STURM-LIOUVILLE DIFFERENCE EQUATIONS (1.1.7) V(c„Ay„) = anw2yn , n = l , . . . , m - l . 1.2 Network Theory 1 Consider a cascade of LC circuits with inductance an, capacitance —, and loop current un in the successive meshes (see Figure 1.2). «-. • • t, J* V^-d=r y Ce -d=r v ^ V c ^ Vc J= & t • • Figure 1.2: LONetwork 1 . The current in the branch containing — is un — tin+i in the sense of un. cn Assume that un is of the form ynctwtj where yn is a complex constant, and, for the nonhomogeneous problem, assume that a generator in the first mesh supplies a voltage Eeiwt. Then, by KirchofTs rule, it follows that (1.2.1) E + c-iyo(iw)~l + aoyoiw + c0(y0 - yi)(ttn)"1 = 0 , and, for n = 1,..., m — 1, (1.2.2) cn-x(yn - yn-i)(iw)~l + anyniw + cn(yn - yn+i)(iw)~l = 0. Note that y_i = ym = 0, so that multiplication by iw in (1.2.1) and (1.2.2) gives V(c0Ay0) = aow2yo - #

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