Optimization of flat top of high pulsed power waveform by wavelet analysis

Accelerator physics for elementary particle colliders : topics linked with time frequency domain tuning of Pulse Forming Network in High Pulsed Power sources Particle physics needs high quality beams for reaching a minimum of energy/position dispersion (low emittance)colliding bunches. It means in first, constant flat top of pulsed power sources, like High Voltage modulators. Conventional micro seconde one, already lie on Pulse Forming Network (PFN), which proper tuning influence the minimum ripple on wave form. That tuning is not trivial. tuning of Pulse Forming Network in High Pulsed Power sources The goal of the machines like Large hadron Collider LHC at CERN, is to generate collision of packets of particles with a high definition in energy/momentum, hence a small dispersion. In fact, the ''emittance'', which is the volume of X,P in phase space, must be the smallest as possible. For that, the impulse power sources must deliver an electrical, ideally ''flat top'' (inside that plateau, the energy of emitted particles is constant), so unlike signal recognition or transmissions for example, for today accelerators the performances are not yet axed on rising or falling time (however it could be the case in future high frequency machines). The resulting spectrum of power waveforms is then dominated by its pass-band character. To improve the flat top, we need to develop and tune some passive Pulse Forming Network PFN, and their performance must reach some per mill of the High voltage, regarding the relative amplitude of the ripples in the flat top. The problem of optimal tuning seems to lie in ''control and optimization'' class. In our machine, it depends on 12 continuous variables, who are the values of tunable power inductances. The PFN is a 12-strongly coupled harmonic oscillator (many bodies problem), and the optimal tuning state is not described by a trivial algorithm. What we need is to find a control and optimization method, either by analytical -if possible- approach, either by analysis of measurement during tuning. As classically, the time or frequency waveforms do not help us to improve the tuning, essentially because in time measurements, 2 distant cells of PFN may interfere with unknown phases. Naturally the Frequency transform doesn't allow us to catch that hidden information. We then used a wavelet toolbox, with scilab, and as we met some difficulties in the dynamic range of our waveforms with that toolbox, we are presently conducting some direct investigations with for example complex morlet wavelets. We shall describe the first results. In alternative approach, we try to solve the correspondent evolution equation, which belongs to Hill family, because it is linked with a pseudo transmission line with inhomogeneous characteristic impedance. We have extrapolated the Flocquet method by infinite determinants, but in synthesis like algorithm. We shall also discuss these first progresses. Considerations on mixed/intricate time frequency analysis Inside the same problem, we try to answer to another different question : is there a class of transform which would use a ''mixed variable'' which we call for example w. For sake of simplicity, w is complex and we have w= g(f,t) (or g(a,t) where a is the scale parameter of wavelet formalism). All our measurement V(t) or V(f) should be then transformed/mapped to V(w) where V is the High voltage image of the power source. To make our question or formalism useful, we have to restrict the features of g. We try to discuss these restrictions regarding the general aspects of g, (symmetries, deterministic category then metric abscissa ?...) and also investigation on the ''visibility'' of V(w), as measurement tool in signal recognition. It is not mandatory to closely link that question to precedent theme. We intend to discuss/list some -linear or not- representations different from the couple translation/rotation. That discussion and these ideas could help us in many other topics of the physic, like Electromagnetic field measurements (photo emission assisted by Schottky effect) for femtosecond electrons bunches. That issue could also be beneficial in other scientific researches.