with geometry different from that of the real one, was found.

F. Conclusion

This study dealt with the influence of the distance between a switching cell and its decoupling capacitors.

As expected, minimising inductances requires to put components as close to one another as possible. However, this study showed that the simple layout we used made loop inductances little sensitive to the distance between the cell and the decoupling capacitors.

Using a thin and grounded PCB is of primal importance to limit loop inductances. Furthermore, a capacitor's ESL can account for a major part of the overall loop inductance.

This study shows that, using the layout under study, the loop inductance sensitivity to the distance between a switching cell and its decoupling capacitors is rather low (about 300 pH/cm). Additionally, it showed that the DC-link capacitor can be placed quite far from the cell. These results highlight the relative freedom designers have regarding the arrangement of the components of a switching cell and its bypass network, thus facilitating, in particular, thermal design.

IV. CONCLUSION

The first part of this study shows that, for devices with fast enough switching speeds, two mechanisms must be considered to predict oscillations. The first one results in an overvoltage that is as high as the full bus voltage 𝑉 𝑏𝑢𝑠 whereas the other one induces an overvoltage of amplitude proportional to the load current.

The distance between the switching cell and its decoupling capacitors was then varied and the resulting stray inductance was measured both in frequency-and timedomains, using either an impedance analyser or an oscilloscope. The results show that the stray inductance remains rather low (< 11 nH) even when the capacitor is far away from the switching cell (30 cm). Furthermore, the sensitivity of this inductance vs. the distance is very low: about 0.3 nH/cm. Choosing a capacitor with low selfinductance (ESL) is therefore most important so as to limit the overall loop inductance. Constrains on the compactness may then, in some cases, be loosen so as to improve thermal management and system integrability.

Fig. 1 .

 1 Fig. 1. Schematics of the converter under study: switching cell (𝑄 𝑋 ), bypass capacitors (𝐶 𝑋 ), parasitic inductances (𝐿 𝑋 ), input voltage (𝑉 𝑏𝑢𝑠 ) and output load.

  This work was supported by a grant overseen by the French National Research Agency (ANR-15-CE05-0010)project HIT-TEMS.

Fig. 2 .Fig. 5 .

 25 Fig. 2. Simplified schematic of the converter, with parasitic elements

Fig. 4 .

 4 Fig. 4. Asymptotic impedance diagram of the bypass network (Fig. 3), seen from the load. 𝐿 3 = ∞, Log-log scale.

Fig. 3 .

 3 Fig. 3. Converter schematic with 𝑄 𝐿𝑆 off and 𝑄 𝐻𝑆 on.

Fig. 8 .

 8 Fig. 8. Measured switching node voltage. Logarithmic X-scale.

Fig. 6 .

 6 Fig. 6. Equivalent circuit during the current switching.

4 Ω

 4 close to the value predicted by our model for the current step (√

Fig. 9 .

 9 Fig. 9. Simplified circuit layout (bottom layer ground plane not shown).

Fig. 10 .

 10 Fig. 10. Typical switching waveform: measurement (blue) and fit (red). 𝑛 𝐿 = 3. Logarithmic X-scale.

Fig. 11 .

 11 Fig. 11. HF loop inductance (𝐿 1 ) vs. 𝑑 𝑋7𝑅 and LF loop inductance (𝐿 2 ) vs. 𝑑 𝑏𝑢𝑙𝑘 .

TABLE 1 .

 1 Maximum rise time so that the steep-edge assumption is valid vs (𝐶 𝑜𝑢𝑡 , 𝐿 1 ).

	𝑪 𝒐𝒖𝒕	𝑳 𝟏	𝑡 𝑟,𝑚𝑎𝑥𝑖 = 2√𝐶 𝑜𝑢𝑡 ⋅ 𝐿 1
	100 pF	1 nH	0,63 ns
	1 nF	1 nH	2 ns
	1 nF	10 nH	6,3 ns
	10 nF	100 nH	63 ns

Fig. 7. Measured switching node voltage (zero-padding interpolated). Log. x-scale.

  𝑉 𝑑𝑐 , 𝑉 ̂𝑘, 𝜔 𝑘 , 𝜙 𝑘 , 𝜏 𝑘 ] 𝑘

	447 nF, 𝐶 𝑜𝑢𝑡,30V 𝑑𝑐 = 571 pF 𝐶 𝑏𝑢𝑙𝑘,30V 𝑑𝑐 = 14.88 µF,	(10)
	A model describing the oscillations of 𝑛 𝐿 loops:
	𝑛 𝐿
	𝑉 𝑠𝑤 * = 𝑉 𝑑𝑐 + ∑ 𝑉 ̂𝑘 ⋅ 𝑠𝑖𝑛(𝜔 𝑘 𝑡 + 𝜙 𝑘 ) ⋅ 𝑒 -𝑡/𝜏 𝑘	,
	𝑘=1
	𝛽 = [