Chapter 2 Design Considerations of Switching-Mode Power Amplifiers
2.3 Stability
Although instabilities are encountered in all RF and microwave circuits, power amplifiers, especially switching-mode power amplifiers, have great potential to exhibit one or several types of instabilities simultaneously. In addition to the linear feedback mechanism that makes an oscillation, the strong nonlinearity of power amplifiers pushes them into an unstable region. The large RF signal periodically stimulates nonlinear circuit elements at the operating frequency, in such a way that the time-varying nonlinear elements exhibit negative resistance and induce those instabilities. These parametric instabilities tend to occur more commonly in switching-mode amplifiers, due to the extremely large input-drive level required for saturated operation of transistors. Actually, several instabilities have been observed experimentally during measurements of many switching-mode amplifiers developed at Caltech, some of which have been reported in [35]. Interestingly, the instabilities are observed only under a certain set of operating conditions, which include input-drive power, frequency, bias voltages, and temperature. For example, the switching-mode amplifier in [35] showed oscillations when driven by input power below a
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certain level, while they showed decent amplifier performance without any instability for sufficiently high input-drive power.
2.3.1 Types of Instabilities
The types of instability commonly encountered in power amplifiers or switching-mode amplifiers are illustrated in Figure 2.17.
Figure 2.17 (a) shows sub-harmonic oscillation (particularly, frequency division by two), in which the oscillation frequency is related with the input-drive frequency fin due to frequency division. The power amplifiers that have a binary power combining structure give much possibility to show sub-harmonic oscillation at half of the input-drive frequency, coming from their odd-mode oscillation characteristic [36]. However, sub-harmonic oscillation at the frequency divided by N larger than two is also observed at several switching-mode amplifiers.
The spurious oscillation at frequency fosc not related with the input-drive frequency, as shown in Figure 2.17 (b), is the most commonly observed instability. Usually, the oscillation frequency is lower than the input-drive frequency and intermodulation products between the two are presented in the spectrum, which drives the amplifiers into a quasi-periodic regime.
This type of oscillation originates from a Hopf bifurcation in which a pair of complex-conjugate poles crosses the imaginary axis into the right-hand side of the complex plane [37].
The chaos shown in Figure 2.17 (c) gives a continuous spectrum in the frequency domain, so that it looks like the noise floor of the measurement is arbitrarily boosted for a continuous frequency interval. However, chaos is not a random noise process but a deterministic phenomenon extremely sensitive to its initial condition. There are many routes that lead to chaos [38], including a quasi-periodic route with more than three non-commensurate frequency bases, a period-doubling route with continuous flip
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bifurcations, and a torus-doubling route with frequency division of two non-commensurate frequency components. Although chaos has applications in communication systems [39], it is usually considered an undesirable instability in power amplifiers. Due to its continuous characteristic of spectrum, time-domain techniques are employed to predict the chaos in simulation rather than frequency-domain ones. However, harmonic balance can be efficiently employed to analyze the routes to chaos, i.e., the preceding stages just before evolving into chaos.
Noisy precursors are also observed often in many power amplifiers. They are different from oscillations, in that the circuit still operates in a stable periodic regime and no distinct spectral line is shown in the spectrum other than the input frequency and its harmonics.
Actually, the noisy precursors present spectral bumps with some frequency interval as shown in Figure 2.17 (d), which are caused by noise amplification with reduced stability margin.
When complex-conjugate poles in the left-hand side of the complex plane are located very closely to the imaginary axis, these noisy bumps are shown centered at the frequency of the poles and the intermodulated frequencies with the input signal. As the poles approach the imaginary axis by varying one or more circuit parameters, the bumps become narrower in bandwidth and higher in power [40]. In most cases, these bumps are eventually changed to oscillation spectral lines at a single frequency when the poles cross the imaginary axis into the right-hand side.
Other types of instabilities observed commonly in power amplifiers are hysteresis and jumps of solutions. Those two are related to each other because one is a usual cause for the other. Figure 2.17 (e) shows hysteresis and jumps presented in power-transfer characteristics of amplifiers. Two turning points, T1 and T2, induced by the D-type bifurcation make an unstable section in the amplifier periodic solution curve between the two points. Then, two jumps, J1 and J2, are observed when the input power is increased and decreased, respectively.
The hysteresis and jumps are also observed in the oscillatory solutions as well as the amplifier solutions.
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fin
fin
fin
fin
fin
fin
fin
fin
fin
fosc
fin
fin
fin
fin
Pout
Pin
T1
T2 J1
J2
Figure 2.17: Types of instability commonly observed in power amplifiers. (a) Sub-
harmonic oscillation. (b) Spurious oscillation at frequency unrelated with the input drive. (c) Chaos. (d) Noisy precursors. (e) Hysteresis and jump of solutions.
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2.3.2 Stability Analysis Techniques
Instabilities in power amplifiers degrade amplifier performance such as output power, gain, and efficiency. They also give rise to unwanted interference with adjacent channels for communications. Moreover, active devices may be destroyed during the operation, due to excessive high voltage and current raised by suddenly provoked instabilities. Hence, these instabilities should be analyzed and eventually eliminated in simulation at the design stage or at the modification stage of the circuit after the first testing. The ways to analyze instabilities of RF and microwave circuits are categorized into linear and nonlinear techniques.
The linear techniques include the calculation of k- and Δ-factors of two-port represented networks. The analysis using the stability circles is also in the same category. These techniques are very powerful and simple to apply to any linear circuits. However, due to the fact that they are based on linear S-parameters, it is difficult to extensively apply to nonlinear circuits such as power amplifiers.
The nonlinear techniques are based on bifurcation detection of large-signal steady-state solution of the nonlinear circuits. The large-signal periodic solution can be efficiently obtained by harmonic balance simulation. In order to find the bifurcation of the large-signal solution and also to determine its stability, pole-zero identification and auxiliary generator are employed along with harmonic balance simulation. In Chapter 4, these stability analysis techniques will be described more in detail.
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