By Anthony Croft

Popular electric engineering maths textbook, packed packed with proper smooth purposes and a big variety of examples and exercises.

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Ii) / + 1 ^ 0 and r + 2 < 0. We consider each case in turn. 6 Solution of inequalities Case (i) 35 f + 1 ^ 0 and so f ^ — 1. + 2 > 0 and so r > —2. Hence the inequality is true when t is greater than —2 and less than or equal to — 1. We write this as —2 < r ^ — 1. Case (ii) r + 1 ^ 0 and so r > — 1. r + 2 < 0 and so t < —2. It is impossible to satisfy both r ^ — 1 and t < —2 and so this case yields no values of t. 36 Solve the following inequalities: (a) x2 > 4 Solution 2t + 3 ^ ^ ^ 1 when —2 < / ^ — 1.

S. is 2k. ) has the form (x + k)2, [ coefficient of x \2 that is I x d . The following example illustrates the idea. 22 Solve the following quadratic equations by completing the square: (a) x2 + 8x + 2 = 0 (b) 2x2 - 4x + 1 = 0 Solution (a) By comparing x2 + 8x + 2 with x2 + 2kx + A we see k = A. Thus the squared term must be (x + 4)2. Now (a-+ 4)2 = x2 + 8x + 16 and so x2 + 8x = (x T 4)2 - 16 Therefore x~ -f- 8x + 2 = (x + 4)~ — 16 -(- 2 = (x +4)2 - 14 At this stage we have completed the square.

The depth of penetration of the signal, termed the skin depth, into the conductor depends on the frequency of the signal. 2, is defined as the depth at which the current 9 10 Chapter 1 Review of algebraic techniques density has decayed to 37% of that at the edge. Skin depth is important because it affects the resistance of wires and other conductors: the smaller the skin depth, the higher the resistance and the greater the loss due to heating. At low frequencies, such as those found in the domestic mains supply, the skin depth is so large that often it can be neglected; however, in very large-diameter conductors and smaller conductors at microwave frequencies it becomes important and has to be taken into account.

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