Distributed RLCK Models for Transmission Lines in High Speed Applications

Design engineers frequently struggle with transmission line design and modeling. We can define a length of interconnect that contains more than 1/100th of a wavelength as a transmission line. This seems to be the breakpoint where distributed effects to start to become significant. To improve circuit performance these long runs are usually shielded and drawn on silicon as differential pairs. Transmission lines are used for high-speed signal propagation and occur in Low Noise Amplifiers (LNA’s), Power Amplifiers (PA’s), clock distribution networks and many other kinds of high performance analog circuits.

There are many tools that can provide a small-signal (i.e. an n-port) model for a transmission line. But the needs of high speed analog circuit design usually require transient simulations. So the engineer is presented with the necessity for models compatible with large-signal simulation. Over the years several techniques have been employed to solve this problem.

One common approach is to use rational fitting. An arbitrary network is used to represent the equivalent circuit. Using mathematical methods a fit can be obtained. However, it will frequently sacrifice DC behavior (i.e. the DC operating point) and may contain negative device values – resulting in a model having active behavior (gain). This can cause problems with convergence during transient circuit simulation, defeating the purpose of making the linear model to begin with.

Using a physics based model is generally a better approach. A circuit that represents the physical system topology is used as the target network for the fitting operation. But because transmission lines exhibit distributed effects, a simple lumped RLCK model will not provide a good solution.

To include distributed effects in their models, engineers will often derive a unit circuit and manually cut the transmission line into short segments that can be fitted individually to the unit circuit. Then these segments are joined at the schematic level providing what is hoped to be an equivalent circuit for the transmission line. This is tedious and potentially error prone. Additionally there are several shortcomings with this method.

Usually the ground is assumed to be an ideal ground and any losses produced in the ground are incorporated into the model for the signal lines – improperly distributing the losses and potentially adversely affecting the DC behavior. This makes the model’s accuracy very sensitive to the actual ground signal connections and will likely violate the assumptions made in transferring the ground loss effect to the signal lines. Therefore the model will not properly reflect actual behavior in a circuit.

The second issue is linear coupling along the transmission line. If Telegrapher’s equations are used to calculate the inductance per unit length on a short segment and compared to the same calculation from a long segment, the values will not agree. It is necessary to EM simulate the full structure for a variety of reasons, thus precluding us from using a segmented approach to both EM simulation and fitting.

Below, as shown in figure 3, it is evident how line length affects inductance per unit length. These are PeakView EM simulation results illustrating the variation of inductance per unit length as overall length is increased. The line lengths shown range from 140 microns to 12,000 microns in the upper curve.

Fortunately starting with its n-port EM model for the full system accounting for all EM effects, PeakView can automatically generate a distributed RLCK model that is segmented to get an accurate RLCK model. This model will even properly reflect the effects of various ground connections. PeakView’s patented 3D full wave electromagnetic solver will take layout directly from Virtuoso, or alternatively PeakView’s PCircuits can be used to synthesize and then model optimal structures using the same method.

The transmission line Physics Based Model (PBM) will automatically fit a segmented unit cell that accounts for primary, mutual, shunt and return path elements. The user can select the number of segments to be used. PeakView will automatically fit the complete network and return a SPICE compatible RLCK subcircuit that can be sync’ed back to Cadence ADE for simulation.

Here are some examples of n-port to PBM model comparisons.

Figure 6 shows circuit simulation and silicon measurements for the 60GHz power amplifier in figure 1 using PeakView transmission line and inductor models. Below you can see light blue and red are S21 simulated and measured, respectively. Yellow and green are S11 simulated and measured, respectively. The circuit and measurements are courtesy of TSMC.

In conclusion, as wavelengths get shorter transmission line behavior in interconnect is becoming more significant in high speed analog designs. Despite working at smaller process nodes, signal lines are in many cases are long enough to require full-wave EM modeling. PeakView with its comprehensive EM and simulation model generation capabilities delivers an efficient, reliable and accurate solution for design engineers. PBM (Physics Based Models) are available for single or differential transmission lines. Shielding options include bottom plate and side shields. Ground pins can be specified where needed.

Lorentz Solution’s flagship EM platform, PeakView, provides a comprehensive solution for automatically generating fully passive broadband distributed RLCK models for a wide range of transmission line structures.

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