IR-Drop in on-Chip Power Distribution Networks
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Transcript of 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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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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514 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 21, NO. 3, MARCH 2013
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)
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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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516 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 21, NO. 3, MARCH 2013
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
Points/blocks 16 96 480 1056
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
Points/blocks 16 96 480 1056
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
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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.