Nonlinear Circuit Analysis in Time and Frequency Domain Example: The Forced Van-der-Pol Oscillator
by José Luis Flores, on behalf of AWR Group, NI
This application note introduces the technique known as auxiliary generator (AG). It is very useful for broadening the set of solutions that can be reproduced from a harmonic-balance (HB) analysis and can help study their stability properties in order to optimize a nonlinear circuit for a desired response, while at the same time avoiding undesired modes such as unwanted oscillations. It is of particular interest for the analysis and design of all kinds of nonlinear circuits, including—but not restricted to—oscillators and power amplifiers.
To predict nonlinear circuit behavior, harmonic balance (HB) is a popular and useful frequency-domain technique. But given its mathematical nature, it can only converge to mathematical solutions being harmonically related to the input sources into the circuit. Nevertheless, the mathematical solutions to a nonlinear system do not necessarily need to be harmonically related to the input signals (for instance, autonomous oscillator or non-harmonic spurious from amplifiers) and here is where the conundrum begins.
The only thing that can be said about a converged result from HB is that it is a steady-state mathematical solution to the nonlinear circuit. Now, on the other hand, there is also time-domain integration for the prediction of nonlinear-circuit behavior. While the time-domain approach typically converges to observable solutions, it is not always practical for the optimization of nonlinear circuits under a particular steady-state response, as these methods may require long simulation times before reaching the stationary regime. Also, they can be inhospitable to some circuit elements or functions. This is when/where the auxiliary generator (AG) feature within NI AWR Design Environment™ can be advantageous.
The Forced Van-der-Pol Oscillator
As a practical example to show the power of using the AG technique for more sophisticated simulations, this application note illustrates its use advantage through the simulation of a forced Van der Pol oscillator (Figure 1).
This schematic contains the basic elements found in any nonlinear tuned circuit: a resonance, nonlinearity and a signal source. The nonlinear conductance is of cubic order and shows negative values at low voltage amplitudes, i(v) = -0.03·v + 0.01·v3.
The parallel resonator has the element values L = 1 nH, C = 9 pF and R = 100 Ω. The parallel structure is fed by an independent current source whose equivalent admittance is included in R, together with the resonator loss conductance.
An interesting phenomena can be observed for different values of the current generator’s frequency and amplitude in this forced Van der Pol oscillator example.
Depending upon the input frequency and with an input current of 5 mA amplitude, two different voltage waveforms can be obtained for this example, as shown in Figures 2a and 2b. The waveform in (Figure 2a) cannot be expanded as a harmonic series on fin because another frequency is involved.
Where does this come from? The time-domain integration performed by APLAC transient simulator within NI AWR Design Environment has converged towards a quasi-periodic solution known as self-oscillating mixer (SOM) mode. Self-oscillating mixer modes are produced because an internal autonomous oscillation, fa, has started and mixes with the input signal fin.
In the circuit shown in Figure 1, when the input frequency fin approaches the circuit’s resonance, synchronization occurs and the solution becomes periodic (Figure 2b). The voltage waveform contains only fin and its harmonics. The frequency range in which synchronization takes place depends on the input generator’s amplitude—it is wider for high Ig values, and narrower for lower current values.
When the input frequency is higher than the circuit’s resonance, but close to one of its harmonics, synchronization can also take place in the form fin = N·fa. In this case, the subharmonic fa = fin/N is observed in the periodic solution. This is the working principle of the analog frequency dividers.
The frequency-domain solutions of the circuit in Figure 1 are very different from the time-domain results in Figure 2. Using APLAC harmonic-balance simulator, the harmonic voltage amplitudes have been plotted in the frequency range 1.5-1.85 GHz and for three different input current levels of 5, 10, and 15 mA, as shown in Figure 3.
It can be observed that the solutions with Ig=5 mA at 1550 MHz and 1650 MHz have much lower amplitudes than those simulated in the time-domain (Figure 2). In addition, the solution corresponding to Ig=15 mA shows discontinuities in the blue diamond curves. This still persists even after increasing the frequency resolution (Figure 4).
How can the previous results be explained? The solution with Ig=5 mA has a much lower amplitude than the values simulated in the time domain because it represents the voltage developed by a low-level input current across the equivalent impedance seen at Vo. (The higher amplitude autonomous oscillations observed in the time domain have not shown up here). For the frequency range 1610–1760 MHz, the results do not reflect the natural behavior of physical systems for which the response to an input excitation must be unique and depend only on initial conditions.
Why then are such results being observed? Is there a problem with the simulator or the circuit? The answer is NO. In this example, harmonic-balance techniques are showing different dimensions of a more complex reality. Results in Figure 3 and Figure 4 are a subset of the many possible solutions for this nonlinear circuit, showing a periodic output of the same fundamental frequency as the input source.
Now, let’s take a look at the harmonic-balance simulation results for the same case but with the addition of the AG technique.
Frequency-Domain Analysis with AG
Periodic Solutions: Jumps in a system’s response are observed in simulation as well as in the real world. Nevertheless, the theory on nonlinear dynamics assumes that any continuous variation of an input parameter must produce a continuous response. It is considered that when a jump is produced, the designer is confronted with a response with multivalued sections, showing more than one output for the same input value. Multivalued responses are not strange in nonlinear systems, and they comprise stable and unstable solutions, including points of infinite slope, which cannot be solved by the Newton-Raphson convergence algorithm typically employed by harmonic-balance simulators.
Here, with the aid of an AG, this convergence problem can be overcome by implementing a technique known as parameter switching. In the case of Figure 4, the voltage of the AG source can be swept and solved for its frequency and phase, thus transforming the very high or infinite slopes of the voltage versus frequency curves into very low or zero slopes. This modified problem is now more readily solved by harmonic-balance techniques. By applying the non-perturbing condition (1) during an optimization, the red-circled curve in Figure 5 is obtained.
Of the three parameters defining a single-tone signal (amplitude, frequency, and phase), any one of the parameters can be swept, thereby leaving the remaining two free for optimization. By sweeping the phase, for example, it is possible to solve for the multivalued sections of the curve (pink squares in Figure 5). It can thus be observed that the bottom side of the curve differs slightly from the frequency sweep results obtained without an AG (brown diamonds). This difference is due to numerical errors in simulation as the error function (cost) for these points is much higher than in the rest of the curve. The most likely cause for this is simply that those points are not a valid solution. Sometimes, during an optimization, harmonic balance may converge close to a solution without reaching it— in other words and in this case, the convergence error is high.
Another aspect to keep in mind when using the AG technique is that it requires an interpretation of the convergence (error or cost) after each optimization. Values lower than 10E-10 or 10E-11 are adequate, while values of the order of 1E-8 or higher are not (non-converged). The cost is an internal function defined within NI AWR Design Environment and may vary slightly from one optimizer to another (see AWR documentation for more details on this subject). This should not be confused with the optimization goal which, in this example, is the maximum value permitted for |YS| in order to verify the non-perturbing condition.
Phase sweeps can also be performed to explore new solutions which may be hidden/not shown after a frequency sweep with harmonic balance. Such is the case with synchronization curves, which are typical solutions in forced nonlinear systems, like injected oscillators. A synchronization curve represents the locus of the solutions for which an internal oscillation develops and locks to the input signal for all the possible phase shifts between them. The input signal comes from an independent source and the synchronization may take place with the fundamental or any harmonic of the internal oscillation. Connecting an AG at a given node and imposing fAG = fin enables the designer to sweep the phase difference between these two signals. It then leaves the variables fin and VAG free for optimization. By doing this, the synchronization curves (red) are reproduced and shown in Figure 6.
In this simulation (Figure 6), multi-valued sections are observed as well. Multivalued sections are those showing more than one voltage amplitude corresponding to the same input frequency for which no more than one can have physical existence. From the time-domain analysis in Figure 2, it can be deduced that 1.675V must be the stable solution at 1650 MHz. In forced circuits with greater complexity, when time-domain integration is no longer viable, small-signal perturbation analysis is usually applied to determine the stability of oscillatory solutions.
Now let’s set Ig=0. The result is a free-running oscillator (autonomous system) with two solutions: the oscillation, represented by a single point (to where the closed curves seem to converge), and the trivial DC solution represented by Vo=0 (zero amplitude of the first harmonic). All free-running oscillators have DC as a mathematical solution. In order for the oscillations to start up, the DC must be unstable and small-signal analysis can be applied to verify it. The voltage and frequency of autonomous oscillatory solutions can be determined by nonlinear analysis using either an AG or an OSCAPROBE element. But the stability of the large-signal solution cannot be addressed by small-signal analysis. This requires the application of specific techniques that are not compatible with OSCAPROBE elements and therefore forces the requirement to use AG (one or more may be required, such as for the case of a coupled oscillator system).
In the case of the forced solutions with Ig > 0 (Figure 6), the frequency sweep (with no AG) shows Vo≠0 (Figure 6 [blue]). This case is considered non-oscillating because these forced solutions correspond to the voltage developed by the external independent-current generator. On the other hand, the synchronization curves (Figure 6 [red]), obtained with an AG and imposing fAG = fin represent oscillatory solutions. They also establish the synchronization bandwidth of this circuit, which depends on the input current amplitude Ig. Sub-harmonic solutions can be studied in the same way if fAG = fin/N for N≥2 is imposed. The role of the AG is to stimulate the internal oscillation caused by the circuit’s resonance, which under certain circumstances synchronizes with the external generator.
Quasi-periodic Solutions: This application has thus far illustrated periodic solutions harmonically related to the input source and how AG is useful for studying multi-valued sections and regions of non-convergence for a harmonic-balance engine. However, AG is also very useful for finding quasi-periodic solutions or those that can be expressed by combining a finite number of non-commensurate (non-rationally related) fundamentals or periodic waveforms.
In this next example, a quasi-periodic solution is produced when the internal circuit oscillation is not synchronous with the input generator signal. This quasi-periodic solution with two fundamental frequencies is easily obtained by connecting an AG to the circuit at the independent frequency fa. The AG stimulates an internal oscillation mode and helps the harmonic-balance converge towards solutions with components m·fin ± n·fa, with |m|+|n| limited by the maximum harmonic order specified in the harmonic-balance engine settings.
Looking at the case Ig=15 mA with fin = 1450 MHz and with the AG’s amplitude, frequency and phase optimized for |Ys|<1e-7 (using the simplex optimizer with 100 iterations followed by a 10-iteration gradient with convergence tolerance better than 1e-5), the resultant voltage waveform and frequency spectrum obtained with both transient- and frequency-domain simulators are shown in Figure 7 and reveal excellent agreement.
While harmonic-balance and time-domain simulation techniques are often tried-and-true methods for nonlinear analysis, they can be inhospitable to some circuit elements or functions. This is when/where the AG feature being employed within NI AWR Design Environment can be advantageous.
This application note discussed the advantage AG techniques bring to nonlinear behavior and demonstrates its use for a Van der Pol oscillator. It further showed how, with the aid of AG techniques, a perfect agreement can be found between frequency-domain and time-domain simulations.
(*) All simulations performed and displayed within this application note relied upon V9.x of NI AWR Design Environment software.
Flores, J. L. Suárez, A., “On the Use of the Auxiliary Generator Technique to Improve Non-linear Microwave Circuit Analysis and Design,” ARMMS RF & Microwave Society Conf., 22-23 April 2013. Milton Hill House, Steventon, Oxfordshire (UK). http://www.armms.org
AWR Group, NI would like to thank José Luis Flores, microwave & RF engineer, for his contributions to this application note.