3.1 Introduction
Efficient analysis of power distribution network (power grid) is essential for the correct functionality of a chip to different load conditions and variability of elements. Nowadays, it has become critical to analyze power distribution networks as the power supply voltage continues to drop because of technology scaling resulting in higher chip density. With the increase in demand for complex low voltage integrated circuits (ICs), issues related to power distribution network have become of vital importance. One of the important issue for a chip designer is to design a reliable power distribution network for a wide variety of workloads to minimize the fluctuations in voltage levels across different functional blocks of a chip caused by increased interconnect resistances, large sizes of networks and, increased element and operation variability in lower silicon technology nodes. Therefore, it is necessary to analyze the power distribution networks for both voltage behavior and grid safety.
Today, power distribution network analysis is typically done by simulating the entire power distribution network to evaluate voltage drop at every node. Due to large sizes of power grid networks, it has become critical to analyze the whole power distribution network on a sin- gle computer due to lack of memory resources. Several attempts have been made to address the issues related to memory using parallel computing strategies [20, 24] and model order re- duction (MOR) techniques [103, 104] including a number of partitioning schemes [21, 105].
However, statistical techniques like RW [82, 83] are proved to be efficient while solving large power distribution networks. Therefore, we start the analysis of power distribution network using RW method.
3.2 Power Distribution Network Analysis
3.1(b) and after applying Kirchhoff’s current law and Kirchhoff’s voltage law at nodex, the algebraic equation can be formulated as [82],
Vx =
∑k y=1
gy
∑k
y=1gyVy− Ix
∑k
y=1gy, (3.1)
where gy is the conductance of yth connected nodes; k represents the number of adjacent nodes ofx; Vy is the potential of the yth adjacent node; Ix is the current drawn from node x, andVx represents the potential at node x. Steady state analysis of VLSI power grid net-
Vdd Vdd
(a)
Ix g1
g2 g3
g4 x
2
3 1
4 (b)
Figure 3.1:(a) A resistive model of power distribution network. (b) A single power distribution node.
work formulates a large linear system with a conductance matrixAmodeling impedances, an unknown voltage vectorx modeling the node voltages and a right hand side vectorb mod- eling the independent sources, i.e.,Ax = b. Since the model of power distribution network consists of resistances, voltage sources (with one end connected to ground node) and current sources, we use nodal or modified nodal analysis method to analyze power distribution net- work. Modified nodal analysis method is not suitable for this purpose because it generates a system matrixA of size (n+V −1)×(n+V −1), where n andV are the number of nodes and number of voltage sources present in the circuit, respectively, and the system ma- trixAalso has a zero diagonal sub-matrix, which makes it unsuitable to be used by iterative solvers. Since all the voltage sources in power grid network are connected to the ground, nodal analysis method can be used to analyze such networks. The system matrix A of the
order (n −V − 1)× (n −V −1) generated by nodal analysis for power grid network is symmetric and positive definite in nature [106].
Theorem 1. The system matrixAgenerated by nodal analysis of the linear system in power grid network formulation, is of the order of(n−V −1)×(n−V −1), wherenandV are the total nodes and total voltage sources in the circuit, respectively.
Notations used :G(diagonal matrix),i(current vector) Proof. Step 1:
Ari = 0
Ari ≡ [ArGArJ]i [ArG ArJ
][ iG
iJ
]
= 0
ArGiG = −ArJiJ (KCE) (3.2) Step 2:
LetiG = GvG (Device Characteristics)
T hen, ArGGvG = −ArJiJ (3.3)
Step 3:
vG
. . . vJ
=
ATrG
. . . ATrJ
vn (KV E) (3.4)
Substituting (3.4) in (3.3), we have,
(ArGGATrG)vn = −ArJiJ (3.5)
It can be proved that the size of system matrixA=ArGGATrG, can be(n−1)×(n−1)[106], where G is the diagonal matrix and ArG and ArJ denote the reduced incidence matrices corresponding to conductance type branches and current source type branches, respectively.
For the nodes of power grid network, whose potentials are known, it is not necessary to formulate Kirchhoff current equations (KCE). This will reduce the effective number of power grid nodes ton−V −1, for which KCE equations need to be formulated, whereV represent the nodes whose potentials are known. Thus, the size of system matrix being generated from
3.2 Power Distribution Network Analysis
3.2.2 Network Formulation for Transient Analysis
For transient analysis of power distribution network, we consider the effects of capaci- tances and time-varying current waveforms. In this section, during implementation, the back- ward Euler approximation with timestephis considered to transform the differential equations to linear equations. We assume timestephto be constant during transient analysis of power distribution network. The capacitances are modeled to be connected to ground along with the metal resistances of the power distribution network, which is shown in the Figure 3.2(a). The current sink used in the transient formulation is a time-varying current source.
Vdd Vdd
Vdd
Vdd
(a)
Ix g1
g2
g3
g4 C x
x 2
3 1
4 (b)
Ix g1
g2 g3
g4 C /h x
C /h xV (t−h)x
x 2
3 1
4
(c)
Figure 3.2: (a) RC modeled power distribution network, (b) A single power distribution node, (c) power distribution node with Norton representation of backward Euler model for capacitorCx.
Therefore, the system of equations for RC-modeled power distribution network can be written as follows,
Ax(t) +C x′(t) = b(t), (3.6)
whereAis the conductance matrix;x(t)is the vector of time-varying node potentials;x′(t)is the vector of rate of change of the node potentials; Cis the impedance matrix corresponding to the capacitances of the power distribution network, and b(t) is the time-varying current sinks. Using backward Euler’s method with a timestep h, the linearized form of system of equations can be written as,
(A+ C
h)x(t) =b(t) + C
hx(t−h) (3.7)
Applying Kirchhoff’s current law and Kirchhoff’s voltage law at a nodex(as shown in Figure 3.1(b)), we have [82],
∑k y=1
gy(Vy(t)−Vx(t)) = Cx
h (Vx(t)−Vx(t−h)) +Ix(t), (3.8) wheregy,Vy,Ix andVxare as defined in (3.1);Cxrepresents the capacitance between nodex and ground. For a network having RC components, where capacitors are connected between two non-ground nodes, those capacitors can be replaced by a network component having resistors and current sources connected in parallel (Norton equivalent as shown in Figure 3.2(c)) or a network component having resistors connected in series with voltage sources (Thevenin equivalent). In this section, we discuss the case described in (3.8). Equation 3.8 can be rewritten as,
Vx(t) =
∑k y=1
gy
Gtotalx Vy(t) +
∑k y=1
Cx
h
Gtotalx Vx(t−h)− Ix(t)
Gtotalx , (3.9)
whereGtotalx =∑k
y=1gy +Chx,gy is the conductance ofythconnected nodes;krepresents the number of adjacent nodes of x; Vy is the voltage of theyth adjacent node; Ix is the current drawn from nodex, andVx represents the voltage at nodex.