3.2 Frequency-Domain Techniques for Small-Signal Distortion
3.2.5 Limitations of Volterra Series Techniques
Although Volterra series techniques are the preferred method for nonlinear distor- tion modeling, they present some important limitations.
The first problem we should point out refers to the Volterra series as a macro- modeling tool: it is the inherent difficulty one faces when measuring the system’s Volterra kernels, directly in time-domain, or their correspondent frequency-domain nonlinear transfer functions. Because this book is mainly devoted to analog and RF circuit designers, we will focus this brief discussion in the NLTF’s extraction.
To put it in simple and practical terms, let us imagine we would like to measure the firstn Hn(1, . . . ,n) of the one-node circuit example we have been using.
The first-order NLTF of this circuit is nothing but the small-signal frequency- domain impedance seen into the node. Thus, measuring H1() in Q frequency points requires Q different tests, which constitutes the usual one-port network analysis.
Extracting any other nth-order NLTF is incomparably more difficult. First of all, we should remember thatHn(1, . . . ,n) is an-dimensional transfer function that can only be uniquely identified by simultaneously exciting the circuit with n sources of distinct (and uncorrelated) frequencies. The experimental setup needs,
therefore,ndifferent signal generators. Also, the same requirement ofQfrequency points considered above now implies a rapidly impossible-to-handle amount of tests, one for each of the possible distortion products arising from mixingnfrequen- cies from the set of Q.4
Beyond this, it is worth mentioning that we would need a quite uncommon, and hard to build, laboratory setup. Indeed, since we are dealing with weakly nonlinear circuits, any distortion product would be easily masked by the much stronger linear responses. Furthermore, the fact that, except for the special case of the harmonics of the input frequencies, the wanted signals are not correlated in phase with any of the sources, obviates, in principle, the use of high dynamic range synchronous receivers.
The second group of problems associated with Voterra series is its inability to handle strong nonlinear circuits. Because this is the most important limitation of the technique, we will explore it in more detail.
First, let us clarify what we mean by the words ‘‘weakly,’’ or ‘‘mildly nonlinear’’
and ‘‘strongly nonlinear.’’ In the same way we said that a system should be consid- ered as nonlinear if it could not be accurately treated as linear, we now say that a certain system is in a ‘‘strong nonlinear regime’’ if it can not be accurately represented by a Volterra series with a practically small number of terms. And this may happen, either because the system incorporates nonlinear elements, which do not have continuous characteristic functions or derivatives, or because the signal excursion is such that the maximum order considered for the series is no longer enough for the desired results’ accuracy. In certain special cases, the Volterra series presents a limited radius of convergence, and is simply not applicable if the input signal exceeds that range [4]. In practical terms, the series loss of accuracy is, by far, the most interesting situation for two orders of reasons. First, we should realize that it is not possible, in general, to say a priori (i.e., without comparing Volterra series results with other simulation means, or measurement data) when the trun- cated series fails in producing useful results. This indicates that when you simulate a circuit with Volterra series, you will not get any warning, or noticeable strange outcomes, if the excitation is increased up to a level corresponding (in the real world) to a strong nonlinear regime. The second, and probably more omitted in published works, is that you may reach a situation where any practical Volterra series becomes hopelessly inaccurate. That is, the additional range of signal excur- sion you may gain is insufficient to compensate the increased computation effort required by raising the maximum order of the series.
Figure 3.18 illustrates this idea using our circuit example. It represents the static node current
4. Obviously, these are general remarks valid for any weakly dynamic system. A mildly nonlinear memoryless system can be represented by a power series. Its NLTFs are constants, and thus, a single frequency is enough for their complete characterization.
Figure 3.18 Power series approximation of the linear current, GvO(t), plus nonlinear voltage dependent current source,iNL(vO), of our circuit example resistive current.
iN(vO) =GvO+iNL(vO)=GvO +I0tanh (␣vO) (3.189) and its first to seventh-order Taylor series expansions around VO =0. Having in mind the amount of labor necessary to obtain the output components for orders higher than three or five, a glance onto Figure 3.18 will discourage any attempt to use Volterra series when the input level gets higher than about 35 mA. Indeed, it seems that beyond this limit all orders begin to simultaneously have a nonnegligi- ble effect. In the circuit behavior this corresponds to the observation that every mixing product begins to reflect the presence of higher order contributions. For example, the fundamentals no longer present a linear behavior. Their output power versus input power patterns begin to depart from the small-signal 1 dB per dB slope, and the device reaches its 1-dB compression point. That is why this point is some times used as a rough estimate of the Volterra series utility limit.
Figure 3.19 illustrates the relation between the 1-dB compression point and the validity limit of a Volterra series description of our example circuit. It depicts the fundamental and third-harmonic’s output power, in a 50-⍀load, versus source available power, calculated by the third-order Volterra series expansion around VO = 0V andIS = 0 mA, and from a large-signal simulator. Note that the odd symmetry of iNL(vO) around the (0V, 0 mA) quiescent point determines no
Figure 3.19 Small- and large-signal response of our example circuit.
even-order output voltage components, whereas the compression behavior of iNL current for increased node voltage vO, is the responsible for the observed gain expansion of the fundamental ofvO when the circuit is excited byiS.
The validity limit of the third-order Volterra series is clearly associated to the deviation of the output responses from the small-signal 1-dB/dB, and 3-dB/dB straight lines, being related to the onset of Inl(0) current saturation, as seen by the 1-dB expansion level of Vo(0).