8.3 NONLINEAR ANALYSIS, STABILITY AND BIFURCATIONS .1 Stability and Bifurcations
8.3.2 Nonlinear Algorithms for Stability Analysis
In this paragraph, the nonlinear algorithms for the analysis of nonlinear circuits with autonomous oscillations and external excitation are described.
Circuits where an oscillation coexists with a forced periodic state are, in principle, quite naturally analysed by means of time-domain algorithms. No special modification is required in the formulation of the algorithm with respect to the analysis of autonomous circuits; however, the considerations made both for the case of two-tone analysis and for the case of oscillator analysis hold.
It is worth repeating, however, that the time-domain analysis always yields a sta- ble solution, that is, the solution actually present in the circuit; moreover, the transient behaviour is correctly found. This is important, especially in the cases in which the state actually reached by the circuit is uncertain because of the presence of possible instabilities.
Harmonic or spectral balance is another viable method for this case. Formally, the electrical quantities are expressed as in the case of a two-tone analysis (see Chapter 1), with the first frequency being that of the external signal sourceωextand the second being that of the oscillating signalωosc; in this case, the second basis frequency is unknown and must be added to the vector of the unknowns of the problem; however, the phase of the oscillation is undetermined and can be set to zero (see Chapter 5). Therefore, the number of unknowns is again equal to the number of equations, and the system can be solved by a numerical procedure. The problem always has a trivial solution, where the amplitudes of all phasors relative to the second basis frequency ωosc and of the intermodulation
frequencies nextωext+noscωosc are zero, corresponding to absence of oscillation. This solution can be real or can be unstable and only mathematical. With the aid of the same procedures as those described for oscillator analysis, the trivial solution can be avoided.
For example, a port of the circuit where non-zero amplitude of the signal is expected to appear at the oscillation frequency is selected. Then, Kirchhoff’s equation at that port and frequency can be replaced by the Kurokawa condition:
Y (ωosc)= I (ωosc)
V (ωosc)=0 (8.28)
Alternatively, Kirchhoff’s equations at all nodes and frequencies can be rewritten as IL(Vnosc)+INL(Vnosc)
n
|Vnosc|2
=0 (8.29)
Both the approaches remove the trivial solution, being in fact extensions of the previously described methods.
An alternative point of view that is an extension of that illustrated in Figure 5.23 in Section 5.5 is now described [10]. A probing voltage or current at frequencyωis intro- duced at a single port of the nonlinear circuit driven by the large signal at frequencyω0
(Figure 8.14). The nonlinear circuit is analysed by means of a non-autonomous, two-tone harmonic or spectral balance algorithm. Frequency and amplitude of the probing signal are swept within a suitable range; an oscillation is detected when the control quantity (the probing current or voltage respectively) is zero, indicating that an autonomous oscillating signal is present in the circuit and that the removal of the probing signal does not perturb the circuit.
+
− VLO(w0)
+
− Vprobe(w) Z (w)= 0
Z (nw)= ∞ Z (kw0)= ∞
Icontrol(w)
ZL Nonlinear circuit
+
− VLO(w0)
+ +
− −
Iprobe(w) Vcontrol(w)
Y(w)= 0 Y(nw)= ∞ Y(kw0)= ∞ ZL
Nonlinear circuit
Figure 8.14 Voltage and current probes for instability detection
A filter ‘masks’ the presence of the probe at all other frequency components;
we remark that harmonics of the probing frequency ω can be present, since a two- tone harmonic or spectral balance analysis is performed, and arbitrary amplitude of the probing tone is accounted for. As seen in the case of oscillators, both probing amplitude and frequency area priori unknown; they are found from the real and imaginary parts of either of the following complex equations:
Iprobe(Vprobe, ω)=0 or Vprobe(Iprobe, ω)=0 (8.30) Quite naturally, the same problems in identifying a suitable starting point for easing the convergence of the analysis are present in this case also.
Volterra analysis could be used for this type of analysis; however, the authors are not aware of such an algorithm being proposed so far.
The methods described have been extensively used for the analysis of two-tone mixed autonomous/non-autonomous circuits, and in particular, they have been used for the determination of the bifurcation diagram of the circuits. To achieve this goal, continuation methods are applied, in order to ‘follow’ the solution of the circuit as a parameter is varied and to detect the qualitative changes in its behaviour at bifurcation points [11–13].
As stated above in Section 8.3.2, the harmonic balance method can also find unstable solutions and is therefore ideal for a complete study of the behaviour of a circuit; however, the stability of branches or solutions must be verified. In general, this is straightforwardly done by application of Nykvist’s stability criterion as described above.
The analysis along a branch of the bifurcation diagram requires, in principle, simply the repeated application of the methods described above, as the value of the parameter is varied. However, problems arise both at turning points and at Hopf bifurcations. Referring to Figure 8.12, the diagram is, in principle, computed by selecting the gate bias voltage Vgs as a parameter, and the input power to the FET as one of the problem unknowns that identifies the branch. However, near the turning pointC, the curve becomes multi- valued, and numerical problems arise. When the turning point is approached, therefore, it is advantageous to switch the role of the two axes in the plot, use the amplitude of a given frequency component at a given port (e.g. the fundamental-frequency component at the gate port, related to the input power) as a parameter and set the gate bias voltage as a problem unknown. This requires a modification of the analysis algorithm, as sketched in Section 5.5. The analysis then gets through the turning point, and the whole branch can be followed. The approaching of the turning-point bifurcation can be detected by inspection or automatically by monitoring the quantity dPout
dVgs
, and setting a maximum value for it.
At the turning point, the derivative becomes infinite. The two quantities can be switched back to the original role when the derivative becomes reasonably small again.
Another problem arises when a Hopf of a flip bifurcation is encountered along a branch. Referring to Figure 8.13 for a frequency divider-by-two, the diagram is plotted by starting from a low input power, where the solution is quasi-linear and no subharmonic is present; the solution is a Fourier expansion on the basis frequency ω0. When the input powerPI is reached by stepping the input power, a second branch appears on the
diagram, representing a frequency-dividing solution, with power at frequencyω0and also at frequency ω0
2 ; however, the previous type of solution is also present as a continuation of the branch, but becomes unstable. Therefore, if the bifurcation is not detected on the way, the stable branch is overlooked and a non-physical result is found. A natural approach consists of monitoring the stability of the solution at the stepping values of the parameter by checking Nykvist’s plot. AfterPI, the solution becomes unstable, indicating that a bifurcation is present at a lower power (Figure 8.15).
The bifurcation can be accurately located, and the frequency of the new fre- quency component approximately determined, by repeating Nykvist’s analysis in smaller steps around PI, until a sufficient approximation is obtained. Then, the stable branch is followed by an analysis with a basis frequencyω= ω0
2 , which includes the frequency- divided component.
The bifurcation can be directly located in a way similar to what has been described above, once the frequency of the new branch is known from a Nykvist’s plot. For a flip- type bifurcation, an analysis based on the frequency ω= ω0
2 is performed, where the amplitude of the frequency-divided component is set to a very small value and therefore is no more an unknown, while the amplitude of the fundamental-frequency component at the bifurcation, which is related to the input powerPIatω0, is unknown. In this way, the bifurcation is located with a single analysis. If the bifurcation is a Hopf-type one, where the frequency of the autonomous oscillation is not exactly known but only approximately determined from Nykvist’s plot (Figure 8.15), the autonomous frequency is included in the vector of the unknowns, while the phase of the relevant phasor is arbitrarily set to a fixed value, for example, zero. The input power at the bifurcation and the autonomous frequency are therefore simultaneously determined. Obviously, a good starting point must be used for all these analyses, given the critical behaviour of the circuits.