The natural frequencies of uniform LC ladder pulse-forming networks

A complete understanding of the way in which charged uniform LC ladder pulse-forming networks (PFNs) discharge into a resistive load remains elusive due to the difficulty of the mathematical analysis. However, a solution to this problem is important in that it will allow designers of PFNs to be in complete control of the design rather than having to design empirically and be dependent on adjusting the waveforms produced either experimentally or in simulation using circuit analysis packages such as PSpice or microcap. Furthermore, a complete understanding will allow PFNs to be designed which generate pulsed wave shapes other than just the usual rectangular shape. The problem stems from the difficulty in inverting the Laplace transform equation for the wave shape of the voltage pulse delivered into a load by a PFN with a finite number of stages P.W. Smith (2002). Such an inversion then leads to the sum of a series of exponentially damped sinusoidal waveforms from which the wave shape generated is composed. Whereas the natural frequencies of an undamped uniform LC ladder are relatively easy to find, the natural frequencies of a ladder damped by a resistive load become very complex even for PFNs with just 2 stages. It is possible to solve the problem for PFNs with an infinite number of stages P.W. Smith (2002) and the solution takes the form of the sum of a series of integrated Bessel functions. The wave shape generated is then built up from the leading edge onwards as more terms in the series are added. This paper will discuss the nature of the problem and the progress made on understanding the way in which PFNs discharge into resistive loads.