Electromagnetic Fields and Energy - Chapter 5 ... - MIT

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Sec. 5.10 Three Solutions 49 The integral can be carried out for any given distribution of potential. In this particular situation, the potential of the surface at y = b is uniform. Thus, integration gives 16v 1 A for m and n both odd mn = mnπ 2 sinh(k mn b) (15) 0 for either m or n even The desired potential, satisfying the boundary conditions on all six surfaces, is given by (10) and (15). Note that the first term in the solution we have found is not the same as the first term in the twodimensional field representation, (5.5.9). No matter what the ratio of a to w, the first term in the threedimensional solution has a sinusoidal dependence on z, while the twodimensional one has no dependence on z. For the capacitive attenuator of Fig. 5.5.5, what output signal is predicted by this three dimensional representation? From (10) and (15), the charge on the output electrode is where a w ∂Φ q = − o 0 0 dxdz ≡ −C M v (16) ∂y y=0 ∞ ∞ 64 C M = oaw k mn π 4 m 2 n 2 sinh(k mnb) m=1 n=1 odd odd With v = V sin ωt, we find that v o = V o cos ωt where V o = RC nωV (17) Using (16), it follows that the amplitude of the output voltage is ∞ ∞ V o ak mn = U 2πm 2 n 2 b sinh m=1 n=1 (kmn a) a odd odd (18) where the voltage is normalized to and U = 128 o wRωV π 3 k mna = (nπ) 2 + (mπ) 2 (a/w) 2 This expression can be used to replace the plot of Fig. 5.5.5. Here we compare the twodimensional and threedimensional predictions of output voltage by considering (18) in the limit where b a. In this limit, the hyperbolic sine is dominated by one of its exponentials, and the first term in the series gives b ln V o → ln 1 + (a/w) U 2 − π 1 + (a/w) 2 (19) a In the limit a/w 1, the dependence on spacing between input and output electrodes expressed by the right hand side becomes identical to that for the twodimensional model, (5.5.15). However, U = (8/π 2 )U regardless of a/w. This threedimensional Cartesian coordinate example illustrates how the orthogonality property of the product solution is exploited to provide a potential
50 Electroquasistatic Fields from the Boundary Value Point of View Chapter 5 Fig. 5.10.3 Twodimensional square wave function used to represent electrode potential for system of Fig. 5.10.2 in plane y = b. that is zero on five of the boundaries while assuming any desired distribution on the sixth boundary. On this sixth surface, the potential takes the form ∞ ∞ Φ = Vmn F mn (20) m=1 n=1 odd odd where mπ nπ F mn ≡ X m Z n ≡ sin x sin z a w The twodimensional functions F mn have been used to represent the “last” boundary condition. This twodimensional Fourier series replaces the onedimensional Fourier series of Sec. 5.5 (5.5.17). In the example, it represents the twodimensional square wave function shown in Fig. 5.10.3. Note that this function goes to zero along x = 0, x = a and z = 0, z = w, as it should. It changes sign as it passes through any one of these “nodal” lines, but the range outside the original rectangle is of no physical interest, and hence the behavior outside that range does not affect the validity of the solution applied to the example. Because the function represented is odd in both x and y, it can be represented by sine functions only. Our foray into threedimensional modal expansions extends the notion of orthogonality of functions with respect to a onedimensional interval to orthogonality of functions with respect to a twodimensional section of a plane. We are able to determine the coefficients V mn in (20) as it is made to fit the potential prescribed on the “sixth” surface because the terms in the series are orthogonal in the sense that a w 0 m = i or n = j F mn F ij dxdz = aw m = i and n = j (21) 0 0 4 In other coordinate systems, a similar orthogonality relation will hold for the product solutions evaluated on one of the surfaces defined by a constant natural coordinate. In general, a weighting function multiplies the eigenfunctions in the integrand of the surface integral that is analogous to (21). Except for some special cases, this is as far as we will go in considering threedimensional product solutions to Laplace’s equation. In the remainder of this section, references to the literature are given for solutions in cylindrical, spherical, and other coordinate systems.
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