what you need to know about volterra- series...

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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 for these 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’s limitations 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 2f2 - f1, where f1 and f2 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, this is 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 mf1 nf2 pf3 …+ + +=

O m n p …+ + +=

f f f f–+=

2f 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 2f2 - f1. Obvious-ly, the circuit must first be analyzed at f1 and f2. The simulator then an-alyzes the circuit at the second-order components, 2f2 and f2 - f1. Finally, it has all the pieces necessary to find the third-order component of interest, 2f2 - f1. 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 (V0, I0),

(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 fR V( )=

I0 i+ I0 Vd

dfR

V V0=v

12---

V2

2

d

d fR

V V0=

v2 16---

V3

3

d

d fR

V V0=

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, so the expression (7) for the small-signal current is valid only when v << V0 and i << I0. 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, and similar 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 complicated than this. Because of practical limitations in the formation of the nonlinear currents, Volterra-series analysis becomes very cumbersome above third order, and above fifth order it loses most of its advantages over harmonic balance analysis.

Why Not Just Use Harmonic Balance?

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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 only weakly. 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 iterative process. There are a number of possible ways to 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 it 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 included Volterra 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 in an 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-


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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 2f2-f1 (box). The input level is -20 dBm.


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Figure 4. 1-12 GHz four-FET distributed amplifier. The capacitors in the drain line equalize the phase velocity on the two lines, maximizing the amplifier’s bandwidth.

Figure 5. Output power of the distributed amplifier at the fundamental frequency (triangle), second harmonic (X), and 2f2-f1 (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), and 2f2-f1 (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 Receivers, 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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