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Dr. Hugh Blanton ENTC 3331

ENTC 3331 RF Fundamentals. Dr. Hugh Blanton ENTC 3331. Measurement Units. The System of International Units (SI units) was adopted in 1960. The use of older systems still persists, but it is always possible to convert non-standard measurements to SI units.

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Dr. Hugh Blanton ENTC 3331

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  1. ENTC 3331 RF Fundamentals Dr. Hugh Blanton ENTC 3331

  2. Dr. Blanton - ENTC 3331 - Math Review 2

  3. Measurement Units • The System of International Units (SI units) was adopted in 1960. • The use of older systems still persists, but it is always possible to convert non-standard measurements to SI units. Dr. Blanton - ENTC 3331 - Math Review 3

  4. SI (International Standard) Base Units • meter (m) = about a yard • kilogram (kg) = about 2.2 lbs • liter (l) = about a quart • liter (l) = 1000 mL Dr. Blanton - ENTC 3331 - Math Review 4

  5. Fundamental Units Seven Fundamental physical phenomena. Dr. Blanton - ENTC 3331 - Math Review 5

  6. Unit Conversions • When converting physical values between one system of units and another, it is useful to think of the conversion factor as a mathematical equation. • In solving such equations, one must only multiply or divide both sides of the equation by the same factor to keep the equation consistent. Dr. Blanton - ENTC 3331 - Math Review 6

  7. æ æ ö ö 1 1 yd yd = ç ç ÷ ÷ 23 23 23 ft ft ft yd 7.6 3 3 ft ft è è ø ø æ ö 1 yd 23 = ç ÷ 23 ft yd 3 ft 3 è ø Same Quantity Unit Conversions • Example: 23 feet = ? yards Dr. Blanton - ENTC 3331 - Math Review 7

  8. Unit Conversions • Example: 5 yd2 = ? ft2 Dr. Blanton - ENTC 3331 - Math Review 8

  9. Units • Fundamental Units • The SI system recognizes that there are only a few truly fundamental physical properties that need basic (and arbitrary) units of measure, and that all other units can be derived from them. Dr. Blanton - ENTC 3331 - Math Review 9

  10. Derived Units • The funadmental units are used as the basis of numerous derived SI units. • Note that derived SI units are sometimes named after famous physicists. Dr. Blanton - ENTC 3331 - Math Review 10

  11. Derived Units Dr. Blanton - ENTC 3331 - Math Review 11

  12. Unit Multiplication Factors • An additional letter that denotes a multiplying factor may prefix fundamental or derived units. • The more common multiplying factors increase or decrease the unit by powers of ten. Dr. Blanton - ENTC 3331 - Math Review 12

  13. Unit Multiplication Factors • An additional letter that denotes a multiplying factor may prefix fundamental or derived units. • The more common multiplying factors increase or decrease the unit by powers of ten. Dr. Blanton - ENTC 3331 - Math Review 13

  14. Powers of Ten (big) • 101 = 10 • 103 = 1000 (thousand) • 106 = 1,000,000 (million) • 109 = 1,000,000,000 (billion) Dr. Blanton - ENTC 3331 - Math Review 14

  15. Powers of Ten (small) • 100 = 1 • 10-3 = 0.001 (thousandth) • 10-6 = 0.000001 (millionth) • 10-9 = 0.000000001 (billionth) Dr. Blanton - ENTC 3331 - Math Review 15

  16. Scientific Notation • 7,000,000,000 • = 7 billion • = 7  109 • 7,000,000 • = 7 million • = 7  106 Dr. Blanton - ENTC 3331 - Math Review 16

  17. 3 significant digits Scientific Notation • 7,240,000 • = 7.24 million • = 7.24  106 Dr. Blanton - ENTC 3331 - Math Review 17

  18. 6 decimal places Very Large Quantities • 7,240,000 = 7.24  106 Dr. Blanton - ENTC 3331 - Math Review 18

  19. Very Small Quantities • 0.0000123 = 1.23  10-5 5 decimal places Dr. Blanton - ENTC 3331 - Math Review 19

  20. Engineering Notation • Exponents = 3, 6, 9, 12, . . . • Instead of5.32  107 • we write • 53.2  106 Decimal part got bigger Exponent got smaller Dr. Blanton - ENTC 3331 - Math Review 20

  21. Adding and Subtracting • Exponents must be the same! • (1.2  106) + (2.3  105) • change to • (1.2  106) + (0.23  106) • = 1.43  106 Dr. Blanton - ENTC 3331 - Math Review 21

  22. Multiplying • Exponents Add • (3.1  106)(2.0  102) • = 6.2  108 Dr. Blanton - ENTC 3331 - Math Review 22

  23. Dividing • Exponents Subtract • (3.8  106) • (2.0  102) • = 1.9  104 6 - 2 = 4 Dr. Blanton - ENTC 3331 - Math Review 23

  24. Adding Fractions • You can only add like to like • Same Denominators Dr. Blanton - ENTC 3331 - Math Review 24

  25. Different Denominators • Make them the same • find a common denominator • The product of all denominators is always a common denominator • But not always the least common denominator Dr. Blanton - ENTC 3331 - Math Review 25

  26. Finding the LCD • Example: Dr. Blanton - ENTC 3331 - Math Review 26

  27. Factor the Denominators Dr. Blanton - ENTC 3331 - Math Review 27

  28. Assemble LCD Dr. Blanton - ENTC 3331 - Math Review 28

  29. ×5 ×4 ×5 ×4 Build up Denominators to LCD Dr. Blanton - ENTC 3331 - Math Review 29

  30. Add Numerators And Reduce if Needed Dr. Blanton - ENTC 3331 - Math Review 30

  31. Rational Expressions • Example: Dr. Blanton - ENTC 3331 - Math Review 31

  32. Factor the Denominators Dr. Blanton - ENTC 3331 - Math Review 32

  33. DENOMINATORS Assemble LCD Dr. Blanton - ENTC 3331 - Math Review 33

  34. ( x 1 ) ( ( ( x x x 1 1 1 ) ) ) + - - + ( ) FACTORED LCD = ( x + 1 )( x - 1 )( x - 1 ) Build up Fractions to LCD Dr. Blanton - ENTC 3331 - Math Review 34

  35. Add Numerators Dr. Blanton - ENTC 3331 - Math Review 35

  36. Simplify Numerator Dr. Blanton - ENTC 3331 - Math Review 36

  37. Radical Index n x Radicand Radicals Dr. Blanton - ENTC 3331 - Math Review 37

  38. Meaning Dr. Blanton - ENTC 3331 - Math Review 38

  39. Example Dr. Blanton - ENTC 3331 - Math Review 39

  40. An Ambiguity • but it’s also true that. . . Dr. Blanton - ENTC 3331 - Math Review 40

  41. So why not say It’s also true that • ? Dr. Blanton - ENTC 3331 - Math Review 41

  42. Two Answers? • Roots with an even index always have both a positive and a negative root • Because squaring either a negative or a positive gives the same result Dr. Blanton - ENTC 3331 - Math Review 42

  43. Principal Root • To avoid confusion we define the principal root to be the positive root, so: Dr. Blanton - ENTC 3331 - Math Review 43

  44. The Negative Root • If we want the negative root we use a minus sign: Dr. Blanton - ENTC 3331 - Math Review 44

  45. Negative Radicands • Do Not Confuse • !!! • With • Does not exist Dr. Blanton - ENTC 3331 - Math Review 45

  46. Negative Radicands • You cannot take an even root of a negative number • Because you cannot square any number and get a negative result Dr. Blanton - ENTC 3331 - Math Review 46

  47. Odd Roots of Negative Radicands • You can take odd roots of negative numbers: Dr. Blanton - ENTC 3331 - Math Review 47

  48. for all non-negative x Some Square Root Identities • for all non-negative x • for all x Dr. Blanton - ENTC 3331 - Math Review 48

  49. for example, you cannot say A Common Error • What is the correct result? Dr. Blanton - ENTC 3331 - Math Review 49

  50. First Evaluate Inside Dr. Blanton - ENTC 3331 - Math Review 50

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