Mathematics

Competency 9.0 is the arithmetic every other competency leans on: Ohm's law and the other formulas on ETA's sheet, wavelength and frequency, number systems, Boolean algebra, decibels, graphs, SI prefixes and powers of ten. This page follows ETA's list in order, with section 10 as job practice. It teaches each idea once and links the site's tools for practice: the converters, calculators and trainers do the drilling, so this page does not repeat them.

Ohm's law: solving for circuit values

Competency 9.1 asks you to quote Ohm's law formulas for power, voltage, current and resistance and solve for circuit values. ETA's sheet prints all twelve as a wheel, with E for voltage; the full table, with a worked example for every form, is in section 13 of the Electrical Theory guide, and the Ohm's Law Wheel solves any of them. The method is always the same: find the two quantities you know, pick the form that uses them, and put every value in base units first.

Rearranging formulas step by step is drilled in the Electronics Algebra trainer, and the numeric drills in the Study Hub generate new Ohm's law problems every time.

Other common formulas

Competency 9.2 asks you to list other common basic electronic formulas. ETA lets you use its Common Formulas sheet on the exam, so the skill is knowing which formula fits the question and using it correctly. Beyond Ohm's law, the sheet gives: resistors, inductors and capacitors in series and parallel; RC and RL time constants; resonant frequency; inductive and capacitive reactance; series and parallel impedance; the sine-wave conversions; decibel gain (section 6); inverting and non-inverting op-amp gain; frequency and wavelength (section 3); capacitor charge and energy; inductor voltage and energy; battery internal resistance; the resistor color code; and the SI prefixes (section 8). The Formula Cheat Sheet and the Formula Matrix list them, and the Formula Trainer drills them.

The sheet also prints the PEMDAS rule (parentheses, exponents, multiplication, division, addition, subtraction). In the reciprocal formulas it means: work out each 1/R, add them, and only then take the reciprocal of the sum. For exactly two parts the same formula reduces to product over sum, R1 × R2 / (R1 + R2), which is algebra rather than a separate formula on the sheet.

With R_f = 47 kΩ and R_i = 4.7 kΩ, the inverting amplifier's gain is −(47 / 4.7) = −10 (the minus sign means the output is inverted) and the non-inverting amplifier's is 1 + 10 = 11. The Calculator Suite works each of these formulas.

Wavelength, frequency and power

Competency 9.3 asks you to explain how to calculate wavelength, frequency and power values. NEETS defines the period as the time of one complete cycle, the frequency as the number of cycles in one second, and the wavelength as the distance the wave travels during one period; the period is the reciprocal of the frequency, and the wavelength is the velocity divided by the frequency. For a radio wave in free space the velocity is the speed of light, c. ETA's sheet gives c = f × λ in three working forms:

The sheet writes the metric form as λ = (3 × 10⁸) / f with f in hertz; dividing both by 10⁶ gives the 300 / f_MHz shortcut. The exact value of c is 299 792 458 m/s, fixed by definition (NIST), so ETA's 3 × 10⁸ is rounded up by about 0.07%. In feet per second c is 299792458 / 0.3048 ≈ 983571056 (the foot is exactly 0.3048 m, per NIST), which is where ETA's 984 comes from, rounded from 983.6. The sheet's 1 m = 3.28 ft is rounded the same way (1 / 0.3048 ≈ 3.2808). NEETS's warning applies to every form: keep the units consistent, meters with hertz or megahertz with the matching shortcut, never a mixture.

Power values come from Ohm's law with the impedance the power is delivered into. RF power is commonly stated into 50 Ω (Kuphaldt references RF dBm levels to a 50 Ω load), so the square-root forms give the voltage and current a power produces:

Antenna lengths, velocity factor and the frequency bands are in section 2 of the RF & Communications guide; the RF / Antenna calculator converts frequency and wavelength.

Binary, decimal, octal and hexadecimal

Competency 9.4 asks you to convert binary, decimal, octal and hexadecimal numbers. All four are place-weighted systems (Kuphaldt): each digit is multiplied by the weight of its place, and each place is worth the base times the place to its right. NEETS calls the base the radix: the number of symbols the system uses.

• To decimal: multiply each digit by its place weight and add (Kuphaldt). His examples: octal 245.37 is 165.484375, and hexadecimal 30F.A9 is 783.66015625.
• From decimal: NEETS divides the decimal number by the radix of the target system again and again; the remainders, read from the last to the first, are the digits. Kuphaldt's alternative is trial and fit: set a 1 in the largest place weight that does not overshoot, and repeat with what is left.
• Binary to octal or hex and back: group the bits in threes (octal) or fours (hex) from the binary point outward, padding with zeros, and translate each group to one digit. Kuphaldt: octal and hex exist as shorthand for binary, and hex is the more popular because digital equipment groups bits in multiples of eight, which split evenly into fours but not into threes.

The Number Systems lesson in the Study Hub adds binary addition, BCD and register overflow; the Binary Converter and Digital / Logic convert any value, and section 2 of the Computer Applications guide shows where bits and bytes meet storage.

Boolean algebra

Competency 9.5 asks you to explain Boolean algebra and its use in digital circuitry. Kuphaldt: George Boole gave symbolic form to Aristotle's system of logic, built on two kinds of proposition, true and false, in 1854, and Claude Shannon's 1938 thesis applied it to relay and switching circuits, where every signal is either high (1) or low (0). Every Boolean quantity is a single 1 or 0, and there are three operations:

• Addition is OR (parallel switch contacts): A + B is 1 when either input is 1. Because nothing can exceed 1, 1 + 1 = 1.
• Multiplication is AND (series contacts): A · B, or just AB, is 1 only when both are 1.
• Complementation is NOT (a normally closed contact): the complement of A is its opposite. Books draw it as a bar over the letter; this page writes it A' (read "A-not"), the prime notation Kuphaldt also mentions.

There is no subtraction or division. NEETS gives the purpose: a Boolean expression describes the inputs needed for an output, and its laws let a designer get the same output from the fewest gates, saving space, weight and cost. Kuphaldt adds reliability: fewer components, fewer failures. NEETS's laws:

DeMorgan's theorem in hardware. Kuphaldt: inverting a gate's output gives the same function as the opposite type of gate with inverted inputs. A NAND gate behaves as an OR gate with both inputs inverted, and a NOR gate as an AND gate with both inputs inverted. Every law in the table can be checked with a truth table; each one was checked that way for this page.

The Logic Gates lesson in the Study Hub teaches the seven gates and their truth tables; try the expressions in the Logic Gates lab and Digital / Logic.

Decibels

Competency 9.6 asks you to explain decibels and show the reasons for using them in signal level, power and audio calculations. Kuphaldt: the bel, named for Alexander Graham Bell, was devised to express power loss in telephone lines and is the base-10 logarithm of a power ratio; it proved too large, so the decibel, one-tenth of a bel, is used. ETA's sheet gives both forms:

The decibel is fundamentally a unit of power. Power goes with voltage squared, so for a constant load impedance a voltage gain of 2 is a power gain of 4, and the 20 in the voltage form absorbs that square (Kuphaldt). The voltage form therefore gives the same answer as the power form only when both voltages are measured across equal impedances; ETA's sheet prints it without that condition, but the Decibel tool and the glossary state it, and so does this page. A gain is a positive figure, a loss negative, and no change is 0 dB.

Why decibels are used.

• Range: a logarithm fits a huge range of ratios into small numbers: a millionfold power ratio is just 60 dB.
• Gains and losses add: stage gains multiply as ratios but add in decibels. Kuphaldt's example: stages of ×3 and ×5 give ×15, and 4.77 dB + 6.99 dB = 11.76 dB, which is 10 log 15.
• Audio: Kuphaldt: human hearing is logarithmic; to double the perceived intensity of a sound, the sound power must be multiplied by a factor of ten, which is why the unit suited telephone work.
• Signal level: with a fixed reference, a decibel figure becomes an absolute level. Kuphaldt: 0 dBm is 1 mW in any impedance (telephone work measured it across 600 Ω, where it is 0.775 V, and RF work across 50 Ω, where it is about 0.224 V), and 0 dBW is 1 W.

The Decibel tool converts in both directions and the dB Spectrum Analyzer shows levels on a dB scale; section 4 of the RF & Communications guide applies dB to gain and loss along a signal path.

Reading graphs

Competency 9.7 asks you to explain how graphs are used to demonstrate electronics functions. A graph shows how one quantity responds to another. Read the axes first: what each measures, its unit and prefix, and whether the scale is linear or logarithmic. Then read the point, slope or shape the question asks about.

Linear and logarithmic axes. On a linear axis, equal distances are equal differences: 0, 2, 4, 6 V. On a logarithmic axis, equal distances are equal ratios: 10 Hz, 100 Hz, 1 kHz, 10 kHz, each step a factor of ten (a decade). A value's position within a decade is its logarithm, so the gridlines crowd together toward the top of each decade. The audio range, 20 Hz to 20 kHz, is a ratio of 1000, exactly three decades, which a log axis shows evenly; a linear axis would squeeze everything below 2 kHz into the first tenth of the width. Frequency-response plots use a log frequency axis with gain in decibels, which is already logarithmic.

Characteristic curves. A characteristic curve plots a component's current against its voltage. Kuphaldt: a function that plots as a straight line is linear, and a straight current-versus-voltage line means a stable resistance, which is how a resistor behaves (I = E/R at every point). A lamp's curve rises steeply and then flattens, because its filament's resistance increases as it heats. An air gap or a varistor passes almost no current until a breakdown voltage, then conducts heavily, and a few devices show a falling curve, called negative resistance. With current on the vertical axis, a steeper line means a lower resistance.

The Mathematics lesson in the Study Hub has a Reading graphs section that works the other curves the exam uses: a sine wave against time, an RC charge curve (63% at one time constant), a frequency response with its −3 dB cutoff, a resonance curve and a diode's knee. The bank's graph questions drill them. See live plots in the Scope Lab and the Visual Learning Lab.

SI prefixes and conversions

Competency 9.8 asks you to explain the SI prefixes and conversions. A prefix is a power of ten written as a letter (Kuphaldt): 2.5 GB is 2.5 × 10⁹ bytes. ETA's sheet lists the nine you will use:

NIST recognizes 24 prefixes, from quetta (10³⁰) to quecto (10⁻³⁰); the newest four, ronna, quetta, ronto and quecto, were added in 2022. Converting means moving the decimal point (Kuphaldt): each step between the prefixes above is three places, left toward a larger prefix and right toward a smaller one. NIST also asks for a leading zero before a decimal point (0.47, not .47), so the quantity is read correctly.

• Case matters (NIST SP 811, 6.2.2, and NIST's prefix table): the multiples from mega up (M, G, T and larger) are capitals; kilo (k) and every submultiple (m, µ, n, p) are lower case. 1 mW and 1 MW differ by a factor of 10⁹.
• No space, no stacking (6.2.1, 6.2.4): the prefix joins its unit with no space (kHz, MΩ, µF), and compound prefixes are not allowed: nm, never mµm.
• Prefix and unit are one symbol (6.2.3), so a power applies to both: 1 cm² is (10⁻² m)² = 10⁻⁴ m².
• Schematic shorthand: Kuphaldt notes the style that puts the prefix where the decimal point would be, 4k7 for 4.7 k, because a printed decimal point is easily lost.

Competency 9.8.1 points to NIST Special Publication 811, section 9.3, on spelling unit names with prefixes. When a prefixed unit name is spelled out, there is no space or hyphen between prefix and unit (milligram, kilopascal), and the name is lower case like any noun (section 9.1). Three names drop the prefix's final vowel: megohm (not megaohm), kilohm (not kiloohm) and hectare. In every other name both vowels stay, so 1 GΩ is a gigaohm (SP 811 spells it so in section 6.2.1) and 1 mA a milliampere.

Drill conversions on the Decimal Ladder and the Unit Conversion Hub; component markings are decoded in Component Values.

Powers of ten

Competency 9.9 asks you to define exponents (powers of ten) and their calculations. 10ⁿ means n tens multiplied together, a 1 followed by n zeros; 10⁻ⁿ is its reciprocal, 1 / 10ⁿ; and NEETS reminds you that any number to the power 0 is 1, so 10⁰ = 1. Kuphaldt's rules for calculating:

• Multiply: multiply the leading numbers and add the exponents: (6.24 × 10¹⁸) × (3.6 × 10³) = 22.464 × 10²¹ ≈ 2.25 × 10²². (6.24 × 10¹⁸ is the number of electrons in a coulomb from the exact SI value of e; Kuphaldt's example uses the older rounding, 6.25.)
• Divide: divide the leading numbers and subtract the exponents.
• Scientific notation keeps the leading number at least 1 and less than 10; engineering notation uses only exponents that are multiples of 3, so each one matches a prefix (10⁻⁶ is micro). A calculator's EE key enters the exponent, and its ENG display mode shows engineering notation.
• Add or subtract: only with equal exponents. Rewrite one number first: 3 × 10³ + 2 × 10² = 3 × 10³ + 0.2 × 10³ = 3.2 × 10³.

The Engineering Notation converter and the Scientific Calculator practise both notations; the Mathematics lesson in the Study Hub works a reactance problem with exponents.

On the job: numbers at the radio site

Job practice, not exam content. These are the same calculations as they turn up in the field. The equipment manual and your employer's procedures come first.

1. Turn watts into dBm and dBW. A 100 W transmitter is 10 log (100 / 0.001) = 50 dBm, or 10 log 100 = 20 dBW. Because 1 W is 1000 mW, dBW is always dBm minus 30.
2. Add up losses in decibels, then convert once. Losses in cable, connectors and combiners add in dB. A total of 3 dB between a 50 dBm transmitter and the antenna leaves 47 dBm, about 50.1 W: half the power, gone before the antenna.
3. Read tiny levels in dBm. A received signal of −100 dBm is 10⁻¹⁰ mW, or 10⁻¹³ W. Writing it in dBm keeps such numbers manageable; Kuphaldt notes that signal generators for testing receivers are set in dBm too.
4. Read a hex value. Where a programming tool or service menu shows a value in hex, 3E8 is 3 × 256 + 14 × 16 + 8 = 1000 in decimal, and 0011 1110 1000 in binary, one nibble per digit.
5. Read binary switch settings. If a bank of switches is weighted 1, 2, 4 and 8 (check the manual for the order), switches 1, 3 and 4 on set 1 + 4 + 8 = 13.