Slow-Wave Causal Model for Multi Layer Ceramic Capacitors

DesignCon 26 Slow-Wave Causal Model for Multi ayer Ceramic Caacitors Istvan Novak Gustavo Blando Jason R. Miller Sun Microsystems, Inc. Tel: (781) 442 34, e-mail: istvan.novak@sun.com

Abstract There is an ongoing interest in refining the simulation models for assive comonents in electronic circuits. For simle analyses, byass caacitors are modeled by a series C-R- equivalent network. To cature the frequency deendency of the circuit arameters, more comlex equivalent circuits can be used: ladder -R networks to model the frequency deendent inductance and resistance and/or C-R networks to model the frequency deendent caacitance. These equivalent circuits have the advantage of being comatible with both time-domain and frequency domain SPICE simulations, but the otimum toology of the equivalent circuit may deend on the tye and construction of caacitor. This aer first summarizes the current distributions inside MCC arts simulated with a bedsring model and we make some counter-intuitive observations about the frequency deendency of inductance and resistance in tall MCC arts. Based on those observations we then derive a slow-wave causal model, which reresents the caacitor with a eriodically loaded lossy transmission line. It is shown that the load circuit in the unit cell corresonds to the waveguide formed by two adjacent caacitor lates. The unit cells are further simlified and lumed together into one lossy, oen-ended transmission line with a series R- circuit caturing the imedance of the cover layer of the caacitor. The arameters of the single lossy transmission line are derived from the geometry and material roerties. It is shown that the counterintuitive features in the model are catured by a virtual caacitance and dielectric loss tangent, which combine the dielectric loss of the ceramic and the resistive losses of the caacitor lates. The model is shown to cature the imortant characteristics of measured data, and it is simle enough to be used in multile coies in circuit simulators. Author Biograhies Istvan Novak is signal-integrity senior staff engineer at SUN Microsystems, Inc. Besides signalintegrity design of high-seed serial and arallel buses, he is engaged in the design and characterization of ower-distribution networks and ackages for mid-range servers. He creates simulation models, and develos measurement techniques for ower distribution. Istvan has twenty lus years of exerience with high-seed digital, RF, and analog circuit and system design. He is Fellow of IEEE for his contributions to signal-integrity and RF measurement and simulation methodologies. Gustavo Blando is a signal Integrity engineer with over 1 years of exerience in the industry. Currently at Sun Microsystems he is resonsible for the develoment of new rocesses and methodologies in the areas of broadband measurement, high seed modeling and system simulations. He received his M.S. from Northeastern University. Jason Miller is currently a Staff Engineer at Sun Microsystems where he works on ASIC develoment, ASIC ackaging, interconnect modeling and characterization, and system simulation. He received his Ph.D. in Electrical Engineering from Columbia University. 2

I. Introduction: Present modeling otions When considering the arasitics of byass caacitors, a widely used simle model is a series C-R- network, where C is the caacitance of the art, R is the Equivalent Series Resistance (ESR) and is the Equivalent Series Inductance (ES), as shown in Figure 1. In its simle form, C, ESR and ES are assumed to be frequency indeendent constants. However, measured data indicates [1] that all three of these arameters are eventually frequency deendent and furthermore may be inter-related through the alication geometry. C ESR ES cover thickness (V2) body length (H1) lane sacing (V4) via sacing (H2) stack height (V1) mounting height (V3) Figure 1. Simle RC equivalent circuit of a caacitor. Figure 2. Vertical cross section of an MCC mounted to PCB lanes. The caacitance may be frequency deendent, rimarily because of dielectric losses [2]. ESR is the result of transforming the arallel dielectric losses and series conductive losses into a single series resistance value. As long as tangent delta of the dielectric material varies little with frequency, the arallel loss resistance dros inversely with frequency. The series resistance of the art comes from the terminals and conductive layers on the dielectric sheet(s). Aart from bulk caacitor constructions, like tantalum brick caacitors and alike, the caacitor lates in high-cv MCCs are thin enough that in the frequency range of interest their thickness is less than the skin deth, and therefore the AC resistance contributions of the lates themselves do not vary much with frequency. Overall ESR still varies at high frequencies, due to non-uniform current distribution in the lates [4] Inductance deends both on the internal construction of the art and the external geometry forming the closed current loo. As illustrated in Figure 3 for the case of a reverse-geometry Multi-ayer Ceramic Caacitor (MCC) attached to a air of lanes on a PCB, the measured imedance of the art exhibits strong frequency deendency on all three of the arameters. When extracting caacitor arameters from measured data, we face a further comlication: ESR is simly the real art of the measured imedance (after the roer calibration and/or deembedding rocess), but the caacitive and inductive reactances show u in a suerimosed way in the imaginary art of the measured imedance. If caacitance and inductance were frequency indeendent, extracting them from the imaginary art of measured imedance would be easy. Because the caacitive and inductive reactances change with frequency in the oosite way, we know that at frequencies much below the Series Resonance Frequency (SRF) the inductive reactance is negligible and from the measured reactance we can calculate the caacitance. Similarly from a measured reactance value at a frequency much above SRF we could calculate the inductance. This aroach is assumed for instance in [3] and [4]. With relatively strong frequency deendency of caacitance and/or inductance, which is the case of tall caacitor stacks with lossy dielectrics and aggressive mounting, using a low-frequency 3

caacitance and a high-frequency inductance can not uniquely resolve the frequency deendent caacitance and inductance values around SRF. One ste further is to iteratively aroximate the caacitance and inductance close to SRF and use those (constant) values to extract the frequency deendent caacitance and inductance curves [5]. This imroves accuracy and the range of validity for the extracted caacitance and inductance curves, but unless we have further data oints or constraints, we still cannot uniquely resolve the two unknowns, C(f) and ES(f), from one data oint of Im{Z(f)}. Instead of trying to blindly extract the arameters from measured data, more sohisticated equivalent circuits can also be used to fit the measured data on the model. Equivalent circuits comosed of frequency indeendent resistors, caacitors and inductors unconditionally guarantee causality and easy comatibility to circuit simulators. To describe frequency deendent caacitance and ESR of bulk caacitors, [6] suggests a comosite RC network. To cature the frequency deendent ESR and ES of MCCs, [7] uses a resonant ladder network, while [8] rooses a transmission-line model. 1.E-1 Imedance magnitude and real art [ohm] magnitude Extracted caacitance and indutance [F, H] 1.6E-5 8.E-1 6H 66H 1.2E-5 6.E-1 Inductance 1.E-2 8.E-6 4.E-1 real 4.E-6 Caacitance 2.E-1 2.8 mohm 3.6 mohm 1.E-3 1.E+4 1.E+5 1.E+6 1.E+7 1.E+8.E+ 1.E+2 1.E+4 1.E+6 1.E+8.E+ Figure 3. Measured imedance magnitude of a 1uF 58 MCC with the real art of the imedance (on the left), and extracted caacitance and inductance versus frequency (on the right). For MCC arts, it is customary to assume that the caacitive and inductive currents balance themselves at SRF and therefore at that frequency the current uniformly enetrates all lates and for this reason the lowest ESR value occurs at SRF. Similarly, it is usually assumed that inductance monotonically dros from its low-frequency value towards the high-frequency loo inductance. In contrast to usual exectations, data on Figure 3 indicates that the minimum of the imedance real art is not at SRF: at 6kHz ESR is 2.8 milliohms; whereas at the 2.1MHz SRF the ESR reading is 3.6 milliohms. Moreover, the ES(f) value extracted according to the rocedure in [5] results in 6H at SRF, but the inductance first increases with frequency, instead of decreasing, reaching a 66H eak at 4.2MHz before it starts going down. Is this due to measurement errors or a deficiency in the extraction rocedure, or really ESR and ES behave contrary to common assumtions? As it was shown in detail in [9], this counter intuitive behavior comes from the vertical resonances (and at higher frequencies, to a lesser degree, from horizontal resonances) along the caacitor body. These features can be catured by using a two-dimensional transmission-line or RGC bedsring array. 4

Figure 4 shows the artial schematics of the bedsring model, where the caacitor lates are groued into ten horizontal layers, each layer broken down into ten segments. Note that the grid size of ten by ten is somewhat arbitrary, but roved to be sufficient to cature the major features. Plate 1 Plate 9 Plate 8 Plate 3 Plate 2 Plate 1 caacitor connections Figure 4. Partial schematics of the bedsring caacitor model. The model consists of ten caacitor lates, 1 through 1. The lowest caacitor late, connecting to the PCB, is Plate 1. Plates 1, 3, 5, 7 and 9 are connected to the left terminal. Plates 2, 4, 6, 8 and 1 are connected to the right terminal. Each caacitor late is divided into ten equal segments (lus an end iece), reresented by series R networks. At each internal late node, a caacitor reresents the dielectrics. Figure 5 shows the imedance magnitude and real art of imedance simulated at the connection terminals. The schematics entries used for the simulation are listed to the right of the chart. Note that the bedsring model catures all of the imortant features that we want to study: it shows that imedance real art starts to increase below SRF, and there is a set of damened, but ronounced secondary resonances. Because we assume no dielectric loss, the imedance real art at low frequencies does not rise. The current-distribution lots in [9] showed also the reason why the inductance starts to rise around SRF before it eventually goes down. For a fully animated illustration of currents in the caacitor lates and in the dielectrics, see [1]. Here we reroduce the current distribution lots only at SRF. On the left chart of Figure 6 we see the current distribution along the caacitor lates, whereas on the right-hand lot the current distribution in the dielectric layers is shown. The nonlinear current distribution is a clear indication that both ESR and ES are increased. 5

While the bedsring model is unconditionally causal, and it is useful to study the vertical and horizontal resonances in MCC arts, the model is clearly too comlicated to be used in PDN simulations, where we may need dozens or hundreds of such caacitor models in the same network. 1.E+ Imedance magnitude and real art [ohm] magnitude 1.E-1 1.E-2 real C [F]: [H]: R [ohm]: t [H]: Rt [ohm]: c [H]: Rc [ohm]: 5.E-9 1.E-11 1.E-3 1.E-1 1.E-4 1.E-1.E+ 1.E-3 1.E+6 1.E+7 1.E+8 Figure 5. Simulated imedance magnitude and imedance real art at the caacitor connections terminals of the circuit shown in Figure 4. Circuit arameter values are shown in the table on the right..3.25.2.15.1.5.2.15.1.5 Figure 6. Current distribution along the caacitor lates (on the left) and inside the dielectrics (on the right) at the 1MHz SRF. Vertical axis unit: A. The black-box behavioral model introduced in [9] offers a unified model for byass caacitors, simle enough to use many of them in frequency-domain PDN simulations. The model uses only three comonents, a series C-R- circuit, but all three elements are frequency deendent. The frequency deendency of each element is catured and described by seven arameters, resulting in a total of 21 arameters for one model. The exressions caturing the frequency deendencies are based on the behavior of measured caacitor ieces. The arameters can be obtained either by manual or by semiautomatic curve fitting. However, the black-box model does not guarantee causal behavior, and because the three comonents are frequency deendent, it is not well suited for time-domain simulations. 6

II. Slow-wave causal model The unit-cell model The slow-wave causal model is built on the realization that a multi-layer ceramic caacitor is a eriodically loaded lossy transmission line. The unloaded transmission line is formed by the two vertical terminals of the caacitor, by removing the caacitor lates, but leaving the dielectric material in lace. This vertically oriented unloaded transmission line in itself is already lossy: the terminals have finite resistance, and the dielectric material has finite dielectric loss tangent. It is also known that causality dictates caacitance to change with the log of frequency in roortion to the dielectric loss tangent. In an MCC art, the large caacitance is achieved by inter digitated caacitor lates, attached alternating to the oosite terminals. These caacitor lates form a set of eriodically arranged lossy transmission lines, attached orthogonally to the caacitor terminals. As it will be shown, the multitude of caacitor lates will not only increase the total caacitance of the art, but it also behaves like a dielectric material with increased loss tangent and additional frequency deendency of caacitance. The virtual loss tangent is a mix of the loss tangent of the original dielectric material and the resistive loss of the caacitor lates. The exectation is that if we roerly assign the dimensions and material constants, or if we do a blind otimization of these arameters to match the measured behavior of a caacitor, all of the major features will be catured simultaneously, without the need to change and otimize indeendently caacitance, resistance and inductance values (which was the case in [9]). Also, the model is based on the hysical roerties of the structure, and it guarantees that the model will be causal. Terminal TH_t Plates H TH_ TH_d Turn 9 degrees W1 H1 W Terminal Figure 7. eft: side view of MCC with the imortant dimensions. Right: reresentation of the same MCC by turning it 9 degrees. The sketch on the left side of Figure 7 defines the major dimensions of an MCC, relevant to our calculations. We assume that the rectangular caacitor body is W wide, long and H high. The illustration here shows a reverse-geometry caacitor, because for the correlation we will use the measured data on the revious 1uF 58 MCC examle. The calculations and methodology, however, do not mandate a reverse-geometry caacitor. For other geometries, such as regular or interdigitated caacitors,, W and H can be changed aroriately as needed. To somewhat simlify the calculations, the caacitor cross section is assumed to be symmetrical, both horizontally and vertically. We assume the same H1 cover thickness both on to and bottom. A similar horizontal symmetry assumes that caacitor lates are stoing W1 distance from the unconnected 7

terminals at both sides. This symmetry is assumed here only for sake of convenience; the rocedure can be easily extended to different to and bottom cover thicknesses and/or for different end gas at the left and right terminals. The caacitor late thickness is TH_, each dielectric layer between the lates is TH_d thick. The vertical caacitor terminals are assumed to be TH_t in thickness. We further assume that the dielectric material has ε d dielectric constant and tan_δ loss tangent at a given working frequency, and their frequency deendent values are inter-related through the causal requirement [2]. The caacitor lates have a conductivity of σ, the terminal material has a conductivity of σ t. The eriodically loaded transmission-line model becomes aarent when we turn the caacitor sideways and distort the asect ratio. As shown on the right side of Figure 7, the height (H) of the original caacitor body becomes the length of the transmission line, and the caacitor lates will reresent a eriodical loading along the transmission line. The caacitance of the unloaded transmission line equals the caacitance of the caacitor body between the vertical terminals, without the caacitor lates: * H C = ε ε d (1) W The roagation delay of the unloaded transmission line equals the roagation delay along the emty vertical caacitor body, with the caacitor lates removed but dielectric material in lace: H t d = = H ε ε d µ (2) v With C and t d known, we can calculate the characteristic imedance and the inductance of the unloaded vertical transmission line: 2 t d H * W 12π W = = µ and Z = = (3) C C ε The resistance of each terminal along its entire vertical length is: 1 H Rt = (4) σ t * TH _ t From (1) through (4), we can calculate the arameters of the two end ieces, simly by scaling the unloaded transmission line length by the ratio of H1/H for each end iece: H1 t d _ end = t d (5) H The end-iece transmission lines are denoted by suffix _end. As shown on the left of Figure 8, adjacent caacitor lates along the terminal will create the eriodical loading. The imedance of the entire caacitor is observed at the left end of the transmission line; the x at the right end of the structure indicates that the right-hand side end of the structure is oen. The loading is created by the lossy, oen-ended transmission lines formed by adjacent lates. Following (1) through (4), we can calculate the arameters of a transmission line formed by adjacent caacitor lates. This transmission line is denoted by suffix -. 8

C W 2* W1 = ε ε d (6) TH _ d t d W 2* W1 = = ( W 2 * W1) * ε ε d µ v (7) 2 t d TH _ d *( W 2* W1) 12π TH _ d = = µ and Z = = C C ε (8) R 1 W W1 = σ * TH _ (9) Terminal Plates Zin Zin X Z, t d_end Unit cell (1) Unit cell (2) Unit cell (N) Z, t d_end X Terminal Figure 8. Slow-wave eriodically loaded model of MCC. The eriodically loaded transmission line is broken down into symmetrical unit cells, each cell reresenting a length equal to the caacitor-late itch. The end ieces, corresonding to the bottom and to dielectric covers, are reresented by unloaded transmission-line sections. From N caacitor lates, we get N-1 airs to create the eriodical loading. For N>>1, we can aroximate the number of cells in the eriodically loaded structure with N. The number of lates, the dielectric and late thicknesses and the to/bottom cover thicknesses are interrelated through the following formula: H 2* H1 N = (1) TH _ + TH _ d The conductive and dielectric losses of the transmission lines are, in theory, frequency deendent. The skin deth in conductors is defined as: 1 skin _ deth = (11) πfσµ The skin deth in coer reaches 1µm at 1GHz frequency. To meet the requirements of the high ceramic firing temerature, caacitor lates use materials with conductivity lower than that of coer. 9

The lower conductivity increases the skin deth. This means the resistance of individual caacitor lates is skin-deth limited and is relatively frequency indeendent u to hundreds of MHz frequencies. The terminals are usually much thicker than the caacitor lates, and therefore their resistance and inductance may show somewhat more frequency deendency. The dielectric losses are reresented by a arallel conductance G in the transmission-line model: G = 2π f C tan_ δ (12).5*TH_d Z, t d_uc Z, t d_uc UNIT CE Z t d X TH_d TH_ Z, t d_uc R Z, t d_uc UNIT CE C R d Figure 9. Generating unit-cell arameters from the geometry. In (12), we can substitute the aroriate caacitance and loss-tangent values for the unloaded transmission line of terminals or the lossy transmission line of caacitor lates. This generic model links the geometry and material roerties of an MCC to a causal electrical model, which can be directly used to calculate the imedance of the caacitor. Though this model is still too comlex to include in an actual circuit simulator in multiles of coies, it is very suitable for correlation uroses. We can use any of the comuter math ackages to obtain the inut imedance of the eriodically loaded transmission-line circuit, which reresents the imedance of the caacitor. The lossy transmission-line model The model derived in Figures 7 through 9 is generic, and as such, it is valid over a wide range of arameters. When we look at the actual geometry and resulting model numbers for a tyical MCC, we can achieve substantial simlifications without major loss of accuracy. 1

Since we are interested in the inut imedance of the structure with oen at its end (on the to), the second end iece of unloaded transmission line can be totally neglected. Because it is oen terminated on the right, only the static caacitance of the end iece would matter anyway. Not having caacitor lates in the end iece, for large N, its static caacitance is orders of magnitudes lower than the total caacitance, and therefore it can be rightfully ignored. The end iece on the left is in series to the external connections, and therefore cannot be comletely ignored. We can still, however, simlify the left end iece by using the earlier arguments, and neglect its arallel caacitance and conductance. This leaves us with its series inductance and resistance. These values can be obtained from (3) and (4), by substituting H1 for H. H1* W _ end = µ (13) R t _ end 1 H1 = (14) σ * TH _ t t The unit cells can be simlified in a similar way. Caacitances and conductance of the series unloaded transmission-line ieces can be ignored, leaving only a series -R term. For one unit cell, using the suffix uc, and substituting TH_ + TH_d for H1, we get: ( TH _ + TH _ d) * W _ uc = µ (15) R t _ uc 1 = σ t TH _ + TH _ d * TH _ t (16) The oen-ended loading transmission line formed by adjacent caacitor lates can be simlified by neglecting its inductance. The C caacitance and R series resistance of the transmission line were already given in (6) and (9). There is one more element though that we don t want to ignore: the arallel conductance of the loading oen-ended transmission line. It can be calculated from (12), by substituting the values for one late air: G = π f C tan_ δ (17) 2 These simlifications lead to the equivalent circuit of the unit cell shown in Figure 1. The shunt caacitance has its own G conductive loss term originated from the dielectric loss tangent, and an R series resistive loss term originated from the resistance of the adjacent caacitor lates. At any given frequency, the series and arallel loss terms can be combined into a single term. The schematics on the right of Figure 1 shows a arallel equivalent, where G reresents a combination of R and G. 11

R t_uc /2 _uc /2 _uc /2 R t_uc /2 R t_uc /2 _uc /2 _uc /2 R t_uc /2 R C G C G Figure 1. Equivalent schematics of the simlified unit cell. We get the circuit on the left by neglecting the caacitance and conductance of the series transmission line and by neglecting the inductance of the arallel transmission line. We get the circuit on the right by combining the series and arallel loss terms around the shunt caacitance. Note that during the transformation, in a general case, both the caacitance and conductance will change. Assuming that tan_δ is small, we get: C ' ' ' C = and G = G + C (18) 2 1+ ( ) If τ << 1, the formulas can be further simlified to: C ' C and G ' G + C (19) By combining (17) and (19), we get: G ' = C tan_ δ + C τ (19) ' ' From (19) and (12) we can calculate a virtual loss tangent for the lossy transmission line: tan_ δ + ' tan_ δ = (2) 2 1+ ( ) In the final ste we can realize that the cascaded unit cells reresent the ladder equivalent of a uniform lossy transmission line. To obtain the er-unit-length transmission-line equivalent arameters, we multily the unit-cell arameters by N. The inductance and the resistance are then simly the inductance and resistance of the (H-2*H1) section of the vertical terminals. The caacitance will become aroximately the full caacitance of the art itself, though as indicated by (18), it dros sharly above the corner frequency. Moreover, to obey causality, the caacitance also dros slightly with frequency due to the finite loss tangent. The arallel conductance can be calculated from the full caacitance and the virtual loss tangent. 12

This eventually leads us to the simlified equivalent circuit of Figure 11. Note that this simle circuit is causal, and works on many time-domain and frequency-domain simulators. Zin R t_end _end COVER AYER R,, G, C OSSY OADED TRANSMISSION INE X R G 2 H 2* H1 = σ * TH _ t t ( H 2* H1)* W = µ C C = 2 1+ ( ) = C(tan_ δ + ) Figure 11. Simlified causal equivalent circuit of a Multi-ayer Ceramic Caacitor. Note that the imortant asect of this simlified model is not the lossy transmission line itself, rather the unique way that it catures the convoluted effect of conductive and dielectric losses in the frequency deendent caacitance and conductance er unit length. III. Correlations Characterization of test fixture Now we return to the examle shown in Figure 3. The 1uF 58 MCC art was measured in a small fixture, shown in Figure 12. The fixture has 22 layers. Caacitor sites are on to and bottom, connecting to 4x6 mil rectangular lane shaes further down in the stack. The horizontal layout of the site used for this test is shown on the right of Figure 12. The caacitor ads for the 58 site are on the bottom side (layer 22), connecting to ower lanes on layers 2 and 21 with a set of blind vias. Three 7-mil blind vias connect to layer 21 with 25-mil center-to-center sacing. Three 12-mil blind vias connect to layer 2, with 25-mil center-to-center sacing. Horizontal sacing between the two columns of vias is 5 mils. The caacitor ads are 8x35-mil rectangular shaes with 2-mil air ga. 4x6 mil lane shaes with 2.1-mil searation on layers 2 and 21. Through holes for test site on layers 2-3. Through holes on 5-mil center-to-center sacing for connecting semirigid robes. Figure 12. Geometry of test fixture for measuring the 1uF 58 MCC samle. 13

First the test site was characterized with the caacitor ads oen and shorted with two Vector Network Analyzers (VNA): 4395A in the 1Hz 1MHz frequency range and 4396A in the 1kHz to 18MHz frequency range. Figure 13 shows the measured imedance magnitude and hase of the oen test site and the extracted caacitance versus frequency. Note the straight sloe of the caacitance curve on the linear-logarithmic scale; this indicates an aroximately 2% of loss tangent of the FR4 material. The average caacitance is around 12F at 1MHz. Figure 14 shows the measured data and extracted arameters with the caacitor ads shorted. The grahs show comosite data from both VNAs. The blue line on the left grah is the measured real art of the imedance; the green line is an aroximation curve. The blue line on the right grah is the inductance extracted from the imaginary art of the measured imedance. The green line is a curve aroximating the extracted inductance. The exressions aroximating the real art and inductance: R f = + (21) f N 1+ ( ) f N ( f ) RDC (1 + ( ) ) and ( f ) = inf f R R Where R DC = 1.7E-3 [ohm], f R = 1.2E7 [Hz], inf = 1.4E-1 [H], = 8E-11 [H], N =.8. 1.E+3 Imedance magnitude and hase [ohm, deg].e+ 1.4E-1 Equivalent caacitance [F] 1.E+2 1.E+1-2.E+1-4.E+1-6.E+1-8.E+1-1.E+2-1.2E+2-1.4E+2 1.3E-1 1.2E-1 1.1E-1 1.E+ -1.6E+2 1.E+6 1.E+7 1.E+8 1.E+9 1.E-1 1.E+6 1.E+7 1.E+8 Figure 13. eft: imedance magnitude and hase measured on the test site with oen ads. Right: extracted caacitance of the test site versus frequency. The arallel caacitance of the oen test site and the series resistance and inductance of the shorted test site have to be included when we are looking for correlation of the lossy transmission-line caacitor model to measured data. Note, however, that the shorted test site data also reflects resistance and inductance of the short itself, not only the test fixture. Since the short later is relaced with the caacitor, this small art of resistance and inductance will be double counted. 14

1.E-1 Imedance real art [ohms] 3.E-1 Inductance [H] 2.5E-1 2.E-1 1.E-2 1.5E-1 1.E-1 1.E-3 1.E+2 1.E+4 1.E+6 1.E+8 5.E-11.E+ 1.E+6 1.E+7 1.E+8 1.E+9 Figure 14. Imedance real art and extracted inductance of the shorted test site. Blue lines are measured data. Green lines are aroximating curves based on (21). Checking the convergence of unit-cell model The unit-cell model catures the hysical roerties of the caacitor, and converts them into an electrical model. Strictly seaking we would need to know how many caacitor lates the art has, and concatenate the same number of unit cells. On the other hand, many times we do not know the number of caacitor lates in the art. And, clearly it is not even necessary. The er-unit electrical arameters of concatenated unit cells saturate beyond a number, resulting in diminishing change as we add more cells. The saturation curve does deend somewhat on the characteristics of the unit cell, but usually 5-1 cells will result in a sufficiently accurate aroximation. Figure 15 illustrates the convergence with the unit cells used to describe the caacitor samle in Figure 3. For this exercise, the unit cells have been readjusted so that the given number of unit cells always added u to the same fixed characteristics of the samle caacitor. 1 Percentage incremental change [%] 1 1.1 5 1 15 2 Number of cells [-] Figure 15. Saturation curve of concatenated unit cells. Vertical axis: ercentage change between consecutive iteration with different number of unit cells. Horizontal scale: number of unit cells. 15

Correlation with unit-cell model As oosed to the caacitance curve of the FR4 material of the test fixture, the extracted caacitance versus frequency on the right-hand grah of Figure 3 exhibits multile sections of frequency ranges with aroximately constant sloe. This behavior suggests that for each of these frequency ranges a different dielectric loss tangent should be alied. This is illustrated in Figure 16, where we comare the correlations with a single dielectric loss tangent value (.15) versus three different loss tangent values (.25,.135,.1). With three loss tangent values, alied searately for each frequency range, and combined linearly to continue to enforce causality, we can roerly cature the shae of the imedance real art (and also the frequency deendency of the caacitance) over three decades of frequencies. Below the Series Resonance Frequency (SRF) the caacitance and imedance real art are rimarily coming from the dielectric material, and therefore the extracted caacitance versus frequency curve gives useful guidance how many segments we may need to use to roerly describe the frequency deendency of the dielectric material. Near to and above SRF, on the other hand, we have no direct indication about the ossible change of the loss tangent. We can, for instance, assume that the loss tangent just below SRF continues unchanged above SRF as well. This aroached was followed in the correlation results shown here. As an alternative solution, we can assume additional frequency segments above SRF with their resective and unknown loss tangent values, and get the values by otimized curve fitting to measured data. 1.E+1 Imedance real art [ohm] 1.E+1 Imedance real art [ohm] 1.E+ 1.E+ 1.E-1 1.E-1 1.E-2 1.E-2 1.E-3 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 1.E-3 1.E+2 1.E+3 1.E+4 1.E+5 Figure 16. Correlation of imedance real art below the series resonance frequency. Measured: blue lines. Aroximation: red lines. On the left: aroximation using one loss tangent value. On the right: aroximation using three different dielectric loss tangent values. Once we have the caacitor arameters below SRF, we can ut together the full model: cascaded unit cells, end iece, fixture. These models are based on the hysical arameters of the caacitor. In this case, however, the measured art was not cross sectioned, and the internal geometry and material constant data was not available from other sources either. The correlation was, instead, done by manual and automated otimization of the model, with seed values based on reasonable assumtions. The number of arameters to be otimized is large enough that a reasonable correlation should be ossible to achieve. However, not having the true numbers for the caacitors, a very accurate correlation was not the goal at this time. Instead, different arameter settings have been tried to see if the model can roerly describe and cature the imortant characteristics of the measured data lots: increased ESR at SRF, sudden increase of ESR above SRF, slow rise of ESR above the secondary resonances. 16

As an illustration, Figure 17 shows the correlation after a brief otimization. 1.E-1 Imedance magnitude [ohm] 1.E-1 Imedance real art [ohm] 1.E-2 1.E-2 1.E-3 1.E+5 1.E+6 1.E+7 1.E+8 1.E-3 1.E+4 1.E+5 1.E+6 1.E+7 1.E+8 Figure 17. Correlation with unit-cell model. Measured: blue lines. Aroximation: red lines. Correlation with lossy transmission-line model The same set of measured data was also correlated to the lossy transmission-line model. Figure 18 shows the correlation after a brief manual otimization of arameters. 1.E+ Imedance magnitude [ohm] 1.E-1 Imedance real art [ohm] 1.E-1 1.E-2 1.E-2 1.E-3 1.E+4 1.E+5 1.E+6 1.E+7 1.E+8 1.E-3 1.E+4 1.E+5 1.E+6 1.E+7 1.E+8 Figure 18. Correlation with lossy transmission-line model. Measured: blue lines. Aroximation: red lines. Conclusions and future work It was shown that a causal model can be constructed for MCC, which can cature the rimary and secondary resonances of the art. The model is based on hysical arameters of the caacitors, but the exact knowledge of these arameters is not a must: the arameters can be obtained by fitting the model to measured data. The arameters of cascaded unit cells can be combined into a single lossy, frequency and arameter-deendent transmission-line equivalent circuit. The simlified model is suitable many frequency and time-domain simulators, and simle enough to be used in multile coies in Power Distribution Network simulations. Future work will examine correlation to arts with known internal geometry. 17

References [1] Istvan Novak, Jason R. Miller, Frequency-deendent characterization of bulk and ceramic byass caacitors, Proceedings of EPEP23, October 27-29, 23, Princeton, NJ. [2] A. Djordjevic, et al., Wideband Frequency-Domain Characterization of FR-4 and Time-Domain Causality, IEEE. Tr. EMC, Nov. 21,.662. [3] Michael J. Hill, eigh Wojewoda, Caacitor Parameter Extraction Techniques and Challenges, Intel Technology Symosium, Fall 23. [4] arry Smith, MC Caacitor Parameters for Accurate Simulation Model, in TF7 Inductance of Byass Caacitors; How to Define, How to Measure, How to Simulate Proceedings of DesignCon 25, January 31 February 3, 25, Santa Clara, CA. [5] Istvan Novak, Frequency-Domain Power-Distribution Measurements An Overview, HP-TF1 TecForum, DesignCon East, June 23, 23, Boston, MA. [6] Hideki Ishida, Measurement Method of ES in JEITA and Equivalent Circuit of Polymer Tantalum Caacitors, in TF7 Inductance of Byass Caacitors; How to Define, How to Measure, How to Simulate Proceedings of DesignCon 25, January 31 February 3, 25, Santa Clara, CA. [7].D.Smith, D.Hockanson, K.Kothari, A Transmission-ine Model for Ceramic Caacitors for CAD Tools Based on Measured Parameters, Proc 52 nd Electronic Comonents & Technology Conference, San Diego, CA., May 22,.331-336. [8] Charles R. Sullivan and Yuqin Sun, Physically-Based Distributed Models for Multi-ayer Ceramic Caacitors, Proceedings of EPEP23, October 23. [9] Istvan Novak, " A Black-Box Frequency Deendent Model of Caacitors for Frequency Domain Simulations, DesignCon East 25, Setember 19-22, 25, Worcester, MA. [1] htt//home.att.net/~istvan.novak/aers/grid_swee.zi 18