Interconnect simulation issues

Modem circuits are extremely complex and comprise o f hundreds of thousands of interconnects and non-linear lumped elements. Simulation of such large systems is associated with two major problems: the mixed time/frequency nature of the simulation and the computational expense.

2 .4 .1 . M ix e d tim e /f r e q u e n c y d o m a in

Including distributed interconnect models in a transient simulation in a general- purpose circuit simulator is very difficult. Circuit simulators such as SPICE [N75] are time-domain based since circuits containing devices with non-linear or time-dependent characteristics must be characterised in the time domain [BS97]. If the lumped RLCG model is sufficient to describe interconnect behaviour, a SPICE like simulator may be used for simulation purposes. This usually involves high CPU cost as SPICE does not handle large linear RLCG networks efficiently. Furthermore, as shown earlier, simple lumped models are inadequate to accurately describe the behaviour of modem high­ speed interconnects and consequently, frequency dependent distributed models must be used raising the problem of mixed time/frequency domain.

The distributed models of interconnect are formulated in terms of time-domain partial differential equations (Telegrapher’s Equations) but obtaining solution to them is very difficult if not impossible. However, in the frequency domain, the corresponding

CHAPTER 2 Simulation o f high-frequency integrated circuits

description is a set of linear equations whose solution is straightforward to obtain. Additionally, if an interconnect has frequency-dependent parameters it is best described in the frequency domain [WW92] since dispersion, conduction and dielectric losses are relatively simple functions of frequency and are generally time invariant.

Therefore, in order to incorporate the transmission-line behaviour of interconnects into a general-purpose circuit simulator, it is necessary to convert frequency-domain results for interconnects into a time-domain description (Fig 2.11). Several approaches have been proposed in the literature, e.g. [XLW+00] and [BOO].

LUMPED ELEMENTS

(non-linear and/or time-varying)

DISTRIBUTED ELEMENTS

(frequency dependent)

(Time domain) (Frequency domain) (Time domain) (Frequency domain)

(non)linear ODE

n/

not available for nonlinear circuits

X

PDE

(no frequency-dependence parameters!

X

Linear eq u a tio n s

\ X

Time dom ain nonlinear ODE

Hx(t ) + Wx(t ) + F(x(t )) = b(t )

F requency dom ain linear equation I(s) = Y(s)V(s)

X /

Time dom ain m acrom odel

Fig. 2.11. Mixed time/frequency domain problem

CHAPTER 2 Simulation o f high-frequency integrated circuits

2 .4 .2 . C o m p u t a t i o n a l e x p e n s e

The first step in the simulation process is to write a set of circuit equations that describe the circuit behaviour. These equations may be written either in the time- domain or in the frequency domain but, due to the mixed time/frequency issue, in the majority of cases the simulation has to be performed in the time domain. For the purpose of obtaining a numerical solution, integration techniques are used to convert a set o f time-domain differential equations into a set of difference equations. Then the Newton iteration process is applied in order to obtain simulation results at each time point. However, the matrices that ensue from the set of difference equations describing the interconnect network are usually very large and thus LU decompositions performed as part of the Newton algorithm place a heavy demand on CPU processing time. Additionally, memory requirements may be overwhelming for large networks. To address this problem, model order reduction techniques are introduced. They enable a speed up o f calculations but introduce new problems regarding ill-conditioning of large matrices and preservation of the stability and passivity of the reduced model.

2.5. Summary

As VLSI feature sizes reach deep sub-micron dimensions and clock frequencies approach the gigahertz range, interconnect effects such as propagation delay, attenuation, crosstalk, signal reflection, ringing and current distribution effects become an increasingly significant factor in determining overall system performance. Hence, the ability to describe high-frequency interconnect effects in an effective and accurate manner is a must for any state-of-art interconnect model.

An interconnect model can be a lumped model (RC or RLCG), a distributed model (with or without frequency-dependent parameters), a model based on a tabulated data set or a fiill-wave model. The interconnect length, cross-sectional dimensions, signal rise time and the clock speed are factors which should be examined when deciding on the type of model to be used for modelling high-speed interconnects. In addition, it might be necessary to take into consideration other factors such as logic levels, dielectric materials and conductor resistance.

With the trend of ever-rising operational frequencies and ever-shrinking feature sizes, lumped models became insufficient to adequately describe the behaviour of modem high-speed interconnects. The full-wave model, although very accurate, is too

CHAPTER 2 Simulation o f high-frequency integrated circuits

computationally involved and cannot produce simulation results in a reasonable amount of time. Therefore, this thesis will focus on distributed interconnect models described in terms o f the Telegrapher’s Equations and models based on a tabulated data set. The aim is to obtain interconnect models that are capable of describing non-uniform and frequency-dependant interconnects with reasonable accuracy and in a computationally efficient manner.

CHAPTER 3 Interconnect simulation techniques

C H A P T E R 3

I n t e r c o n n e c t S i m u l a t i o n T e c h n i q u e s

Except for very simple interconnect networks and structures (e.g. short lossless lines), accurate simulation of interconnects is not a simple task. SPICE-like simulators cannot handle the large numbers of state variables associated with the description of an interconnect in terms lumped resistors, inductors and capacitors [CC98]. In particular, the extensive mutual inductive and capacitive coupling present in the equivalent model, makes SPICE-based simulation prohibitively slow if at all possible [CCP+98]. Therefore, during the last twenty years, substantial research into developing accurate and efficient techniques for modelling and simulation of interconnects has been carried out. The resulting interconnect simulation techniques can be broadly classified into two main categories [AN01]: approaches based on macromodelling of each individual transmission line set and approaches based on model order reduction (MOR) of the entire linear network containing both lumped and distributed subnetworks.

The goal o f this Chapter is to review some of the existing interconnect

simulation techniques and highlight their merits and demerits. The basic properties of a

distributed network are first introduced followed by a short description of the most widely used macromodelling and model order reduction strategies.