By Anders Bostrom
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Additional resources for Review of hypersingular integral equation method for crack scattering and application to modeling of ultrasonic nondestructive evaluation
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87) Write s − jω0 = rejθ , X ∗ KX = keiφ . 89) for −π/2 < θ < π/2. 5). This means that the jω-axis poles of September 9, 2015 32 8:50 Broadband Matching... - 9in x 6in b1862-ch01 BROADBAND MATCHING A(s) are simple, and that K denotes the matrix of the residues of the elements of A(s) evaluated at the pole jω0 . 6). 8). Thus, condition (iii) is satisfied. To show that condition (iv) cannot be violated, we observe that since A(s) is positive-real, X ∗Ah (s)X is nonnegative, and is a continuous function of σ and ω for all s in the open RHS.
Thus, the indefinite-admittance matrix is singular. of Y(s) ˆ Next we show that each row sum of Y(s) is also zero. Without loss of generality, we assume that the network is connected. For, if not, we can apply the following procedure to each of its connected subnetworks, yielding the desired property. To justify the zero-rowsum property, we apply a voltage generator V k at the kth terminal and set all other terminals open. Then, obviously, all the terminal currents are zero except the kth one, and all the terminal voltages are equal to V k .
Referring to Fig. 9, let I(s) and V(s) be the (n + 1)-vectors denoting the terminal curents I k and voltages V k being measured between terminals k and some arbitrary but unspecified reference point. Since the network is linear, terminal currents can be expressed in terms of terminal voltages by ˆ = Y(s) ˆ V(s). 29). We now proceed to discuss several properties ˆ associated with the matrix Y(s). page 16 September 9, 2015 8:50 Broadband Matching... - 9in x 6in FOUNDATIONS OF NETWORK THEORY b1862-ch01 page 17 17 ˆ First, we show that each column sum of Y(s) is equal to zero.
Review of hypersingular integral equation method for crack scattering and application to modeling of ultrasonic nondestructive evaluation by Anders Bostrom
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