Envelope equations for transients in linear chains of resonators

Abstract

Transient signals in strings of resonators consist of regimes with different time constants: high frequency oscillations, beat signals and exponentials. If one is interested only in the signals envelope one can transform the system of second order differential equations into a system of first order differential equations. The later carries fast varying terms, which are averaged out, and slowly varying terms. The resulting equations are well behaving and can be integrated numerically. Results are shown for the filling process under beam loading of the superconducting nine-cell TESLA cavity.Let us assume a single resonator which is driven by a generator via some coupling device, Fig. 1 i*=i:sin w, t m Figure 1: transformed current i* and internal impedance R*. Single resonator driven by a generator with I IntroductionTransients in strings of resonators are usually calculated by means of a Laplace transform in matrix notation or by a discrete Laplace transform (see for instance [l]). Both approaches become quite awkward if the string is not h e mogenous and/or has branches. Also, one is often not interested in the full time response but only the signals envelope. Then, it may be convenient to take advantage of the fact that the system consists of three regimes with normally very different time constants: First, the high frequency oscillations with the time constant TRF of one period. Second, beating signals with time constants T R F / k where k is the coupling between resonators. Third, signals which are related to the filling time Q/wRF. In the following it is shown how to transform the system of second order differential equations (DE) describing the individual resonators into a system of first order DE's of twice the size. The system is written in a way that the fast varying terms can be averaged out and only slowly varying terms remain. The left over system of DE'S is integrated numerically yielding the signal envelopes.The method is applied to the filling process of the superconducting TESLA cavity consisting of nine resonators.Due to the high Q of the cavity the filling time is of the order of one ms whereas the R F period is less than one ns. The coupling between cells is in the percent region. Thus, the time constants are well separated and the proposed method is ideally suited. i* and R* are the generators current and impedance transformed by the coupling device. The loop equation of the circuit can be written as + 2 4 + wGq = fo sinwot (1) W O L f -&=e* q = J i d t , W O = & , & L = -, 0 -L * o .For q we try an ansatz called variation of constants(2) (l), (2) are two equations for three unknown functions q, a , b. Hence, we can impose a third condition which we choose as U coswot + b sinwot = 0.(3)Differentiation of (2) while considering (3) and substituting into (1) gives Now, multiplyjng (4) with sinwot and (3) with coswot we can eliminate b through substraction. In a similar way we eliminate a and obtain a system of first order DE's a+*.+&= 0-7803-1203-1/93$03.00 0 1993 IEEE 901

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