Physically-Based Distributed Models for Multi-Layer Ceramic Capacitors

Transcription

1 Physically-Based Distributed Models for Multi-Layer Ceramic Capacitors Charles R Sullivan and Yuqin Sun Thayer School of Engineering Dartmouth College

2 Introduction Why an RLC model won t do.

3 Standard RLC Model for a Cap Simple but not accurate (as much as 5X impedance error). 0.1 Ω Z 1 mω 10 khz Calculated Curve 100 µf MLC EIA 2220 X5R 400 khz 10 MHz Frequency

4 Measurements vs. RLC Model 100 µf MLC EIA 2220 X5R RLC Model Z 5X 0.1 Ω Z, ESR Measured Z Measured ESR 1 mω 100 khz RLC Model ESR 1 MHz 10 MHz 100 MHz Frequency

5 What s Going On? Inductance is distributed effect; different for each plate. High Inductance Low Inductance Behavior is like a transmission line.

6 Measurements

7 What to Measure Need to be sure measurement technique captures real in-circuit behavior. Inductance is only defined for a closed loop. Z undefined Z defined.. But includes some Z interconnect!

8 Defining Impedance Z total Z board Z cap = Z total Z board The relevant impedance for application with adjacent ground plane. Same inductance as Z total with zero board thickness. Can correct for resistance. How to measure: ph accuracy test fixture: Session 35, paper 2 here this afternoon.

9 New Low-Impedance Test Fixture Based on Agilent 4TP (four terminal pair) configuration. Less than 100 ph stray inductance, ~3 ph repeatability A V ~ DUT 3 mil (75 µm) polyimide (Kapton) Ground traces underneath

10 A Better Model

11 I. Simple Transmission Line Model T Actual system: hundreds of plates. Model as continuous distributed transmission line. Z 0

12 Two Parameters Describe the Transmission Line Model l w h T Parameters linked to geometry. = 1 µ 0 Z0 w C v Z 0 T = 2 hl µ C 0 v where C v is capacitance per unit volume.

13 Ideal Transmission-Line Behavior Calculated from geometry and ESR Z 1 Ω 0.1 Ω 10 mω Real behavior exhibits: Some ESL Much greater damping Shifted resonance frequencies Lumped Model Measured Distributed Model 1 mω 100 khz 1 MHz 10 MHz Frequency 100 MHz

14 II. Improving the Model External L: Models Coating Effect Damping: Real damping effects include: Series R of plates. Eddy-current losses in plates. Must model both effects

15 Distributed Model with Added External L 1 Ω 0.1 Ω Lumped model Measured Z 10 mω Distributed model with L ext 1 mω 100 khz 1 MHz 10 MHz 100 MHz Frequency

16 Model with Damping from Series R of Plates 1 Ω Z 0.1 Ω One section of transmission line Peak too high Distributed model Measured 10 mω 1 mω 100 khz 1 MHz 10 MHz 100 MHz Frequency

17 Add Effect of Eddy Currents in Plates 1 Ω 0.1 Ω One section of transmission line Distributed model Z 10 mω Peak positions don t match Measured 1 mω 100 khz 1 MHz 10 MHz 100 MHz Frequency

18 III. Final Model Improvements Damping model works OK by including Series R of plates. Eddy-current losses in plates. Last remaining discrepancy: non-uniform spacing of resonant peaks. Two possible causes: Non-uniform distributed inductance. Mutual Inductance. LM LE LD1 LD1 RM RSD1 RED RSD2 Model including mutual inductance CD RPD

19 Model with Both Damping Effects and Non-Uniform Inductance 1 Ω Z 0.1 Ω Distributed model Measured 10 mω 1 mω 100 khz 1 MHz 10 MHz Frequency 100 MHz

20 Coating Effect

21 Why Does Cap Have Extra Series L? Thickness of coating Coating Cap Element Opportunity: Reduce coating thickness to reduce external L and HF Z Extra Loop

22 Measured Effect of Coating Thickness 0.1 Ω Standard X5R 100 µf ~0.25 mm thick coating 356 ph theory 385 ph meas. Z 1 mω 100 khz Same cap with coating reduced by 0.11 mm Inductance reduction: 156 ph theoretical 169 ph actual Actual HF inductance 216 ph 1 MHz 10 MHz 100 MHz Frequency

23 Observations

24 Simple Frequency Domain Model All parameters needed to sketch impedance can be simply calculated 0.1 Ω Z Z = 1/(jωC LF ) Z = Z 0 f 0 = 1/(2T round trip ) Z = jωl EXT 1 µ 0 Z 0 w C v T = 2 hl µ 0C v 1 mω 100 khz 1 MHz 10 MHz Frequency 100 MHz = L EXT = µ 0 t cl / w

25 Frequency Effect of Plate Orientation 22 µf MLC capacitor square ends 0.1 Ω Z 10 mω 2 mω 0.1 Ω ESR 10 mω 1 mω 100 khz 1 MHz 10 MHz 100 MHz

26 Application

27 DC-to-DC Converter Output Filter PWM waveform and output voltage waveform. Measured V=37.6mV New model V=37.7mV Lumped RLC model V=66.4mV 12 V to 1.2 V, 1 MHz buck converter with 2 x 22 µf caps Distributed model is much better than RLC model.

28 Conclusions MLC capacitors exhibit distributed behavior. LRC model can have factor-of-five error. Improved distributed model can Fit measurements precisely. Match observed in-circuit behavior. Simple model is also useful conceptually. Parameters are easily obtained from geometry. High-frequency impedance: Dominated by L EXT, due to coating thickness. Reducing coating thickness can greatly reduce highfrequency impedance.