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Math for the General Class Ham Radio Operator A Prerequisite Math Refresher For The Math-Phobic Ham Why is This Lesson for You? Math Vocabulary • What are equations and formulas? • What do variables mean? • What is an operator? 2 C 2 =A + 2 B Math Vocabulary What is an operator? • Math operations: – Add: + – Subtract: − – Multiply: X or – Divide: ∕ or – Exponents: YX – Roots: or n ⃰ Math Vocabulary • What does solving an equation mean? • Getting the final answer! Getting the Final Answer: Tricks of the Trade: • Opposite math operations: Addition Subtraction Multiplication Division Roots Exponents • A number divided by the same number is 1, X = 1 X • A number multiplied by 1 is that number, Y * 1=Y If you do something to one side of the equation, do exactly the same thing to the other side of the equation to keep everything equal What does solving an equation mean? Example #1 2 C 2 =A 2 B + 2 2 2 C = A + B 2 2 2 C = A + B 2 2 C = A + B Assume A and B are known Want to solve for C. Apply same operation to both sides Opposite operations cancel each other Voila!!! What does solving an equation mean? Example #2 • The equation for Ohm’s Law is: E=I*R • The variables mean: – E represents voltage – I represents current – R represents resistance • The math operator is multiplication. What does solving an equation mean? Example #2 • E=I*R – Current is 10 (we will disregard units for now) – Resistance is 50 • Therefore: E = 10*50 • E = 500 (in this case volts) Math Vocabulary What does solving an equation mean? • What if we know the voltage and the current and want to find the resistance? E=I*R R=E/I Let’s do some math! R1 R2 R3 RN RT • Simple addition Let’s do some math! R1 R2 RT R1 R2 • Multiply R1 times R2 – Write the number down • Add R1 and R2 • R1 = 50 – Write the number down • R2 = 200 • RT = Total Resistance = ? • Divide the first number by the second to find the answer. Let’s do some math! R1 R2 RT R1 R2 • R1 = 50 • R2 = 200 • RT = ? • R1 * R2 = ? 50 * 200 = 10,000 • R1 + R2 = ? 50 + 200 = 250 • RT = 10,000/250 = 40 Let’s do some math! 1 1 1 1 1 R1 R2 R3 R N • Do each fraction in the denominator in turn 1/Rn RT – Write the number down • Add all fraction results together. – Write the number down • Divide 1 by the sum of the fractions. Let’s do some math! 1 1 1 1 1 R1 R2 R3 R N RT • 1/R1 = ? 1/50 = 0.02 • 1/R2 = ? 1/100 = 0.01 • R1 = 50 • R2 = 100 • R3 = 200 • 1/R3 = ? 1/200 = 0.005 • Sum of fractions = ? 0.02 + 0.01 +0 .005 =0.035 • 1/Sum of fractions = ? RT = 1/0.035 = 28.6 Let’s do some math! 2 E P R • E = 300 • R = 450 • Square the numerator E – Same as E * E – Write the number down • Divide the squared number by R. Let’s do some math! 2 E P R • E = 300 • R = 450 • E2 = ? (square E) 3002 = 90,000 • 90,000/R = ? P = 90000/450 = 200 Let’s do some math! VPeak 1.414VRMS • VPeak = 100 • VRMS = ? • Solve for VRMS VRMS = VPeak / 1.414 • Plug in value for VPeak VRMS = 100/1.414 100/1.414 = 70.7 Let’s do some math! VPeak 1.414VRMS PEP VRMS 2 R • VPeak = 300 • R = 50 • PEP = ? • Sometimes two formulas need to be used to come to a final answer. • Solve equation 1 for VRMS • Plug the value of VRMS into equation 2. Let’s do some math! VPeak 1.414VRMS PEP VRMS 2 R • VPeak = 300 • R = 50 • PEP = ? • Solve for VRMS VRMS = 300 / 1.414 300/1.414 = 212.2 Write the number down • Plug the value into VRMS. VRMS2 = 45,013.6 Write the number down • Divide the square by 50 45,013.6 /50 = 900.3 Let’s do some math! ES N S EP N P • • • • NS = 300 NP = 2100 EP = 115 ES = ? • Solve for ES – Multiply both sides by EP – The EP values on the left cancel ES N S EP EP EP N P • Solution is N S EP ES NP Let’s do some math! N S EP ES NP • • • • NS = 300 NP = 2100 EP = 115 ES = ? • NS * EP = ? 300 * 115 = 34,500 Write the number down • Result / NP = ? ES = 34500/2100 = 16.4 Let’s do some math! ZP NP ZS NS • The right side of this equation is a ratio. • Ratios are numbers representing relative size • A ratio compares two numbers. – Just a fraction with the two numbers being compared making up the fraction. Let’s do some math! ZP NP ZS NS • ZP / ZS = ? 1600/8 = 200 Write the number down • 2001/2 = ? • ZP = 1600 • ZS = 8 • Ratio of NP to NS = ? 2001/2 = 14.1 • Ratio of NP to NS = 14.1 / 1 Ratio is 14.1 to 1 Let’s do some math! L log 10 N N 10 L ←Logarithms – “the log of N is L.” – Or “What power of 10 will give you N?” ←Anti-log: Reverse or opposite of the log. Making Sense of Decibels Ratio of the Power Out to the Power In P2 dB 10 * log 10 P1 Examples of Power Ratios commonly expressed in dB: • Gain of an amplifier stage • Pattern of an antenna • Loss of a transmission line Common Decibel Tables 1dB 3dB 6dB 7dB 9dB 10dB 13dB 17dB 20dB = = = = = = = = = 10 x log101.26 10 x log102 10 x log104 10 x log105 10 x log108 10 x log1010 10 x log1020 10 x log1050 10 x log10100 -1dB -3dB -6dB -7dB -9dB -10dB -13dB -17dB -20dB = = = = = = = = = 10 x log101/1.26 10 x log101/2 10 x log101/4 10 x log101/5 10 x log101/8 10 x log101/10 10 x log101/20 10 x log101/50 10 x log101/100 Let’s do some math! P2 dB 10 log 10 P1 • P2 = 200 • P1 = 50 • dB = ? • Divide P2 by P1. – Write the number down. • Press the log key on your calculator and enter the value of P2/P1. – Write the number down. • Multiply the result by 10. Let’s do some math! P2 dB 10 log 10 P1 • P2/P1 = ? 200/50 = 4 Write the number down. • Log 4 = ? • P2 = 200 • P1 = 50 • dB = ? Log (4) = 0.602 Write the number down. • 0.602 * 10 = ? 0.602 * 10 = 6.02 Thank goodness it’s over!