# Nonlinear Transmission Lines Improve Digital Speeds

### Key Takeaways

• Nonlinear transmission lines differ from linear ones due to current- or voltage-defined elements.

• The effects of nonlinearity requires a vastly different approach to solve the system.

• Nonlinearity effects have been shown to reduce rise/fall times – a pending boon to digital electronics.

Nonlinear transmission lines can use inductors and capacitors to vary rise/fall times.

In electronics design, linearity is preferable when evaluating circuits, as it greatly simplifies the solution set for a system. While a genuinely linear circuit is an idealized form, treating many circuits within a linear framework is reasonable, provided their parameters fit into some well-defined criteria. Most circuits actively evolve in response to changing currents and voltages at any instantaneous moment, and those circuits that do not express linearity require a significantly altered approach to solve during design prototyping. Nonlinear transmission lines have many uses at the board and IC levels, but incorporating them requires sophisticated computations.

Comparing Nonlinearity and Dispersion

 Linearity Nonlinearity Dispersion A system that scales proportionally with voltage/current. Capacitance and inductance become voltage- and current-dependent. Waves of different frequencies travel with different velocities.

## What Changes With Nonlinear Transmission Lines

Transmission lines are the bare board printed circuit's essential function, offering conductive signal pathways. As data speeds increase, the need for transmission lines becomes more apparent for signal quality: high-frequency conditions invite considerable interference to surrounding conduction pathways that can negatively impact performance. One of the advantages of transmission line implementation at high speeds is the ability to recreate lumped element models with conductor elements alone, a necessary substitution due to losses occurring at the junction between component and board (i.e., the solder joint). Varying the physical dimensions of the transmission line creates fundamental circuit elements, both passive and active, that operate in the linear region (a proportional relationship between voltage and current).

Unlike standard transmission lines, nonlinear transmission lines (NLTLs) are locally voltage- or current-defined for at least one of the components in the signal path. While NLTLs once used shunt diodes regularly spaced within a transmission line, this effect can use fabrication processes that reduce cost and board space requirements instead. Embedding materials replicating the distributed shunt characteristics also improve the power capabilities of NLTLs by shifting away from the semiconductor materials necessary in the older method.

Correctly understanding the effects of nonlinearity requires a short discussion on dispersion. Dispersion and nonlinearity can be simultaneously present in a transmission line, with the former representing a change in the velocity according to frequency and the latter altering the waveform's amplitude depending on the nonlinear parameters. Mathematically, combining these effects complicates solution sets, as they generally require numerical solutions rather than analytical ones. Because of this restriction, the best approach to modeling and designing NLTLs relies heavily on circuit modeling and simulation to analyze design parameter changes. These models focus on the state parameter variables that describe the circuit in the fewest parameters possible, as there is no universal problem-solving application for nonlinear systems.

State variables can still convey the full complexity of the circuit without adding unnecessary parameters that can slow computations. For example, a simple pulse-driven circuit that consists of only two time-varying components needs only two differential equations to describe the circuit, regardless of how many additional linear elements there are. Inductors and capacitors, which constantly store and transform current and magnetic energy or charge and electric energy (respectively), can use voltage and current relationships alongside other time-invariant circuit elements to build a state vector that completely describes the circuit.

## Signal Performance Application of NLTLs

Besides modeling more complex circuit behavior, NLTLs hold significant promise for pulse sharpening in digital circuitry. Improving rise and fall times associated with new technology and the maturation of manufacturing quality have greatly improved system speeds. However, improvements will likely proceed much slower due to material and physical limitations. An innovative solution simultaneously sharpens the rise and fall waveforms; non-return-to-zero binary encodings that are popular due to larger bandwidths over traditional 0V low-level encodings would specifically benefit. Circuit applications like timing and sampling systems can also benefit from sharpened rise/fall times for greater precision.

While pulse sharpening via NLTLs is necessary to improve rise/fall times, the nonlinear nature means the effect is felt more strongly with the fall time than the rise time. Consider a voltage-controlled NLTL: the voltage will scale positively with any capacitance in the system. At the outset, the voltage is low as the pulse propagates through the medium until the voltage rises to the pulse's apex. Due to the nonlinear effect, the voltage increases as the capacitance increases, which increases the propagation speed. The result is a “quickening” of the wave forward compared to its original time frame, with most of the increase at the falling edge.

Furthermore, pulse sharpening circuit elements can be daisy-chained to shape the waveform consecutively. Sequential pulse sharpening stages narrow the pulse width and increase the amplitude, making the pulse appear more instantaneous while conserving the total area under the curve. The values of the inductors and capacitors used for pulse narrowing must vary with each iteration to ensure only one pulse passes through each stage, preventing degeneration and unintentional feedback to the input. Notably, pulse narrowing also gains some of the performance benefits of pulse sharpening.

## Cadence Solutions for Complex Circuit Design

Nonlinear transmission lines are not a new technology, but continual advancements have endeared them to digital electronics. The major difficulty of working with nonlinear circuits is the additional complexity required to solve the system; robust circuit analysis and simulation tools are necessary for modeling. Cadence’s PCB Design and Analysis Software suite allows electronics design teams to accelerate designs with comprehensive simulation tools as part of a greater ECAD package. Alongside the fast and powerful OrCAD PCB Designer, electronic layout has never been easier or more capable.

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