3.3 Frequency-Domain Techniques for Large-Signal Distortion
3.3.2 Harmonic Balance by Newton Iteration
In the following, we will describe an alternative way to extend Volterra series’
maximum excitation level.
Consider again the dc problem of finding the VO output for an IS=100 mA excitation. Contrary to what was done in the previous section, where the solution
for thatiSwas calculated from successively solving partial linear problems obtained by increasing the level of a reduced version of the excitation, we will now attempt to solve the circuit for the full excitation amplitude, by successively refining coarse linear solutions. This is illustrated in Figure 3.22, in which the determination of node voltagevO[iN(t)] for aiS(t)=100 mA, based on a first-order Taylor expan- sion around iN[vO(t)] for vO(t) = 0, is sought. A first tentative voltage 1vO is clearly too low, as the first-order approximation does not account for theiNL(vO) hyperbolic tangent saturation characteristic. As shown in Figure 3.22, this voltage solution can be much better represented if we now change the expansion point to that1vO. Thus, a new2vO is obtained, which can be used again to further refine the solution, until the desired error level is met.
Passing from dc to the ac problem corresponds to solving successively linear time-varying equations that are dealt, in the frequency-domain, with the appropriate conversion matrices. Therefore, the application of this form of harmonic balance iteration to find the outputvO(t) of our example circuit, when subject to iS(t)= I0+ IP cos0t (A), can be again handled in much the same way as before. We start by computing a better approximation to the dc operation point, and then proceed to calculate the complete dc plus ac solution. The HB equation for dc is GVo0+iNL(Vo0)−Is0=0 (3.211a)
Figure 3.22 Bias point calculation by successive first-order model approximation refinement.
or
F0(Vo0)=0 (3.211b)
which can be solved iteratively using the Newton-Raphson iteration algorithm,
i+1Vo0=iVo0−冋dFdv0(vOO)|vO=iVo
0册−1F0(iVo0) (3.212)
where the initial solution is again 0Vo0=0.
After having determined the dc solution up to a desired approximation,
fVo0, the full dc plus ac HB equation can be iteratively solved for all the considered harmonics. We again use a Newton iteration scheme, in which the starting solution is that dc bias pointfVo0,
F(iVo)≡GiVo+ j⍀Qnl(iVo) +Inl(iVo)−Is=0 (3.213) and the next improved solution can be computed as
i+1Vo=iVo−[J(iVo)]−1[F(iVo)] (3.214) where the Jacobian matrix J(Vo) has the same conversion matrix form given by (3.204).
An example for iteratively determining the various 20 harmonics ofvO[iS(t)]
for our previous time-varying source ofiS(t)=100 cos0tmA using this harmonic balance scheme for an error ceiling of ⑀=1 mA, is shown in Figure 3.23.
3.3.2.1 Concluding Remarks
The similarities between this harmonic-Newton algorithm and the previous one are evident, and far from being accidental. In fact, when we first discussed the source-stepping procedure, we mentioned that we could have decided to use bigger steps against the close samples initially considered, at the expense of loosing accu- racy. That is, the HB equation could no longer be verified with a single linear solution, for each source increment, and consequently a nonlinear solver like the Newton iteration would be required. The bigger the steps, the larger the number of Newton iterations to be undertaken for a certain allowed error. In the limit, we could try only one step (i.e., attempt to directly determine the solution for the full excitation, but still anchoring our expansion atVo0=0V) if we accepted the larger number of Newton-Rapson iterations needed. That is exactly the harmonic-Newton scheme just described.
Figure 3.23 Frequency-domain large-signal calculation by successive model approximation refinement.
Whether to use one method or the other depends on the specific problem to be solved. As explained below in more detail, the approach utilized in general- purpose microwave nonlinear circuit simulators is a combination of both. Usually, the problem is attacked with a harmonic-Newton scheme for both the dc and the complete excitation. Nevertheless, if the considered initial condition for the New- ton-Raphson iteration is not close enough to the solution, it is likely that conver- gence problems will be faced. In that case, a source-stepping procedure is used to find a closer initial condition, and the harmonic-Newton is resumed.
Achieving Convergence in the Harmonic-Newton
Unfortunately, the harmonic-Newton is an iterative procedure that has not guaran- teed convergence (i.e., in which we can not be sure that we will obtain the desired
||F(fVo)|| <⑀). This may happen because the circuit is unstable or presents a nonunique stable point, the functions describing the nonlinearities are not continu- ous or show discontinuities in their derivatives, the initial estimate is far from the
exact solution, or simply because the selected number of harmonics in the Fourier expansions is insufficient to represent the signals with the required accuracy.
The first type of convergence problems is usually not important when simulating nonlinear distortion phenomena, since the circuits considered are normally stable.
An eventual exception could be the analysis of the distortion performance of voltage-controlled oscillator–based FM/PM modulators. In this case, probably the best alternative would be to simulate first the free oscillator with an appropriate HB machine, and then perturbing that stable point with the modulating signal.
The second type of convergence problems must be circumvented by a proper selection of the nonlinear model expressions. As we saw from Volterra series analysis, the ability of the device model in representing well, not only the function, but also its higher order derivatives, is crucial for achieving accurate small-signal intermodulation distortion prediction. Therefore, if one is seeking good IMD simu- lations, he should not face any harmonic-Newton convergence problems due to nonlinear device model format.
The third origin of convergence failure refers to the initial estimate selection.
To understand this, we must remember that the harmonic-Newton algorithm searches the solution of the HB equation by sensing the gradient of its error function
||F(iVo)||. Thus, it is likely to fail convergence whenever||F(iVo)||does not decay monotonically to zero, in its trajectory from the initial solution iVo to the final solution offVo. This is shown in Figure 3.24 for the one-dimensional case, where an initial estimate of0Vowould lead to an iteration trajectory of0Vo,1Vo, 2Vo,
3Vo=0Vo, . . . that wanders around the local minimum of F(Vo).
Figure 3.24 Harmonic-Newton convergence sensitivity to initial estimate selection.
The possibility that harmonic-Newton leads to such an anomalous behavior is evident from Figure 3.23. Due to the fact that the initial solution is taken as the linearized system response around the quiescent point to the full excitation, it may be quite far from the desired solution. The harmonic-Newton may even start from a reasonable low error and then temporarily (or definitely) increase the error, if the iterative trajectory passes through a zone of low gradient. In this case, an appropriate way for a successful simulation consists of choosing another initial solution, like the one referred to as0Vo′in Figure 3.24. And this may be obtained by switching from the harmonic-Newton to the source-stepping HB. The procedure followed by commercial HB simulators in the presence of such convergence diffi- culties consists of reducing the excitation to a value where convergence is guaran- teed, and using this stimulus level backed-off solution as the alternative harmonic- Newton initial estimate. Obviously, if the simulation already consists of a source power sweep (like in common output power versus input power transfer characteris- tics) the initial solution to the next amplitude point is taken as the preceding calculated result.
The source-stepping procedure just described pertains to a much broader strat- egy known as continuation methods. Continuation methods rely on varying a circuit parameter from a situation where the solution is easily found, to the desired circuit condition. In principle, any circuit parameter is amenable for use as a continuation parameter, as long as the circuit behavior responds smoothly to its changes. So, beyond the natural excitation level, we could also use the values of some critical circuit components, or even some parameters of the nonlinear device models. Normally, the idea is to convert the nonlinear circuit facing convergence difficulties into another one nearly linear, for which the solution can be obtained in one iteration. This solution is then used as the first estimate to another simulation where the circuit is weakly nonlinear, and the process repeated until the solution is found for the original circuit [1].
Finally, the last referred origin of harmonic-Newton convergence failure is related to an insufficient number of harmonics used to represent the signals. The only remedy for such a situation is to try another simulation run with a more conservative harmonic truncation. In practical RF or microwave circuits, a number of harmonics on the order of eight to ten is generally enough. This is specially true in the nonlinear distortion simulation field, not only because the nonlinearities usually encountered tend to produce small harmonic amplitudes with increasing order, as all circuits behave in an asymptotic lowpass manner when excitation frequency goes to infinity. Also, the package or intrinsic device reactances contribute to soften the impact of higher frequency components.
Summarizing Algorithm of the Harmonic-Newton Method
In order to close this section, we will now summarize the underlying concepts of the harmonic-Newton method by presenting a flow chart of its algorithm. For the
sake of generality, it is assumed that our circuit example is now composed of a linear subcircuit that comprises all linear dynamic elements, beyond the previous nonlinear subcircuit including only a memoryless current source and charge models.
It all works as if the linear conductanceGof our example circuit (see Figure 3.1) were substituted by an admittance matrix,Ycl(), while the nonlinear subcircuit still involves a voltage-dependent current source, iNL[vO(t)], and a nonlinear capacitance, whose current is given by d/dt{qNL[vO(t)]}. So, the harmonic balance equation is formed by imposing Kirchoff’s current law to the circuit’s node—that is, F(Vo) =Icl+ Inl+j⍀Qnl −Is= 0, whereIs() is again our excitation source vector, the linear subcircuit’s current is given by Icl() =Ycl().Vo(), and the nonlinear currents are computed by Fourier transforming the nonlinear current and charge previously evaluated in time-domain.
As seen in Figure 3.25, the algorithm starts by estimating an initial solution
0Vo, which is then used to formulate the harmonic balance equation in that time- varying quiescent point. In the following, a new solution estimate,1Vo, is generated by the Newton-Raphson nonlinear solver, unless the harmonic balance equation is already approximately verified within an error level not greater than a prescribed
⑀. In this case, it is considered that the solutionfVohas been reached.