PEEC Macromodels for Above Plane Decoupling Capacitors X. Fang, T. Makharashvili, A. E. Ruehli J. Fan, J. Drewniak and B. Archambeault
M. Cocchini IBM System and Technology Group Poughkeepsie, NY 12601 Email:
[email protected]
UMR/MST EMC Laboratory Missouri University of Science and Technology Rolla, MO 65401 Email:
[email protected]
Abstract—Decoupling capacitors perform an important function in the impedance reduction of power distribution systems. Hence, they are a key part of an electrical model required for the design of such systems. In this paper, we construct circuit models for the capacitors which include the local environment such that the overall PDN model is simplified and it can be considered as decoupled macromodels.
I. I NTRODUCTION Integrated circuits of all types need low impedance voltage supplies. Decoupling capacitors perform an important function in the impedance reduction of Power Distribution Network (PDN) systems. Hence, they are a key part of an electrical model required for the design of such systems. The circuit models provided by the manufacturers of the component, which usually consist of an inductor (ESL), a resistor (ESR) and a capacitance (ESC). We show that this data is insufficient for obtaining useful simple models. The key problem is that the above plane decoupling capacitors are strongly coupled to the local environment. Hence, it is impossible to represent the capacitors without the knowledge of the details of the environment, e.g., the spacing to the planes as well as the connection wires. The important issue is to find connections which allow a system to be subdivided into parts which are sufficiently decoupled. This is an on-going area of research [1]. In this paper, we construct circuit macromodels for the decoupling capacitors, which include the local environment, such that the overall PDN model is simplified. Such a macromodel must be sufficiently decoupled from the rest of the PDN system such that the coupling can be ignored with a sufficiently small error. The macromodel for the capacitance environment will simplify the modeling of the rest of the PDN system. Simple PEEC models of the local couplings have been considered before [2]. This work represents a more detailed model which also considers partitioning aspects. Conventionally, the model for the inductance of the decoupling capacitors is specified for the capacitors alone [3]. The problem is that the impedance of the above plane decoupling capacitor model is highly influenced by the connections and other mounting details such as the distance to the nearest groundplane. Hence, even a good PEEC equivalent circuit for the capacitor itself is insufficient since all the couplings to the mounting environment must be taken into account. For this reason, we pick a reasonable connection structure to the capacitor.
z y
x
Fig. 1.
Simplifies three layer model for decoupling capacitors. Capacitor Connections 15 20
z
25
y x
15 1 0 0 1
11 00 00 11
15
Fig. 2. a) Top view of connections to the capacitor with ground plane underneath.
The internal structure of most high capacitance decoupling capacitors consist of multiple layers. Representing all the layers would be computationally expensive. To keep the compute time reasonably low, we use the simplified 3 layer model shown in Fig. 1 such that the main properties of the model are preserved like the capacitor resistance R – also known as ESR. This value is less dependent on the environment than other element values. However, R is in general dependent on frequency [3]. We note that the PEEC preserves the frequency dependence. A model is needed which includes the strong couplings to the local PDN system. Hence, the model needs to include these additional local connections. Figure 2 shows connections where the capacitor is connected to the 20 × 20 mil pads. The connections to the macromodel are established to the 10 × 10 mil pads via wires or vias. Also shown is a plane which usually is located 5 mils to 30 mils underneath the contacts. An important quantity in the model is the distance between the capacitor plates and the contacts which we measured to be 9 mils. The result we are looking for is a simple capacitor environment which can be used for the PDN analysis to reduce complexity of the overall models. However, the PEEC model
must take all the local couplings into account between the capacitor in Fig. 1 and the connections in Fig. 2 as well as the ground plane underneath. Of course, the coupling of the capacitor macromodel to the rest of the PDN system must be weak. II. PEEC MODELS FOR DISTRIBUTED AND LUMPED MODEL
The transformation of the EM model into PEEC equivalent circuits has been published in many papers, e.g., [4]. Hence, we present only the issues which are key for this short paper. For the PDN model, a frequency content from dc to 2 GHz covers the spectrum of interest. Hence, a quasi-static PEEC model is sufficient for the decoupling capacitors, which can be assumed to be smaller than 1 cm, – less than λ/10 for fmax = 15 cm. The PEEC model used is based on the total electric field inside of a conductor is given by E(r, t) =
J (r, t) ∂A(r, t) − − ∇Φ(r, t) σ ∂t
(1)
where in our case E i (r, t) = 0 since we assume that the applied electric field is zero. Further, J (r, t) is the current density in the conductors and A(r, t), Φ(r, t) are the vector and scalar potentials respectively. Also, σ is the electrical conductivity. The second MNA equation is derived in its conventional form in the frequency domain as AT V (s) + (R + sLp)I(s) = 0.
(2)
For the first MNA equation, we want to represent the high permittivity between the inner layers shown in Fig. 3 a) with the lumped inter-layer capacitances shown in Fig. 3 b). Usually, a few capacitors are sufficient.
x
.
where the circuit matrix is −1 s(Ad Cd AT ) −Al d + Pp (5) [ M ]= −AT − (R + sLp) l The incidence matrices A can also be called matrix Kirchoff current laws. Specifically, Al is the matrix for the partial inductance connections. Also, Ad represents the matrix KCL for the dielectric capacitances Cd for the dielectrics. Hence, this is used to insert lumped capacitances Cd into the PEEC circuit. We should note that all the non-orthogonal partial inductances are coupled. This results in important inductive couplings between the capacitor in Fig. 1, the connections in Fig. 2 as well as the ground plane. The result obtained without these couplings is inaccurate. Hence, the ESL value specified by a manufacturer only gives an order of magnitude estimate. The terminals in Fig. 2 represent a much better location for partitioning the system. An assumption concerning the accuracy of the model is based on the loop opening between the terminal connections in Fig. 2 where the model is connected to the rest of the PDN circuit. The distance between the terminals needs to be small enough such that only a small fraction of the total inductance is not included between the terminals. III. N UMERICAL AND TEST RESULTS 60
y
a)
00 11 00 11 1 1111 0000 0 0 1 Cd 0 11 1 0 00 1 00 11 0 1 Cd 0 1 11 00 00 11 1111 0000 0 1 00 11 00 0 11 1 1 0 0 1 0 1 0 1 11 00 0 1
11 00 00 11
11 00 00 11
Cd 1 0 0 1 1 0
Cd
Input Impedance [dBOhm]
z
Finally, we can set up the MNA equations for the model. The resultant pcEFIE (potential-charge) integral equation is best described in the circuit domain by the MNA equations in terms of the node potentials Φ(s) and the source currents I s (s) Φ(s) I s (s) [ M ] = (4) I l (s) 0
11 00 1 0 00 11
Cd 11 00 00 11 0 1 1111 0000 0 1 Cd C d 1 0 0 11 1 00 0 1 0 1 0 1
1 0 0 1
Cd
b)
1 0 1 00 1
40
2 Lumped Capacitances 6 Lumped Capacitances 14 Lumped Capacitances 18 Lumped Capacitances
20 0 −20 −40 −60
0
10
Fig. 3. a) Side view of simplified cross-section model for decoupling capacitor. b) Single layer of Combined lumped capacitor and distributed PEEC model for decoupling capacitor.
The sum of all N internal layer capacitors is N X
Cd,k = ESC
(3)
k=1
where ESC is the value of the capacitance. The conductivity σ of the plated in Fig. 3 is chosen to represent the vendor specified resistance ESR of the capacitor. Here, we assume that a short exists at the center in the y-direction between the outside of the plates and the center conductor.
Fig. 4.
2
10 Frequency [MHz]
4
10
Response as a function of the number of Cd capacitances.
In this short paper, we consider the modeling of a 0402, 1 µF decoupling capacitor with the connection model such that a partitioned macromodel can be defined. The couplings between the local connections have a large impact on the circuit model used for the capacitor. Hence, we tested the macromodel unit consisting of the capacitor in Fig 1 connected to the interconnects in Fig. 2. We should note that the inductance in the macromodel does not have a relation to the ESL inductance value specified for capacitor mainly due to the partial inductance which is absent in this model.
Input Impedance [dBOhm]
80 60 40
0402 Cap − 5 mils above the ground 80
Input Impedance [dBOhm]
Our first test represents the hybrid PEEC-circuit model for the capacitor shown in Fig. 3. The first test checks how the number of discrete capacitors Cd representing the high value dielectrics impacts the response of the macromodel in Fig. 4. We also observe that 14 capacitors which are uniformly distributed between the plates give a good answer even for the parallel resonance observed at about 90 MHz. Next, we consider the impact of the groundplane under the capacitor macromodel, which is 5 mils for close planes and to 30 mils for planes which are further away. Figure 5 shows the behavior of the macromodel in the frequency range of interest also as a function of the ground plane spacing which is computed using the PEEC model. These inductances
60 40 20 0 −20 −40 −60
0
2
10
Fig. 6. PEEC − No Ground PEEC − 30 mils above the GND plane PEEC − 5 mils above the GND plane Measured
PEEC ADS − Macromodel with Parallel Resonance ADS − Simple Macromodel
10 Frequency [MHz]
4
10
Impedance curve comparison for the PEEC and macromodel.
Rp = 0.491 Ohm Rs = 5.6 mOhm
Cs = 1µF
Lp = 0.18 nH
20
Ls = 0.34 nH
0
Cp = 80 nF
−20 −40 −60
Fig. 7. 0
10
2
10 Frequency [MHz]
Macromodel which includes parallel resonance.
4
10
Fig. 5. Approximate inductance of the model for different ground plane distances.
TABLE I C OMPARISON OF MACROMODEL INDUCTANCES Model Details Spacing to ground Total Inductance Measured result 5 mils 0.17 nH With ground plane 5 mils 0.34 nH With ground plane 30 mils 0.39 nH No ground under capacitor ∞ 0.57 nH
are derived as a simple model at about 500 MHz. These calculations show that the macromodel must be specified as a function of the groundplane distance as the most important parameter. We also included a measurement in the results. However, the probes were placed close to the center part of the connections in Fig. 2. We learned from the measurement, that our parallel capacitor loss model should include dielectric losses. The measured resonance effect in the measurement in Fig 5 shows the presence of the parallel resonance. However, the damping is strong due to the dielectric loss [3]. The inclusion of the dielectric loss is one of the next steps in this work. The input impedance obtained from the PEEC solution can be synthesized into an appropriate circuit macromodel for the frequency range of interest. We note that the connection terminals to the model form an open loop. However, the distance is sufficiently small such that the error is also small. We provide two different equivalent circuits, one with a parallel resonance model and one without it. Finally, we show that the circuit model in Fig. 6 without the parallel circuit part leads to the dotted response in Fig. 6 which is a good approximate simpler macromodel.
IV. C ONCLUSIONS In this work, we show that a good macromodel for a decoupling capacitor, which includes a local connection model, provides a good solution for accurate PDN models. Such models are computationally very efficient. Alternatively, capacitor manufacturers need to specify internal partial inductance values and detailed physical shapes for the couplings to the capacitors such that more accurate models like the PEEC model given in this paper can be built. We give a specific example in this short paper for a good solution. Aspects for future work have been exposed, like macromodels which include the distance to the groundplane and the variability due to the exact placement of the decoupling capacitor on the local connection pads. V. ACKNOWLEDGMENT The authors would like to thank Michael Cracraft for his expertise in the measurement. R EFERENCES [1] A. E. Ruehli, D. Johns, and K. Shingarpure, “Partitioning systems into smaller sub-sytems for EM/CKT modeling,” in Proc. of the IEEE Int. Symp. on Electromagnetic Compatibility, Raleigh, NC, Aug. 2014. [2] C. Wang, J. Mao, G. Selli, S. Luan, L. Zhang, J. Fan, D. J. Pommerenke, R. E. DuBroff, and J. L. Drewniak, “An efficient approach for power delivery network with closed-form expressions for parasitic interconnect inductances,” IEEE Transactions on Advanced Packaging, vol. 29, no. 2, pp. 320–334, May 2006. [3] G. M. Harayda, A. Omi, and A. Yamamoto, “Improve your designs with large capacitance value multi-layer ceramic chip (mlcc) capacitors,” Panasonic, Electronic Component Brosure, pp. 1–5, 2003. [4] A. E. Ruehli, G. Antonini, J. Esch, J. Ekman, A. Mayo, and A. Orlandi, “Non-orthoganal PEEC formulation for time and frequency domain EM and circuit modeling.” in Proc. of the IEEE Int. Symp. on Electromagnetic Compatibility, vol. 45, May 2003, pp. 167–176.
