Chapter 4 Nonlinear Stability Analysis Techniques
4.2 Nonlinear Stability Analysis Techniques
4.2.2 Auxiliary Generator
An auxiliary generator (AG) is another versatile method to analyze the stability of steady-state solutions obtained from harmonic balance simulation. As mentioned earlier,
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harmonic balance has difficulty in predicting an oscillation at unknown frequency due to the failure to provide the frequency basis of the oscillation. So, harmonic balance simulation converges to an amplifier periodic solution that is unstable and so unobservable in real. The auxiliary generator is a technique to circumvent this difficulty by assigning the oscillation frequency and amplitude to harmonic balance simulation. The implementation of the auxiliary generator is shown in Figure 4.4. A nonlinear circuit is driven by an input signal of fin and Vin, and is assumed to give an oscillation at a different frequency. The auxiliary generator consists of an ideal voltage source and an ideal bandpass filter in series [38]. The amplitude VAG and frequency fAG of the voltage source correspond to those of the oscillation signal. The ideal bandpass filter passes only the oscillation signal at fAG and prevents the other signals from flowing into the original nonlinear circuit. By connecting this auxiliary generator to a circuit node in shunt, the frequency basis and amplitude of the oscillation can be loaded into harmonic balance simulation.
Figure 4.4: Implementation of an auxiliary generator in harmonic balance simulation. A nonlinear circuit is driven by a large input signal of Vin and fin.
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The next step is to determine the oscillation parameters, VAG and fAG, because they are still unknowns. Since the auxiliary generator is an artificial source added into the original circuit, it must not perturb the original harmonic balance solution that exists before connecting the auxiliary generator. This non-perturbation condition is fulfilled when the input admittance looking into the auxiliary generator is zero at fAG:
0
AG AG
AG = =
V
Y I , (4.6)
where IAG is current flowing into the auxiliary generator. The equation (4.6) is solved in combination with harmonic balance equations of
0 ) (X =
H , (4.7)
where X is a vector of state variables. Since equation (4.7) is a complete system of equations, which means the number of real variables is the same as that of real equations, two more real variables are needed to solve the simultaneous equations of (4.6) and (4.7). The auxiliary generator parameters, VAG and fAG, serve as those two real variables, and can be determined.
Note that the phase of the ideal voltage source is not a variable in case of non-synchronized circuits [38]. In commercial harmonic balance simulators, the non-perturbation condition is solved by optimization or error-minimization, in combination with harmonic balance as the inner loop.
Using an auxiliary generator, a bifurcation locus, which is a boundary of stability, can be traced in a plane of two circuit parameters η1 and η2. At a bifurcation point, an oscillation just starts up, so that the amplitude of oscillation VAG will be very small. Hence, the non-perturbation condition is solved in terms of fAG, η1, and η2, with fixed VAG = ε (tiny amplitude):
0 ) , , ( AG 1 2
AG f η η =
Y . (4.8)
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Here, η1 and η2 can be any circuit parameters that designers are interested in: input-drive power, input-drive frequency, bias voltages, circuit elements, etc. By sweeping one of the variables in equation (4.8), a locus is traced in terms of the other two variables. When the locus has a turning point or an infinite-slope point, the sweep parameter is changed to another and the equation is solved for the other two as well [38], [55], so that a complete locus is traced. An illustrative bifurcation locus is shown in Figure 4.5. In this locus, there are two infinite-slope points in terms of η1 (marked by two dots), so the sweep parameter has to be changed to fAG or η2 at those points.
An oscillating solution curve inside the unstable operating region of Figure 4.5 can also be traced through the auxiliary generator technique. The oscillating solution curve is necessary for investigating the oscillation characteristics such as oscillating mode or possible hysteresis phenomenon. In order to trace it, the non-perturbation is solved for VAG, fAG, and one circuit parameter η1:
0 ) , ,
( AG AG 1
AG V f η =
Y . (4.9)
Note that the oscillation amplitude VAG is not a tiny value any more and has to be considered as one of the variables in equation (4.9). A switching-parameter algorithm should also be applied when sweeping a parameter. Figure 4.6 shows an illustrative oscillating solution curve obtained by solving equation (4.9), in which two bifurcations, hysteresis, and resulting jumps of solution are presented. Oscillation is generated or extinguished at the values of η1
corresponding to the two bifurcation points, when the circuit parameter is increased. However, when the circuit parameter is decreased, the value of η1 at which the oscillation is extinguished by Jump2 is lower than the bifurcation point, which suggests a hysteresis phenomenon. It is very important to figure out the oscillation characteristics in order to devise an efficient way to eliminate the oscillation.
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Circuit parameter η2
Figure 4.5: Illustrative bifurcation locus traced by the auxiliary generator technique.
Circuit parameter 1
Oscillating amplitude VAG
Bifurcation Bifurcation
Jump1
Jump2 Hysteresis
Figure 4.6: Illustrative oscillating solution curve traced by the auxiliary generator technique.
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