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This is not a test! It s a review of important general concepts you learned in the previous nine chapters Read it though slowly and let it sink in If you re confused about anything here, or about anything in the section you ve just finished, go back and study that material some more

These equations are valid at the frequency fo for which the line length measures 1 4 wavelength Sometimes, the word wavelength is replaced by the lowercase Greek letter lambda ( ), so you will occasionally see the length of a quarter-wave section denoted as (1 4) or 025 Neglecting line losses, the preceding relations hold at all odd harmonics of fo, that is, at 3fo, 5fo, 7fo, and so on At other frequencies, a quarter-wave section of line does not act as a transformer Instead, it behaves in a complex manner that is beyond the scope of this discussion Quarter-wave transmission-line transformers are most often used in antenna systems, especially at the higher frequencies, where their dimensions become practical A quarter-wave matching section should be made using unbalanced line if the load is unbalanced, and balanced line if the load is balanced A disadvantage of quarter-wave sections is the fact that they work only at specific frequencies But this is often offset by the ease with which they are constructed, if radio equipment is to be used at only one frequency, or at odd-harmonic frequencies

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Actually the TSP demands a cycle, but one can de ne an alternative version that seeks a path, and it is not hard to see that this is just as hard as the TSP itself

11

.

Question 11-1

Problem 18-5 Suppose an antenna has a purely resistive impedance of 100 It is connected to a 1 4-wave section of 75- coaxial cable What is the impedance at the input end of the section Use the formula from above:

What things can we do with the equation if we want to change its form, but be sure that the result is equivalent to the original equation and still valid

This graph has multiple edges between two vertices a feature we have not been allowing so far in this book, but one that is meaningful for this particular problem, since each bridge must be accounted for separately We are looking for a path that goes through each edge exactly once (the path is allowed to repeat vertices) In other words, we are asking this question: When can a graph be drawn without lifting the pencil from the paper The answer discovered by Euler is simple, elegant, and intuitive: If and only if (a) the graph is connected and (b) every vertex, with the possible exception of two vertices (the start and nal vertices of the walk), has even degree (Exercise 326) This is why K onigsberg s park was impossible to traverse: all four vertices have odd degree To put it in terms of our present concerns, let us de ne a search problem called E ULER PATH: Given a graph, nd a path that contains each edge exactly once It follows from Euler s observation, and a little more thinking, that this search problem can be solved in polynomial time Almost a millennium before Euler s fateful summer in East Prussia, a Kashmiri poet named Rudrata had asked this question: Can one visit all the squares of the chessboard, without repeating any square, in one long walk that ends at the starting square and at each step makes a legal knight move This is again a graph problem: the graph now has 64 vertices, and two squares are joined by an edge if a knight can go from one to the other in a single move (that is, if their coordinates differ by 2 in one dimension and by 1 in the other) See Figure 82 for the portion of the graph corresponding to the upper left corner of the board Can you nd a knight s tour on your chessboard Figure 82 Knight s moves on a corner of a chessboard.

Answer 11-1

R in = Z o2/R out = 752/100 = 5625/100 = 56

We can do any of these things: Switch the left and right sides Add a quantity to the left side, and add the same quantity to the right side Subtract a quantity from the left side, and subtract the same quantity from the right side Multiply the left side by a nonzero quantity, and multiply the right side by the same nonzero quantity Divide the left side by a nonzero quantity, and multiply the right side by the same nonzero quantity

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