# The W5JCK Guide to the Mathematic Equations Required for the Amateur Extra Class Exam

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1 The W5JCK Guide to the Mathematic Equations Required for the Amateur Extra Class Exam This document contains every question from the Extra Class (Element 4) Question Pool* that requires one or more mathematical formulas to answer. Each question illustrates every formula necessary to answer the question in an easy to understand, step-by-step process. *Question Pool Element 4 - Extra Class, as released by the Question Pool Committee of the National Conference of Volunteer Examiner Coordinators November 30, Question Pool Element 4 was updated effective December 15, 2006 to reflect the FCC Report and Order (R&O) on docket The current question pool is valid until June 30, I designed this document as an aid for reviewing the mathematics necessary to solve the exam questions. It does not cover the mathematical or electrical theories behind the equations. To study the theory, I highly recommend The ARRL Extra Class License Manual and Gordon West s Extra Class FCC License Preparation for Element 4. Both of these publications were of great help to me in preparing for my Amateur Extra Class exam. This document is copyright 2007 by Ceburn Jack Swinden, W5JCK, and may be freely distributed. Good luck on your Amateur Extra exam. 73 de W5JCK CONTENTS Section E4: Amateur Radio Practices... 2 Section E5: Electrical Principles... 4 Section E6: Circuit Components Section E8: Signals and Emissions Section E9: Antennas... 25

3 E4B10: If a 100 Hz signal is fed to the horizontal input of an oscilloscope and a 150 Hz signal is fed to the vertical input, what type of Lissajous figure will be displayed on the screen? E4C08: What is the blocking dynamic range for a receiver that has an 8-dB noise figure and an IF bandwidth of 500 Hz when the blocking level (1-dB compression point) is -20 dbm? Note that the use of vertical bars (for example, ) denotes absolute value. E4C14: If a receiver tuned to MHz receives an intermodulation-product signal whenever a nearby transmitter transmits on MHz, what are the two most likely frequencies for the other interfering signal? E4C26: In a receiver, if the third-order intermodulation products have a power of-70 dbm when using two test tones at -30 dbm, what is the third-order intercept point? E4C27: In a receiver, if the second-order intermodulation products have a power of-70 dbm when using two test tones at -30 dbm, what is the second-order intercept point? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 3 of 26

4 Section E5: Electrical Principles E5A12: What is the half-power bandwidth of a parallel resonant circuit that has a resonant frequency of 1.8 MHz and a Q of 95? E5A13: What is the half-power bandwidth of a parallel resonant circuit that has a resonant frequency of 7.1 MHz and a Q of 150? E5A14: What is the half-power bandwidth of a parallel resonant circuit that has a resonant frequency of MHz and a Q of 150? E5A15: What is the half-power bandwidth of a parallel resonant circuit that has a resonant frequency of MHz and a Q of 95? E5A16: What is the half-power bandwidth of a parallel resonant circuit that has a resonant frequency of 3.7 MHz and a Q of 118? E5A17: What is the half-power bandwidth of a parallel resonant circuit that has a resonant frequency of MHz and a Q of 187? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 4 of 26

5 E5A18: What is the resonant frequency of a series RLC circuit if R is 47 ohms, L is 50 microhenrys and C is 40 picofarads? E5A19: What is the resonant frequency of a series RLC circuit if R is 47 ohms, L is 40 microhenrys and C is 200 picofarads? E5A20: What is the resonant frequency of a series RLC circuit if R is 47 ohms, L is 50 microhenrys and C is 10 picofarads? E5A21: What is the resonant frequency of a series RLC circuit if R is 47 ohms, L is 25 microhenrys and C is 10 picofarads? E5A22: What is the resonant frequency of a series RLC circuit if R is 47 ohms, L is 3 microhenrys and C is 40 picofarads? E5A23: What is the resonant frequency of a series RLC circuit if R is 47 ohms, L is 4 microhenrys and C is 20 picofarads? E5A24: What is the resonant frequency of a series RLC circuit if R is 47 ohms, L is 8 microhenrys and C is 7 picofarads? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 5 of 26

6 E5A25: What is the resonant frequency of a series RLC circuit if R is 47 ohms, L is 3 microhenrys and C is 15 picofarads? E5B06: What is the time constant of a circuit having two 100-microfarad capacitors and two 470-kilohm resistors all in series? E5BO7: What is the time constant of a circuit having two 220-microfarad capacitors and two 1-megohm resistors all in parallel? E5B08: What is the time constant of a circuit having a 220-microfarad capacitor in series with a 470-kilohm resistor? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 6 of 26

7 E5B09: How long does it take for an initial charge of 20 V DC to decrease to 7.36 V DC in a 0.01-microfarad capacitor when a 2-megohm resistor is connected across it? E5B10: How long does it take for an initial charge of 20 V DC to decrease to 0.37 V DC in a 0.01-microfarad capacitor when a 2-megohm resistor is connected across it? E5B11: How long does it take for an initial charge of 800 V DC to decrease to 294 V DC in a 450-microfarad capacitor when a 1-megohm resistor is connected across it? E5C10: In rectangular coordinates, what is the impedance of a network comprised of a 10-microhenry inductor in series with a 40-ohm resistor at 500 MHz? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 7 of 26

8 E5C11: In polar coordinates, what is the impedance of a network comprised of a 100- picofarad capacitor in parallel with a 4,000-ohm resistor at 500 khz? E5D01: What is the phase angle between the voltage across and the current through a series R-L-C circuit if XC is 25 ohms, R is 100 ohms, and XL is 100 ohms? E5D02: What is the phase angle between the voltage across and the current through a series R-L-C circuit if XC is 500 ohms, R is 1 kilohm, and XL is 250 ohms? E5D03: What is the phase angle between the voltage across and the current through a series R-L-C circuit if XC is 50 ohms, R is 100 ohms, and XL is 25 ohms? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 8 of 26

9 E5D04: What is the phase angle between the voltage across and the current through a series R-L-C circuit if XC is 100 ohms, R is 100 ohms, and XL is 75 ohms? E5D05: What is the phase angle between the voltage across and the current through a series R-L-C circuit if XC is 50 ohms, R is 100 ohms, and XL is 75 ohms? E5D08: What is the phase angle between the voltage across and the current through a series RLC circuit if XC is 25 ohms, R is 100 ohms, and XL is 50 ohms? E5D09: What is the phase angle between the voltage across and the current through a series RLC circuit if XC is 75 ohms, R is 100 ohms, and XL is 100 ohms? E5D10: What is the phase angle between the voltage across and the current through a series RLC circuit if XC is 75 ohms, R is 100 ohms, and XL is 50 ohms? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 9 of 26

10 E5D11: What is the phase angle between the voltage across and the current through a series RLC circuit if XC is 250 ohms, R is 1 kilohm, and XL is 500 ohms? E5E01: In polar coordinates, what is the impedance of a network comprised of a 100- ohm- reactance inductor in series with a 100-ohm resistor? E5E02: In polar coordinates, what is the impedance of a network comprised of a 100- ohm- reactance inductor, a 100-ohm-reactance capacitor, and a 100-ohm resistor all connected in series? E5E03: In polar coordinates, what is the impedance of a network comprised of a 300- ohm- reactance capacitor, a 600-ohm-reactance inductor, and a 400-ohm resistor, all connected in series? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 10 of 26

11 E5E04: In polar coordinates, what is the impedance of a network comprised of a 400- ohm- reactance capacitor in series with a 300-ohm resistor? E5E05: In polar coordinates, what is the impedance of a network comprised of a 400- ohm- reactance inductor in parallel with a 300-ohm resistor? E5E06: In polar coordinates, what is the impedance of a network comprised of a 100- ohm- reactance capacitor in series with a 100-ohm resistor? E5E07: In polar coordinates, what is the impedance of a network comprised of a 100- ohm- reactance capacitor in parallel with a 100-ohm resistor? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 11 of 26

12 E5E08: In polar coordinates, what is the impedance of a network comprised of a 300- ohm- reactance inductor in series with a 400-ohm resistor? E5E16: In polar coordinates, what is the impedance of a circuit of 100 -j100 ohms impedance? E5E17: In polar coordinates, what is the impedance of a circuit that has an admittance of 7.09 millisiemens at 45 degrees? Note that. E5E18: In rectangular coordinates, what is the impedance of a circuit that has an admittance of 5 millisiemens at -30 degrees? Note that. Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 12 of 26

13 E5E19: In rectangular coordinates, what is the admittance of a circuit that has an impedance of 240 ohms at 36.9 degrees? E5E20: In polar coordinates, what is the impedance of a series circuit consisting of a resistance of 4 ohms, an inductive reactance of 4 ohms, and a capacitive reactance of 1 ohm? Figure E5-2 for E5E21, E5E22, and E5E23 Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 13 of 26

14 E5E21: Which point on Figure E5-2 best represents the impedance of a series circuit consisting of a 400 ohm resistor and a 38 picofarad capacitor at 14 MHz? E5E22: Which point in Figure E5-2 best represents the impedance of a series circuit consisting of a 300 ohm resistor and an 18 microhenry inductor at MHz? E5E23: Which point on Figure E5-2 best represents the impedance of a series circuit consisting of a 300 ohm resistor and a 19 picofarad capacitor at MHz? E5G01: What is the Q of a parallel R-L-C circuit if the resonant frequency is MHz, L is 2.7 microhenrys and R is 18 kilohms? E5G02: What is the Q of a parallel R-L-C circuit if the resonant frequency is MHz, L is 47 microhenrys and R is 180 ohms? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 14 of 26

15 E5G03: What is the Q of a parallel R-L-C circuit if the resonant frequency is MHz, L is 8.2 microhenrys and R is 1 kilohm? E5G04: What is the Q of a parallel R-L-C circuit if the resonant frequency is MHz, L is 12.6 microhenrys and R is 22 kilohms? E5G05: What is the Q of a parallel R-L-C circuit if the resonant frequency is MHz, L is 42 microhenrys and R is 220 ohms? E5G10: What is the power factor of an R-L circuit having a 60 degree phase angle between the voltage and the current? E5G11: How many watts are consumed in a circuit having a power factor of 0.2 if the input is 100-V AC at 4 amperes? E5G13: What is the Q of a parallel RLC circuit if the resonant frequency is MHz, L is 4.7 microhenrys and R is 18 kilohms? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 15 of 26

16 E5G14: What is the Q of a parallel RLC circuit if the resonant frequency is MHz, L is 3.5 microhenrys and R is 10 kilohms? E5G15: What is the Q of a parallel RLC circuit if the resonant frequency is MHz, L is 10.1 microhenrys and R is 100 ohms? E5G16: What is the Q of a parallel RLC circuit if the resonant frequency is MHz, L is 3 microhenrys and R is 2.2 kilohms? E5H01: What is the effective radiated power of a repeater station with 50 watts transmitter power output, 4-dB feed line loss, 2-dB duplexer loss, 1-dB circulator loss and 6-dBd antenna gain? E5H02: What is the effective radiated power of a repeater station with 50 watts transmitter power output, 5-dB feed line loss, 3-dB duplexer loss, 1-dB circulator loss and 7-dBd antenna gain? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 16 of 26

17 E5H03: What is the effective radiated power of a station with 75 watts transmitter power output, 4-dB feed line loss and 10-dBd antenna gain? E5H04: What is the effective radiated power of a repeater station with 75 watts transmitter power output, 5-dB feed line loss, 3-dB duplexer loss, 1-dB circulator loss and 6-dBd antenna gain? E5H05: What is the effective radiated power of a station with 100 watts transmitter power output, 1-dB feed line loss and 6-dBd antenna gain? E5H06: What is the effective radiated power of a repeater station with 100 watts transmitter power output, 5-dB feed line loss, 3-dB duplexer loss, 1-dB circulator loss and 10-dBd antenna gain? E5H07: What is the effective radiated power of a repeater station with 120 watts transmitter power output, 5-dB feed line loss, 3-dB duplexer loss, 1-dB circulator loss and 6-dBd antenna gain? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 17 of 26

18 E5H08: What is the effective radiated power of a repeater station with 150 watts transmitter power output, 2-dB feed line loss, 2.2-dB duplexer loss and 7-dBd antenna gain? E5H09: What is the effective radiated power of a repeater station with 200 watts transmitter power output, 4-dB feed line loss, 3.2-dB duplexer loss, 0.8-dB circulator loss and 10-dBd antenna gain? E5H10: What is the effective radiated power of a repeater station with 200 watts transmitter power output, 2-dB feed line loss, 2.8-dB duplexer loss, 1.2-dB circulator loss and 7-dBd antenna gain? E5H13: What is the power factor of an RL circuit having a 45 degree phase angle between the voltage and the current? E5H14: What is the power factor of an RL circuit having a 30 degree phase angle between the voltage and the current? E5H15: How many watts are consumed in a circuit having a power factor of 0.6 if the input is 200V AC at 5 amperes? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 18 of 26

19 E5H16: How many watts are consumed in a circuit having a power factor of 0.71 if the apparent power is 500 watts? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 19 of 26

20 Section E6: Circuit Components E6B11: What voltage gain can be expected from the circuit in Figure E6-4 when R1 is 10 ohms and RF is 470 ohms? E6B14: What will be the voltage of the circuit shown in Figure E6-4 if R1 is 1000 ohms and RF is 10,000 ohms and 0.23 volts is applied to the input? E6B15: What voltage gain can be expected from the circuit in Figure E6-4 when R1 is 1800 ohms and RF is 68 kilohms? E6B16: What voltage gain can be expected from the circuit in Figure E6-4 when R1 is 3300 ohms and RF is 47 kilohms? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 20 of 26

21 E6D13: How many turns will be required to produce a 1-mH inductor using a ferrite toroidal core that has an inductance index (A L) value of 523 millihenrys/1000 turns? E6D14: How many turns will be required to produce a 5-microhenry inductor using a powdered-iron toroidal core that has an inductance index (A L) value of 40 microhenrys/100 turns? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 21 of 26

22 Section E8: Signals and Emissions E8A08: What is the peak voltage at a common household electrical outlet? E8A09: What is the peak-to-peak voltage at a common household electrical outlet? E8A11: What is the RMS value of a 340-volt peak-to-peak pure sine wave? E8A16: What is the approximate DC input power to a Class B RF power amplifier stage in an FM-phone transmitter when the PEP output power is 1500 watts? E8A17: What is the approximate DC input power to a Class AB RF power amplifier stage in an unmodulated carrier transmitter when the PEP output power is 500 watts? E8B11: In an FM-phone signal having a maximum frequency deviation of 3000 Hz either side of the carrier frequency, what is the modulation index when the modulating frequency is 1000 Hz? E8B12: What is the modulation index of an FM-phone transmitter producing a maximum carrier deviation of 6 khz when modulated with a 2-kHz modulating frequency? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 22 of 26

23 E8B13: What is the deviation ratio of an FM-phone signal having a maximum frequency swing of plus or minus 5 khz and accepting a maximum modulation rate of 3 khz? E8B24: What is the deviation ratio of an FM-phone signal having a maximum frequency swing of plus or minus 7.5 khz and accepting a maximum modulation rate of 3.5 khz? E8C07: What is the necessary bandwidth of a 13-WPM international Morse code emission A1A transmission? E8C08: What is the necessary bandwidth for a 170-hertz shift, 300-baud ASCII emission J2D transmission? E8C09: What is the necessary bandwidth of a 1000-Hz shift, 1200-baud ASCII emission F1D transmission? E8C10: What is the necessary bandwidth of a 4800-Hz frequency shift, 9600-baud ASCII emission F1D transmission? E8D05: What is the PEP output of a transmitter that has a maximum peak of 30 volts to a 50-ohm load as observed on an oscilloscope? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 23 of 26

24 E8D06: If an RMS reading AC voltmeter reads 65 volts on a sinusoidal waveform, what is the peak-to-peak voltage? E8D16: What is the average power dissipated by a 50-ohm resistive load during one complete RF cycle having a peak voltage of 35 volts? E8D17: If an RMS reading voltmeter reads 34 volts on a sinusoidal waveform, what is the peak voltage? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 24 of 26

25 Section E9: Antennas E9A17: How much gain does an antenna have over a 1/2-wavelength dipole when it has 6 db gain over an isotropic radiator? E9A18: How much gain does an antenna have over a 1/2-wavelength dipole when it has 12 db gain over an isotropic radiator? E9D03: What is the approximate beamwidth of a symmetrical pattern antenna with a gain of 20 db as compared to an isotropic radiator? E9D19: What is the beamwidth of a symmetrical pattern antenna with a gain of 30 db as compared to an isotropic radiator? E9D20: What is the beamwidth of a symmetrical pattern antenna with a gain of 15 db as compared to an isotropic radiator? Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 25 of 26

26 E9D21: What is the beamwidth of a symmetrical pattern antenna with a gain of 12 db as compared to an isotropic radiator? E9E04: What should be the approximate capacitance of the resonating capacitor in a gamma matching circuit on a Yagi beam antenna for the 20-meter band? E9E05: What should be the approximate capacitance of the resonating capacitor in a gamma matching circuit on a Yagi beam antenna for the 10-meter band? E9E10: What would be the physical length of a typical coaxial transmission line that is electrically one-quarter wavelength long at 14.1 MHz? (Assume a velocity factor of 0.66.) E9E11: What is the physical length of a parallel conductor feed line that is electrically one-half wavelength long at MHz? (Assume a velocity factor of 0.95.) E9E16: What would be the physical length of a typical coaxial transmission line that is electrically one-quarter wavelength long at 7.2 MHz? (Assume a velocity factor of 0.66.) Courtesy W5JCK, Copyright 2007 by Ceburn Jack Swinden Page 26 of 26

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