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Nonlinear signal processing is necessary in many emerging applications where form factor and power are at a premium. In order to make such complex computation feasible under these constraints, it is necessary to implement the signal processors as analog circuits. Since analog circuit design is largely based on a linear systems perspective, new tools are being introduced to circuit designers that allow them to understand and exploit circuit nonlinearity for useful processing. This paper discusses two such tools, which represent nonlinear circuit behavior in a graphical way, making it easy to develop a qualitative appreciation for the circuits under study.

Portable and implantable, always-on electronics stand to benefit from analog signal processing, when only low levels of precision are necessary [

In order to make nonlinear circuit design relevant to an engineer, it must be taught intuitively enough to foster creativity, yet rigorously enough to be of practical benefit. The two most popular tools for studying nonlinear circuits are harmonic balance and Volterra series. Within certain limitations, they are rigorous, but not necessarily intuitive. Since harmonic balance is simulation-based, it can be used to predict nonlinear behavior without ever requiring a deep understanding [

To bridge the gap between rigor and intuition, we can use visual representation techniques. If the appropriate visualization is formed from rigorous definitions of a circuit's dynamics, then the human vision system, with its pattern recognition ability, will perceive the circuit's qualitative behavior [

We will present a filter block diagram for analyzing harmonic distortion that is derived from perturbation analysis. Unlike Volterra series kernels, our filter block diagram does not include multidimensional Fourier transforms, and so is accessible to an introductory-level engineering audience. In the second part of this paper, we will discuss the creation and use of phase plane plots of nonlinear circuits. We will describe how to rapidly create the phase plane plots with a reconfigurable hardware platform, instead of with a numerical simulator.

This paper is an expansion of the work presented in [

Whenever designers want to get an analytical handle on the sources and causes of distortion, the most commonly-used tool is Volterra series analysis. If a problem is tractable using Volterra series, then it can also be solved with perturbation theory, which will yield asymptotically-identical results [

There are certain problems for which Volterra series are ill-suited—multiple-time-scale behavior and multiple steady states, for instance [

We therefore find it worthwhile to give a basic treatment of regular perturbation—the simplest perturbation method—as applied to distortion analysis of first-order analog circuits. In addition, a filter block diagram representation of the circuit will naturally evolve from our analysis, making it visually clear how the distortion terms are manifested, and how well-known tenets of low-distortion design, such as feedback, come about.

Consider the initial value problem

Most common first-order analog ciruits (simple amplifiers, buffers, switches, etc.) are of the form depicted in Figure

General block diagram form of a first-order circuit. The primary processing block is

In order to apply perturbation analysis to (

With the introduction of the perturbation parameter

If

Substituting (

The

The

The inputs of the

Raising

Analogous to that of the

We now make some observations about the harmonic distortion results that were discussed in the previous section.

In the

Magnitude-frequency plots of the third harmonic. The “gain”,

Notice from the figure that if

These two notions—that frequency and feedback gain can be sacrificed for higher linearity—conform with the traditional rules-of-thumb for low-distortion design.

According to KCL, the circuit equation of the source follower amplifier in Figure

Source follower amplifier. (a) Circuit schematic. (b) Block diagram representation of source follower output. The fundamental harmonic is a low-pass filtered version of the input. The second order terms are generated by high-pass filtering the input, squaring and then low pass filtering. The total output is a power series of

Note that

First, define

Magnitude-frequency response of source follower. Analytical prediction is in bold, and experimental data is plotted as “x”s and “o”s. The fundamental harmonic is a low-pass filtered version of the input. The second harmonic has a bandpass shape, as predicted by perturbation analysis.

Consider the unity-gain buffer depicted in Figure

Unity gain buffer. (a) Circuit schematic. (b) Block diagram representation of output. The fundamental harmonic is a low-pass filtered version of the input. The third-order terms are generated by high-pass filtering the input, cubing and then low pass filtering. The total output is a power series of

We can define a characteristic voltage,

To calculate distortion terms, assume

Magnitude-frequency response of unity-gain buffer. Analytical prediction is in bold, and experimental data is plotted as “x”s and “o”s. The fundamental harmonic is a low-pass filtered version of the input. The third harmonic has a bandpass shape, as predicted by perturbation analysis.

The harmonic behavior of a circuit is similar for above- and subthreshold operation. In absolute numbers, however, above threshold operation yields less distortion. This is because the parameter

We have developed a field programmable analog array (FPAA) that can be configured to synthesis and analyze a vast variety of circuits [

Compiled Circuit on an FPAA. Component terminals can be connected or disconnected using the switch matrix. In this illustration, connections are indicated by solder dots. The gain of each amplifier is programable over a continuous range of values over a few decades.

Consider the simple current mirror depicted in Figure

Simple current mirror. (a) Circuit that was compiled onto the FPAA. (b) Measured trajectories for different initial conditions. (c) Vector field derived from trajectory measurements. The origin is an unstable equilibrium point, while

For subthreshold operation in saturation, the current through transistors M1 and M2 is [

Equation (

Since the simple current mirror is a one-dimensional system, its vector field is represented as a flow on a line. The direction and speed of the flow are dictated by the right hand side (RHS) of (

The vector field provides clear, qualitative information about the behavior of

The vector field in Figure

The geometric analysis predictions can be checked against experimental measurements of a current mirror that was compiled onto an FPAA. Figure

Assuming subthreshold operation, the KCL equation for the source follower amplifier of Figure

Source Follower Amplifier Acting as a Simple Peak Detector. (a) Circuit that was compiled onto the FPAA. (b) Measured step responses. (c) Vector field derived from step response measurements. The point

The vector field of the logistic equation is represented as a flow on the

The time that it takes for

For a large positive step input,

For a large negative step input,

One way of avoiding the math is to employ intuitive descriptions of the charging action of the active device (i.e., the transistor) versus the discharging action of the current source [

Figure

Second-Order Section. Varying the bias currents of the various amplifiers leads to interesting dynamics.

If we define

We can linearize (

For certain values of

For very large values of

Sketch of SOS Phase Plane for Large Signals.

Figure

SOS Experimental Phase Plane Results for Various Values of

In this paper, we have introduced visual and graphical techniques for analyzing nonlinear circuit dynamics. Our approach to studying harmonic distortion yields information about the various processing flows that are responsible for each harmonic term. The FPAA was used to rapidly create phase plane plots, which concisely encapsulate the nonlinear dynamics of the circuit under study. We have provided various examples of our techniques and have compared our predictions to experimentally-measured data.