On the Size and Weight of Passive Components: Scaling Trends for High-Density Power Converter Designs

High-performance power electronics design requires a firm characterization of active and passive components. This work presents a framework for quantifying passive component performance by reviewing both existing methods and robust device figures-of-merit (FOM). A comprehensive survey yields aggregated data for nearly 700 000 commercial capacitors and inductors of all types. To supplement deficiencies in this data, this work proposes and validates several empirical expressions to estimate passive component energy storage and mass. The estimation of volumetric mass density per component type allows the approximation of component mass from accessible box volume. The estimation of energy-equivalent capacitance in nonlinear Class II ceramic capacitors facilitates the evaluation of stored energy and related energy density FOM. A phenomenological analysis of the comprehensive component data produces several conclusory determinations about peak energy density capabilities—with respect to volume, mass, and cost—across capacitor and inductor technologies.

passive devices to electromagnetically store and release energy (e.g., capacitors, inductors, and transformers).A comprehensive understanding of the breadth and capabilities of these devices is required to design and realize a physical converter of desired conversion efficiency, volume, mass, cost, lifetime, and dynamic performance.Both novel and mature component technologies constantly improve over time, and the best-suited device for a particular application depends on the system specification, and even more intricately, on other selected devices.
This work extends our previous conference paper [4] by aggregating a comprehensive set of capacitor and inductor device data and motivating useful figures-of-merit (FOM) for comparison and extension to design.The rest of this article is organized as follows.Section II describes general methods for device-level characterization and introduces a framework for producing robust, or maximally applicable, device FOM.Employing comprehensive data collection, this work surveys a useful breadth of over 606 000 commercial capacitors and 88 000 commercial inductors as detailed in Section III.Additional sampled data are garnered to supplement and augment particular deficiencies in this large dataset.To enable complete characterization of energy storage metrics-the critical benchmark of capacitive and inductive energy storage elements-a sampled set of capacitor data is collected and extrapolated to the full dataset as described in Section IV.To estimate component mass, a sampled set of components is measured and extrapolated to the full dataset as described in Section V.After applying the supplements to energy storage and mass, Section VI explores the veracity of the data through visualization, and investigates several device FOM for all surveyed components.As a demonstration, the analysis compares the energy densities of various capacitor and inductor technologies and additionally specifies the conditions for voltage overrating in capacitors and current overrating in inductors.Finally, Section VII concludes this article.

II. CHARACTERIZING COMPONENTS AND DEFINING PERFORMANCE
Making reductive determinations from millions of passive components requires careful consideration of how data are collected, and then, manipulated into useful quantitative metrics that describe comparative performance tradeoffs.The goal is to produce founded statements such as "based on the present available technology, the smallest possible capacitor solution for this application has volume X." To substantiate these claims, this work first introduces viable methods of data aggregation and analysis, then motivates useful device FOM.

A. Data Analysis Methods
There are three general methods to determine the broad capabilities of a set of circuit components, where sets are classified by the distinctive "type" or "technology" of the component.
1) Analytically Derived Performance: Utilizing a firstprinciples approach based on physics, one can derive analytical expressions for the lumped circuit model from its internal geometries and constituent material properties.Further determinations are either made directly from the analytical expressions, iteratively fit to measured data, or generated using a probabilistic Monte Carlo simulation.Examples of analytic methods include estimation of the "macroscopic" hysteretic losses of Class II ceramic capacitors from "microscopic" properties in [24], [25], and [26]; quality factor prediction of air core inductors in [27]; derivation of frequency-dependent volume and loss scaling trends in inductors in [1]; and estimation of parasitics for transistors in [28].
2) Sampled Data and Extrapolation: Data collection, often measured, can completely characterize circuit phenomenon for small datasets.However for large sets of intractable or unknown data, intelligent data sampling can yield exceedingly meaningful qualitative and quantitative insight.With additional care, the sampled data extrapolate to larger supersets of data-especially with the advent of recent machine-learning (ML) techniques.This sampled data method is applied to predict capacitor equivalent series resistance (ESR) and quality factor in [3] and [29]; capacitor lifetime in [30], [31], and [32]; power loss in inductors in [27] and [33]; and power field-effect transistor (FET) losses in [28] and [34].Another variation of this sampled data approach trains ML models to predict core losses in broadly excited inductors [35], [36] and to predict losses in transistors [37].
3) Comprehensive Data Collection: Electronics distributors and even some manufacturers/suppliers have in recent years drastically broadened the interactivity of their consumer-facing interfaces and design tools.This proliferation of digitized data and its increasing ease of access enable large-scale, comprehensive, and practically exhaustive collection of component information.A comprehensive consideration of data necessarily enables conclusive interpolative-rather than extrapolativequantification of performance, even notoriously mercurial metrics such as cost.A comprehensive data collection approach in [38] is used to benchmark performance among commercial high power transistor technologies (> 1 kV and > 1 kA).A recent approach in [37] uses data from commercial capacitors, inductors, and transistors to train ML models and produce optimal converter designs [39].
This work primarily employs the comprehensive data collection approach for device characterization.In aspects where this method stalls, sampled data and extrapolation is utilized to augment the comprehensive dataset.

B. Defining Useful Device FOM
Meaningful device metrics, deemed FOM, must be developed in conjunction with bulk device characterization.A good FOM is a quantitative measure of performance and indisputably indicates better performance for larger (or smaller) values, similar performance for equivalent values, and worse performance for smaller (or larger) values.Although not often accentuated, the FOM philosophy intrinsically permeates the field of engineering and enables quantitative benchmark and comparison of complex systems [40].These metrics provide a common language for engineers to judge a solution's capabilities or the evolution of a technology [41].Applied to power electronics-a system comprised of many smaller subsystems-it is possible in principal to relate converter-level FOM to constituent device-level FOM to constituent material-level FOM [28], [42].
The "system scope" in this work is a discrete passive component.Others have motivated specific FOM for quantifying the performance of individual electronic devices: capacitors [43], inductors [33], and transistors [28], [44], with the intention of characterizing component application to larger circuits, however, no work has generalized a method for conceiving useful device FOM.
To accomplish this, consider how these devices are utilized.Two-terminal passive devices are commonly configured into series and/or parallel connected component banks to meet specified requirements often relating to energy storage or power throughput.Series-parallel modularity is even common for electrical systems such as power conversion circuits [40], [45] and photovoltaic panels [46], [47].Consequently, any metric defined as an FOM in this work critically adheres to the following proposed property: Property of series-parallel modular invariance: A device metric invariant to series and/or parallel configuration of the device.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I GENERIC CAPACITOR BANK SPECIFICATIONS AND CONSTRUCTED FOM
If a metric adheres to this property, then it is an FOM that can fairly compare devices of various voltage and current ratings and is deemed "robust." The series-parallel modular invariance property is demonstrated schematically in Fig. 1 and generalized in Table I for a series and parallel configuration of a capacitor with specified capacitance C, rated dc voltage V r , rated rms current I r , ESR, box volume, cost, and mass.In this context, combinations of these base attributes yield some robust FOM, which satisfy this property and are agnostic to both series and parallel component configurations-e.g., volumetric energy density γ v , gravimetric power density ρ m , and loss tangent tan δ [or dissipation factor (DF)].
Because these capacitor and inductor FOM are suitable candidates for component comparison across the entire device subspace, future research explorations of these metrics could consider tradeoffs and derive connections to the devices' broader system application [64].

D. Limited or Operating Condition Specific Capacitor Metrics
Some common-use device metrics, especially for capacitors, are only parallel modular, and thus, partially satisfy the seriesparallel modular invariance property.These metrics can be utilized for device comparison, but require a more restrictive and judicious context, most commonly by only comparing capacitors of a specific rated voltage V r .Some examples include current density I r Vol and I r Mass [48], [74]; capacitance-related current density C I r [74]; volumetric efficiency or charge density C•V r Vol [48]; capacitance density, volumetric capacitance, or capacitance volumetric efficiency C Vol [31], [60], [75], [76], [77]; capacitance voltage product per rated current C•V r I r [48]; cost per farad Cost C [61]; and specific capacitance C•V r Mass [41], [43], [52].

E. Limitations of Series-Parallel Device Configurations
Arbitrarily configuring discrete components in series and in parallel has associated practical limitations: increased layout inductance, asymmetrical current distribution, unbalanced voltage distributions, and lower packing factor [60], [78], [79].For the purposes of this FOM analysis, these shortcomings-which can be mitigated with conscientious design-are neglected.

III. COMMERCIAL CAPACITOR AND INDUCTOR DATA
Desired device FOM are constructed from base metrics, and thus, the greater acquisition of base metrics directly enables the determination of more FOM.This section describes the availability and extent of the surveyed data for discrete commercial passive components.It also delineates the capacitor and inductor typologies used throughout this work.

A. Extent of Available Data-Capacitors
The present distributor datasets contain certain practicable information with varying degrees of consistency: (near) fully available, partially available, or not available.Satiating a partially or unavailable component attribute can enable the determination of secondary metrics (e.g., rated stored energy E r and volume) and tertiary FOM (e.g., energy and power density).As discussed in Section II-A, this requires either supplemental measured data with extrapolation, or theoretic generalization of the component derived from material properties.
The comprehensive dataset consists of an aggregation of roughly 606 000 distinct capacitors from the prominent distributor Digikey Electronics.
1) Fully Available Data: Readily available data include several primary attributes: capacitor rated voltage V r ; the zerovoltage differential capacitance C(0) [24]; the dimensional parameters: length, width, height, and diameter; and the cost per unit.The secondary attribute "box" or enclosure volume is computed from the dimensional attributes.For linear capacitors, the secondary attribute of rated stored energy E r is calculable in aggregate from these base attributes.However, the prominent Class II ceramic capacitor technology has a nonlinear voltagedependent capacitance characteristic [24], and thus, E r cannot be calculated for all capacitor technologies without additional analysis presented in Section IV.
2) Partially Available Data: The ESR; loss tangent or dissipation factor tan δ = DF; rated rms current I r ; and lifetime L 0 are all critical attributes for any quantitative performance analysis of capacitor loss and reliability, however, these metrics are unavailable in the overall distributor dataset for most capacitor technologies except for some aluminum electrolytic capacitors.Some of these base attributes also maintain a pertinent frequency and temperature dependence, and are inconsistently standardized across manufacturers.Thus, even though correction factors are sometimes disclosed, the oftentimes singular values for ESR, tan δ, I r , and L 0 common to catalogs and datasheets are often inadequate for involved electronics design.
To generalize and predict losses, some have collected sampled measurements for realistic, large-signal operating conditions and the abstracted broad capacitor trends as a function of frequency, dc voltage bias, and temperature [3].Others propose an empirical loss equation for capacitors [24], [25], [26], being the dual of the venerable Steinmetz equation for inductors [80], [81].These models depend on the capacitor's excitation waveforms, and could directly integrate with a comprehensive component dataset to generalize loss, however, this requires more extensive data sampling and more study.
3) Unavailable Data: Some primary attributes are essentially unavailable in the distributor dataset.For instance, the component mass would be invaluable for quantitative evaluation of weight-optimized power conversion systems, however, presently this information is digitally available for only a select few suppliers.To reconcile the deficiency in mass data, Section V applies sampled measured data to estimate the mass of all capacitor components as a function of its type, rated voltage V r , and capacitance C.

B. Surveyed Technologies-Capacitors
Commercially viable capacitors are constructed in a variety of technologies best suited for particular electrical applications [48], [82].Technologies are most easily distinguished by the dielectric material where certain capacitor types excel in cost, reliability, high-frequency capability, voltage and current ratings, mass, and volume.In this work, the capacitors in the comprehensive dataset are classified into subsets with distinct elements: aluminum electrolytic, tantalum electrolytic, Class I and Class II ceramic, film, and electrolytic double layer.Brooks and Pilawa-Podgurski [83] clarify the precise distinctions between these capacitor technologies as applied in this work.Fig. 2 illustrates, for the surveyed data, a conventional differentiator between capacitor technologies: the range of possible capacitance C and rated dc voltage V r .
Niobium electrolytic, mica, and silicon capacitors are other notable capacitor technologies.However for these types, the quantity of commercial devices is small and the relevant data are sparse, thus, they are not considered in greater detail within this work.

C. Extent of Available Data-Inductors
Similar to capacitor components, the base attributes for inductors have varying degrees of availability in the comprehensive dataset.Data on roughly 88 000 distinct inductors were aggregated in total from the distributor Digikey Electronics.
1) Fully Available Data: Readily available data include several primary attributes: inductance at zero bias current L; inductor thermal rated rms current I rms (conventionally at a 40 • C increase in temperature); peak saturation current limit I sat (often defined at either 20% or 40% inductance derating); the dc resistance DCR; the dimensional parameters: length, width, height, and diameter; and the cost per unit.The "box" volume and the rated stored energy E r are calculable in aggregate from these base attributes.
2) Partially Available Data: The ac quality factor Q ac = 2πf L R ac FOM quantifies the ideality and damping of the inductor and predicts losses [66].Manufacturers measure and report Q ac for a small-signal excitation at a singular test frequency f , however, this attribute is only partially available (roughly 60% of aggregated inductors) in the comprehensive dataset.In addition, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.roughly 20% of surveyed inductors have undefined or poorly defined core material; these were discarded from the analysis.
3) Unavailable Data: As with capacitors, the component mass of inductors is largely absent from the comprehensive dataset.In Section V, sampled measured data are used to estimate the mass of all inductor components as a function of its type, rated current I r , and inductance L.

D. Surveyed Technologies-Inductors
Analogous to capacitors, inductors are most distinguishable by the type of core material: ferrite, metal, or nonmagnetic/air.The core material significantly impacts suitable operating frequency ranges, core losses, and saturation limits for magnetic flux.Again, Brooks and Pilawa-Podgurski [83] clarify the precise distinctions between these inductor technologies as applied in this work.Fig. 3 illustrates the inductance L and rated current I r for each commercial inductor technology.

IV. SAMPLED DATA AND EXTRAPOLATION: STORED ENERGY OF CLASS II CERAMIC CAPACITORS
Class II ceramic capacitors are a unique component type that requires special consideration in data analysis.Their significant voltage, temperature, and age dependence makes the determination of certain metrics circuitous compared to stable capacitor technologies.However, Class II multilayer ceramic chip (MLCC) capacitors are a particularly good capacitor choice for electronics due to their comparatively low losses, high energy density, and widespread applicability [3], [25].A discussion of the best capacitor technologies is markedly incomplete without the inclusion of this capacitor type, thus intentional effort is exerted to determine their rated stored energy E r and compute energy density FOM γ.
In this section, an empirically derived fit is shown to accurately estimate the voltage-dependent stored-energy-equivalent capacitance C E (v) at any dc voltage v, specifically at the rated dc voltage v = V r .This fit only depends on the differential capacitance C(v) known at two values: v = 0 V and v = V r .The approximation is validated by using datasheet information from a sampling of 2 550 MLCC capacitors manufactured by the TDK Corporation.

A. Standards
Class II ceramic capacitors are primarily distinguished by an associated alphanumeric code indicating some information about the temperature characteristic (TC) or temperaturevoltage characteristic (TVC).There are three primary standards codifying these nonlinear characteristics.
1) EIA RS-198: The most common in use (e.g., X6S, Y5V, and C0G) but only specifies TC information [84].2) IEC/EN 60384-1: Less commonly used (e.g., NP0, 2X1) and specifies information about TC and TVC [85].3) MIL-C-11015: Military standard that specifies TVC information [86].For all standards, codes indicate information about temperature range, expected capacitance derating at these temperature limits, and expected capacitance derating at rated voltage.Unfortunately, when using the prominent EIA standard, any two capacitors with the same code (e.g., X6S), and thus, similar TC do not necessarily have similar or even necessarily correlated TVC [24].

B. Defining Capacitance and Stored Energy
Regardless of the TC or TVC, a coherent definition of energy storage for nonlinear capacitors is necessary to eventually deduce energy-related FOM.For a general capacitor, the stored energy E is completely defined at an applied dc voltage V a as where C(v) is the characteristic incremental, small-signal, or differential capacitance [24], [25], [87], [88] defined as For a linear capacitor, the differential capacitance C is constant with applied voltage (and temperature).Thus, by evaluating (1), the integral equation for stored energy at an applied dc voltage V a simplifies to which is notably invalid for voltage-dependent capacitors since the C(v) varies with voltage.
The stored energy E a of a nonlinear capacitance at an applied voltage V a can instead be equivalently defined using an effective energy-equivalent capacitance Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Equating (1) and ( 5), the energy-equivalent capacitance C E (v) [87] can be computed as a function of the differential capacitance curve C(v) as follows: For linear capacitors with voltage-invariant capacitance C, the energy-equivalent capacitance reduces simply to C E = C.

C. Data Acquisition
Within the context of programmatic data acquisition, there are varying degrees of information available that can help determine the energy-equivalent capacitance C E (V r ), and thus, rated stored energy E r and energy density γ. 1) The first three attributes, C 0 , V r , and TC code, are defining characteristics of every capacitor and are readily available in the comprehensive distributor dataset.The full C(v) curve as well as differential capacitance C r at rated voltage are not directly available from distributors, however they are often available on datasheets.The TVC code is rarely available anywhere, including datasheets.
The TDK Corporation, a prominent capacitor manufacturer, publicly provides digitized differential capacitance C(v) data for their Class II MLCC components in conjunction with C 0 , V r , and TC.Although not a sufficiently comprehensive survey of all Class II ceramics, a sampling of roughly 2 550 TDK Class II MLCC components informs several meaningful insights for the Class II ceramic capacitor technology as a whole.

D. Stored Energy Approximation: Using Temperature Characteristic
It would be convenient to approximate rated energy E r in (1) or the rated energy-equivalent capacitance C E (V r ) in ( 6) without the express requirement of the entire C(v) characteristic curve, which cannot presently be attained en masse.One potential method is to identify general trends in the C(v) curves of capacitors with specific temperature characteristics (e.g., X6S and X7R).For instance, X6S capacitors could have an approximate 70-90% capacitance derating at V r , whereas X5R capacitors could have an approximate 60-70% capacitance derating.Such an identifiable relationship would aid estimation of E r with sparse information.Prior work has investigated the existence of a practicable linkage between TC and capacitance-voltage dependence [89], [90].
The relative shapes and values of the C(v) curves are aggregately visualized in Fig. 4 to identify patterned correlations with the TC and determine whether a TC-dependent scheme has plausible utility.All C-V curves are normalized as C(v)/C(0), and the differential capacitances generally derate with a characteristic logistic or mirrored "S" shape (for a log-linear plot).Visual inspection yields some groupings or families of C(v) curves for similar TC indicating similar dielectric materials, however they are not visibly distinctive enough to determine any generalized correlations between TC and C(v); consequently, a different method must be employed to approximate E r .

E. Stored Energy Approximation: Using C(0) and C(V a )
The effective energy-equivalent capacitance C E (V a ) at applied voltage V a could possibly be estimated knowing only at most two values: C(0) and C(V a ).Different approximations of C E (V a ) are presented and evaluated compared to the exact value in (6).
1) Zeroth-Order Approximation: Most simply, the energyequivalent capacitance C E at an applied voltage V a can be approximated evaluating C(v) at its limits as or as These estimates roughly serve as upper and lower bounds, respectively, on the actual C E (V a ).
2) First-Order Approximation: For this approximation of C E (v), the differential capacitance C(v) is approximated by a linear fit, or first-order approximation, between the zero-voltage capacitance C(0) and the differential capacitance at the applied voltage C(V a ) as follows: Directly evaluating (6) using (9) yields Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.3) Power Mean Approximation: Finally, C E (V a ) is approximated with a special average function.The power mean (or Hölder mean) M p is a family of functions, which averages n positive numbers x 1 , x 2 , . .., x n as where the exponent p is some real nonzero number [91], [92].This power mean is equivalent to other well-known means for particular values of p: the arithmetic mean for p = 1, the quadratic mean or root mean square for p = 2, and the harmonic mean for p = −1; it is also related to the p -norm of a vector for integer p ≥ 1 [93].
A good value of p results in the best approximation of C E (V a ) at every applied dc voltage 0 < V a < V r utilizing only the endpoints of the differential capacitance curve C(0) and C(V a ) or as 6), the energy-equivalent capacitances at rated voltage C E (V r ) are computed for all 2 550 sampled Class II MLCC capacitors from TDK.Then, regression is applied to fit this data to (11) yielding a best fit value p = −0.504≈ −0.5 and an empirically derived approximation 4) Results: All C E (v) approximations are graphically compared for a particular device in Fig. 5 across applied voltage V a .From inspection, the proposed first-order and power mean approximations in (10) and (12) very nearly match the general waveshape of the actual C E (v) curve evaluated from (6).
To prove their efficacy, these estimates must also be validated for the entire sampled dataset-not just a single component.The relative accuracy of an estimate is judged by its mean percentage

TABLE II PERCENTAGE ERROR
for N elements in a set where x = C E (V r ).Fig. 6 presents the percentage error of each C E (v) approximation for all 2 550 sampled TDK components; the resulting mean and median percentage error of each approximation are tabulated in Table II.The zeroth-order approximations ( 7) and ( 8) result in prohibitively high estimation inaccuracy, but the power mean approximation in (12) has a low MPE of 3.1% (median percentage error of 1.8%), sufficiently validating its usage among the others.This method suggests if ever a manufacturer, supplier, or distributor reports the differential capacitance values of Class II ceramic capacitors at both zero dc voltage bias C(0) and at rated dc voltage bias C(V r ), then the rated stored energy E r , and consequently, energy densities γ, at rated voltage V r could be estimated with a relatively high degree of accuracy using (12) without requiring the full C(v) curve, temperature characteristic, or temperature-voltage characteristic.

F. Investigating Energy Density FOM of Sampled Data
The FOM framework proposed in Section II can be applied to the sampled TDK dataset.The exact rated stored energy E r , as well as the volumetric energy density Vol , of each capacitor is computed as (1).Recall that the series-parallel modularity invariance property of an FOM allows every capacitor, regardless of rated voltage, to be fairly compared.Fig. 7 shows the consequent impact of TC and energyequivalent capacitance C E (V r ) approximation on γ v with respect to rated voltage V r .Despite some TC clustering, the relationship between TC and γ v in Fig. 7(a) is not correlative enough to predict a TC code that has the smallest volume nor the γ v of any individual capacitor based on its TC.The associated Pareto fronts for each approximation method are also included relative to the exact Pareto front of volumetric energy density γ v in Fig. 7(b).If using the simplest zeroth-order approximation in (7), the highest energy density capability of Class II ceramic capacitors can be overestimated by as much as 1000×.The recommended power approximation in (12) sufficiently estimates γ v .

G. Extrapolating Sampled Data to the Full Dataset
This analysis concludes that energy storage estimation of Class II ceramic capacitors is possible if the differential capacitance at two voltages, 0 V and V a , are known.However, as previously mentioned, the comprehensive dataset only contains C(0) data for all capacitors.Thus, for the remainder of this work, the energy equivalent capacitance C E (V r ) at rated voltage is uniformly approximated as 60% of C(0)-an improvement over the zeroth-order estimate of (7)-to estimate the rated energy storage E r of Class II ceramic capacitors.This approximation has an MPE of 40% applied to the TDK dataset.This is an inaccurate and dissatisfying estimate, but reasonable considering the logarithmically wide breadth in energy densities across the technology as shown in Fig. 7.If capacitor manufacturers, suppliers, and distributors begin to report values for C(V r )-doubly useful because it conveys small-signal capacitance at rated voltagethen C E (V r ) can be estimated with a high degree of accuracy using the power mean in (12), fully enabling estimation of rated energy storage E r for all capacitors, both linear and nonlinear, and satisfyingly quantifying those device FOM related to energy.

H. Application to Nonlinear Inductors
Analogous to Class II ceramic capacitors, ferrite and metal core inductors have a nonuniform differential inductance L(i) dependent on the applied current i.Consequently, the peak energy density of inductors reported in this work (see Figs. 11, 13, and 14) utilizes an approximation for energy-equivalent inductance L E similar to the zeroth-order approximation for energy-equivalent capacitance (8).Calculable exact peak energy storage at rated current I r -or even more accurate estimationis not possible without a sufficiently large sampled dataset of L(i) curves; unfortunately, no present manufacturers have digitized L(i) curves available en masse.

V. SAMPLED DATA AND EXTRAPOLATION: CAPACITOR AND INDUCTOR MASS
Although minimized volume and cost are often desired, minimizing the system mass can also be a critical need for electronics applications.In particular, innovations in the mass reduction Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Vol of each capacitor in the comprehensive dataset is empirically estimated with the power fit described in (21) and parameters in Table III.The gravimetric energy density γ m is then computed by transforming the volumetric energy density γ v in (15).The resulting transformation is subtle, yet significant at the highest energy densities.for mobile electronics-most commonly electric aircraft or automobiles-are driven by sustainable energy targets for the growing electric propulsion industry [94], [95].Electronics for space applications also prioritize lightweight designs.In these applications, even the choice of converter topology is informed by the achievable mass of the system and the requisite masses of the constituent devices [96], [97], [98], [99].The mass of commercially available capacitors and inductors remains largely indeterminable en masse as few manufacturers supply this information.Thus, an insufficient proportion of component mass is known for the comprehensive dataset surveyed in this work.
This section expands on the analysis introduced in [4].Determining the volumetric mass density (mass per volume) D of a broadly sampled set of passive devices through measurement yields sufficient information to extrapolate and transform readily available volume data to an estimated value of mass for each component.This work presents generic fits for both capacitor and inductor density D, with relevant empirical parameters provided for all major capacitor and inductor technologies as described in Section III.

A. FOM Transformation-Theory
The volumetric mass density (or density) of any component is an intrinsic FOM relating its mass to its volume.Incidentally, the density D (as well as mass) for every component in the Vol of each inductor in the comprehensive dataset is empirically estimated with the power fit described in (22) and parameters in Table III.The gravimetric energy density γ m is then computed by transforming the volumetric energy density γ v in (15).The resulting transformation is subtle, yet significant at the highest energy densities.comprehensive dataset is unknown.However, a single estimated value of D may apply to an entire component technology by relying on homogeneity in material composition and construction.This value of D can be further refined by empirically fitting its dependence to known component attributes: the capacitance C and rated dc voltage V r for capacitors, and inductance L and rated dc current I r for inductors.
Once a reliably accurate density mapping D is known, then a volume-related FOM for an individual component can be transformed into a mass-related FOM as The volumetric energy density FOM γ v for each passive component is defined as a ratio of rated energy E r and volume as follows: whereas the gravimetric energy density FOM γ m is defined as a ratio of rated energy E r and mass as follows: The rated stored energy metric E r is computed for capacitors from the energy-equivalent capacitance C E in (6) and rated dc voltage and is computed for inductors from the inductance L and rated dc current The inductor's rated dc current I r is defined as the minimum of the thermal rms current rating I rms and the peak saturation current rating I sat as specified on the datasheet, or Fig. 8 illustrates this transformation principal in (15) by mapping a set of capacitor volumetric energy density data γ v to a set of gravimetric energy density data γ m utilizing a density D that is dependent on C and V r .

B. Volumetric Mass Density Measurement
The mass and box volume of 315 unique capacitors and 132 unique inductors were measured to construct empirical estimates for the density D. To capture a sufficiently diverse spread of device variants, capacitors were selected to encompass a vast range of package sizes, rated dc voltages V r , capacitance C, and manufacturer.Inductors were similarly selected to encompass a breadth of package sizes, rated current I r , inductance L, and manufacturer.For each unique component, multiple samples were weighed based on the accuracy of the electronic scale to limit the relative measurement error to a maximum of 0.5%.Fig. 9 shows the measured density D as a function of C and V r for each capacitor in Fig. 9(a) and as a function of L and I r for each inductor in Fig. 9(b).

C. Empirically Fitting Density Measurements
Two empirically derived fits are proposed by utilizing the measured density data: 1) a mean or constant fit and 2) a power fit dependent on base electrical attributes.
1) Mean Fit Estimate: Passive components are separated by type, then a mean fit for density D is calculated for each with a zeroth-order linear regression.These mean densities are indicated in Table III along with their 95% confidence intervals Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE III RESULTS OF PROPOSED MEAN FIT AND EMPIRICAL POWER FIT FOR CAPACITOR/INDUCTOR VOLUMETRIC MASS DENSITY (MASS/VOLUME)
assuming a normal distribution of the measured data.By ranking of least to most dense, the component technologies are ordered as film capacitor, air core inductor, aluminum electrolytic capacitor, ferrite core inductor, tantalum electrolytic capacitor, Class I ceramic capacitor, metal core inductor, and Class II ceramic capacitor.
The relative accuracy of a fit is judged by its MPE defined in (13), where x is the density D. The computed MPE is less than 30% for every mean fit tabulated in Table III, indicating a fairly accurate fit between the measured and the mean fit approximation.The MPE can be further reduced by utilizing a more accurate power fit model dependent on base component attributes (e.g., C, V r , L, and I r ) fully available in the comprehensive dataset as described in Section III.
2) Power Fit Estimate: A possible correlation between density D and rated dc voltage V r and capacitance C (for capacitors) emerges from inspection of the measured densities in Fig. 9. Taking inspiration from classical empirical fits for loss in passive components [25], [81] while remaining conscientious of statistical uncertainty in inferential models [100], this work proposes a power fit expression for density D as with inputs V r and C, and empirical parameters k, α, and β.An analogous power fit expression is proposed for inductors with inputs rated dc current I r and inductance L, and similar empirical parameters k, α, and β.Linear regression, applied to the logarithm of ( 21) and (22), is used to determine the best fit parameters k, α, and β, which minimize the MPE [101].Table III tabulates the resultant fit parameters (including the 95% confidence interval of each in parenthesis) of the power fit for each component technology as well as the associated statistical p-values and MPE.
The p-value indicates the occurrence probability of the best-fit parameters when assuming the null hypothesis-in this case, a constant fit D with α = 0 and β = 0-to be true.All statistical p-values-except for ferrite and air core inductors-are much lower than a typical significance threshold of 0.05, confirming the modeled power fits, with the specified parameters, are statistically significant [102].
Observing the empirical fit results, the densities of some capacitor technologies have a positive correlation with rated voltage V r (e.g., tantalum electrolytic, Class I ceramic, and film), while some have a negative correlation (e.g., aluminum electrolytic).Empirical trends also exist with respect to the capacitance C: positive correlation for tantalum electrolytic, Class I ceramic, Class II ceramic, and film; and negative correlation for aluminum electrolytic.Similarly, metal core and air core inductor density correlates with respect to either rated dc current I r or inductance L. The fit for ferrite core inductors insufficiently describes the density of the measured data; this likely results from wide variability in the constructions of this inductor type as well as high variance in the fill factor of the box volume used for the density calculation.In general, the typical intrinsic materials and construction for each component technology cause these parametric dependencies in capacitance C, voltage V r , inductance L, or current I r .These correlations produce interesting implications for present and future innovations in material science and packaging, however, this work does not explore these implications further and merely suggests perceived trends from phenomenological observation of the measured data.
3) Comparison Between Fits: The power fit estimation further reduces the prediction error compared to mean fit estimation.As a result, designers can either utilize the mean fit estimation to quickly predict a component's density D, and thus, its mass and gravimetric energy density γ m , or utilize the power fit estimation to improve the predictive accuracy without much added inconvenience.

D. FOM Transformation-Application
Now buttressed with substantive estimates for component density D, the transformation theory introduced in Section V-A is applied to extrapolate box volume to mass for the comprehensive passive component dataset introduced in Section III.
The visualization in Fig. 10 applies the FOM transformation in (15) and Fig. 8 to the comprehensive dataset by utilizing the power fit expression for density D in ( 21) and (22) with best fit empirical parameters in Table III.In Fig. 10(a), the rated dc voltage V r versus volumetric energy density γ v of every  commercially surveyed capacitor is distinguished by technology.Fig. 10(b) depicts the gravimetric energy density γ m , after applying the capacitance and voltage-dependent estimation for D(V r , C).After the density transformation, the shape of each component technology set marginally distorts and the sets shift relative to each other.
An analogous volume-to-mass transformation is performed for the commercial inductors in the comprehensive dataset.The volumetric energy density γ v in Fig. 11(a) maps to gravimetric energy density γ m in Fig. 11(b) with the inductance and currentdependent power fit estimation for density D(I r , L) in (22).
With the analysis proposed in this section, all device FOM comprised of mass can be estimated to a high degree of accuracy using the expressions in ( 21) and ( 22) with empirically derived parameters in Table III, fully enabling estimation of mass for all capacitors and inductors.The energy density FOM of the comprehensive data are more deeply investigated in Section VI.

VI. ANALYZING THE DATA
The comprehensive dataset of commercial passive components has been clearly defined and its energy and mass deficiencies bolstered in Sections IV and V, respectively.With this foundation set and a utilitarian FOM framework established, the data can be freely interpreted and analyzed.It is possible to infer fundamental limitations intrinsic to a component technology as well as quantified comparisons between technologies, even between capacitors and inductors.The prototypical question posed in Section II ("Based on the present available technology, the smallest possible capacitor solution for this application has volume X.") now has a determinable answer.From Fig. 10(a), Class II ceramic capacitors are the capacitor technology with the smallest volume (with respect to energy storage), since they have the highest volumetric energy density.Similar quantifiably supported claims will be made throughout this section, including application to optimally choosing capacitors (or inductors) with an overrated voltage (or current).

A. Capacitor Voltage Overrating
For an applied capacitor voltage V a , selecting a capacitor with a greater rated voltage V r -called "voltage overrating"-is often necessary depending on temperature and lifetime requirements [103] but, interestingly, can also be worthwhile to improve the realized capacitor volume, mass, or cost.The following analysis quantitatively determines when this design strategy has likely volume, mass, or cost benefit.
Recall from (3), the stored energy in a particular (linear) capacitor at an applied voltage V a is expressed as thus the consequent applied volumetric energy density γ v,a of the underutilized capacitor is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE IV SUMMARY OF CRITICAL VOLTAGES FOR CAPACITOR OVERRATING
This derated energy density γ v,a scales quadratically with the applied voltage V a justifying its +40 dB/decade slope indicated by the dashed lines in Fig. 12(a).
Besides enabling juxtaposition of individual capacitors, the empirical data also indicate the peak performance capability of the whole technology.
The "best" capacitor has the best FOM and lies on the Pareto front of the comprehensively surveyed dataset.Fig. 12(a) illustrates the empirically derived Pareto fronts f P (V r ) for volumetric energy density γ v as a function of rated voltage V r for major capacitor technologies: aluminum electrolytic, tantalum electrolytic, Class I ceramic, Class II ceramic, and film.For each capacitor dataset, the derivative of its Pareto curve f P (V r ) can exceed γ v,a (V a ) the derivative of the derated energy density curve in ( 24) is below a specific critical rated voltage V r,crit : Evaluating this inequality for components on the Pareto curve (γ v = f P (V r )) and at a rated voltage application (V a = V r ) yields The critical rated voltage V r,crit is the rated voltage satisfying this equality comprised of the Pareto curve and its gradient.In Fig. 12(a), every capacitor (not just Pareto optimal components) with voltage rating less than the critical inflection point V r < V r,crit has a larger volume than the highest volumetric energy density γ v capacitor with rated voltage V r = V r,crit when derated to any applied voltage.In summary to minimize volume, an overrated capacitor with V r = V r,crit should always be sought when V a < V r,crit .For capacitors with rated voltages above this critical inflection point V r > V r,crit , it is preferable to avoid voltage derating beyond that which is practically necessary; the volume-minimized capacitor has a voltage rating nearer to the applied voltage or V r = V a .
A similar process yields the critical rated voltage for the gravimetric (or specific) energy density γ m = E r Mass and the energy per cost γ c = E r Cost as shown in Fig. 12(b) and (c), respectively.All critical rated voltages V r,crit for energy density γ with respect to volume, mass, and cost are tabulated in Table IV.Inspection of Fig. 12 reveals aluminum electrolytic capacitors notably do not have a critical rated voltage with respect to volume or mass, thus efficient component selection should nearly always adhere to V r = V a for this capacitor technology.Interestingly, Fig. 12(c) reveals that film capacitors with V r < 300 V should not be selected to minimize cost since the best V r = 300 V device proves to be more cost effective, even when derated to lower voltages.
Finally, practical margins of rated voltages (per each capacitor technology) can be incorporated into the analysis of the data.This includes innate self-healing and overvoltage properties in film and ceramic capacitor technologies, as well as conventional voltage derating such as for tantalum electrolytic capacitors [52].

B. Capacitor Energy Density
The comprehensive data allow consideration of energy density trends in particular capacitor technologies.For instance, as mentioned in [48], the achievable volumetric and gravimetric energy density γ v and γ m of aluminum electrolytic capacitors approximately increases linearly with rated voltage V r ; this trend is confirmed by the Pareto fronts (for V r < 450 V) in Fig. 12.
The acquired data also allow comparison in the energy storage capabilities of different capacitor technologies.The Pareto curves in Fig. 12 indicate certain capacitor technologies dominate at various rated voltages with respect to volume, mass, and cost.For V r < 10 V, tantalum electrolytic capacitors have the largest energy densities, and thus, the smallest volume and mass.In the 10 V < V r < 700 V range, aluminum electrolytic capacitors are the lightest and Class II ceramic capacitors are the smallest.Above V r > 1 kV, both Class II ceramic and film capacitors are superior with respect to mass, while only Class II ceramic capacitors are superior with respect to volume.Aluminum electrolytic capacitors are the lowest cost solution to fulfill system energy requirements for V r < 700 V as claimed in [41] and [48], whereas film capacitors are the lowest cost for 700 V < V r < 6 kV, and ceramics are the lowest cost for V r > 6 kV.Class I ceramic capacitors are ubiquitously poor performers with respect to energy density except at the highest rated voltages V r > 10 kV.

C. Inductor Current Overrating
The comprehensive dataset can also produce the conditions for useful inductor current overrating.By following an analogous derivation as Section VI-A, the critical rated current I r,crit is derived for ferrite, metal, and air core inductor technologies since the energy stored in a linear inductor at an applied dc current I a is Fig. 13 illustrates the technological subsets and their Pareto fronts for rated current I r versus volumetric energy density γ v , gravimetric energy density γ m , and energy per cost γ c , respectively.Table V tabulates the resulting critical current ratings I r,crit for each technology, i.e., the lower limit below which no advantage is gained for devices with a reduced current rating.

D. Inductor Energy Density
The comprehensive data reveal the energy storage capabilities of inductors with respect to core material.Ferrite cores are the most energy-dense core type with respect to mass.With respect to volume and cost, ferrite core inductors are the most energy dense for rated current I r < 1 A, while above I r > 1 A, both ferrite and metal core technologies dominate.The best air core inductors have a consistent 10-100× lower energy density (with respect to volume, mass, and cost) than the best-performing ferrite and metal core inductors, although they compete equally with metal core inductors with respect to mass for current ratings below I r < 200 mA.

E. Comparing Capacitors and Inductors
Modern power converter topologies increasingly leverage the performance of capacitors versus inductors: hybrid switchedcapacitor converters such as the FCML converter [11], [97], [104], series-capacitor buck converter [16], [17], [18], and switching bus converter [105], [106].The fundamental tradeoff between these energy storage elements requires the careful quantification of realizable device performance.Some passive component FOM are jointly applicable to both capacitors and inductors-two fundamentally different circuit elements-and once these FOM are identified, these components can be justly compared.Energy density γ is one such generic FOM, as rated energy storage is well defined for both capacitors in (18) and inductors in (19).From the comprehensive dataset, Fig. 14 conveys that there is a marked difference in the energy density capabilities of commercial capacitors and inductors.As a whole, the highest volumetric energy density γ v commercial capacitor devices are nearly 1000× greater than that of the highest γ v commercial inductors.For volume-sensitive applications, choosing a circuit topology that heavily utilizes capacitors as energy transfer elements can result in a more volume efficient design [11], [13], [104], [107], [108].
Similarly for gravimetric energy density γ m and energy per cost γ c , the best commercial capacitors outperform the best commercial inductors by a factor of nearly 2000 and 10 000, respectively.Thus, capacitor-dominant circuit topologies are even more heavily favored for cost and mass-optimized designs than for volume-optimized designs.
It should be noted that, unlike capacitors, custom inductors are capable of achieving markedly higher performance than their commercial counterparts [1], [70], [71], [109].Further work may subsequently explore the capabilities of custom inductor constructions, with the benchmarks for commercial inductors firmly established in this work as reference.

F. Future Applications
Now armed with precisely quantitative intuition for inductor and capacitor performance, various device FOM can be further incorporated into analytical and procedural design methodologies for power electronics design.This work compliments recently developed analyses allowing for the design of optimized hybrid switched-capacitor converters [15], [107], [108], [110].Brooks et al. [64] analytically links capacitor energy density FOM and power density FOM to the operating waveforms of an aluminum electrolytic dc-link capacitor bank designed for single-phase twice-line frequency power buffering.Both of these applications, and many others, require actual device FOM for optimal design and demonstrate the utility of the data presented in this work.
Finally, the comprehensive dataset can continue to be improved as manufacturers and distributors standardize and digitally disseminate more passive component information.

VII. CONCLUSION
The true technological capabilities of electrical devices are historically difficult to quantify due to the immense variety of components currently in development, commercially available to consumers, or entirely obsolete.Utilizing modern advancements in the accessibility of public, large-scale, and digitized component data, a phenomenological framework is developed yielding definitive and quantified component performance.This work reviews methods of broad passive component characterization and it presents a methodology for developing robust device FOM to benchmark and compare passive-capacitor and inductor-components.This framework is directly applied to the data for 606 000 commercial capacitors and 88 000 commercial inductors.
Sampled data collection and measurement are used to supplement deficiencies in the comprehensive dataset.To generate information on energy storage, this work presents an empirical fit for estimating the energy-equivalent capacitance of Class II (and Class III) ceramic capacitors.The fit was validated on 2550 datasheet characteristic capacitance (or C-V) curves and produces a mean error of 3.1% for energy-equivalent capacitance at rated voltage C E (V r ).This work also presents empirical constant and power expression fits in Table III for estimating the volumetric mass density D = Mass Vol of a passive component from its rated voltage and capacitance for capacitors in (21), and rated current and inductance for inductors in (22).The empirical fits are generated from 447 device volume and mass measurements across various capacitor and inductor technologies, and produce a worst-case MPE of 20%.
This work then specifically investigates trends in capacitor and inductor stored energy density with respect to volume, mass, and cost.Analysis of the entire dataset for each component technology reveals the minimum useful capacitor rated voltage for the purposes of voltage overrating; similar critical rated currents are derived for inductor technologies.Recommendations for minimum useful component overvoltage/overcurrent ratings are tabulated in Tables IV and V.

Fig. 1 .
Fig. 1.Schematic denoting series and parallel bank configurations of a capacitor component.

Fig. 2 .
Fig. 2. Survey of component rated capacitance C versus rated DC voltage V r across all major capacitor technologies including aluminum electrolytic, tantalum electrolytic, Class 1 ceramic, Class II ceramic, film, and electrolytic double-layer capacitors (EDLC).

Fig. 3 .
Fig.3.Survey of component rated inductance L versus rated DC current I r across all major inductor technologies including ferrite core, metal (including powdered) core, and air core.

Fig. 4 .
Fig. 4. Applied DC voltage V a versus normalized differential capacitance C(v)/C(0) and distinguished by temperature characteristic.Data are sampling of 2 550 TDK Class II MLCC capacitors.

Fig. 6 .
Fig. 6.Percentage error for approximations of energy-equivalent capacitance C E (V r ) at rated DC voltage V r .Solid horizontal lines indicate the MPE for each corresponding approximation.Data are sampling of 2 550 TDK Class II MLCC capacitors.

Fig. 7 .
Fig. 7.Rated DC voltage V r versus volumetric energy density γ v for TDK Class II MLCC capacitors where (a) components are delineated by temperature characteristics and (b) Pareto front varies depending on the approximation for C E (V r ).

Fig. 8 .
Fig. 8. Portrayal mapping capacitor volumetric energy density γ v to gravimetric energy density γ m using a density transformation D(V r , C) dependent on rated DC voltage V r and capacitance C.

Fig. 9 .
Fig. 9. Measured volumetric mass density D for (a) capacitors as a function of rated DC voltage V r and capacitance C and (b) inductors as a function of rated DC current I r and inductance L. A best power fit surface in (21) and (22) is visualized for each component technology: (a) aluminum electrolytic, tantalum electrolytic, Class I ceramic, and Class II ceramic capacitors film capacitors; and (b) ferrite, metal, and air core inductors.

Fig. 10 .
Fig. 10.Energy density FOM of commercial capacitors across common technologies.The density D = MassVol of each capacitor in the comprehensive dataset is empirically estimated with the power fit described in(21) and parameters in TableIII.The gravimetric energy density γ m is then computed by transforming the volumetric energy density γ v in(15).The resulting transformation is subtle, yet significant at the highest energy densities.(a) Rated DC voltage V r versus volumetric energy density γ v .(b) Rated DC voltage V r versus gravimetric energy density γ m .
Fig. 10.Energy density FOM of commercial capacitors across common technologies.The density D = MassVol of each capacitor in the comprehensive dataset is empirically estimated with the power fit described in(21) and parameters in TableIII.The gravimetric energy density γ m is then computed by transforming the volumetric energy density γ v in(15).The resulting transformation is subtle, yet significant at the highest energy densities.(a) Rated DC voltage V r versus volumetric energy density γ v .(b) Rated DC voltage V r versus gravimetric energy density γ m .

Fig. 11 .
Fig. 11.Energy density FOM of commercial inductors across common technologies.The density D = MassVol of each inductor in the comprehensive dataset is empirically estimated with the power fit described in(22) and parameters in TableIII.The gravimetric energy density γ m is then computed by transforming the volumetric energy density γ v in(15).The resulting transformation is subtle, yet significant at the highest energy densities.(a) Rated DC current I r versus volumetric energy density γ v .(b) Rated DC current I r versus gravimetric energy density γ m .
Fig. 11.Energy density FOM of commercial inductors across common technologies.The density D = MassVol of each inductor in the comprehensive dataset is empirically estimated with the power fit described in(22) and parameters in TableIII.The gravimetric energy density γ m is then computed by transforming the volumetric energy density γ v in(15).The resulting transformation is subtle, yet significant at the highest energy densities.(a) Rated DC current I r versus volumetric energy density γ v .(b) Rated DC current I r versus gravimetric energy density γ m .

Fig. 12 .Fig. 13 .
Fig. 12.Commercial capacitor energy density FOM-(a) volume, (b) mass, and (c) cost-of the comprehensive dataset across common technologies.The Pareto fronts are highlighted as well as the critical rated voltage V r,crit where capacitor voltage overrating becomes useful.(a) Rated DC voltage V r versus volumetric energy density γ v .(b) Rated DC voltage V r versus gravimetric energy density γ m .(c) Rated DC voltage V r versus energy per cost γ c .Fig. 13.Commercial inductor energy density FOM-(a) volume, (b) mass, and (c) cost-of the comprehensive dataset across common technologies.The Pareto fronts are highlighted as well as the critical rated current I r,crit where inductor current overrating becomes useful.(a) Rated DC current I r versus volumetric energy density γ v .(b) Rated DC current I r versus gravimetric energy density γ m .(c) Rated DC current I r versus energy per cost γ c .

Fig. 14 .
Fig. 14.Commercial capacitor versus inductor energy density FOM-(a) volume, (b) mass, and (c) cost-of the comprehensive dataset.From the highlighted Pareto fronts, commercial capacitors have a roughly 1000× better maximum energy density capability compared to commercial inductors with respect to volume, a roughly 2000× improvement with respect to mass, and a roughly 10 000× improvement with respect to cost.(a) Rated DC voltage V r / current I r versus volumetric energy density γ v .(b) Rated DC voltage V r / current I r versus gravimetric energy density γ m .(c) Rated DC voltage V r / current I r versus energy per cost γ c .

TABLE V SUMMARY
OF CRITICAL CURRENTS FOR INDUCTOR OVERRATING