3.7 Switching Circuits 163 Drawing Switching Circuits That Represent Symbolic Statements We will next study how to draw a switching circuit that represents a given symbolic statement. Suppose we are given the statement p q r ( ) ∧ ∨ and are asked to construct a circuit corresponding to it. Remember that ∧ indicates a series branch and ∨ indicates a parallel branch. Working first within parentheses, we see that switches p and q are in series. This series branch is in parallel with switch r, as indicated in Fig. 3.19. Occasionally, it is necessary to have two switches in the same circuit such that when one switch is open, the other switch is closed; and when one switch is closed, the other switch is open. Therefore, the two switches will never both be open together and never be closed together. This situation can be represented by using p for one of the switches and p for the other switch. The switch labeled p corresponds to p∼ in a logic statement. For example, in a series circuit, p p ∧ ∼ would be represented by Fig. 3.20. In this case, the light would never go on; the switches would counteract each other. When switch p is closed, switch p is open; and when p is open, p is closed. p r q Figure 3.19 p p Figure 3.20 Example 3 Representing a Symbolic Statement as a Switching Circuit Draw a switching circuit that represents p q r q s ( ) ( ) . ⎡⎣ ∧ ∼ ∨ ∨ ⎤⎦ ∧ Solution In the statement, p and ∼q have ∧ between them, so switches p and q are in series, as represented in Fig. 3.21. Also in the statement, r and q have ∨ between them, so switches r and q are in parallel, as represented in Fig. 3.22. Now try Exercise 17 p q Figure 3.21 q r Figure 3.22 Because p q ( ) ∧ ∼ and r q ( ) ∨ are connected with ,∨ the two branches are in parallel with each other. The parallel branches are represented in Fig. 3.23. Finally, s in the statement is connected to the rest of the statement with .∧ Therefore, switch s is in series with the entire rest of the circuit, as illustrated in Fig. 3.24. p q r q Figure 3.23 p q s r q Figure 3.24 7 Profile in Mathematics Nikola Tesla Nikola Tesla (1856–1943) was a Serbian-American inventor and engineer who discovered the rotating magnetic field, the basis of most alternating current (AC) machinery. He designed the AC electrical system currently used in our homes and the three-phase system of electrical power transmission. Tesla designed one of the first AC hydroelectric power plants in the United States at Niagara Falls. Tesla was born in the Austrian Empire in modern-day Croatia. He attended the Austrian Polytechnic Institute and the University of Prague, where he initially studied mathematics and physics but soon became fascinated with electricity. After leaving school, Tesla worked with the telephone, the telegraph, and electricity at various locations across Europe. In 1884, he moved to the United States where he worked for Thomas Edison, with whom he eventually became a fierce competitor. He then worked with George Westinghouse, who licensed many of Tesla’s patents. Tesla holds more than 300 patents, including the Tesla Coil, which laid the foundation for wireless technologies that are still in use today. In addition to his vast influence on electricity, Tesla made significant contributions to the discovery of radar, X-ray technology, radio technology, and remote-control devices. Today, Tesla’s name and likeness are still featured throughout pop culture including science fiction literature, music, video games, plays, television shows, and movies. He is the namesake of the carmaker Tesla, and the rock band Tesla. IanDagnall Computing/Alamy Stock Photo
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