By Kenneth S. Kundert

The motivation for beginning the paintings defined during this publication was once the curiosity that Hewlett-Packard's microwave circuit designers had in simulation thoughts that may take on the matter of discovering regularĀ country options for nonlinear circuits, relatively circuits containing disbursed parts corresponding to transmission strains. interpreting the matter of computing steady-state suggestions during this context has ended in a suite of novel numerical algorithms which we have now accrued, besides a few historical past fabric, into this publication. even though we wanted to attract as vast an viewers as attainable, to regard the topic extensive required protecting a slender concentration. Our compromise used to be to imagine that the reader knows uncomplicated numerical equipment, resembling may be present in [dahlquist74] or [vlach83], yet now not imagine any really good wisdom of equipment for steady-state difficulties. even if we specialise in algorithms for computing steady-state recommendations of analog and microwave circuits, the equipment herein are normal in nature and will locate use in different disciplines. a couple of new algorithms are offered, the contributions basically centering round new methods to harmonic stability and combined frequency-time tools. those equipment are defined, in addition to applicable heritage fabric, in what we are hoping is a fairly pleasant mix of conception, perform, and effects. the speculation is given in order that the algorithms should be totally understood and their correctness established.

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Introduction transform is used to convert the coefficient representation of the stimulus into a sampled-data representation. In the sampled-data representation the nonlinear devices can be evaluated easily and then the results can be converted back into response coefficients using the forward Fourier transform. Frequency-domain methods applied to nonlinear circuits where the computations are performed using the trigonometric-series coefficients directly are referred to as harmonic-balance methods, mostly for historical reasons.

1: Let the function f (x ,t) be continuous in t over the finite interval [O,T] for all x and Lipschitz continuous in x, uniformly in t. That is, II f (x ,t) - f (y ,t) II ~ K II x II - y for all x ,y E JRN and t E [O,T]. K is known as the Lipschitz constant. Then the initial-value problem x(t)=f(x(t),t), x(O)=xo has a unique solution x = x (t ,x 0) over the interval 0 ~ t ~ T. Furthermore, the solution is Lipschitz continuous in xo, uniformly in t and satisfies I x(t ,xo) for all t o E x(t ,Yo) II ~ e Kt II Xo - Yo II [O,T] and Xo,YoE JRN.

2 + ... + kd Ad; k l' k 2, . . 4) 29 3. "1, A2, ... , Ad} are referred to as the fundamental frequencies and form a basis (called the fundamental basis) for A. For each ro E A to correspond uniquely to a sequence of harmonic indices {kj }, the sequence of fundamental frequencies {Aj} must be linearly independent d over the rationals (that is 'LkjAj j=l =0 implies kl = k2 = ... = kd = 0). If A is a module, then AP(A;E) is also denoted QP (Al> A2, ... , Ad; E). Waveforms belonging to such a set are referred to as d -fundamental quasiperiodic or simply dquasiperiodic.