Subharmonic instability

Figure 2-10 Subharmonic Instability in Current Mode Control, and Avoiding It By Slope Compensation...

This technique is called slope compensation, and is the most recognized way of quenching the alternate wide and narrow pulsing (or subharmonic instability ) associated with current-mode control (see lower half of Figure 2-10). 

It can be shown that to avoid subharmonic instability, we need to ensure that the amount of slope compensation (expressed in A/s) is equal to half the, slope of the falling inductor current ramp, or more. Note that in principle, subharmonic instability can occur only if D is (close to or) greater than 50%. So slope compensation can be applied either over the full duty cycle range, or just for D > 0.5 as shown in Figure 2-10. Note that subharmonic instability can also occur only if we are operating in continuous conduction mode (CCM). 

As a result of more detailed modeling of current-mode control, optimum relationships for the minimum inductance required (to avoid subharmonic instability) have been generated as follows... 

More details on subharmonic instability and slope compensation can be found in Chapter 7. 

Ensure that the crossover frequency is well below any troublesome poles or zeros — like the RHP zero in continuous conduction mode (boost and buck-boost — with voltage mode or current mode control), and the subharmonic instability pole in continuous conduction mode (buck, boost, and buck-boost — with current mode control). The latter pole is discussed later. 

For subharmonic instability to occur, two conditions have to be met simultaneously — the duty cycle should be close to or exceed 50%, and simultaneously, we should be in CCM... 

If we take the Bode plot of any current mode controlled converter (one that has not yet entered this wide-narrow-wide-narrow state), we will discover an unexplained peaking in the gain plot, at exactly half the switching frequency. This is the source of subharmonic instability. Because, though this point is much past the crossover frequency, it is potentially dangerous because of the fact that if it peaks too much, it can end up intersecting the 0 dB axis again — which we know is one of the prescriptions for full instability. 

Subharmonic instability is nowadays being modeled as a pole at half the switching frequency. Note that in any case we never consider setting the crossover frequency higher than half the switching frequency. So in effect, this subharmonic pole will always occur at a... 

The original models for current mode control did not predict this half-switching-frequency peaking (i.e. subharmonic instability). But it has been well known that we need to set a minimum amount of slope compensation — the value of which depends on the slopes of the up-ramp and down-ramp of the inductor current. But the criteria used for setting the precise amount of slope compensation, have been slightly differing. Our approach, outlined later, is based on more recent trends. 

S. Douady, S. Fauve, and C. Laroche. Subharmonic instabilities and defects in a granular layer under vertical vibrations. FPL Europhysics Letters), 8(7) 621, 1989. 

The frequencies of a spectrum can be divided into two parts subharmonic and harmonic (i.e., frequencies below and above the running speed). The subharmonic part of the spectrum may contain oil whirl in the journal bearings. Oil whirl is identifiable at about one-half the running speed (as are several components) due to structural resonances of the machine with the rest of the system in which it is operating and hydrodynamic instabilities in its journal bearings. Almost all subharmonic components are independent of the running speed. 

This chapter reports experimental and analytical results for pulsed control of combustion instabilities at both fundamental and subharmonic frequencies. Two suites of control algorithms have been developed one based on least-mean-square (LMS) techniques that is suitable for inner-loop stabilization of combustion instabilities, and one bcised on direct optimization that can be used either for stabilization or outer-loop optimization of combustion process objectives, such as flame compactness or emissions. 

Control and optimization of combustion processes will require controllers capable of reacting to fast processes, such as combustion instabilities, as well as controllers capable of optimizing the combustion parameters on a slower scale to achieve desired performance objectives, such as flame compactness or minimal emissions. The focus of research has been on refining two suites of control tools one based on LMS techniques and suited to control of instabilities, and another based on direct descent techniques and suited for either instability control or combustion process optimization. In addition, investigation of the performance consequences of using pulsed (on-off) as opposed to proportional actuators has been completed, and this chapter presents experimental results of the effect of varying the subharmonic order of the control pulses on instability suppression. 

Carson, J.M. 2001. Subharmonic and non-subharmonic pulsed control of thermoacoustic instabilities Analysis and experiment. M.S. Thesis. Blacksburg, VA Virginia Polytechnic Institute and State University. 

So far we have discussed EC instabilities driven by a sinusoidal AC voltage. When the AC driving voltage U t) with period T is asymmetric , i.e. U(t -h T/2) —U t), besides the conductive and the dielectric symmetries there is a subharmonic pattern where the director dynamics is 2T-periodic in time. The effect of flexopolarization on standard and non-standard EC for an asymmetric driving voltage has been analysed. One recovers in principle the scenarios with symmetric driving described... 

In an empty collapsing bubble, the amplitude of the spherical harmonic components increases proportionally to and oscillates with increasing frequency as R tends towards zero. This is a situation for which the sinklike nature of the flow gets the better of stabilization due to acceleration. In the case of a gas bubble, the situation is quite different, however. The problem, approched independently by Benjamin, 2 h s received considerable attention from experimentalists in order to quantify the heat transfer, to link the non-radial instability and subharmonics (acoustic) emission,determine the threshold for triggering off these instabilities. ... 

The experimental investigation of three-dimensional instabilities of film flows is presented in the paper by Liu, Schneider and Gollub (1995) and several distinct transverse instabilities are found to deform the travelling waves a synchronous mode (in which the deformations of adjacent wave front are in phase) and a subharmonic mode (in which the modulations of adjacent wave front are out of phase - in this case the herringbone patterns result). 

The 3D subharmonic weakly nonlinear instability is due to the resonant excitation of a triad of waves consisting of the fundamental two-dimensional wave and two oblique waves. The evolution of wavy films after the onset of either of these 3D instabilities is complex - however, sufficiently far downstream, large-amplitude solitary waves absorb the smaller waves and become dominant. In Liu, Schneider and Gollub (1995) a detailed study of these instabilities is then presented, along with a qualitative treatment of the further evolution toward an asymptotic turbulent regime. In recent review paper by Oron, Davis and Bankoff (1997), the long-scale evolution of thin (macroscopic) liquid films is considered. 

Cheng, M., and Chang, H.-C. (1995). Competition between subharmonic and sideband secondary instabilities on a falling film. Phys. Fluids 7(l) 34-54. 

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