Interconnect Modeling and Simulation:Interconnect Models

Interconnect Models

High-speed system designers are driven by the motivation to have signals with higher clock and slew rates while at the same time to innovate on reducing the wiring cross-section as well as packing the lines together. Reducing the wiring dimensions results in appreciably resistive lines. In addition, all these interconnections may have non-uniform cross-sections caused by discontinuities such as con- nectors, vias, wire bonds, flip-chip solder balls, redistribution leads, and orthogonal lines. Interconnections can be from various levels of design hierarchy (Figure 14.4) such as on-chip, packaging structures, multi-chip modules, printed circuit boards, and backplanes. On a broader perspective, interconnection technology can be classified into five categories, as shown in Table 14.1 [6], namely, on-chip wiring, thin-film wiring, ceramic carriers, thin-film wiring, printed circuit boards, and shielded cables. For the categories shown in Table 14.1, the wavelength is of the order of 1 to 10 cm. The propagated signal rise times are in the range 100 to 1000 ps. Hence, the line lengths are either comparable or much longer than the signal wavelengths. Depending on the operating frequency, signal rise times, and the nature of the structure, the interconnects can be modeled as lumped (RC or RLC), distributed (frequency-independent/dependent RLCG parameters, lossy, coupled), full-wave models, or measured linear subnetworks.

Lumped Models

In the past, interconnect models have been generally restricted to RC tree models. RC trees are RC circuits with capacitors from all nodes to ground, no floating capacitors, no resistors to ground. The signal delay through RC trees were often estimated using a form of the Elmore delay [8,34], which provided a dominant time constant approximation for monotonic step responses.

Elmore Delay

There are many definitions of delay, given the actual transient response. Elmore delay is defined as the time at which the output transient rises to 50% of its final value. Elmore’s expression approximates the mid-point of the monotonic step response waveform by the mean of the impulse response as

Since v(t) is monotonic, its first derivative (the impulse response) will have the form of a probability density function. The mean of the distribution of the first derivative is a good approximation for the 50% point of the transient portion of v(t). For an RC tree, Elmore’s expression can be applied since step responses for these circuits are always monotonic.

However, with increasing signal speeds, and in diverse technologies such as bipolar, BiCMOS, or MCMs, RC tree models are no longer adequate. In bipolar circuits, lumped-element interconnect models may require the use of inductors or grounded resistors, which are not compatible with RC trees. Even for MOS circuits operating at higher frequencies, the effects of coupling capacitances may need to be included in the delay estimate.

RLC circuits with non-equilibrium initial conditions may have responses that are non-monotonic. This typically results in visible signal ringing in the waveform. A single time constant approximation with Elmore delay is not generally sufficient for such circuits. Usually, lumped interconnect circuits extracted from layouts contain large number of nodes, which make the simulation highly CPU intensive. Figure 14.5 shows a general lumped model where R, L, C, and G correspond to the resistance, inductance, capacitance, and conductance of the interconnect, respectively.

Distributed Transmission Line Models

At relatively higher signal speeds, the electrical length of interconnects becomes a significant fraction of the operating wavelength, giving rise to signal-distorting effects that do not exist at lower frequencies. Conse- quently, the conventional lumped impedance interconnect models become inadequate, and transmission line models based on quasi-TEM assumptions are needed. The TEM (Transverse Electromagnetic Mode) approxi- mation represents the ideal case, where both E and H fields are perpendicular to the direction of propagation and it is valid under the condition that the line cross-section is much smaller than the wavelength. However, in practical wiring configurations, the structure has all the inhomogeneities mentioned previously. Such effects

give rise to E or H fields in the direction of propagation. If the line cross-section or the extent of these non- uniformities remain a small fraction of the wavelength in the frequency range of interest, the solution to Maxwell’s equations are given by the so-called quasi-TEM modes and are characterized by distributed R, L, C, G per unit length parameters (Figure 14.6).

The basic quasi-TEM model is the simple “delay” line or lossless line (R = G = 0). In this case, a signal traveling along a line has the same amplitude at all points, but is shifted in time with a propagation delay per unit length (t) given by

Distributed Models with Frequency-Dependent Parameters

As the operating frequency increases, the per unit length parameters of the transmission line can vary. This is mainly due to varying current distribution in the conductor and ground plane caused by the induced electric field. This phenomenon can be categorized as follows: skin, edge, and proximity effects [8,20,56].

Edge and Proximity Effects

The edge and proximity effects influence the interconnect parameters in the low to medium frequency region. The edge effect causes the current to concentrate near the sharp edges of the conductor, thus raising the resistance. It affects both the signal and ground conductors, but is more pronounced on signal conductors. The proximity effect causes the current to concentrate in the sections of ground plane that are close to the signal conductor. This modifies the magnetic field between the two conductors, which in turn reduces the inductance per unit length. It also raises the resistance per unit length as more current is crowded in the ground plane under the conductor. The proximity effect appears at medium frequencies. While both effects need to be accounted for, the proximity effect seems to have more significance, especially in its effect on the inductance.

Skin Effect

The skin effect causes the current to concentrate in a thin layer at the conductor surface. It is pronounced mostly at high frequencies on both the signal and ground conductors. The current distribution falls off exponentially as we approach the interior of the conductor. The average depth of current penetration is a function of frequency and is known as skin depth, which is given by

where w is frequency (rad/s), r is the volume resistivity, and m is the magnetic permeability. This results in the resistance being proportional to the square root of the frequency at very high operating frequencies. The magnetic fields inside the conductors are also reduced due to skin effect. This reduces the internal inductance and therefore the total inductance. At even higher frequencies, as the internal inductance approaches zero, the edge, and proximity effects being fully pronounced, the inductance becomes essentially constant.

Typical Behavior of R and L

The frequency plots of R and L of the microstrip in Figure 14.7 are shown in Figure 14.8 and Figure 14.9. These plots present a typical behavior of R and L in general. L starts off as essentially constant until the edge and proximity effects get into effect. The edge effect causes the resistance to increase, and the current crowding under signal conductor (due to the proximity effect) causes the inductance to decrease and resistance to increase. As we go higher in frequency, the edge and proximity effects become fully pronounced and will cause little additional change in R and L, but the skin effect becomes significant. Initially, the inductance is reduced due to skin effect because of the reduction of magnetic fields inside the conductors; but as the contribution of those magnetic fields to the overall inductance becomes insignificant, L becomes essentially constant at very high frequency. The resistance, on the other hand, becomes a direct function of the skin depth and therefore varies with the square root of the frequency.

Full-Wave Models

At further subnanosecond rise times, the line cross-section or the non-uniformities become a significant fraction of the wavelength and, under these conditions, the field components in the direction of propa- gation can no longer be neglected (Figure 14.10). Consequently, even the distributed models based on A full-wave analysis is required if the dielectric heights are comparable to the wavelength of propagation

quasi-TEM approximations become inaccurate in describing the interconnect performance [12–19]. In such situations, full-wave models, which take into account all possible field components and satisfies all boundary conditions, are required to give an accurate estimation of high-frequency effects.

The information that is obtained through a full-wave analysis is in terms of field parameters such as propagation constant, characteristic impedance, etc. A typical behavior of the modal propagation constant and characteristic impedances obtained using the full-wave spectral domain method for the structure shown in Figure 14.11 is given in Figure 14.12. The deviation suffered by the quasi-TEM models with respect to full-wave results is illustrated through a simple test circuit shown in Figure 14.13. As seen from Figure 14.14 for the structure under consideration, quasi-TEM results deviated from full-wave results as early as 400 MHz. The differences in the modeling schemes with respect to the transient responses are illustrated in Figure 14.15 and Figure 14.16. In general, an increase in the dielectric thickness causes quasi-TEM models to become inaccurate at relatively lower frequencies. This implies that a full-wave analysis becomes necessary as we move up in the integration hierarchy, from the chip to PCB/system level. It is found that depending on the interconnect structure, when the cross-sectional dimensions approach 1/40 to 1/10 of the effective wavelength, quasi-TEM approximation deviates considerably from full-wave results. (The effective wave- length of a wave propagating in a dielectric medium at a certain frequency is given by

Also, a reduction in the separation width between adjacent conductors makes the full-wave analysis more essential, especially for crosstalk evaluation. The same is true with an increase in the dielectric constant values.

However, circuit simulation of full-wave models is highly involved. A circuit simulator requires the information in terms of currents, voltages, and circuit impedances. This demands a generalized method to combine modal results into circuit simulators in terms of a full-wave stencil. Another important issue involved here is the cost of a full-wave analysis associated with each interconnect subnetwork at each frequency point of analysis. For typical high-speed interconnect circuits which need thousands of

frequency point solutions to get accurate responses, it would become prohibitively CPU-expensive to simulate because of the combined cost involved (i.e., evaluation of a full-wave model to obtain modal parameters and computation of circuit response at each point).

Measured Subnetworks

In practice, it may not be possible to obtain accurate analytical models for interconnects because of the geometric inhomogeneity and associated discontinuities. To handle such situations, modeling techniques based on measured data have been proposed in the literature [22,24,59–65]. In general, the behavior of high-speed interconnects can easily be represented by measured frequency-dependent scattering param- eters or time-domain terminal measurements. Time-domain data could be obtained by time-domain reflectometry (TDR) measurements [59] or electromagnetic techniques [23,24]. One important factor to note here is that the subnetwork can be characterized by large sets of time-domain data, making the system overdetermined. This kind of measurement data in large number of sets is essential in a practical environment as the measurements are usually contaminated by noise, and the use of only a single set of

measurements may lead to inaccurate results. Including all available sets of measurements in the simu- lation helps to reduce the impact of noise on network responses. However, handling such overdetermined situations in circuit simulation is a tedious and a computationally expensive process [22].

EMI Subnetworks

Electrically long interconnects function as spurious antennas to pick up emissions from other nearby electronic systems. This makes susceptibility to emissions a major concern to current system designers of high-frequency products. Hence, the availability of interconnect simulation tools, including the effect of incident fields, is becoming an important design requirement. In addition, analysis of radiations from interconnects is also becoming increasingly important in high-furnace designs [9,57,58].