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VLSI Design
Volume 2010 (2010), Article ID 687498, 8 pages
http://dx.doi.org/10.1155/2010/687498
Research Article

Nonlinear Circuit Analysis via Perturbation Methods and Hardware Prototyping

1Thayer School of Engineering, Dartmouth College, Hanover, 03755, USA
2School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, 30332-0250, USA

Received 3 June 2009; Accepted 8 December 2009

Academic Editor: Benjamin J. Blalock

Copyright © 2010 K. Odame and P. E. Hasler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

Portable and implantable, always-on electronics stand to benefit from analog signal processing, when only low levels of precision are necessary [1, 2]. To achieve sophisticated signal processing with low power and area overhead, an analog processor can exploit the fundamental nonlinear dynamics that are found in devices and simple circuits. So, the circuit designer must depart from the traditional linear systems paradigm, and learn to analyze and understand circuits from a nonlinear dynamical systems theory perspective.

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 [3] of the circuit. Volterra series is an analytical tool that quickly leads to high entropy [4] mathematical expressions, from which the causative physical phenomena are hard to discern, much less purposefully manipulate.

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 [5].

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 [6].

2. Regular Perturbation

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 [7].

There are certain problems for which Volterra series are ill-suited—multiple-time-scale behavior and multiple steady states, for instance [8]—that can be solved with perturbation theory. Despite the power of perturbation theory, it is still a relatively obscure concept in discussions about nonlinearity and distortion in analog circuits.

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 probleṁ𝑥=𝑓(𝑡,𝑥,𝜖);𝑥𝑡0=𝑥0(𝜖),(1) where 𝜖 is a small perturbation parameter such that 𝜖=0 yields an analytically-soluble equation. If 𝑓 is sufficiently smooth (the specific smoothness requirements of 𝑓 are discussed in [9]), then the problem has a unique solution 𝑥(𝑡,𝜖). As the solution for 𝜖0 may not be analytical, it can be approximated as a power series in 𝜖 to an accuracy of O(𝜖𝑛+1). That is, we can write the solution as 𝑥(𝑡,𝜖)=𝑛𝑖=0𝑥𝑖(𝑡)𝜖𝑖̂𝑥(𝑡,𝜖)+O𝜖𝑛+1,(2) where ̂𝑥(𝑡,𝜖) is the approximate solution. To conduct regular perturbation, we apply the substitution 𝑥(𝑡,𝜖)̂𝑥(𝑡,𝜖) to (1). The resulting system is then solved by equating like powers of 𝜖. The following sections will illuminate this idea.

3. The Basic First-Order Circuit

Most common first-order analog ciruits (simple amplifiers, buffers, switches, etc.) are of the form depicted in Figure 1. The governing equation iṡ𝑥=𝑔(𝑢𝑥)+(𝑥),(3) where 𝑢 is the a.c. input signal, 𝑥 is the a.c. output signal and 𝑔() and () are nonlinear functions. The dependence of the system on the output, other than through feedback to the input, is modeled by (𝑥). In practice, (𝑥) is typically some nonideality such as finite output resistance.

687498.fig.001
Figure 1: General block diagram form of a first-order circuit. The primary processing block is 𝑔(), which is a nonlinear function of the input 𝑢 and of 𝑥 via feedback. The nonlinearity (𝑥) models such nonidealities as finite output impedance.

In order to apply perturbation analysis to (3), we begin by assuming that the input signal has a small amplitude. This is expressed as 𝑢=𝜖𝑣, where 𝜖 is a small perturbation parameter and 𝑣 is a suitably-scaled version of the input signal. Note that with the definition of 𝑢, (3) is solvable via separation of variables for the special case 𝜖=0.

With the introduction of the perturbation parameter 𝜖, we can approximate the solution to (3) with the power series𝑥(𝑡)𝑛𝑖=1𝜖𝑖𝑥𝑖(𝑡).(4) Note the 𝜖0 term of (4) is set to 0. This corresponds to analyzing a circuit about its d.c. bias point, where the d.c. bias point is shifted to the origin. For ease of notation, define 𝑧=𝑢𝑥. The approximation of 𝑧 is defined similarly to (4), with 𝑧1=𝑣𝑥1 and 𝑧𝑖=𝑥𝑖,for all 𝑖>1.

If 𝜖 is sufficiently small, then the functions 𝑔(𝑧) and (𝑥) can be approximated by their truncated Taylor series as𝑔(𝑧)𝑔1𝑧+𝑔𝑛1𝑧𝑛1+𝑔𝑛𝑧𝑛,(𝑥)1𝑥+𝑛1𝑥𝑛1+𝑛𝑥𝑛.(5) Functions 𝑔 and are assumed to be dominantly (𝑛1)th-order nonlinearities, with 𝑔𝑖=𝑔(𝑖)(0)/𝑖! and 𝑖=(𝑖)(0)/𝑖!. Equation (5) assumes 𝑔(0)=(0)=0, which, again, corresponds to analyzing a circuit about its d.c. bias point.

Substituting (4) and (5) into (3) and collecting powers of 𝜖, we get the following set of first-order linear equationṡ𝑥1+𝑔11𝑥1=𝑔1𝑣,̇𝑥𝑘+𝑔11𝑥𝑘=0𝑘<𝑛1̇𝑥𝑛1+𝑔11𝑥𝑛1=𝑔𝑛1𝑧𝑛11+𝑛1𝑥𝑛11̇𝑥𝑛+𝑔11𝑥𝑛=𝑔𝑛𝑧𝑛1𝑛𝑔𝑛1𝑧𝑛11𝑥2+𝑛𝑥𝑛1𝑛𝑛1𝑥𝑛11𝑥2.(6)

The 𝜖1 equation is the linearized portion of (3) with input 𝑣. Taking the Laplace transform of this equation, we write𝑋1(𝑠)=𝑔1𝐻(𝑠)𝑉(𝑠),(7) where 𝐻(𝑠)=1/(𝑠+𝑔11).

The 𝜖𝑘 equations (𝑘<(𝑛1)) are filters with 0 input. As such, the steady state solutions of these equations is 0.

4. Harmonic Distortion Terms

The inputs of the 𝜖𝑛1 equation are terms of 𝑧𝑛11 and 𝑥𝑛11. To understand the implications of these terms to harmonic distortion, assume a single-tone input, 𝑣=cos(𝜔𝑡). This elicits the signals𝑥1=𝑔1||𝐻(𝑗𝜔)||cos(𝜔𝑡+𝜙(𝑗𝜔)),𝑧1=||1𝑔1𝐻(𝑗𝜔)||cos𝜔𝑡+𝜙𝑧1(𝑗𝑤),=||||||𝑠1𝐻(𝑗𝜔)𝐻𝑧1(𝑗𝜔)||||||cos𝜔𝑡+𝜙𝑧1(𝑗𝑤),(8) Here we have defined 𝐻𝑧1(𝑠)=(1𝑔1𝐻(𝑠)). The phases 𝜙(𝑠) and 𝜙𝑧1(𝑠) are the arguments of 𝐻(𝑠) and 𝐻𝑧1(𝑠), respectively. The signals 𝑥1 and 𝑧1 are single tones of frequency 𝜔 as well, since they are merely linearly-filtered versions of 𝑣.

Raising 𝑧1 and 𝑥1 each to the (𝑛1)th power produces harmonics as follows. If (𝑛1) is odd(even), then odd(even) harmonics up to the (𝑛1)th harmonic are generated. The amplitude of the 𝑚𝜔 frequency term in 𝑥𝑛11 is(𝑛1)!𝑔1((𝑛+𝑚1)/2)!((𝑛𝑚1)/2)!2𝑛2||𝐻(𝑗𝜔)||,(9) while that of the 𝑚𝜔 frequency term in 𝑧𝑛11 is(𝑛1)!((𝑛+𝑚1)/2)!((𝑛𝑚1)/2)!2𝑛2||𝐻𝑧1(𝑗𝜔)||.(10) After filtering in the 𝜖𝑛1 equation, the amplitudes of these terms will be, respectively,(𝑛1)!𝑛1𝑔1((𝑛+𝑚1)/2)!((𝑛𝑚1)/2)!2𝑛2||𝐻(𝑗𝜔)||||𝐻(𝑗𝑚𝜔)||,(11)(𝑛1)!𝑔𝑛1((𝑛+𝑚1)/2)!((𝑛𝑚1)/2)!2𝑛2||𝐻𝑧1(𝑗𝜔)||||𝐻(𝑗𝑚𝜔)||.(12)

Analogous to that of the 𝜖𝑛1 equation, the input to the 𝜖𝑛 equation has terms in 𝑧𝑛1 and 𝑥𝑛1. In general, the 𝑥2 terms are identically zero, except for the special case 𝑛=3.

5. Feedback and Distortion

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

In the 𝜖𝑛1 equation, the amplitude of the 𝑚th harmonic that the 𝑧𝑛11 term contributes is given by (12). We plot this amplitude expression, along with that of (11), as a function of frequency in Figure 2 for the third-order harmonic generated by a dominantly-third order nonlinearity. That is, 𝑛=4 and 𝑚=3. Also, we chose 1=1, 3=1/3, 𝑔1=𝐺, 𝑔3=𝐺/3, where 𝐺 was varied from 10 to 1000.

fig2
Figure 2: Magnitude-frequency plots of the third harmonic. The “gain”, 𝐺 of the 𝑔(𝑧) function is varied from 10 to 1000. This causes the band-pass shape of the 𝑧31-contributed harmonic to shift to the right, while that contributed by 𝑥31 falls in magnitude.

Notice from the figure that if 𝑔11, then, for a given frequency, the amplitude of the 𝑧𝑛11-contributed harmonic is greatly reduced. In fact, if we ensure 𝑔𝑖𝑖forall𝑖, then the harmonic contribution of the 𝑥𝑛11 terms is negligible. This would mean that the distortion is effectively due only to 𝑧1, whose associated harmonics are band-pass filtered. This in turn means that the distortion can be kept small if the circuit is operated well below the corner frequency.

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.

6. Illustrative Examples

6.1. Source Follower Amplifier

According to KCL, the circuit equation of the source follower amplifier in Figure 3(a) is𝐶𝑑𝑉out(𝑡)𝑑𝑡=𝐹𝑉in,𝑉out𝐼bias,(13) where the function 𝐹 is defined as𝐹𝑉in,𝑉out=𝐾2𝜅𝑉in(𝑡)𝑉out(𝑡)𝑉th2,(14) if 𝑀1 is in above-threshold saturation, and𝐹𝑉in,𝑉out=𝐼o𝑒(𝜅𝑉in(𝑡)𝑉out(𝑡))/𝑈T,(15) if it is in subthreshold saturation. The parameter 𝐾 depends on transistor dimensions and doping and 𝑉th is the threshold voltage. Also, 𝜅, 𝐼o, and 𝑈T have their usual meanings from the EKV MOSFET model [10].

fig3
Figure 3: 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 𝜖 terms.

Note that 𝐼bias=𝐹(𝑉g,𝑉S), where 𝑉g and 𝑉S are the d.c. bias-points of the gate and source of 𝑀1, respectively. Let us define a characteristic voltage, 𝑉c, as𝑉c=𝜅𝑉g𝑉S𝑉th2,abovethreshold𝑈T,subthreshold.(16) Now, (13) can be nondimensionalized [9] by making the substitutions𝜏=𝐼bias𝐶𝑉c𝑡;𝑢=𝜅𝜈in𝑉c;𝑥=𝜈out𝑉c,(17) where 𝜈in and 𝜈out are the a.c. portions of 𝑉in and 𝑉out. This gives the state-space equation of the source follower as𝑑𝑥𝑑𝜏=𝑢𝑥+(𝑢𝑥)24,(18) for above threshold, and𝑑𝑥𝑑𝜏=𝑢𝑥+(𝑢𝑥)22,(19) for the truncated Taylor expansion in subthreshold. The point is that, regardless of region of operation of 𝑀1, the nonlinear equation that describes the source follower has the same functional form. Relating the source follower equations to (3), we have 𝑔(𝑧)𝑧+𝑧2 and (𝑥)=0. As such, we expect the harmonic distortion terms to have a band-pass-like dependence on frequency. To show this, we will apply regular perturbation to (18).

First, define 𝑢=𝜖𝑣, where the small parameter 𝜖 is a scaled version of the input amplitude. That is, 𝜖=𝐴in/𝑉c. Also, taking 𝑥=𝜖𝑥1+𝜖2𝑥2 and 𝑧=𝑢𝑥 and equating like powers of 𝜖 up to 𝜖2, we have 𝜖1̇𝑥1=𝑣𝑥1,(20)𝜖2̇𝑥2=𝑧214𝑥2,(21) as depicted in Figure 3(b). Assume a pure-tone input, 𝑣=cos(𝜔𝑡). Equation (20) is the linear portion of the amplifier. Equation (21) is a linear filter with input 𝑧21/4. The squaring produces a second-harmonic term as well as a d.c. offset. In addition, since 𝑧1=𝑣𝑥1, the second harmonic generated by the squaring is high-pass filtered. The overall effect is that 𝑥2 is a band-pass filtered version of a second harmonic of 𝑣. Figure 4 is a plot of experimental data that corroborates our analysis. There is error in the second harmonic measurement due to the small amplitudes involved.

687498.fig.004
Figure 4: 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.
6.2. Unity Gain Buffer

Consider the unity-gain buffer depicted in Figure 5(a). It is formed by placing an operational transconductance amplifier (OTA) in negative feedback. If we operate the OTA above threshold, the describing equation is𝐶𝑑𝑉out𝑑𝑡=𝜅𝛽𝐼bias𝑉in𝑉out1𝜅𝛽𝑉in𝑉out24𝐼bias,(22) while it is 𝐶𝑑𝑉out𝑑𝑡=𝐼biastanh𝜅𝑉in𝑉out2𝑈T,(23) for subthreshold operation. Notice that we have ignored the output conductance term, which is considered very small for OTAs.

fig5
Figure 5: 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 𝜖 terms.

We can define a characteristic voltage, 𝑉c, as𝑉c=2𝑈T𝜅,subthreshold𝐼bias𝜅𝛽,abovethreshold.(24) Then, with the following definitions𝜏=𝐼bias𝐶𝑉c𝑡;𝑢=𝜈in𝑉c;𝑥=𝜈out𝑉c,(25) the nondimensional form of the unity-gain buffer's describing equations (taken to the first few Taylor series terms) is𝑑𝑥𝑑𝜏=(𝑢𝑥)(𝑢𝑥)34,abovethreshold(𝑢𝑥)(𝑢𝑥)33,subthreshold.(26) Again, the functional form of the equations is identical, regardless of region of operation.

To calculate distortion terms, assume 𝑢=𝜖𝑣 is a pure-tone signal and proceed as usual. For subthreshold, the separated equations of 𝜖 are 𝜖1̇𝑥1=𝑣𝑥1,(27)𝜖2̇𝑥2=0𝑥2,(28)𝜖3̇𝑥3=𝑧313𝑥3.(29) These equations are depicted in the block diagram of Figure 5(b). Equation (27) is the linear portion of the amplifier. Equation (28) is a linear filter with 0 input; it contributes no harmonics at steady state. Equation (29) is a linear filter with input 𝑧31/3. The cubing produces a third-harmonic term as well as a fundamental-frequency term (this fundamental-frequency term will cause gain compression, which is not discussed in this paper). Since 𝑧1=𝑣𝑥1, the overall effect is that 𝑥3 is a band-pass filtered version of a third harmonic of 𝑣, as shown in Figure 6.

687498.fig.006
Figure 6: 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.
6.3. Note on Above-Threshold versus Subthreshold Operation

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 𝜖=𝐴in/𝑉c is much smaller for above threshold than for subthreshold. Since the harmonics are multiplied by 𝜖𝑖, the smaller 𝜖 seen in above threshold operation translates to lower distortion.

7. Field Programmable Analog Array

We have developed a field programmable analog array (FPAA) that can be configured to synthesis and analyze a vast variety of circuits [11]. Figure 7 depicts a simple, second-order filter compiled on our FPAA. More complex circuit configurations are possible, and would involve a larger number of the over 400 components in the FPAA. In this part of the paper we demonstrate the utility and versatility of the FPAA in analyzing the dynamics of a number of fundamental circuit blocks.

687498.fig.007
Figure 7: 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.

8. One-Dimensional Systems

8.1. Simple Current Mirror

Consider the simple current mirror depicted in Figure 8(a). From Kirchhoff's Current Law (KCL), it obeys the following differential equation:𝐶𝑑𝑉g𝑑𝑡=𝐼b𝑓𝑉g,(30) where 𝑓(𝑉g) is the drain current of transistor M1. Assuming M1 and M2 are identical and are both saturated, we have 𝑓(𝑉g)=𝐼out, which gives𝐶𝑑𝑉g𝑑𝑡=𝐼b𝐼out.(31)

fig8
Figure 8: 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 5 is stable.

For subthreshold operation in saturation, the current through transistors M1 and M2 is [10] 𝑓𝑉g=𝐼out=𝐼o𝑒(𝜅𝑉g𝑉S)/𝑈T,(32) where 𝐼o is a pre-exponential constant dependent on the transistor's size and on doping concentrations. Also, 𝜅 is the body-effect coefficient and 𝑈T is the thermal voltage. 𝑉S is the source voltage, which, for this case, is zero. Setting 𝑉S=0 and taking the derivative of (32) with respect to time, we get 𝑑𝐼out𝑑𝑡=𝜕𝜕𝑉g𝐼o𝑒𝜅𝑉g/𝑈T𝑑𝑉g𝑑𝑡,(33)=𝜅𝑈T𝐼out𝑑𝑉g𝑑𝑡,(34) which allows us to rewrite (31) as𝐶𝑈T𝜅𝐼b𝑑𝐼out𝑑𝑡=𝐼out1𝐼out𝐼b,𝜏𝑑𝐼out𝑑𝑡=𝐼out1𝐼out𝐼b.(35) The time constant is identified as 𝜏=(𝐶𝑈T)/(𝜅𝐼b).

Equation (35) happens to be the logistic equation, a simple model of population dynamics. It can be solved exactly either by separation of variables followed by partial fractions, or by solving it as Bernoulli's equation. The solution is 𝐼out(𝑡)=𝐼b𝑒𝑡/𝜏𝑒𝑡/𝜏1+𝐼b/𝐼out0,(36) where 𝐼out0 is the initial value of 𝐼out. We are lucky to have an exact solution to (35), given that it is a nonlinear differential equation. Even so, it is difficult to discern much useful information about 𝐼out's qualitative behavior from (36). For instance, it is not clear how the behavior of 𝐼out might change with different initial conditions. To answer questions of this sort, it is helpful to do geometric analysis on the system's corresponding vector field.

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 (35). It is a quadratic, as shown in Figure 10. The 𝐼out-intercepts are 0 and 𝐼b. There is a maximum at 𝐼out=𝐼b/2. The vector field is depicted as the arrows on the 𝐼out axis. For positive values of 𝑑𝐼out/𝑑𝑡, 𝐼out is increasing, meaning the arrows point to the right. For negative values of 𝑑𝐼out/𝑑𝑡, 𝐼out is decreasing, meaning the arrows point to the left. When 𝑑𝐼out/𝑑𝑡=0, there is no change in 𝐼out and the circuit is said to be at equilibrium.

The vector field provides clear, qualitative information about the behavior of 𝐼out. There are two equilibrium points, namely 𝐼out=0 and 𝐼out=𝐼b. Note that the vector field flows away from 𝐼out=0. This equilibrium point is unstable, since the system will not recover from slight disturbances away from it. The vector field flows towards 𝐼out=𝐼b, implying that this is a stable equilibrium point. If the system is initially at 𝐼out=𝐼b and then experiences a small disturbance, it will tend back to the 𝐼out=𝐼b point.

The vector field in Figure 10 also gives information about the acceleration of 𝐼out as it approaches the 𝐼out=𝐼b equilibrium point. For 0<𝐼out<𝐼b/2, the rate of change of 𝐼out increases until it reaches a peak at 𝐼out=𝐼b/2. Between 𝐼b/2 and 𝐼b, the system decelerates until the rate of change of 𝐼out eventually becomes zero. For 𝐼out>𝐼b, the rate of change of 𝐼out steadily decreases until 𝐼out=𝐼b. It is interesting to note that, for 𝐼out<𝐼b, the rate of change of 𝐼out is limited to a maximum of 𝐼b/(4𝜏).

The geometric analysis predictions can be checked against experimental measurements of a current mirror that was compiled onto an FPAA. Figure 8(b) depicts various trajectories, or solutions, of the system of (35) for different initial conditions. Notice that trajectories that start at values lower than 𝐼out=𝐼b/2 have a sigmoidal shape, with the point of inflection corresponding to the maximum rate of change of current 𝑑𝐼out/𝑑𝑡=𝐼b/(4𝜏). The parabolic shape of 𝑑𝐼out/𝑑𝑡 can be extracted from these trajectories, and it is shown in Figure 8(c).

8.2. Simple Peak Detector

Assuming subthreshold operation, the KCL equation for the source follower amplifier of Figure 9(a) is the following.𝐶𝑑𝑉out𝑑𝑡=𝐼o𝑒(𝜅𝑉in𝑉out)/𝑈T𝐼b.(37) Note that𝑑𝑑𝑡𝑒𝑉out/𝑈T=𝑒𝑉out/𝑈T𝑈T𝑑𝑉out𝑑𝑡,(38) in which case, the solution to (37) is𝑉out=𝜅𝑉in+𝑈Tlog𝐼o𝐼b𝐼o𝐼b𝑒(𝑉out0𝜅𝑉in)/𝑈T𝑒𝑡/𝜏,(39) where 𝜏=𝐶𝑈T/𝐼b and 𝑉out0 is the initial value of 𝑉out.

fig9
Figure 9: 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 (0,0.7) is a stable fixed point.
687498.fig.0010
Figure 10: The vector field of the logistic equation is represented as a flow on the 𝐼out axis. For positive values of 𝑑𝐼out/𝑑𝑡, 𝐼out is increasing and the flow is to the right. For negative values of 𝑑𝐼out/𝑑𝑡, 𝐼out is decreasing and the flow is to the left.

The time that it takes for 𝑉out to be within 10% of its final value is 𝑡10=𝜏log||||𝐼o/𝐼b𝑒(𝑉out0𝜅𝑉in)/𝑈T𝐼o/𝐼b𝑒0.1𝜅𝑉in/𝑈T||||.(40)

For a large positive step input, 𝑒(𝑉out0𝜅𝑉in)0, and (40) is approximately𝑡10=𝑡10+𝜏log||||𝐼o𝐼o𝐼b𝑒0.1𝜅𝑉in/𝑈T||||.(41)

For a large negative step input, 𝑒(𝑉out0𝜅𝑉in)𝐼o/𝐼b, and (40) becomes𝑡10=𝑡10𝜏log||||𝐼b𝑒(𝑉out0𝜅𝑉in)𝐼o𝐼b𝑒0.1𝜅𝑉in/𝑈T||||=𝑡10++𝜏𝑉out0𝑈T𝜅𝑉in𝑈Tlog𝐼b𝐼o.(42) Equations (41) and (42) indicate that the response of the peak detector is slower for a negative input step than it is for a positive input step. We surmise that if the input is continuously varying at a rate faster than 1/(𝑡10), then the output will be a reasonable representation of the input's peak values. Explaining the peak detector's behavior with (41) and (42) is rigorous, but depends on having to manipulate the expression of (39).

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 [12]. A more rigorous approach is to apply nonlinear geometric analysis to the problem. Consider the plot of 𝑑𝑉out/𝑑𝑡 versus 𝑉out shown in Figure 9(c). We constructed it from a number of step response measurements (Figure 9(b)) that we took after compiling the source-follower amplifier onto the FPAA. A large negative input step corresponds to an initial value of 𝑉out0𝑉in. The rate of growth of 𝑉out is bounded by 𝐼b/𝐶. For a large positive input step, however, 𝑉out0𝑉in, and the maximum rate at which 𝑉out approaches 𝑉in can be much greater than 𝐼b/𝐶. The maximum rate of approach in this case is limited only by the initial value, 𝑉out0. As such, there is an asymmetry in the speed of the circuit's response to up-going versus down-going movements on the input. The effect of this asymmetry is that 𝑉out tracks increasing 𝑉in and not decreasing 𝑉in, which is the behavior of a peak detector.

9. A Two-Dimensional System

Figure 11 depicts Lyon and Mead's classic second-order section [13]. It is a Gm-C filter with two poles that can be placed anywhere on the real/imaginary plane. We begin our analysis by writing down the governing equations for the circuit, assuming that the OTAs are based on subthreshold MOS transistor differential pairs: 𝐶1𝑑𝑉2𝑑𝑡=𝐼2𝑘tanh𝜅𝑉1𝑉22𝑈T,𝐶1𝑑𝑉1𝑑𝑡=𝐼1tanh𝜅𝑉in𝑉12𝑈T𝐼3tanh𝜅𝑉2𝑉12𝑈T,(43) where 𝐼1,2,3 are the bias currents of the OTAs. Also, 𝑘 is the ratio of the 𝐶2 to 𝐶1.

687498.fig.0011
Figure 11: Second-Order Section. Varying the bias currents of the various amplifiers leads to interesting dynamics.

If we define𝑥=𝜅𝑉1𝑉in2𝑈T,𝑦=𝜅𝑉2𝑉12𝑈T,(44) then (43) become2𝑈T𝐶1𝜅𝑑𝑥𝑑𝑡=𝐼1tanh(𝑥)𝐼3tanh(𝑦),2𝑈T𝐶1𝜅𝑑𝑦𝑑𝑡=𝐼1tanh(𝑥)+𝐼3𝐼2𝑘tanh(𝑦).(45) Further defining𝐼1=𝐼bias,𝐼2=𝑔𝑘𝐼bias,𝐼3=2𝑟𝐼bias,𝑡=𝜏2𝑈T𝐶1𝜅𝐼bias,(46) where 𝑔0, we get the following dimensionless equation𝑑𝑥𝑑𝜏=tanh(𝑥)2𝑟tanh(𝑦),𝑑𝑦𝑑𝜏=tanh(𝑥)+(2𝑟𝑔)tanh(𝑦).(47)

9.1. Small Signal Analysis

We can linearize (47) by replacing the RHS with its Jacobian, givinġ𝑥̇𝑦12𝑟12𝑟𝑔𝑥𝑦.(48) The origin is a fixed point. In fact, it is a unique fixed point, since (from (47)) tanh(𝑥)=2𝑟tanh(𝑦)𝑥=𝑦=0. The origin is stable for𝑟<1+𝑔2,(49) and unstable otherwise. It is a spiral for1+𝑔2𝑔<𝑟<1+𝑔2+𝑔,(50) and a node otherwise.

9.2. Large Signal Analysis

For certain values of 𝑟 and 𝑔, the nonlinearities of the second-order section causes it to suffer instability. In this region of parameter space, linear analysis accurately predicts that the circuit is small signal stable, but completely fails to recognize that instability would occur for large signals. Mead addresses this issue in [14], but we will present a somewhat more thorough treatment of the problem, using phase-plane analysis and experimental verification with the FPAA.

For very large values of 𝑥 and 𝑦, the tanh functions get saturated, and can each be approximated with a signum function. Equation (47) becomes𝑑𝑥𝑑𝜏=sgn(𝑥)2𝑟sgn(𝑦),𝑑𝑦𝑑𝜏=sgn(𝑥)+(2𝑟𝑔)sgn(𝑦).(51) Figure 12 shows the phase plane (𝑥 versus 𝑦) that corresponds to (51). The depicted motion is valid only if 1/2<𝑟<1. The gradient in the first and third quadrants is𝑎=2𝑟1+𝑔1+2𝑟,(52) and that in the second and fourth quadrants is𝑏=2𝑟1𝑔12𝑟.(53) Observe that, with an initial condition of (1,0), (51) predicts that the positive 𝑥-axis will again be intercepted at (𝑎2/𝑏2,0). If 𝑎2/𝑏2>1, then 𝑥 and 𝑦 will grow without bound. Stated in terms of the 𝑟 and 𝑔 variables, there is large signal instability if𝑟>𝑔+𝑔2+44.(54) Our analysis of the second-order section can readily be verified experimentally. We compiled the filter on the FPAA, as shown in Figure 7. The bias currents of all three OTAs are user-programmable, and varying them corresponds to varying the values of 𝑟 and 𝑔. The FPAA thus allows us to explore the parameter space of the filter, and to observe changes in its qualitative behavior. It can effectively be used for bifurcation analysis.

687498.fig.0012
Figure 12: Sketch of SOS Phase Plane for Large Signals.

Figure 13 shows the filter's phase plane plots for various values of 𝑟, with 𝑔 kept fixed. Just as we predicted, there is a unique fixed point, which is initially stable, and gradually changes from a node to a spiral (Figures 13(a) and 13(c)). While linear analysis would predict these three responses as damped, slightly underdamped, and very underdamped, it fails to recognize the possibility of the fourth response, which is large-signal instability. In the fourth panel, 𝑟 meets the criterion derived from nonlinear analysis, (54), and we observe oscillation. Further analysis and exploration of parameter space reveals that this second-order section is capable of low-distortion sinusoidal oscillation [15]. Such functionality is valuable in communication systems.

fig13
Figure 13: SOS Experimental Phase Plane Results for Various Values of 𝑟.

10. Conclusion

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.

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