Awr vs General

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1

What You Need To

Know About Volterra-

Series Analysis

Stephen Maas

Applied Wave Research, Inc.

Copyright (C) 1999 Applied Wave

Research, Inc.; All Rights Reserved

Abstract

Volterra series analysis is finally becoming available in mainstream circuit

simulators. It has distinct advantages of speed and accuracy compared to

harmonic-balance analysis for modeling weakly nonlinear circuits. In order

to obtain optimum results from Volterra methods, it helps to understand a

little about the method and the way nonlinear circuits are modeled. This pa-

per describes these subjects.

Introduction

You’ve probably used nonlinear circuit-analysis programs quite a bit with-

out knowing that there are more ways to do nonlinear circuit analysis than

so far have been available to you. Most RF and microwave circuit simulators

are based on harmonic-balance analysis. Harmonic balance has been sold as

a “one size fits all” method; it supposedly does everything.

In fact, there are a lot of things that harmonic balance doesn’t do very well.

One of these is to analyze circuits that are only weakly nonlinear or are ex-

cited by weak signals; for example, small-signal FET amplifiers. Although

one might think that analyzing small-signal circuits would be easy, it actual-

ly is pretty tricky; most harmonic-balance solid-state device models are not

designed for this type of analysis, and harmonic balance itself is exasperat-

ingly slow, cumbersome, and prone to subtle errors.



What, Exactly, is Volterra-Series Analysis?

2

Volterra-series analysis is a superior method for analyzing weakly nonlinear

circuits. It is ideal for analyzing intermodulation distortion (IM), weak satu-

ration, (that is, up to the 1-dB compression point) and AM-to-PM conver-

sion. Volterra analysis is generally more accurate than harmonic balance forthese problems, has much wider numerical range, and is several orders of

magnitude faster. Additionally, it integrates well with linear analysis, allow-

ing simultaneous optimization of nonlinear phenomena and noise, gain, and

return. Because of these advantages, it should be considered the method of

choice for analyzing weakly nonlinear circuits.

Unfortunately, until now no full-featured nonlinear circuit simulator has of-

fered Volterra analysis. Recently a new product, VoltaireXL from Applied

Wave Research [1], has offered this capability, so it is now in the technolog-

ical mainstream. Now that Volterra analysis is available to everyone, it is

worthwhile to examine its capabilities, advantages, and disadvantages, and

to show how it can be used to design and optimize RF and microwave cir-cuits.


Volterra-series analysis is one of the oldest methods of nonlinear circuit

analysis, dating from the 1960s. It was developed as a method of system

analysis by Norbert Weiner, who used the work of the Italian mathematician,

Vito Volterra, to develop input/output relations for nonlinear system compo-

nents. Weiner showed that the so-called Volterra functional series could be

used to describe such systems, as long as they were not very strongly nonlin-

ear.

Later, this idea was applied to circuits. By the 1960s, distortion in receiver

circuits had become a serious problem, and academic researchers looked for

ways to improve their understanding of distortion. Much of this work was

performed at the Rome Air Development Center in the US, and the work re-

sulted in several classic publications [2-4]. Volterra theory seemed to be the

ideal tool. However, at the same time, the linearity of solid-state devices was

improving dramatically, and it wasn’t long before the “hardware fix” sup-

planted the theoretical one. This situation lasted about twenty years. The

communications systems in use today, however, are again becoming pro-

gressively more sensitive to distortion, and the need to design circuits and

systems to minimize distortion effects has again become painfully neces-

sary.

In this paper, you will be pleased to know, I will not get into the messy busi-

ness of describing the Volterra functional series. Fortunately, understanding

the mathematics of the Volterra series is not really necessary to make good

use of it; in fact, the algorithms used in Volterra-series software look very




3

different from the functional series itself, anyway, so there the basic mathe-

matics provide little intuitive understanding of the method’s implementa-

tion. It is much more important to understand how to make accurate Volterra

calculations, how to model devices, and how to work within the method’slimitations so that your results are valid.

In Volterra-series analysis, we calculate the orders of various intermodula-

tion products, more generally called mixing products. Usually, we are inter-

ested in such two-tone products as the infamous third-order IM product at

2 f 2 - f 1, where f 1 and f 2 are the frequencies of two excitation tones. However,

Volterra analysis is not limited to IM analysis. Harmonics, AM-to-PM con-

version, and saturation all can be described in terms of the mixing of excita-

tion frequencies. The order of the mixing, or IM, product is simply the sum

of the coefficients of the frequencies; that is, if a mixing product is

(1)

then the order of the mixing product is

(2)

Note that a high-order product can occur at a low-order frequency. For ex-

ample, one third-order product is

(3)

Because it involves three frequencies, this product is indeed third order.

However, it occurs at a fundamental frequency. Strange as it may seem, thisis an important product: the interference between this component and the

true fundamental output can be used to describe saturation and AM-to-PM

conversion. Also, harmonics can be treated as mixing products, since

(4)

and we see that a harmonic is just a tone mixing with itself.

The most common implementation of Volterra analysis uses a technique

called the method of nonlinear currents. In this method, each nonlinear cir-

cuit element is converted into a linear element in parallel with a number of

current sources. As an example, the equivalent circuit of a nonlinear resistor

is shown in Figure 1. The element’s nonlinearity is expressed entirely by

these current sources. The current at each order of nonlinearity depends on

the element’s voltage at all lower orders, so the currents are calculated re-

cursively. The first-order element voltage is calculated, then the second-or-

der current. The circuit is then excited by the second-order current and the

second-order element voltage is obtained. The third-order current is ob-

tained from the first- and second-order voltages, and the process repeats un-

f mf 1 nf 2 pf 3 …+ + +=

O m n p …+ + +=

f f f –+=

2 f f f +=




4

til all mixing products of all orders of interest are found. For example,

suppose we want to know the circuit’s performance at 2 f 2 - f 1. Obvious-

ly, the circuit must first be analyzed at f 1 and f 2. The simulator then an-

alyzes the circuit at the second-order components, 2 f 2 and f 2 - f 1.

Finally, it has all the pieces necessary to find the third-order component

of interest, 2 f 2 - f 1. In the end, we obtain the solution at the frequency of

interest, but also the linear performance and a couple of second-order

frequencies as by-products.

In Volterra analysis, a circuit is described as a combination of linear and

nonlinear elements. The linear elements are the usual ones: resistors, ca-

pacitors, transmission lines, etc. The nonlinear elements are described

by Taylor-series expansions of their current-voltage ( I/V ) or charge-volt-

age (Q/V ) characteristics. For example, a nonlinear resistor that has the

I/V characteristic

(5)

is described by the Taylor series in the vicinity of its bias point (V 0, I 0),

(6)

where i is the small-signal current and v is the small-signal voltage. We

are interested only in the small-signal quantities v and i, and the deriva-

tives at any bias point are constants. Therefore, at a specified bias point,

(6) can be written

(7)

I f R V ( )=

I 0 i+ I 0 V d

df R

V V 0

=

v1

2---

V 2

2

d

d f R

V V 0=

v21

6---

V 3

3

d

d f R

V V 0=

v3 …+ + + +=

i g1v g2v2 g3v3 …+ + +=

Figure 1 The nonlinear resistor can be converted to a linear resistor in

parallel with two “nonlinear” current sources. The values of

these sources can be rearranged so that each represents a sin-

gle order and depends only the lower-order voltages.

i = g1v + g2v2

+ g3v3

+

v

–

i

g2v2

g3v3

+

v

–

i

g1v



Advantages and Disadvantages of Volterra-Series Analysis

5

Note that g1 = 1/ R, where R is the small-signal, linear resistance at the bias

point.

A Taylor series is valid only for small deviations from its central value, sothe expression (7) for the small-signal current is valid only when v << V 0and i << I 0. This requirement is a fundamental limitation of Volterra-series

analysis: Volterra analysis is valid only when the excitation is small enough

that the derivatives of the I/V characteristic remain constant over the AC

voltage and current deviation from the bias point. We say that such a circuit

element is weakly nonlinear ; that is, the nonlinearity must be weak enough,

and the excitation level small enough, that a Taylor series is a valid repre-

sentation of the element’s I / V or Q / V characteristic†.

In practice, the excitation level must be kept well below saturation of the

component being analyzed. For this reason, Volterra analysis is not appro-

priate for mixers, frequency multipliers, saturated power amplifiers, andsimilar strongly driven circuits. It is ideal, however, for calculating IM lev-

els in small-signal amplifiers, electronic switches, phase shifters, attenua-

tors, and similar circuits.

Advantages and Disadvantages of Volterra-Series Analysis

Equation (7) is obviously pretty simple, so it does not require a lot of effort

for evaluation. Furthermore, and somewhat less obviously, it can be evaluat-

ed entirely in the frequency domain; there is no need for the repeated Fouri-

er transforms required by harmonic-balance analysis. As a result, Volterra

analysis is not limited by the numerical range of multitone Fourier trans-

forms, and since it is not iterative, it is extremely fast. Most of the computa-

tion time is used by the linear analyses of the circuit; the time required for

generating the nonlinear currents is invariably negligible.

Since Volterra-series analysis consists largely of linear analyses, it inte-

grates very easily with linear analysis. This property allows simultaneous

optimization of intermodulation distortion, noise, gain, and other linear-cir-

cuit properties. To do this with harmonic balance analysis requires repeated

switching between an optimized harmonic-balance and linear circuit simula-

tors, a clumsy, slow, and error-prone process. In Volterra, it is effortless.

† The limitation to weakly nonlinear cases is actually a little more complicatedthan this. Because of practical limitations in the formation of the nonlinear currents,Volterra-series analysis becomes very cumbersome above third order, and abovefifth order it loses most of its advantages over harmonic balance analysis.



Why Not Just Use Harmonic Balance?

6

The worst disadvantage of Volterra-series analysis is the occasional difficul-

ty in deciding whether the limitation to weakly nonlinear operation has been

exceeded. The method gives few clues when the limits are broken; the IM

products do not saturate, and the fundamental frequencies saturate onlyweakly. A little experience can be very helpful here; after all, anyone who

has spent a little time measuring small-signal amplifiers knows approxi-

mately where the IM levels start to saturate. Just stay well below those lev-

els.

Finally, it is important to note the pleasing complementarity between har-

monic-balance analysis and Volterra analysis: Volterra works well where

harmonic balance is poor, and harmonic balance can be used where Volterra

is inappropriate. Clearly, a modern nonlinear circuit simulator should offer

both methods.

Why Not Just Use Harmonic Balance?

Analyzing weakly nonlinear circuits with harmonic balance is a deceptively

tricky business. On one hand, it seems to work well—you just “push the but-

ton” and it converges easily to a reasonable-looking result. Unfortunately,

that “reasonable-looking” result is usually wrong. Usually, the cause is one

or more of three problems: (1) incomplete convergence, (2) numerical noise,

and (3) modeling difficulties.

In harmonic balance analysis, the simulator estimates a solution and im-

proves it iteratively. At some point, it has to decide that the solution is “good

enough,” and stop the i terative process. There are a number of possible waysto make this decision, and usually the simulator has a default criterion,

which the user can modify. If this criterion is too loose, the error at termina-

tion may be too great, perhaps even much larger than the very weak IM

products of interest. Result: the answer is wrong because convergence, al-

though seemingly complete, really was incomplete.

For example, suppose you tell the simulator to quit when the largest error at

any mixing product is 10-5 amps, a common value. IM products on the order

of -80 dBm, however, have currents on the order of 10-6 amps, so the current

error, when the simulator terminates, can be ten times the signal level. This

is a 20 dB error in your results!

The solution to the incomplete convergence is simple: tighten up the conver-

gence criterion. However, the convergence criterion cannot be tightened ar-

bitrarily, because the huge amount of “number crunching” required for

harmonic-balance analysis reduces numerical precision or, to put i t another

way, generates numerical noise. Fourier transforms, although not the most

numerically intensive part of the process, generate disproportionate levels of



A Modern Nonlinear Circuit Simulator

7

numerical noise. If you try to put the convergence criterion below the noise

level (e.g., make the maximum current error 10-12 amps when the noise is on

the order of 10-10 amps), the simulator simply will not converge. You’re

caught between a rock (a low signal level) and a hard place (a high numeri-cal noise level), and it can be a tight squeeze. This kind of problem does not

occur in Volterra analysis, because Volterra is not iterative and doesn’t need

Fourier transforms. As a result, the numerical range is much greater.

By far the worst problem, however, is that some of the device models used in

harmonic-balance simulators are completely inappropriate for modeling

weak nonlinearities. The general rule is implied by Eq. (6): the model must

accurately reproduce the derivatives of the device’s I/V or Q/V characteris-

tic. Only a few of the many available harmonic-balance models do this well,

and the rest require careful parameter-extraction techniques. Volterra mod-

els, on the other hand, meet this requirement automatically.

In essence, harmonic balance was designed for large-signal circuits; Volter-

ra, for small-signal circuits. Doesn’t it make sense to get them together in

the same circuit simulator?


The first commercially available Volterra simulator was C/NL2, produced

by Nonlinear Technologies, Inc., and marketed by Artech House Publishers.

Although low-cost, C/NL2 was very limited in its capabilities: it was netlist-

driven and did not include the wide range of models and other features re-

quired for the design of modern hybrid and MMIC circuits. C/NL2 includedVolterra analysis but not harmonic balance. Still, it developed a small but

loyal group of users.

A recent release of VoltaireXL, from Applied Wave Research (AWR), in-

cludes a Volterra simulator as well as a harmonic-balance simulator. This is

the first mainstream microwave circuit-analysis program to include Volterra

analysis. The AWR simulator also includes tightly integrated electromagnet-

ic (EM) analysis and linear analysis in a graphically driven environment. As

such, it is a powerful tool for designing RF and microwave circuits.

VoltaireXL has some significant advantages over C/NL2. Like most of to-

day’s circuit-analysis software, the architecture of C/NL2 was conceived inan era when memory was expensive. For this reason, C/NL2 analyzed the

complete circuit at each frequency, printed the results, cleared its memory,

and reused it at the next frequency point. In VoltaireXL, all calculations are

vectorized; for example, linear-circuit admittance matrices at each analysis

frequency are generated once, are stored, and are reused until some part of

the circuit changes. This capability makes VoltaireXL very fast, even allow-




8

ing real-time manual tuning of large, nonlinear circuits. Similarly, mod-

el data can be cached, so complex models need not be reevaluated

frequently. These capabilities are possible through the use of modern,

object-oriented program structure.

Object-oriented design allows tight integration between the parts of the

simulator. For example, a structure created and analyzed by the EM sim-ulator simply becomes a circuit element in the circuit simulator, and can

be dropped into the schematic like any other element. Again, the EM el-

ement’s data is vectorized, so the element is not reanalyzed until its

structure changes.

All the harmonic-balance models in VoltaireXL can be used by the Volt-

erra simulator. To obtain the Taylor-series coefficients, the Volterra sim-

ulator generates the derivatives of the I/V and Q/V functions in the

nonlinear models analytically, not by taking finite differences. (The

technology for doing this is well known, although not widely used.)

Thus, any nonlinear circuit can be analyzed either by Volterra or har-

monic-balance analysis. In the same way, linear characteristics of non-linear circuits also can be calculated; the simulator linearizes each

Figure 2. This 7-11 GHz amplifier circuit is analyzed by Volterra

series. The amplifier, having lumped-element matching cir-

cuits, is in prototype form. In practice the matching circuits

would be realized by distributed elements.



Examples of Volterra Analysis

9

transistor or other nonlinear element at its bias point. This capability is

similar in concept to SPICE’s AC analysis, but unlike SPICE, the same

model is used for both the linear and nonlinear analysis. This guarantees

consistency between the two analyses.

These are a few examples of the capabilities offered by a combination of

modern software design and nonlinear circuit-analysis technology. We

hope and expect to see such capabilities realized in all circuit simulators

in the future.


Figure 2 shows a single-stage 7-11 GHz small-signal amplifier consist-

ing of a single FET and prototype, lumped-element matching circuits.

The FET model consists of a linear equivalent circuit and a single non-linear controlled source; this is usually adequate to model conventional

MESFETs for Volterra analysis.

Figure 3 shows the amplifier’s calculated gain and third-order intercept

point (IP3). The Volterra calculation is fast enough that it is possible to

tune the circuit, including the nonlinear analysis, in real time. Thus, the

circuit can be optimized manually or by the numerical optimizer almost

as quickly as a linear circuit.

Figure 3. Output power of the 7-11 GHz amplifier at the fundamental

frequency (triangle), second harmonic (X), and 2f 2-f 1 (box).

The input level is -20 dBm.




10

Figure 4. 1-12 GHz four-FET distributed amplifier. The capacitors inthe drain line equalize the phase velocity on the two l ines,

maximizing the amplifier’s bandwidth.

Figure 5. Output power of the distributed amplifier at the fundamental

frequency (triangle), second harmonic (X), and 2f 2-f 1 (box).

The input level is -20 dBm.




11

Figure 6. VHF varactor tuner.

Figure 7. Output power and IM levels of the VHF varactor tuner at the

fundamental frequency (triangle), second harmonic (X), and2f 2-f 1 (box). The input level is -20 dBm.



Conclusions

12

Figure 4 shows a four-FET distributed amplifier. Again, gain and IP3 are

calculated. Even with a larger number of FETs, the speed of the analysis

again allows real-time tuning. The results are shown in Figure 5.

Volterra methods are applicable to passive as well as active devices. Figure 6

shows a varactor-tuned VHF preselector. The results are shown in Figure 7.

This circuit can be tuned in real time by varying the varactor bias. It is inter-

esting to note that the IP3 is worse at lower frequencies, where the varactor

must be biased into a more strongly nonlinear part of its Q/V characteristic.

Also interesting is the fact that the IM levels are relatively flat, compared to

the fundamental-frequency transmission. The tuner is used to reject interfer-

ence from out-of-band signals; however, these results imply that out-of-band

signals can cause substantial distortion within the tuner itself.

Conclusions

Volterra-series analysis is the method of choice for analyzing weakly non-

linear circuits. Modern circuit simulators can perform Volterra analysis, in

conjunction with linear and noise analysis, fast enough to allow real-time

tuning of even very large circuits. Because of its complementarity with other

methods, Volterra analysis should be a part of all modern circuit simulators.

References

[1] Applied Wave Research, 1960 E. Grand Ave., El Segundo, California, 90245,

USA.[2] J. W. Graham and L. Ehrman, Nonlinear System Modeling and Analysis with

Applications to Communications Receiver s, Rome Air Development Center

Technical Report No. RADC-TR-73-178, 1973.

[3] J. J. Busgang, L. Ehrman, and J. W. Graham, “Analysis of Nonlinear Systems

with Multiple Inputs,” Proc. IEEE , vol. 62, p. 1088, 1974.

[4] D. D. Weiner and J. F. Spina, Sinusoidal Analysis and Modeling of Weakly

Nonlinear Circuits, Van Nostrand, New York, 1980.



References

13

Figure 1 The nonlinear resistor can be converted to a linear resistor in parallel with two “nonlinear”

current sources. The values of these sources can be rearranged so that each represents a sin-

gle order and depends only the lower-order voltages.

i = g1v + g2v2

+ g3v3

+

v

–

i

g2v2

g3v3

+

v

–

i

g1v

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