Analysis of Power grid noise using pi-fp and Hspice™

Gayathri Ganapathy 1, Donald Bennett & Raj Nair 2

gganapat@asu.edu

The constant scaling of technology and increase in performance has led to

increased power dissipation and hence greater noise in SoC power grids. Due to

excessive increase in power noise in microprocessors, a need to analyze the noise

voltage variations on the power grid more precisely and quickly has gathered

momentum. In order to make correct architectural, design and chip implementation

trade offs, comprehensive power noise estimation is now essential at various stages of

chip design and implementation.

By conducting several experiments focused on checking correlation in noise

estimation between Pi-fp and Hspice, an effort has been made to prove accuracy,

simplicity and ease of use of Pi-fp in comparison to Hspice. This paper contains

mathematical models and results from experiments performed to correlate the analysis

of on-chip power supply noise using a multi chip power noise simulator “Pi-fp” with

the industry’s gold standard circuit simulator Hspice.

Hspice: It is a circuit simulator widely accepted by the industry owing to its accuracy

and ability to precisely foresee the timing, power consumption and functionality of

their designs. It is able to simulate variability, has parameters to perform signal

integrity analysis, characterize Radio Frequency designs, and allows direct integration

1 ASU Research Intern at Anasim, 2005 – 2006, currently at Agilent Technologies2 Anasim Corporation

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with cadence composer. Hspice is often called an industry standard for circuit

simulation.

Pi-fp: Pi-fp is a powerful tool used to analyze chip level power supply noise. Pi-fp is

a fast system level power grid noise simulator. It calculates resistive and inductive

voltage changes in on-chip power grids as a function of grid parameters, time

dependent current sources and on and off chip capacitance. It helps in identifying the

hotspots where the maximum voltage drops occur. Optimizations such as power bus

sizing, on-chip decoupling capacitor placements etc. can then be done to improve

performance while minimizing noise.

Correlating the power noise variations between Pi-fp and Hspice

Exercise – 1:

Here, the behavior of a power grid having five transmission-lines, created

using Pi-fp, is compared with an equivalent grid created using Hspice. The goal is to

check the amount of correlation existing between the Pi-fp grid model and the Hspice

grid model.

Pi-fp model to analyze a small grid having five transmission lines is given

below:

.TRAN 2000E-12

.PLOT 40

.ACC 0.0080

.PRINTNODE ALLGchip1 0.1 0.1 0.0010 0.0200 0.020 10e-9 5e-9Ichip1 0.04 0.04 0.02 0.02 i1.txt 1T1 1 2 5 20e-9 30e-12 1Nchip1 1 0.02 0.02

Netlist definitions and current profile i1.txt are in the appendix.

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In the corresponding Hspice model, transmission-lines have been created

using the W model where RLC values are defined per unit length (/m).The equivalent

current source profile is created using a piecewise linear current source.

A distributed RLC Hspice model to analyze the power grid having five

transmission lines is also given in the appendix.

Calculation of equivalent unit length RLC values for the distributed RLC model

power grid created using Hspice:

In Pi-fp, the parameter values on the on-chip grid were defined as,

Gchip1 0.1 0.1 0.0010 0.02 0.020 10e-9 5e-9

Here, R = 0.020 ohm/square

L/cm =10 n H

C/cm2 = 5nF

Length of the grid = 0.1cm

Periodic space between transmission line pairs is 0.02 cm

Width of the wire is 0.0010 cm.

In Hspice, the RLC values of the grid were obtained as follows:

The number of t-line pairs = length of the grid / Periodic space between transmission

line Pairs.

= 0.1cm/0.02cm = 5.

Length of each t-line = length of the grid/ number of t-line pairs = 0.1cm/5 = 0.02 cm

= 0.2mm.

Number of squares per t-line = Length of each t-line / width of the wire = 20.

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For a t-line of length 0.02cm, Resistance = resistance/square x Number of squares

per transmission-

line.

= 0.02 x 20 = 0.4Ω.

Resistance/m across each T-line = Rper0.02cm / Length of each t-line = 2000 Ω .

Inductance /m = L/cm x 1e02 = 1u H/m.

The capacitance per unit length associated with each power grid wire is C = sCA,

where CA is the capacitance per unit area in the region surrounding the wire and s is

the wire spacing.

Chip Capacitance/cm2 = 5nF.

Wire spacing = 0.02cm.

So, Capacitance per cm = Chip Capacitance/cm2 x wire spacing

= 5nF/cm2 x 0.02cm

= 0.1nF.

Capacitance per m = 10nF.

The RLC values of the external transmission line connected to the grid were obtained

as shown below:

Since the RLC parameter values for the external transmission-lines are defined

per unit length (/cm), the conversion becomes simple.

Values in Pi-fp are:

R1/cm= 5 ohms.

L1/cm = 20nH

C1/cm = 30PF

Equivalent Values in Hspice are:

4

R1/m= 500 ohms.

L1/m = 2uH.

C1/m = 3nF.

Calculation of equivalent RLC values for the Lumped RLC model grid created in

Hspice:

Grid parameters:

Resistance of each t-line on the grid = Resistance/m x Length of each t-line

= 2000 x 0.2mm = 0.4 Ω.

Inductance of each t-line on the grid = Inductance/m x Length of each t-line

= 1uF x 0.2mm = 0.2nH.

Capacitance of each t-line on the grid = Capacitance/m x Length of each t-line

= 10nF x 0.2mm = 2PF.

Similarly, the external transmission-line RLC values are calculated and found to be

Resistance of transmission-line1 = 5 ohms

Inductance of transmission-line1 = 20nH

Capacitance of transmission-line1 = 30PF.

Hspice Lumped RLC model to analyze a power grid having five transmission

lines is given in the Appendix A.4.

To avoid reflections at node 2 (the end of transmission line 1 which is not

connected to the grid), one more transmission line was added. Once the mathematical

model was prepared, netlists were created in HSpice and were simulated and its

results were compared with that of the grid created using Pi-fp.

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Analysis of the obtained results:

At node1, the noise voltage peaks at 130mv in pi-fp, but in Hspice, it peaks at

140mv.The node 2 noise voltages vary in accordance with the transmission line-2

being connected to transmission line-1 (having same impedances or different

impedances).Here Node1 is the node that connects transmission line1 to a point on the

grid as specified in the netlist. Node2 is the node that connects transmission line-2 to

transmission line-1.The other end of transmission line-2 is left floating.

Case 1: When transmission line-2 is absent (i.e., when the other end of transmission

line-1 is floating):

In Pi-fp model, At node 1 the noise voltage peaks at 130mv.

At node 2 the noise voltage peaks at 190mv.

In Hspice model, At node1, the noise voltage peaks at 140mv.

At node 2, the noise voltage peaks at 230mv.

Figure 3.1: Pi-fp results at node1 and node 2

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Figure 3.2: Hspice results at node1 and node 2

Case 2: When transmission line-1 and transmission line-2 have the same impedances

(i.e., R1/m = R2/m, L1/m = L2/m and C1/m = C2/m):

In Pi-fp model, At node1, the noise voltage peaks at 130mv.

At node2, the noise voltage peaks at 95mv.

In Hspice model, At node1, the noise voltage peaks at 140mv.

At node2, the noise voltage peaks at 110mv.

Figure 3.4: Pi-fp results at node1 and node2

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Figure 3.5: Hspice results at node1 and node2

Case 3: When transmission line-1 and transmission line-2 have different impedances

(i.e.,R1/m != R2/m, L1/m != L2/m and C1/m != C2/m):

In Pi-fp model, At node1, the noise voltage peaks at 130mv.

At node2, the noise voltage peaks at 130mv.

In Hspice model, At node1, the noise voltage peaks at 140mv

At node2, the noise voltage peaks at 150mv

Figure 3.6: Pi-fp results for node1 and node2

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Figure 3.7: Hspice results for node1 and node2

The noise voltages in both the above mentioned models show almost identical

behavior, as per the required expectations. However, slight differences could be seen

between the noise voltage levels at the nodes, node1 and node2. The noise voltage

levels in the Hspice model are slightly higher, when compared to the Pi-fp model.

This could be due to lack of DC supply voltage provided to the grid in pi-fp, which

could lead to more reflections and hence more noise.

The lumped model yields similar results at node 1 for all the three cases, but

varies at node2. Also, oscillations are seen in this case. Here, each transmission line is

assumed to have a single R, L and C parameters. This model is far from realistic

scenario because the behavior of a transmission line is such that, voltages are seen at

all the points across it. But in the lumped model, voltages are seen only at the nodes at

the end of each component, which is not true.

3.1.2 Exercise – 2:

In this exercise, one more power grid having the same features as in the

previous experiment was connected to the first grid at the other end of its external

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transmission line-2. This was simulated using both Pi-fp and Hspice. The pi-fp and

Hspice code for this experiment is given in Appendix A.5 and A.6 respectively.

In Pi-fp, Node1 and Node 2 voltages of the previous Hspice model and the

new Hspice model remain almost the same. Node 3 is the node that connects the

transmission line 2 to the second power grid.

Case 1: In the absence of transmission line-2:

In Pi-fp model, at node2, the noise voltage = 55mv.

Figure 3.8: Pi-fp results at node3

In Hspice model, at node3, the noise voltage = 60mv.

Figure 3.9: HSpice results at node3

Case 2: When T1 line and T2 lines have the same impedances:

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Pi-fp model:

At node3, the noise voltage = 37mv.

Figure 3.10: Pi-fp results at node3

Hspice model:

At node3, the noise voltage = 32mv.

Figure 3.11: HSpice results at node3

Case 3: When Transmission line-1 and Transmission line-2 line have different

impedances:

Pi-fp model:

At node3, the noise voltage = 47mv.

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Figure 3.12: Pi-fp results at node3

Hspice model:

At node3, the noise voltage = 40mv.

Figure 3.13: HSpice results at node3

From the above experiments, it is evident that the power supply noise

simulator, Pi-fp and circuit simulation tool, Hspice correlates well for power grid

analysis. The Correlation between the two tools was seen to be much better when

more than one grid was connected. So, it seems that as the grid becomes more

complicated with dense array of transmission lines, the correlation between pi-fp and

Hspice models improves.

3.2 Most likely reasons for not obtaining the exact correlation between Pi-fp and

Hspice models:

• Pi-fp model assumes that the on-chip grid is comprised of a thick array of

transmission line pairs whereas in exercise-1, the power grid created using

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both the tools were constructed using only five transmission line pairs in both

directions.

• The load current is defined to be distributed across an area in Pi-fp whereas it

is defined as a point load in HSpice .This could lead to small amount of

differences seen between the two models.

• In HSpice, width of the transmission line could not be defined.

In order to relate both models more accurately, in HSpice, multiple current

sources can be connected in parallel. To achieve this, the longer transmission lines

will have to be divided in to multiple smaller transmission lines and then current

sources could be added at each connection node .But, creating such a huge grid (of

more than 20 transmission lines) is a cumbersome and time consuming task.

3.3 Advantages and limitations of Pi-fp and HSpice with reference to power grid

analysis:

Based on several experiments conducted to find out the correlation between

Pi-fp and HSpice, the following observations related to power noise analysis have

been made.

Pi-fp:

Advantages:

• It is fast .Simulation run time can be drastically reduced using this tool.

• It is simple. An entire power grid of a specific size can be defined using a

single statement.

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• It is very easy to make modifications to the on chip grid parameters and

analyze the noise voltage variations accordingly.

• The tool has a feature which allows us to have a look at the potential

variations all along the length of a transmission line, at regular time intervals

as defined in the netlist.

• On-chip grid potential can also be observed at different time instants.

• It is a very good tool for making quick qualitative analysis.

• Most parameters are defined as per unit area, per unit length or ohms per

square. This makes it easier to create a large system model without the need to

consider detailed connections between each wire, capacitors and load currents.

• Power noise estimation is done at system level.

Limitations:

• The tool is Power grid noise analysis specific.

• It is based on the assumption that noise voltage is always differential and that

the supply noise voltage bounce and the ground noise voltage bounce are

always equal.

• There is no definition provided for the addition of external DC supplies.

Hspice:

Advantages:

• It allows us to analyze the power grid noise voltage in a more realistic

environment, with the inclusion of external DC supply voltages.

• Has been recognized as an industry standard for cell designing and process

modeling.

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Limitations:

• Its biggest disadvantage is its slow simulation run time, while analyzing noise

on a power grid. The simulation run time was seen to be inversely proportional

to the complexity of the power grid.

• Constructing a complex power grid using this tool is a difficult task.

• Any slight modifications to the grid are very hard to implement due to the

drastic increase in the size of the netlist.

• Power estimation is done at the design level.

3.4 Conclusion:

Based on the experiments conducted to find out the correlation between Pi-fp and

HSpice, it is evident that Pi-fp happens to be a fast, system level power grid noise

simulator. Since an entire grid can be defined using just a single line, any number of

connected IC’s or transmission lines can be simulated, i.e. it is a multi chip simulation

tool. Tools like Hspice estimates power noise at the circuit level. So, after analysis, if

any changes have to be made to the architecture itself, unfortunately it will be too late.

Thus, a tool like Pi-fp which is a system level power noise analyzer helps in achieving

correct architectural and implementation trade offs thereby helping to achieve the

performance targets of future VLSI designs.

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Bibliography

[8] Chen, H. H. and Ling, D.D., “Power Supply Noise Analysis Methodology for Deep-Submicron VLSI Chip Design”, Design Automation Conference, June1997 Pages: 638-643http://ieeexplore.ieee.org.ezproxy1.lib.asu.edu/iel3/4655/13048/00597223.pdf?tp=&arnumber=597223&isnumber=13048

Bennett, D., “Pi-fp user manual”, Anasim Corporation.

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AppendixA.1 Pi-fp Netlist definitions :

.Tran denotes the simulation run time.

.ACC gives the accuracy parameter. Plot defines the number of plots during the simulation.

Global on-chip grid definition is given by,

Gchipname <chip_width (cm) > <chip_height (cm) > <tline_wire_width(cm) > <tline_periodic_space(cm) >< sheet_resistance(Ohms/sq.cm)> <inductance (H/cm> <Default_cap(F/cm2)>

Example: Gchip1 0.1 0.1 0.0010 0.0200 0.020 10e-9 5e-9

On chip current source is given by

Ichipname <x_location(cm)> <y_location(cm)> <width(cm)> <height(cm)> <filename> <repeats>

Example: Ichip1 0.04 0.04 0.02 0.02 i1.txt 1The current profile is given in the appendix A.2.

A transmission line is defined asTname <nodename1> <nodename2> <R (Ohms/cm)> <L (H/cm)> <C (F/cm)> <length (cm)>

Example: T1 1 2 5 20e-9 30e-12 1

The node where the transmission line is connected to the grid is given byNchipname <nodename> < X (cm)> <Y (cm)>Where X denotes the location on the X-axis of the grid where the transmission line node will be connected. Similarly, Y denotes the location on the Y-axis of the grid where the transmission line node will be connected.

A.2 Current Profile (i1.txt):

0.0 0.020e-11 0.0309024e-11 0.0587785256.00E-11 0.0809016998E-11 0.0951056521.00E-10 0.11.2E-10 0.0951056521.40E-10 0.080901699

17

1.6E-10 0.0587785251.80E-10 0.0409016992E-10 0.042.20E-10 0.0352.4E-10 0.032.60E-10 0.032.8E-10 0.033.00E-10 0.0253.2E-10 0.023.40E-10 0.013.6E-10 03.80E-10 04E-10 0

A.3 RLC distributed model grid (containing five transmission lines) created using Hspice:

*simple power grid model.option POST

W1 N=1 1 0 2 0 RLGCMODEL=tline_1 l=0.0002W2 N=1 2 0 3 0 RLGCMODEL=tline_1 l=0.0002W3 N=1 3 0 4 0 RLGCMODEL=tline_1 l=0.0002W4 N=1 4 0 5 0 RLGCMODEL=tline_1 l=0.0002W5 N=1 5 0 6 0 RLGCMODEL=tline_1 l=0.0002W6 N=1 8 0 7 0 RLGCMODEL=tline_1 l=0.0002W7 N=1 9 0 8 0 RLGCMODEL=tline_1 l=0.0002W8 N=1 10 0 9 0 RLGCMODEL=tline_1 l=0.0002W9 N=1 11 0 10 0 RLGCMODEL=tline_1 l=0.0002W10 N=1 12 0 11 0 RLGCMODEL=tline_1 l=0.0002W11 N=1 13 0 14 0 RLGCMODEL=tline_1 l=0.0002W12 N=1 14 0 15 0 RLGCMODEL=tline_1 l=0.0002W13 N=1 15 0 16 0 RLGCMODEL=tline_1 l=0.0002W14 N=1 16 0 17 0 RLGCMODEL=tline_1 l=0.0002W15 N=1 17 0 18 0 RLGCMODEL=tline_1 l=0.0002W16 N=1 20 0 19 0 RLGCMODEL=tline_1 l=0.0002W17 N=1 21 0 20 0 RLGCMODEL=tline_1 l=0.0002W18 N=1 22 0 21 0 RLGCMODEL=tline_1 l=0.0002W19 N=1 23 0 22 0 RLGCMODEL=tline_1 l=0.0002W20 N=1 24 0 23 0 RLGCMODEL=tline_1 l=0.0002W21 N=1 25 0 26 0 RLGCMODEL=tline_1 l=0.0002W22 N=1 26 0 27 0 RLGCMODEL=tline_1 l=0.0002W23 N=1 27 0 28 0 RLGCMODEL=tline_1 l=0.0002W24 N=1 28 0 29 0 RLGCMODEL=tline_1 l=0.0002W25 N=1 29 0 30 0 RLGCMODEL=tline_1 l=0.0002W26 N=1 32 0 31 0 RLGCMODEL=tline_1 l=0.0002W27 N=1 33 0 32 0 RLGCMODEL=tline_1 l=0.0002W28 N=1 34 0 33 0 RLGCMODEL=tline_1 l=0.0002W29 N=1 35 0 34 0 RLGCMODEL=tline_1 l=0.0002W30 N=1 36 0 35 0 RLGCMODEL=tline_1 l=0.0002

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W31 N=1 1 0 12 0 RLGCMODEL=tline_1 l=0.0002W32 N=1 2 0 11 0 RLGCMODEL=tline_1 l=0.0002W33 N=1 3 0 10 0 RLGCMODEL=tline_1 l=0.0002W34 N=1 4 0 9 0 RLGCMODEL=tline_1 l=0.0002W35 N=1 5 0 8 0 RLGCMODEL=tline_1 l=0.0002W36 N=1 6 0 7 0 RLGCMODEL=tline_1 l=0.0002W37 N=1 7 0 18 0 RLGCMODEL=tline_1 l=0.0002W38 N=1 8 0 17 0 RLGCMODEL=tline_1 l=0.0002W39 N=1 9 0 16 0 RLGCMODEL=tline_1 l=0.0002W40 N=1 10 0 15 0 RLGCMODEL=tline_1 l=0.0002W41 N=1 11 0 14 0 RLGCMODEL=tline_1 l=0.0002W42 N=1 12 0 13 0 RLGCMODEL=tline_1 l=0.0002W43 N=1 13 0 24 0 RLGCMODEL=tline_1 l=0.0002W44 N=1 14 0 23 0 RLGCMODEL=tline_1 l=0.0002W45 N=1 15 0 22 0 RLGCMODEL=tline_1 l=0.0002W46 N=1 16 0 21 0 RLGCMODEL=tline_1 l=0.0002W47 N=1 17 0 20 0 RLGCMODEL=tline_1 l=0.0002W48 N=1 18 0 19 0 RLGCMODEL=tline_1 l=0.0002W49 N=1 19 0 30 0 RLGCMODEL=tline_1 l=0.0002W50 N=1 20 0 29 0 RLGCMODEL=tline_1 l=0.0002W51 N=1 21 0 28 0 RLGCMODEL=tline_1 l=0.0002W52 N=1 22 0 27 0 RLGCMODEL=tline_1 l=0.0002W53 N=1 23 0 26 0 RLGCMODEL=tline_1 l=0.0002W54 N=1 24 0 25 0 RLGCMODEL=tline_1 l=0.0002W55 N=1 25 0 36 0 RLGCMODEL=tline_1 l=0.0002W56 N=1 26 0 35 0 RLGCMODEL=tline_1 l=0.0002W57 N=1 27 0 34 0 RLGCMODEL=tline_1 l=0.0002W58 N=1 28 0 33 0 RLGCMODEL=tline_1 l=0.0002W59 N=1 29 0 32 0 RLGCMODEL=tline_1 l=0.0002W60 N=1 30 0 31 0 RLGCMODEL=tline_1 l=0.0002W61 N=1 26 0 40 0 RLGCMODEL=tline_2 l=0.01

R1 22 99 0.0001

Vsupply 6 0 DC 5V

I1 99 0 PL (0a 0s 31ma 20ps 58ma 40ps 81ma 60ps 95ma 80ps 100ma 100ps 95ma 120ps 81ma 140ps 58ma 160ps 41ma 180ps 40ma 200ps 35ma 220ps 30ma 240ps 30ma 260ps 30ma 280ps 25ma 300ps 20ma 320ps 10ma 340ps 0a 360ps 0a 380ps 0a 2ns)

.tran 0.1ns 2ns

.plot tran V(1) V(26) v(6) V(22)

.plot tran I(R1)

*RLGC matrix.MODEL tline_1 W MODELTYPE=RLGC N=1+Lo=+10e-7+Co=

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+10e-9+Ro=+2000+Go=+0+Rs=+0+Gd=+0

*RLGC matrix.MODEL tline_2 W MODELTYPE=RLGC N=1+Lo=+20e-7+Co=+30e-10+Ro=+500+Go=+0+Rs=+0+Gd=+0

A.4 Lumped RLC model grid containing five transmission lines, created using Hspice is given below:

*simple power grid model.option POST

X1 1 2 tline1X2 2 3 tline1X3 3 4 tline1X4 4 5 tline1X5 5 6 tline1X6 7 8 tline1X7 8 9 tline1X8 9 10 tline1X9 10 11 tline1X10 11 12 tline1X11 13 14 tline1X12 14 15 tline1X13 15 16 tline1X14 16 17 tline1X15 17 18 tline1X16 19 20 tline1X17 20 21 tline1X18 21 22 tline1

20

X19 22 23 tline1X20 23 24 tline1X21 25 26 tline1X22 26 27 tline1X23 27 28 tline1X24 28 29 tline1X25 29 30 tline1X26 31 32 tline1X27 32 33 tline1X28 33 34 tline1X29 34 35 tline1X30 35 36 tline1X31 1 12 tline1X32 2 11 tline1X33 3 10 tline1X34 4 9 tline1X35 5 8 tline1X36 6 7 tline1X37 12 13 tline1X38 11 14 tline1X39 10 15 tline1X40 9 16 tline1X41 8 17 tline1X42 7 18 tline1X43 13 24 tline1X44 14 23 tline1X45 15 22 tline1X46 16 21 tline1X47 17 20 tline1X48 18 19 tline1X49 24 25 tline1X50 23 26 tline1X51 22 27 tline1X52 21 28 tline1X53 20 29 tline1X54 19 30 tline1X55 25 36 tline1X56 26 35 tline1X57 27 34 tline1X58 28 33 tline1X59 29 32 tline1X60 30 31 tline1R1 22 99 0.0001

Vsupply 6 0 DC 5VI1 99 0 PL (0a 0s 31ma 20ps 58ma 40ps 81ma 60ps 95ma 80ps 100ma 100ps 95ma 120ps 81ma 140ps 58ma 160ps 41ma 180ps 40ma 200ps 35ma 220ps 30ma 240ps 30ma 260ps 30ma 280ps 25ma 300ps 20ma 320ps 10ma 340ps 0a 360ps 0a 380ps 0a 2ns)

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.tran 0.1ns 2ns

.plot tran V(1) V(26) v(6) V(22)

.plot tran I(R1)

.subckt tline1 1 5

.param R_per_m = 4000

.param L_per_m = 1e-6

.param C_per_m = 10e-9

.param length = 0.0002

.param rr = R_per_m*length/2

.param ll = L_per_m*length/2

.param cc = C_per_m*lengthR1 1 2 rrL1 2 3 llC1 3 0 ccL2 3 4 llR2 4 5 rr.ends

.subckt tline2 1 5

.param R_per_m = 1000

.param L_per_m = 2e-6

.param C_per_m = 3e-9

.param length = 0.01

.param rr = R_per_m*length/2

.param ll = L_per_m*length/2

.param cc = C_per_m*lengthR1 1 2 rrL1 2 3 llC1 3 0 ccL2 3 4 llR2 4 5 rr.ends

.end

A.5 Pi-fp model for exercise-2 in chapter 3:

.TRAN 2000E-12

.PLOT 40

.ACC 0.0080

.PRINTNODE ALLGchip1 0.1 0.1 0.0010 0.02 0.020 10e-9 5e-9Ichip1 0.04 0.04 0.02 0.02 i1.txt 1T1 1 2 5 20e-9 30e-12 1Nchip1 1 0.02 0.02T2 2 3 5 20e-9 30e-12 2Gchip2 0.1 0.1 0.0010 0.0200 0.020 10e-9 5e-9

22

Nchip 2 3 0.02 0.02

i1.txt shows the current profile and is contained in a separate file and is given in A.2.

A.6 RLC distributed model grid created using Hspice for exercise-2 in chapter 3:

*simple power grid model.option POST

W1 N=1 1 0 2 0 RLGCMODEL=tline_1 l=0.0002W2 N=1 2 0 3 0 RLGCMODEL=tline_1 l=0.0002W3 N=1 3 0 4 0 RLGCMODEL=tline_1 l=0.0002W4 N=1 4 0 5 0 RLGCMODEL=tline_1 l=0.0002W5 N=1 5 0 6 0 RLGCMODEL=tline_1 l=0.0002W6 N=1 8 0 7 0 RLGCMODEL=tline_1 l=0.0002W7 N=1 9 0 8 0 RLGCMODEL=tline_1 l=0.0002W8 N=1 10 0 9 0 RLGCMODEL=tline_1 l=0.0002W9 N=1 11 0 10 0 RLGCMODEL=tline_1 l=0.0002W10 N=1 12 0 11 0 RLGCMODEL=tline_1 l=0.0002W11 N=1 13 0 14 0 RLGCMODEL=tline_1 l=0.0002W12 N=1 14 0 15 0 RLGCMODEL=tline_1 l=0.0002W13 N=1 15 0 16 0 RLGCMODEL=tline_1 l=0.0002W14 N=1 16 0 17 0 RLGCMODEL=tline_1 l=0.0002W15 N=1 17 0 18 0 RLGCMODEL=tline_1 l=0.0002W16 N=1 20 0 19 0 RLGCMODEL=tline_1 l=0.0002W17 N=1 21 0 20 0 RLGCMODEL=tline_1 l=0.0002W18 N=1 22 0 21 0 RLGCMODEL=tline_1 l=0.0002W19 N=1 23 0 22 0 RLGCMODEL=tline_1 l=0.0002W20 N=1 24 0 23 0 RLGCMODEL=tline_1 l=0.0002W21 N=1 25 0 26 0 RLGCMODEL=tline_1 l=0.0002W22 N=1 26 0 27 0 RLGCMODEL=tline_1 l=0.0002W23 N=1 27 0 28 0 RLGCMODEL=tline_1 l=0.0002W24 N=1 28 0 29 0 RLGCMODEL=tline_1 l=0.0002W25 N=1 29 0 30 0 RLGCMODEL=tline_1 l=0.0002W26 N=1 32 0 31 0 RLGCMODEL=tline_1 l=0.0002W27 N=1 33 0 32 0 RLGCMODEL=tline_1 l=0.0002W28 N=1 34 0 33 0 RLGCMODEL=tline_1 l=0.0002W29 N=1 35 0 34 0 RLGCMODEL=tline_1 l=0.0002W30 N=1 36 0 35 0 RLGCMODEL=tline_1 l=0.0002W31 N=1 1 0 12 0 RLGCMODEL=tline_1 l=0.0002W32 N=1 2 0 11 0 RLGCMODEL=tline_1 l=0.0002W33 N=1 3 0 10 0 RLGCMODEL=tline_1 l=0.0002 W34 N=1 4 0 9 0 RLGCMODEL=tline_1 l=0.0002W35 N=1 5 0 8 0 RLGCMODEL=tline_1 l=0.0002W36 N=1 6 0 7 0 RLGCMODEL=tline_1 l=0.0002W37 N=1 7 0 18 0 RLGCMODEL=tline_1 l=0.0002W38 N=1 8 0 17 0 RLGCMODEL=tline_1 l=0.0002W39 N=1 9 0 16 0 RLGCMODEL=tline_1 l=0.0002W40 N=1 10 0 15 0 RLGCMODEL=tline_1 l=0.0002

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W41 N=1 11 0 14 0 RLGCMODEL=tline_1 l=0.0002W42 N=1 12 0 13 0 RLGCMODEL=tline_1 l=0.0002W43 N=1 13 0 24 0 RLGCMODEL=tline_1 l=0.0002W44 N=1 14 0 23 0 RLGCMODEL=tline_1 l=0.0002W45 N=1 15 0 22 0 RLGCMODEL=tline_1 l=0.0002W46 N=1 16 0 21 0 RLGCMODEL=tline_1 l=0.0002W47 N=1 17 0 20 0 RLGCMODEL=tline_1 l=0.0002W48 N=1 18 0 19 0 RLGCMODEL=tline_1 l=0.0002W49 N=1 19 0 30 0 RLGCMODEL=tline_1 l=0.0002W50 N=1 20 0 29 0 RLGCMODEL=tline_1 l=0.0002W51 N=1 21 0 28 0 RLGCMODEL=tline_1 l=0.0002W52 N=1 22 0 27 0 RLGCMODEL=tline_1 l=0.0002W53 N=1 23 0 26 0 RLGCMODEL=tline_1 l=0.0002W54 N=1 24 0 25 0 RLGCMODEL=tline_1 l=0.0002W55 N=1 25 0 36 0 RLGCMODEL=tline_1 l=0.0002W56 N=1 26 0 35 0 RLGCMODEL=tline_1 l=0.0002W57 N=1 27 0 34 0 RLGCMODEL=tline_1 l=0.0002W58 N=1 28 0 33 0 RLGCMODEL=tline_1 l=0.0002W59 N=1 29 0 32 0 RLGCMODEL=tline_1 l=0.0002W60 N=1 30 0 31 0 RLGCMODEL=tline_1 l=0.0002W61 N=1 22 0 40 0 RLGCMODEL=tline_2 l=0.01W62 N=1 40 0 70 0 RLGCMODEL=tline_3 l=0.02

W65 N=1 45 0 46 0 RLGCMODEL=tline_1 l=0.0002W66 N=1 46 0 47 0 RLGCMODEL=tline_1 l=0.0002W67 N=1 47 0 48 0 RLGCMODEL=tline_1 l=0.0002W68 N=1 48 0 49 0 RLGCMODEL=tline_1 l=0.0002W69 N=1 49 0 50 0 RLGCMODEL=tline_1 l=0.0002W70 N=1 50 0 51 0 RLGCMODEL=tline_1 l=0.0002W71 N=1 56 0 55 0 RLGCMODEL=tline_1 l=0.0002W72 N=1 55 0 54 0 RLGCMODEL=tline_1 l=0.0002W73 N=1 54 0 53 0 RLGCMODEL=tline_1 l=0.0002W74 N=1 53 0 52 0 RLGCMODEL=tline_1 l=0.0002W75 N=1 52 0 51 0 RLGCMODEL=tline_1 l=0.0002W76 N=1 57 0 58 0 RLGCMODEL=tline_1 l=0.0002W77 N=1 58 0 59 0 RLGCMODEL=tline_1 l=0.0002W78 N=1 59 0 60 0 RLGCMODEL=tline_1 l=0.0002W79 N=1 60 0 61 0 RLGCMODEL=tline_1 l=0.0002W80 N=1 61 0 62 0 RLGCMODEL=tline_1 l=0.0002W81 N=1 68 0 66 0 RLGCMODEL=tline_1 l=0.0002W82 N=1 67 0 66 0 RLGCMODEL=tline_1 l=0.0002W83 N=1 66 0 65 0 RLGCMODEL=tline_1 l=0.0002W84 N=1 65 0 64 0 RLGCMODEL=tline_1 l=0.0002W85 N=1 64 0 63 0 RLGCMODEL=tline_1 l=0.0002W86 N=1 69 0 70 0 RLGCMODEL=tline_1 l=0.0002W87 N=1 70 0 71 0 RLGCMODEL=tline_1 l=0.0002W88 N=1 71 0 72 0 RLGCMODEL=tline_1 l=0.0002W89 N=1 72 0 73 0 RLGCMODEL=tline_1 l=0.0002

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W90 N=1 73 0 74 0 RLGCMODEL=tline_1 l=0.0002W91 N=1 80 0 79 0 RLGCMODEL=tline_1 l=0.0002W92 N=1 79 0 78 0 RLGCMODEL=tline_1 l=0.0002W93 N=1 78 0 77 0 RLGCMODEL=tline_1 l=0.0002W94 N=1 77 0 76 0 RLGCMODEL=tline_1 l=0.0002W95 N=1 76 0 75 0 RLGCMODEL=tline_1 l=0.0002W96 N=1 45 0 56 0 RLGCMODEL=tline_1 l=0.0002W97 N=1 46 0 55 0 RLGCMODEL=tline_1 l=0.0002 W98 N=1 47 0 54 0 RLGCMODEL=tline_1 l=0.0002W99 N=1 48 0 53 0 RLGCMODEL=tline_1 l=0.0002W100 N=1 49 0 52 0 RLGCMODEL=tline_1 l=0.0002W101 N=1 56 0 57 0 RLGCMODEL=tline_1 l=0.0002W102 N=1 55 0 58 0 RLGCMODEL=tline_1 l=0.0002W103 N=1 54 0 59 0 RLGCMODEL=tline_1 l=0.0002W104 N=1 53 0 60 0 RLGCMODEL=tline_1 l=0.0002W105 N=1 52 0 61 0 RLGCMODEL=tline_1 l=0.0002W106 N=1 51 0 62 0 RLGCMODEL=tline_1 l=0.0002W107 N=1 57 0 68 0 RLGCMODEL=tline_1 l=0.0002W108 N=1 58 0 67 0 RLGCMODEL=tline_1 l=0.0002W109 N=1 59 0 66 0 RLGCMODEL=tline_1 l=0.0002W110 N=1 60 0 65 0 RLGCMODEL=tline_1 l=0.0002W111 N=1 61 0 64 0 RLGCMODEL=tline_1 l=0.0002W112 N=1 62 0 63 0 RLGCMODEL=tline_1 l=0.0002W113 N=1 68 0 69 0 RLGCMODEL=tline_1 l=0.0002W114 N=1 67 0 70 0 RLGCMODEL=tline_1 l=0.0002W115 N=1 66 0 71 0 RLGCMODEL=tline_1 l=0.0002W116 N=1 65 0 72 0 RLGCMODEL=tline_1 l=0.0002W117 N=1 64 0 73 0 RLGCMODEL=tline_1 l=0.0002W118 N=1 63 0 74 0 RLGCMODEL=tline_1 l=0.0002W119 N=1 69 0 80 0 RLGCMODEL=tline_1 l=0.0002W120 N=1 70 0 79 0 RLGCMODEL=tline_1 l=0.0002W121 N=1 71 0 78 0 RLGCMODEL=tline_1 l=0.0002W122 N=1 72 0 77 0 RLGCMODEL=tline_1 l=0.0002W123 N=1 73 0 76 0 RLGCMODEL=tline_1 l=0.0002W124 N=1 74 0 75 0 RLGCMODEL=tline_1 l=0.0002

R1 22 99 .0001

W125 N=1 6 0 100 0 RLGCMODEL=tline_4 l=0.05Vsupply 100 0 DC 5V

I1 99 0 PL (0a 0s 31ma 20ps 58ma 40ps 81ma 60ps 95ma 80ps 100ma 100ps 95ma 120ps 81ma 140ps 58ma 160ps 41ma 180ps 40ma 200ps 35ma 220ps 30ma 240ps 30ma 260ps 30ma 280ps 25ma 300ps 20ma 320ps 10ma 340ps 0a 360ps 0a 380ps 0a 2ns) .tran 0.1ns 8ns.plot tran V(1) V(26) v(6) V(22) .plot tran I(R1)

25

*RLGC matrix.MODEL tline_1 W MoDELTYPE=RLGC N=1+Lo=+10e-7+Co=+10e-9+Ro=+4000+Go=+0+Rs=+0+Gd=+0

*RLGC matrix.MODEL tline_2 W MoDELTYPE=RLGC N=1+Lo=+20e-7+Co=+30e-10+Ro=+1000+Go=+0+Rs=+0+Gd=+0

*RLGC matrix.MODEL tline_3 W MoDELTYPE=RLGC N=1+Lo=+40e-7+Co=+15e-10+Ro=+1000+Go=+0+Rs=+0+Gd=+0

*RLGC matrix.MODEL tline_4 W MoDELTYPE=RLGC N=1+Lo=+20e-7

26

+Co=+30e-10+Ro=+10+Go=+0+Rs=+0+Gd=+0

.end

27