ir-drop in on-chip power distribution networks

7/28/2019 IR-Drop in on-Chip Power Distribution Networks

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512 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 21, NO. 3, MARCH 2013

IR-Drop in On-Chip Power Distribution Networksof ICs With Nonuniform Power Consumption

Josep Rius, Member, IEEE

AbstractA compact IR-drop model for on-chip powerdistribution networks in array and wire-bonded ICs is analyzed.Chip dimensions, size, and location of the supply pads, metalcoverage, piecewise distribution of IC consumption, and theresistance between the pads and the power supply are consideredto obtain closed-form expressions for the IR-drop. The IR-dropmodel is validated by comparing its results with electricalsimulations. The obtained error is in the range of 1%.

Index Terms IC modeling, IR-drop, power distributionnetworks (PDNs), power supply noise (PSN).

I. INTRODUCTION

TO ENSURE a good supply voltage throughout the IC, andfor the high-consumption and high-density ICs availablein current technologies, the on-chip power distribution network

(PDN) is usually organized as a grid of wide parallel wiresin the two or more upper metal layers covering the IC

surface. Connection to the package is currently made by two

approaches: the so-called peripheral bonding, in which the

supply pads are distributed along the sides of the IC, and

array bonding, where the supply pads are distributed in anarray over the whole IC surface, in a flip-chip package.

The PDN behaves as a conductive mesh with resistive,

inductive, and capacitive properties. As a consequence, the

electric current spikes produced during the circuit activity aretransformed into voltage bounces at the supply terminals of the

internal circuits. This power supply noise (PSN) has several

undesirable effects on the performance and reliability of ICs

[1]. A good PDN design is therefore necessary to reduce thePSN below a specified value. The PSN can be roughly divided

into static and dynamic. Static PSN, or IR-drop, is the voltage

drop caused by the DC supply current in the PDN resistances,

whereas dynamic PSN is due to transients exciting the PDN

inductances and capacitances. The analysis of the IR-drop is

important [2], [3], [1] because it allows addressing the most

important issues in PDN design, that is, width and pitch of

the PDN wires [4][8], [9] and size, number, and location of

pads [10], [11], [4], [5], [12][13]. When a dynamic analysisof the PSN is required, there are additional important issues

to solve, such as the impact of on-chip PDN inductance [14],[15] and the amount and distribution of on-chip decoupling

capacitance [15], [1].

Manuscript received June 6, 2011; revised November 9, 2011; acceptedFebruary 16, 2012. Date of publication March 20, 2012; date of current versionFebruary 20, 2013.

The author is with the Department of Electronic Engineering,Universitat Politecnica de Catalunya, Barcelona 08028, Spain (e-mail:[email protected]).

Digital Object Identifier 10.1109/TVLSI.2012.2188918

The design of a good, reliable on-chip PDN of a digital IC

is a very complex task because designers cannot anticipate all

the details of the design. The PSN depends on the location,

size, and activity of the circuit blocks. Therefore, in order to

check that the PSN is below the specified value, it is necessary

to simulate the complete circuit, which is clearly unfeasible

for large ICs. The help of specific CAD tools alleviates thisproblem. However, due to the simulation time, CAD tools are

primarily intended for use in postlayout verification, after the

design is complete. A failure in the design involves a costly

rework of the PDN. This leads to overdimensioning, resulting

in the sacrifice of valuable routing resources. For this reason,the use of prelayout tools in the early stages of the PDN

design, which give approximate results for the PSN expected,

becomes a necessity [16], [9], [17].This paper is exclusively centered on IR-drop. It addresses

the estimation of the PDN performance in the early steps of an

IC design by an analytical approach. As mentioned, this prob-lem can also be tackled with numerical tools. However, the

analytical approach has the advantage that, in addition to pro-viding a numerical solution, it shows the relationships between

the significant parameters, improving the understanding of the

problem. Thus, there is room for an analytical tool that, in an

interactive fashion, rapidly provides approximate results for

the expected IR-drop of a PDN. This tool shows the depen-dency of IR-drop on the number and size of pads and consum-ing blocks, IC dimensions, current density and sheet resistance,

thus allowing rapid optimization of these parameters.In their seminal paper [16], Shakeri and Meindl demonstrate

that the PDN can be approximated as a continuous layer of

conductive material and that IR-drop can be calculated by

solving a partial differential equation, that is, Poisson equation,

with the proper boundary conditions and source function. This

paper takes as the starting point the framework and definitionof the problem as presented in [16] without repeating the

derivation of the Poisson equation and related concepts, whichare extensively discussed in [16]. The organization of this

paper is as follows. Section II presents the problem to solve.

In Section III, expressions to obtain the IR-drop at any pointof an infinite array-bonded PDN are derived for any number

and location of pads and any number of rectangular consuming

blocks. The results are used in Section IV to find the solution

of the same problem in finite PDNs. In Section V, we compare

our formulas with electrical simulations of several PDNs.

Section VI discusses some features of the proposed approach

and finally, Section VII summarizes the conclusions of this

paper.

10638210/$31.00 2012 IEEE


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RIUS: IR-DROP IN ON-CHIP POWER DISTRIBUTION NETWORKS OF ICs 513

Fig. 1. IC with six consuming blocks and an array of power/ground pads.

I I . STATEMENT OF THE PROBLEM

According to the approach of [16], the IR-drop in a PDNfollows the Poisson equation:

2 V = RSJ (1)

where V is the IR-drop (V), RS is the sheet resistance of thePDN (), and J is the current density function (A/m2). The

sheet resistance RS is assumed to be constant in the whole

IC. In the array-bonding configuration, the supply current

drawn by the consuming blocks is supplied by an array of

power/ground pads distributed over the IC surface. Fig. 1

illustrates an IC with six blocks and an array of power/groundpads (small black and white squares).

In the array-bonding configuration, the normal derivative of

the voltage, V/n (where symbol n in a rectangular IC meanseither x or y), at the four sides of the PDN is zero; that is, the

current drawn by the blocks flows from the power to ground

pads through the PDN.

The solution of (1) for the simple case of constant J in an

infinite IC with an infinite PDN and an infinite regular array

of pads is shown in [16]. A solution for the IR-drop at any

point is given in the form of several double and triple infinite

series in [16]. After some approximations and numerical work,

the authors show that the maximum IR-drop (which is at the

center of a square with four supply pads at its vertices) isgiven by

VIR max =RSIPAD

2 ln 0.387a

DPAD

(2)

where a is the distance between adjacent pads, is a correc-tion factor related to the pad shape, and DPAD is the side

length of a square pad. The coefficient 0.387 is obtainedafter a numerical calculation of the double and triple infinite

series and assuming several approximations. Equation (2)

puts together the relevant variables in PDN design: the sheetresistance of the power grid, RS, which is related to the metal

coverage of such grid; the current per pad, IPAD; the pad

density, a, which is related to the distance between the pads;

and the pad size, DPAD.

a

b

PAD

observation

point P

rpp

rpxy

dxdy

A

ap

Rpad

V0

a

b

PAD

observation

point P

rpp

rpxy

dxdy

A

ap

Rpad

V0

Fig. 2. Parameters involved in the analysis of the IR-drop at the observationpoint P in an infinite resistive plane with one pad and one consuming block.

In Sections III and IV, we obtain approximate expressionsfor the IR-drop under more realistic conditions, that is, the

current density J is not constant in the whole IC and/orthe PDN is of finite dimensions. Instead of solving (1)

directly, we use several results from potential theory and

conformal mapping techniques to find the IR-drop in these

cases.

At this point, it is appropriate to say that if the sheet

resistance RS of the PDN is nonisotropic, that is, if the sheet

resistance in the x-direction, RS X, and in the y-direction, RSY,

is different, a change in the independent variables x and y

makes the sheet resistance isotropic at the small price of a

change in the PDN dimensions [16]. Hence, our analysis only

considers the isotropic case, with RS constant.Moreover, our analysis is intended for circular pads but, as

shown in [16] and [18], it can be extended to square pads by

using the concept of a circular pad of equivalent radius havingthe same resistance to the PDN as the square pad.

III. IR-DROP IN AN INFINITE PDN

Let us now attack the following simpler problem: we

consider an infinite PDN as a continuous conductive surfacewith constant sheet resistance RS. A single block A of

dimensions a b m2 and a constant current density J A/m2

is connected to the PDN at an arbitrary place. At another

arbitrary point, there is a circular pad of radius aP that

supplies the current IPAD = abJ required by A. A resistance

Rpad connects the pad to the power supply, which is assumed

to be at a constant voltage V0 = 0. Fig. 2 illustrates thegeometry of the problem. The IR-drop between the pad

(whose voltage is Vpad = J ab Rpad) and the potential VP at

any observation point P over the PDN is found as follows.

We denote the distance between the center of the pad andthe observation point P as rP p, and the distance between

the differential area dxdy inside A and point P as rP x y . The

potential at P is [19]

VP =J RS

2

a0

b0

ln

rP x y

d x d y J ab RS

2ln

rP p

. (3)


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a

b

PAD1

observation

point P

rP1

rpxy

dxdy

A

PAD2

PAD3

rP2

rP3PADN

rPN

ap2

apN

ap1ap3

Rpad

V0 Rpad

V0

Rpad

V0

Rpad

V0

a

b

PAD1

observation

point P

rP1

rpxy

dxdy

A

PAD2

PAD3

rP2

rP3PADN

rPN

ap2

apN

ap1ap3

Rpad

V0

Rpad

V0 Rpad

V0

Rpad

V0

Rpad

V0

Rpad

V0

Rpad

V0

Rpad

V0

Fig. 3. Parameters involved in the analysis of the IR-drop at the observationpoint P in an infinite resistive plane with multiple pads and one consumingblock.

Integrals like the one in (3) are well known in engineeringelectromagnetics. Their explicit solution can be found else-

where [20]. They define the so-called geometric mean distance

(GMD) between a point P and the rectangular block A, asshown in the following equation:

a0

b0

ln

rpx y

d x d y = ab ln (GMDP ) . (4)

Now, (3) can be written as

VP =J RSab

2ln

GMDP

rP p. (5)

If point P is at a distance aP from the center of the pad,

that is, at any point of its circumference, then the following

equalities hold:

rP p = aP , VP = Vpad, GMDP = GMDpad (6)

where GMDpad is the GMD from the center of the pad to

A, which is assumed to be the same as the distance from the

circumference of the pad to A provided that the pad radius a Pis small with respect to the block dimensions.

Now the complete IR-drop, VP , between the power supply

and point P becomes

VP =J RSab

2ln

GMDpad

GMDP

rP p

a P

+ J a b Rpad. (7)

Let us now generalize this result for N pads.

A. Multiple PadsImagine the same block A and N circular pads, PAD1,

PAD2, , PADN, of radius aP1, aP2, , aP N, and equal

resistances Rpad, distributed on an infinite PDN. It is assumed

that the pads are widely separated, that is, the distances

between them are much greater than their radius, ri j >> (a Pi ,

a P j ). Fig. 3 shows the involved geometry.

Each pad supplies a fraction of the total current drawnby A. Thus

IPADi = i J ab,

Ni=1

i = 1. (8)

Now we can write

VP =J ab RS

2ln (GMDP )

1J ab RS

2ln (rP1)

2J ab RS

2ln (rP2)

N J a b RS

2ln (rP N) .

(9)

That is

VP =J ab RS

2ln

GMDPN

i=1

riPi

(10)

where GMDP is the GMD between point P and block A, and

rPi is the distance between point P and pad i , which supplies

the fraction i of the total current.

By applying the above principle, we can find the IR-drop

between point P and the pad voltage. To do so, we place

point P at a distance a Pi from the center of pad i , that is, atits circumference. Thus, the following equalities hold:

VP = Vpadi = i J a b Rpad

GMDP = GMDi

rP1 = ri1, rP2 = ri2, . . . ,

rPi = aPi , . . . , rP N = ri N. (11)

By grouping together all the terms in i , we obtain the

following set of N equations, one for each value of i , with N

unknowns (the values of )

ln GMDi

Nj =i

j ln ri j i

ln a Pi 2

Rpad

RS

= 0,

i = 1, 2, . . . , N. (12)

Such N equations are not linearly independent because

of (8). However, we can subtract each equation in (12) from

its predecessor and build N 1 equations. These, together

with (8) form a system of N linearly independent equations

with N unknowns, as shown in (13)

lnGMDi

GMDi+1= i

ln

a Pi

ri+1,i 2

Rpad

RS

+i+1

ln

ri,i+1

aP,i+1+ 2

Rpad

RS

+

Nj = i

j = i + 1

j lnri j

ri+1,j

Ni=1

i = 1. (13)


4/11


This system can be written in matrix form as

lnaP1r21

2Rpad

RSln

r12aP2

+2Rpad

RS ln

r1,N1r2,N1

lnr1,Nr2,N

lnr21r31

lnaP2r32

2Rpad

RS ln

r2,N1r3,N1

lnr2,Nr3,N

lnrN1,1

rN1ln

rN1,2rN2

lnaP,N1rN,N1

2Rpad

RSln

rN1,NaPN

+2Rpad

RS1 1 1 1

1

2...N

= ln

GMD2GMD1

lnGMD

3GMD2

lnGMDN

GMDN11

and in compact form as

M = B (14)

where M is an N N matrix, and and B are column

vectors of N elements. Now, vector can be easily calculated

with (15) = M1B (15)

and the N elements of are the coefficients we are looking

for. As a simple example, if N = 2, the explicit result is

1 =1

2+

1

2

ln GMD2GMD1

ln r12aP

+ 2RpadRS

2 =1

2

1

2

ln GMD2GMD1

ln r12aP

+ 2RpadRS

. (16)

B. Completing the Solution

The total IR-drop, VP , between the power supply and

point P can be calculated as the sum of the voltage drop

at the Rpad of a reference pad plus the IR-drop from this pad

to point P. As any pad can be selected as the reference, wechoose pad 1. Thus, the formula for VP becomes

VP =J ab RS

2ln

GMD1

GMDP

Nj =1

rjP j

a1P1

Nj =1

rj1j

+1J ab Rpad (17)

which reduces to (7) if N = 1.

As can be seen, the problem of finding the IR-drop at any

point of an infinite PDN having one consuming block and N

pads is solved if the fraction of the current supplied by each

pad (coefficients ) is known.

Because of the linearity of the problem, it is easy to

generalize (17) for M blocks by applying superposition. Thus,

the previous procedure is repeated M times, one for each

block, to calculate vectors 1, 2, , M. Then, the total

IR-drop at any point is found by summing the contribution of

each block: VP(total) =M

j =1 VP j .

C. Flexibility and Generality of (17)

Under the above assumptions, (17) gives the IR-drop at any

point of a PDN with a sheet resistance RS, a number N of

circular pads of radius a P and resistance to power supply Rpad,

and one block of dimensions a b with a current density J.

XX

a

a

2a

aP

Fig. 4. Calculation of the IR-drop at the center of a square with four padsand one square consuming block and an infinite resistive plane.

Note that under the assumption of infinite dimension for the

PDN, (17) is fully flexible, which allows deciding on the

size and location of the consuming block, and the number,radius, and location of pads. As will be shown in Section V,

the IR-drop VP as calculated from (17) provides a very

good approximation of the real IR-drop of finite PDNs if theconsuming block is not very close to the external borders of

the pad array, that is, the IC sides.

Equation (17) can also be used to calculate the maximum

IR-drop under the same conditions as those analyzed by Shak-

eri and Meindl in [16]. In this paper, the maximum IR-drop

(which is placed at the center of the square formed by four

pads) is given by (2), where the numerical coefficient is known

after a long calculation of several double and triple Fourierseries and assuming several approximations. The interested

reader may read [16] for details. As will be shown here, (2)can be derived from (17), when the latter is applied to this

particular case.

Let us consider the square consuming block in Fig. 4, which

is embedded in an infinite PDN with a sheet resistance RS.

In this example, Rpad = 0. The side length of the block is

2a, which is twice the distance between adjacent pads. Ithas four circular pads with the same radius aP symmetrically

distributed in the block. Note that this geometry reproducesthe scenario studied by Shakeri and Meindl, except that in

this case the consuming block is finite. Let us now use (17)

to calculate VP at its center, that is, the point marked with

X in Fig. 4.

In these conditions, (17) becomes

VX =

4J a2RS

2 ln

GMD1 r1

X1r2

X2r3

X3r4

X4

GMDX a1

P r212 r

313 r

414

=J a2RS

2ln

GMD1 r

1X1r

2X2r

3X3r

4X4

GMDX a1P r

212 r

313 r

414

4. (18)

Due to the particular symmetry of the figure, (18) becomes

VX =J a2RS

2ln

GMD1 2 12 a 14

GMDX 2 218 a

14P

4

=IPADRS

2ln

0.3797a

a P(19)


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4 pads

16 pads

36 pads

64 pads

X

4 pads

16 pads

36 pads

64 pads

X

Fig. 5. Shakeri and Meindls problem [16]: calculation of the IR-drop at the

center of a square. The number of pads and the area of the square consumingblock tend to infinity.

where GMD1 and GMDX are calculated as functions ofa fromthe solution of (4), according to [21]. This result is very close

to Shakeri and Meindls formula (2). Now, to reproduce thecase in [16], we increase the size of the block and the number

of pads, as shown in Fig. 5.

In this way, we obtain an asymptotic equation for VX by

generalizing (18)

VX =IPADRS

2ln

coef a

aP

. (20)

We check the coefficient of (20) for different numbers N

of symmetrically distributed pads. The results are shown in

Table I.

As can be seen, when N increases, the numerical coefficient

coef tends to a definite value which is very close to that

reported by Shakeri and Meindl in [16].It is worth pointing out that the method to obtain the

numerical coefficient of (2) presented in our paper is much

simpler than that in [16] and gives practically the same results

under the same conditions. In addition, it is much more flexible

and can be applied to a variety of cases because it does not

impose any restriction on the number, size, or symmetry of

the distribution of the consuming blocks and pads.

IV. IR-DROP IN A FINITE PDN

In Section III, we made a strong assumption of a PDN of

infinite extension. Here, we remove this assumption becauseit gives erroneous results in the estimation of the IR-drop

when the consuming blocks are close to the IC sides. In

fact, on-chip PDNs are on top of dies of finite dimensions,L units wide and H units high. Let us now extend the results

of Section III to obtain the IR-drop for such PDNs. This

extension is based on the conformal transformation of the

interior of a rectangle in a complex plane Z into the upper

TABLE I

COEFFICIENT coefOF (20) AS A FUNCTION OF NUMBER OF PAD IN FIG . 5

N Calculated coef

4 0.3797

16 0.3810

36 0.3813

64 0.3814

100 0.3814

half of another complex plane W. Conformal transformation

is a mathematical technique that uses the functions of com-

plex variables to map complicated boundaries into simpler,

more readily analyzed configurations [21]. After the problem

is solved in the transformed configuration, inverting these

functions allows coming back to the original geometry. This

technique is restricted to 2-D fields satisfying Laplace or

Poisson equation, as in our case, and has been successfully

applied to many engineering problems. A good summary ofthe technique and its applications can be read, for instance, in

the first chapter of [21].

It is well known [21] that the Jacobi elliptic function w =sn(z,k) maps the interior of a rectangle with vertices K, K,

K + j K, K + j K in the complex plane Z into the upper

half of the complex plane W. Here, j = sqrt(1) and K and

K are complete elliptic integrals of the first and second kind

related to the dimensions of the rectangle; the modulus k ofthe elliptic functions can be calculated as follows [22]:

k =

2

3

2(21)

where 2 and 3 are elliptic theta functions of the second and

third kind with zero argument. These functions are calculated

as follows [22]:

2 =

n=0

2q

n+ 12

2

3 = 1 +

n=0

2q n2

q = eL

H . (22)

With this transformation, the side L/2, L/2 of the rectangle

in plane Z becomes the segment 1, 1 of the real axis ofplane W. The side L/2, L/2+jH of the rectangle becomes the

segment 1, 1/k of the real axis of plane W, whereas the side

L/2, L/2+jH becomes the segment 1, 1/k, and the side

L/2+jH, L/2+jH becomes the rest of the real axis of planeW [21]. A sketch of the transformation showing the lines of

constants x and y is shown in Fig. 6(a) and (b).Fig. 6(a) shows a square PDN with L = 1 and H = 1.

This PDN has nine identical pads identified by black circles.The top and bottom sides of the square are drawn in black

and the left and right sides in gray. This square is mapped in

plane Z with its origin at the center of the bottom side. Thetransformation w = sn(z, k) maps points z = x + j y of the

interior of this square into points w = u + j vof the upper half

of W, as drawn in Fig. 6(b). Thus, the origin of plane W is

also the origin of plane Z, and point jK in Z is transformed


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-0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

jy

-0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

jy

(a)

-8 -6 -4 -2 0 2 4 6 80

2

4

6

8

10

12

14

16

jv

u

-8 -6 -4 -2 0 2 4 6 80

2

4

6

8

10

12

14

16

jv

u

(b)

Fig. 6. (a) Square PDN with L = 1, H = 1, nine pads (circles), and a rectangular block (thick line), represented in plane Z. (b) Same PDN, pads, andblock, represented in plane W. Dashed lines are the lines of constants x and y in plane Z (and constants u and vin plane W).

into the infinity point in W. The size of pads is also modified,being greater in W when they are far from the real axis and

smaller when they are close to the real axis. Notice that thepoints of the real axis, v= 0 in W, are the transformed points

of the four sides of the rectangle in plane Z.

The current at the four sides of the PDN (the four sides of

the rectangle in Z) is zero. Therefore, the real axis of plane W

must have the same property, that is, the Neumann boundary

condition V/n = 0 must be satisfied in the real axis of W.

To force this condition, we need to add to W the image of the

upper half-plane [that is, the conjugate of plane W, conj(W)]

including the pads of the original W domain at their conjugate

coordinates.

After this step, we build the infinite domain W = WUconj(W). By including the current sources in W and conj(W),

we can calculate the IR-drop in this infinite domain using themethods in Section III. However, caution must be taken when

including the current sources (rectangular blocks). Equation

(4) for GMD, as derived in [20], is valid only for rectangular

blocks. Therefore, this solution cannot be used directly in

W because a rectangular block in Z transforms into a

nonrectangular figure in W. Similarly, if pads are circlesin Z, in W they take a different shape.

To overcome these restrictions, we use two results fromthe theory of conformal mapping [19], [21]. The first one is

that the regions about the corresponding points z and w areinfinitesimally similar. This means further that angles between

the intersecting lines in plane Z are preserved between the

corresponding lines in plane W [19]. That is, if the circles

or squares in Z are sufficiently small, their transformed

images in W are also circles and squares. The second one

is the invariance of the Poisson equation under a conformal

transformation, in other words, a differential area dx dy at

a point z Z transforms into a differential area du dv at a

point w W with a change of scale equal to |f(z)|2 and arotation of angle equal to the argument of f(z), with f(z)

being the derivative of the transformation f at point z. In our

case, f(z) = sn (z) = cn(z) dn(z), where cn(z) and dn(z)are also Jacobi elliptic functions.

With these results, the application of the methods inSection III to W, including pads and blocks, becomes

possible if the radius of pads are small with respect to L and

H and if the blocks are small. If the blocks are large, they must

be divided into small square sub-blocks, and each transformed

sub-block in W must be considered as a scaled and rotated

square, which is the image of the original sub-block in Z.

Bearing the above in mind, the procedure to find the IR-drop

VP at any point of a finite PDN is as follows.1) Map the PDN in plane Z into the half-space W by the

transformation w = sn(z). This mapping must includethe pads with scaled radius.

2) If necessary, divide the consuming blocks in Z into small

sub-blocks, and map them into W, scaling and rotating

them as required.

3) Add to W the conjugate half-plane conj(W) including

the transformed pads and blocks (or sub-blocks) in

conjugate positions. We now have the infinite domain

W = WU conj(W).4) Obtain the IR-drop VP at any point PW W

by the method described in Section III for an infinite

PDN considering all pads and all blocks (including the

conjugate ones).

5) Finally, come back to plane Z by using the inversefunction z = sn1(w,k) and find the potential at point

PZ Z. The inversion requires calculating an incom-

plete elliptic integral of the first kind, which is a standard

built-in function in any computer algebra system.

V. VALIDATION OF THE RESULTS

The above method was validated by comparing the calcu-

lated IR-drop with electrical simulations of PDNs of array-bonded ICs with a range of values of their parameters.

The error metric is defined as the normalized difference


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Fig. 7. IC with an array of 16 pads and a block within the array.

Fig. 8. IC with an array of four pads and a block at the IC side.

between the value of the maximum IR-drop obtained by

the method described here and the value obtained from thesimulations.

A. Infinite PDN

First, we compare the IR-drop predicted from the results

of Section III (infinite PDN) with the electrical simulation

results. Fig. 7 illustrates the following case: one consumingblock of 12.5 mm2 inside a chip of 10 10 mm2 and an

array of 16 regularly spaced pads of radius 100 m. Here,Rpad = 0.

In this case, the consuming block is fully inside the array

of pads. As expected, the error in the IR-drop at any place

(including the location of its maximum) is small, that is, lessthan 0.5%.

However, when the consuming block is at the chip side (thatis, totally or partially outside the array of pads), the error is

much greater. This is the case, for example, of Fig. 8: oneconsuming block of 8 mm2 inside a chip of 10 10 mm2 and

four regularly spaced pads of radius 200 m. Here, Rpad = 0.

Now the error is as large as 25%, which is an unacceptablevalue. Fig. 9(a) and (b) shows the IR-drop distribution in

the electrical simulation and the calculation, respectively. The

differences resulting from the assumption of an infinite PDN

are clearly visible.

Fig. 9. Difference of IR-drop on the PDN surface between (a) electricalsimulation and (b) calculation when an infinite PDN is assumed in the IC ofFig. 8.

B. Finite Rectangular PDN

To compare our results with the simulations of finite PDNs,we defined chips of different sizes and features, including anumber of pads of different sizes excited by consuming blocks

of different sizes at different places and drawing different

currents.

In the HSPICE simulations, the PDN was defined as an

array of cells modeling the regular grid of metal segments

with the same length in the X and Y directions and samewidth. These interconnected cells form the whole PDN. The

length of each segment was 100 m, and in our simulationsthe square pads had a side length Dpad of 1, 2, or 3 segment

lengths. According to the approach described in Section II,

an appropriate coefficient multiplying Dpad was calculated

to obtain the equivalent radius of the circular pads withthe same resistance to the PDN as the square pads used

in the simulations. This coefficient depends on the numberof segments connected to the square pads in horizontal and

vertical directions. For 1, 2, and 3 segments, its value is0.7071, 0.6334, and 0.6049, respectively. If the number of

segments goes to infinity, this coefficient tends asymptotically

to 0.5903, which is the value given in [18] and used in [16].The simplest check of our formulas is the comparison of

the maximum IR-drop when the consuming block is the whole

chip. The results are summarized in Table II, where the first

column gives the chip size, the second, the length of a side of


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TABLE II

IR-DROP IN IC S WHERE THE CONSUMPTION IS CONSTANT IN THE WHOLE CHI P

Chip size (mm2) Dpad (m) No. of pads rsegment () RS() J(mA/mm2) Vcalc (mV) Vsim (mV) Error (%)

7.2 7.2 200 9 4.4 2.2 25 100.0 99.3 +0.7

7.2 7.2 200 36 2.2 1.1 25 8.23 8.15 +0.98

2.6 2.6 100 4 4.4 2.2 25 28.81 28.73 +0.28

2.6 2.6 300 4 4.4 2.2 25 15.16 15.01 +1.0

10.4 10.4 100 64 2.2 1.1 25 14.40 14.36 +0.3

10.4 10.4 300 64 2.2 1.1 25 7.44 7.51 0.9310.4 10.4 200 16 2.2 1.1 25 61.01 60.61 +0.66

A B C

D E F

J1

J2

J1

J2

J1

J2

J1

J2

J1

J2

J1

J2

J3 J4

A B C

D E F

J1

J2

J1

J2

J1

J2

J1

J2

J1

J2

J1

J2

J3 J4

Fig. 10. Six examples of ICs with nonuniform current distribution, anddifferent sheet resistance and number of pads.

the square pad, and the third, the number of pads. The fourthand fifth columns contain the resistance of a line segment, and

consequently the sheet resistance of our formulas. The sixthcolumn shows the current density, and the seventh and eighthgive the calculated and simulated maximum IR-drop for each

example, respectively. The last column contains the error as

defined before. In these examples, Rpad = 0.

As can be seen in Table II, in all cases the maximum error

is 1%. Interestingly, by applying the result in [16] (2) to the

same examples, the error ranges from 2.8 to 10 %.We also checked our results for a nonuniform current

distribution with two or more consuming blocks, each onedrawing a different amount of current. The six examples

simulated are illustrated in Fig. 10 and their main parameters

are described in Table III.

Here, examples AC illustrate a chip of 7.2 7.2 mm2with nine pads. Example D is of a chip of the same size but

with 36 pads, and examples E and F show a chip of 10.4 10.4 mm2 with 16 pads. The dotted lines in Fig. 10 define the

contour of the separation between the consuming blocks J1,J2, and so on. Again, in these examples, Rpad = 0.

Table III is divided into two parts. The second column in the

top part shows the size of the consuming block. The asterisk(*) for examples C and E indicates that only the size of the

smaller consuming block is given, the other block is the rest

of the chip. The third and fourth columns contain the segment

resistance and sheet resistance, respectively. The fifth and sixth

TABLE III

MAI N PARAMETERS OF THE SIX EXAMPLES OF THE PD N IN FIG . 1 0

ExampleBlock

size(mm2)

rsegment()

RS()J1

(mA/mm2)J2

(mA/mm2)

A 9.60 4.4 2.2 0 100

B 5.76 2.2 1.1 0 100

C 9.60 (*) 2.2 1.1 25 100

D 9.60 2.2 1.1 0 100

E 13.52 (*) 2.2 1.1 25 100

F

13.52,27.04,40.56,27.04

2.2 1.1 100 25

ExampleJ3

(mA/mm2)J4

(mA/mm2)Vcalc(mV)

Vsim(mV)

Error(%)

A - - 195.89 194.80 +0.56

B - - 100.25 99.60 +0.65

C - - 120.74 120.3 +0.37

D - - 28.87 28.42 +1.58

E - - 146.50 145.64 +0.59

F 25 100 233.4 235.31 0.81

show the current density of blocks 1 and 2. The second andthird columns in the bottom part of Table III give the current

of blocks 3 and 4. The fourth and fifth contain the calculatedand simulated maximum IR-drop in millivolts and finally, the

sixth column shows the error, which is below 1% in most

cases.

The influence of Rpad was investigated by repeating the

simulation of example D, but imposing Rpad = 50 m. In this

case, the maximum IR-drop increases to 33.6 mV accordingto our formulas, and to 33.08 mV in the simulations. Thus, the

error is again 1.57%. We also checked the calculated voltagedrop at each pad Vpad. Table IV shows the results for all

36 pads. There, columns 2 and 6 contain the calculated voltagein millivolts of each pad and, columns 3 and 7 the simulated

one. Columns 4 and 8 are the differences between both results

in microvolts.

Finally, Figs. 11 and 12 show a view of the IR-drop of

example F according to electrical simulations (Fig. 11) and

calculation (Fig. 12).

VI . DISCUSSION

A cardinal feature of our approach is that knowing the IR-

drop at a given point only requires knowing its coordinates,

chip size, and location of all pads and consuming blocks.


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TABLE IV

VOLTAGE DROP ACROSS THE RESISTANCE Rpad OF THE 36 PADS OF EXAMPLE D

No. of pad Vpad (mV) (calc) Vpad (mV) (sim) Diff. (V ) No . of pa d Vpad (mV) (calc) Vpad (mV) (sim) Diff. (V)

1 0.0494 0.0648 15.4 19 0.0338 0.0440 10.2

2 0.4851 0.4539 31.2 20 0.3523 0.3912 38.9

3 4.4046 4.3620 42.6 21 3.6979 3.6430 54.9

4 5.6705 5.5930 77.5 22 4.7674 4.6850 82.4

5 1.2613 1.2850 23.7 23 0.9624 0.9790 16.6

6 0.1269 0.1540 27.1 24 0.0882 0.1060 17.8

7 0.0488 0.0640 15.2 25 0.0156 0.0210 5.4

8 0.4832 0.5360 52.8 26 0.1326 0.1483 15.7

9 4.4003 4.3560 44.3 27 0.7089 0.7208 11.9

10 5.6656 5.5860 79.6 28 0.9065 0.9095 3.0

11 1.2582 1.2810 22.8 29 0.2987 0.3072 8.5

12 0.1257 0.1530 27.3 30 0.0386 0.0470 8.4

13 0.0457 0.0590 13.3 31 0.0043 0.0067 2.4

14 0.4660 0.5140 48.0 32 0.0210 0.0279 6.9

15 4.3474 4.2940 53.4 33 0.0630 0.0757 12.7

16 5.6020 5.5120 90.0 34 0.0757 0.0891 13.4

17 1.2274 1.2450 17.6 35 0.0371 0.0452 8.1

18 0.1189 0.1430 24.1 36 0.0091 0.0130 3.9

Fig. 11. IR-drop distribution on the surface of the PDN of example Faccording to electrical simulations.

This is a great advantage over conventional approaches based

on the numerical solution of differential equations (that is,

finite element or finite difference methods), which require thecalculation of the IR-drop at all the points of the PDN surface

to know the IR-drop at a given point. Thus, our approachmakes it possible to obtain a faster response of IR-drop at

specific locations. In the case of searching the IR-drop at allthe points of the PDN surface, then both approaches have a

comparable execution time. Additionally, the execution time

is independent of the size of the consuming block. Let usnow sketch the computational complexity of the approach. At

this point, it is worth mentioning that no effort was made

to optimize the speed of our calculations, which are actually

written as MATLAB scripts.

Fig. 12. IR-drop distribution on the surface of the PDN of example Faccording to calculation.

The algorithm can be roughly divided into three phases:

1) building plane W and calculating the location and size of

the pads and blocks (or sub-blocks, when required), includingtheir images, on it; 2) executing the core of the algorithm,

which is in (15) and (17); and 3) coming back to plane Z,performing the inverse transformation.

Phase 1) is extremely fast because it only requires theconformal mapping of a small number of objects, like blocks

(or sub-blocks, when required) and pads, whose number is

limited. Its computational load depends on the product of thenumber of pads and the number of blocks (or sub-blocks, when

required). In its turn, the computational load of phase 3) is

linearly proportional to the number of observation points

where the IR-drop must be known.


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TABLE V

EXECUTION TIME IN SECONDS AS A FUNCTION OF NUMBER OF PAD

(1, 25, AN D 100), NUMBER OF OBSERVATION POINTS (1, 100, AN D 900),

AN D NUMBER OF BLOCKS (16, 96, 480, AN D 1056)

1 PAD

Points/blocks 16 96 480 1056

1 0.08 0.38 1.85 4.07

100 0.20 1.12 5.51 12.06

900 1.12 6.68 33.32 73.16

25 PADS


1 0.20 1.15 5.69 12.48

100 0.46 2.69 13.33 29.30

900 2.47 14.75 73.75 162.12

100 PADS


1 0.94 5.52 27.47 60.42

100 1.60 9.52 47.53 104.55

900 6.96 41.69 208.37 458.30

Phase 2) has the highest computational load. Equation (15)involves: 1) building matrix M of N N elements (where

N is the number of pads), each one containing the logarithmof the ratio of the distances between two pads, which must

be calculated previously; 2) building vector B of N elements,

each one containing the logarithm of the ratio of the GMD of

two pads to the block, which must be calculated previously;

3) inverting matrix M; and 4) multiplying the inverted matrix

by B. Actions 1) and 3) must be done only once, and actions2) and 4) must be done only once per block (or sub-block).

Thus, computational load of phase 2) depends on the numberof pads only and is independent of the number of points where

the IR-drop must be known.On the other hand, (17) involves the following actions:

1) calculating the ratio between the GMD of the reference

pad and the product of all the distances between the reference

pad and all the pads at the power calculated previously in

(15); 2) multiplying the distances between the observation

point and each pad at the power calculated previously in (15);

3) calculating the GMD between the observation point, forwhich the IR-drop must be known, and the consuming block;

4) dividing the results of actions 2) and 3); and 5) multiplyingthe results of actions 1) and 2) and taking the logarithm.

Action 1) can be precomputed and the result is reused every

time (17) is calculated, but actions 2) to 5) must be executed

for every observation point for which the IR-drop must beknown. The above are operations on scalars, and therefore

the computational load increases linearly with the numberof observation points. For each block (or sub-block, when

required), (17) is executed as many times as the observationpoints we define are executed Thus, the computational load

depends on the product of the number of blocks (or sub-

blocks) and the number of observation points.

To give an idea of the execution time, we executed the

MATLAB script on a standard PC with an Intel Q8200 CPU

with a clock frequency of 2.33 GHz and 3 GB of RAM. Only

a single core was used in the runs. It is worth mentioning

here that in the open literature devoted to PDNs and related

topics, complex ICs are divided into a few tens of functional

blocks (see [23][25]), of known location, size, and averageconsumption. Therefore, it seems reasonable to analyze the

execution times for this number of blocks. However, we also

present the execution time for cases involving a much higher

number of blocks (1056). Thus, Table V shows the execution

time in seconds for several combinations of number of pads,

number of blocks, and number of observation points. In all

cases, the IC size is 10 10 mm2. All blocks are of size

100 100 m2 in order to ensure accurate calculation of theIR-drop, and because no division into sub-blocks is required.

Except for the cases of one observation point, a small

fraction of the execution time is spent in phases 1) and 2) ofthe algorithm. As mentioned, no attempt was made to optimize

the execution time, which can be improved with little effort bytaking advantage of the parallelizable nature of the algorithm,

recoding it in a compiled language and adapting it for parallel

execution in multicore processors.

These execution times, as well as the results of Section IV,

showing a good agreement between the IR-drop calculatedwith our approach and the results obtained by electricalsimulation, demonstrate that our method is useful in exploring

the tradeoffs to optimize the PDN in its early design phase.Parameters like the number, size, and distribution of pads,

metal coverage, or distribution of functional blocks can be

explored in an interactive way to obtain a preliminary view of

the consequences of each decision.

In addition, it is worth pointing out that, although our

approach has been described for PDNs in flip-chip packages,it can also be used for wire-bonded ICs by placing the pads at

the IC periphery instead of over the PDN surface. Moreover, inspite of the fact that this paper assumes a PDN with symmetric

ground and supply grids, the described methodology to get IR-drop can also be applied in nonsymmetrical PDNs with powerand ground grids with different properties and with a different

pad distribution.

VII. CONCLUSION

This paper analyzed the IR-drop in PDNs of array-bondedICs. The PDN is modeled as a conductive surface of

constant sheet resistance. Under this restriction, closed-formexpressions to find the fraction of current supplied by

each pad, given a set of consuming blocks inside the IC,

were derived. The number, size, and location of pads and

consuming blocks and the current drawn by each block arearbitrary. Closed-form expressions to find the IR-drop at

any point of a finite PDN of array-bonded ICs having anynumber of pads were also given. The IC power is consumed

by rectangular blocks of any size, placed in any location anddrawing an arbitrary DC current. The effect of the resistance

between the IC pads and the power supply was also included

in the model. As particular cases, the methodology proposedfor the calculation of pad current and IR-drop is also valid for

wire-bonded ICs and nonsymmetrical PDNs. The analytical

expressions were validated with electrical simulations. The

maximum error found is in the range of 1%. The execution


11/11


time using a single core of an Intel Q8200 CPU, running a

MATLAB script with a clock frequency of 2.33 GHz and 3 GB

of RAM, is of 0.46 s for the calculation of the IR-drop at 100observation points of a PDN of 10 10 mm2, with 25 supply

pads, and 16 consuming blocks. For the same PDN, with

100 supply pads, 1056 consuming blocks, and 900 points for

which the IR-drop must be known, the execution time is 458 s.

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Josep Rius received the M.S. and Ph.D. degrees inelectrical engineering from the Universitat Politc-nica de Catalunya (UPC), Barcelona, Spain.

He has been an Associate Professor with the Elec-tronic Engineering Department, UPC, since 1991.His current research interests include VLSI testing,power estimation, and power/signal integrity.

ir-drop in on-chip power distribution networks

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