5.1 Mathematical Foundations & Transposition for Electricians

5.1 Mathematical Foundations & Transposition for Electricians

Mathematics is the foundational language of electrotechnology. Every task an electrician performs—from sizing circuit protective conductors and verifying maximum allowable voltage drops under BS 7671, to determining fault currents and evaluating motor loads—depends directly on sound mathematical principles. Mastering basic arithmetic operations, algebraic transposition, and scientific notation ensures calculations are accurate, compliant, and safe.

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Core Arithmetic for Electrotechnology

Order of Operations (BODMAS / BIDMAS)

When an electrotechnical equation contains multiple mathematical operations, they must be evaluated in an established mathematical hierarchy. The acronym BODMAS (or BIDMAS) defines this sequence:

1. B — Brackets: Evaluate all expressions contained within brackets $(~)$ first, working from innermost to outermost.
2. O / I — Orders / Indices: Evaluate powers, exponents, squares ($x^2$), and roots ($\sqrt{x}$).
3. D / M — Division and Multiplication: Carry out division and multiplication from left to right. These two operations hold equal priority.
4. A / S — Addition and Subtraction: Carry out addition and subtraction from left to right. These two operations also hold equal priority.

Fractions, Decimals, Percentages, and Ratios

• Fractions to Decimals: Divide the numerator by the denominator. For example, a $\frac{3}{8}\text{-inch}$ conduit entry converts to $3 \div 8 = 0.375\text{ inches}$.
• Percentages: Express parts per hundred. In electrical installation, percentage calculations are critical for voltage drop verification. For example, BS 7671 limits voltage drop on low-voltage lighting installations to $3\%$ of the nominal $230\text{ V}$ supply: $V_{\text{drop(max)}} = 230\text{ V} \times 0.03 = 6.9\text{ V}$ For power circuits, the $5\%$ limit equates to: $V_{\text{drop(max)}} = 230\text{ V} \times 0.05 = 11.5\text{ V}$
• Ratios: Used extensively in transformer turns, voltage scaling, and current sharing. If a single-phase transformer has a primary-to-secondary turns ratio of $10:1$ and the primary voltage is $230\text{ V}$, the secondary voltage is: $V_s = \frac{230\text{ V}}{10} = 23\text{ V}$

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The Golden Rules of Algebraic Transposition

Algebra involves finding the value of an unknown variable by rearranging an equation. The process of rearranging a mathematical formula so that a chosen variable becomes the subject (isolated on the left-hand side) is known as transposition.

The Fundamental Principle of Balance

An algebraic equation is like a balanced pair of scales. The equals sign ($=$) declares that the mathematical value of the left side is identical to that of the right side.

• The Golden Rule: Whatever operation you perform on one side of an equation, you must perform identically on the other side.
• Inverse Operations: To eliminate an unwanted term on one side of the subject, apply its exact mathematical opposite:
  • To cancel addition ($+$), apply subtraction ($-$).
  • To cancel subtraction ($-$), apply addition ($+$).
  • To cancel multiplication ($\times$), apply division ($\div$).
  • To cancel division ($\div$), apply multiplication ($\times$).
  • To cancel a square ($x^2$), apply a square root ($\sqrt{x}$).
  • To cancel a square root ($\sqrt{x}$), apply a square ($x^2$).

               ┌──────────────────────────────────────────────┐
               │    THE GOLDEN RULES OF TRANSPOSITION         │
               └──────────────────────┬───────────────────────┘
                                      │
           ┌──────────────────────────┴──────────────────────────┐
           ▼                                                     ▼
┌─────────────────────────────┐               ┌─────────────────────────────┐
│      RULE OF BALANCE        │               │   RULE OF INVERSE ACTION    │
│ Apply the exact same        │               │ Undo operations with their  │
│ mathematical operation to   │               │ mathematical opposite:      │
│ both sides of the '=' sign. │               │ • (+) cancels (-)           │
│ Keeps the equation true.    │               │ • (×) cancels (÷)           │
└─────────────────────────────┘               │ • (√) cancels (x²)          │
                                              └─────────────────────────────┘

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Transposing Core Electrotechnical Formulas

1. Ohm's Law ($V = I \times R$)

• Subject $I$ (Current): Divide both sides by $R$: $\frac{V}{R} = \frac{I \times R}{R} \implies I = \frac{V}{R}$
• Subject $R$ (Resistance): Divide both sides by $I$: $\frac{V}{I} = \frac{I \times R}{I} \implies R = \frac{V}{I}$

2. Electrical Power Formula ($P = V \times I$)

• Subject $V$ (Voltage): Divide both sides by $I$: $V = \frac{P}{I}$
• Subject $I$ (Current): Divide both sides by $V$: $I = \frac{P}{V}$

3. Power Dissipation with Resistance ($P = I^2 R$)

This formula introduces an index (power of 2), requiring a two-step transposition:

• Subject $R$: Divide both sides by $I^2$: $R = \frac{P}{I^2}$
• Subject $I$: First isolate $I^2$ by dividing by $R$, then take the square root of both sides: $I^2 = \frac{P}{R} \implies I = \sqrt{\frac{P}{R}}$

Similarly, for $P = \frac{V^2}{R}$:

• Multiply both sides by $R$: $V^2 = P \times R$
• Take the square root of both sides: $V = \sqrt{P \times R}$
• Transposing for $R$: multiply by $R$ and divide by $P$: $R = \frac{V^2}{P}$

4. Conductor Resistance and Resistivity ($R = \frac{\rho \times L}{A}$)

Where $R$ is resistance in ohms ($\Omega$), $\rho$ is resistivity in ohm-meters ($\Omega\cdot\text{m}$), $L$ is length in meters ($\text{m}$), and $A$ is cross-sectional area in square meters ($\text{m}^2$).

• Subject $L$ (Length):
  1. Multiply both sides by $A$: $R \times A = \rho \times L$
  2. Divide both sides by $\rho$: $L = \frac{R \times A}{\rho}$
• Subject $A$ (Cross-Sectional Area):
  1. Multiply both sides by $A$: $A \times R = \rho \times L$
  2. Divide both sides by $R$: $A = \frac{\rho \times L}{R}$
• Subject $\rho$ (Resistivity):
  1. Multiply both sides by $A$: $\rho \times L = R \times A$
  2. Divide both sides by $L$: $\rho = \frac{R \times A}{L}$

5. AC Reactance Formulas

• Inductive Reactance ($X_L = 2\pi f L$):
  • To solve for Inductance ($L$): $L = \frac{X_L}{2\pi f}$
  • To solve for Supply Frequency ($f$): $f = \frac{X_L}{2\pi L}$
• Capacitive Reactance ($X_C = \frac{1}{2\pi f C}$):
  • To solve for Capacitance ($C$):
    1. Multiply both sides by $C$: $C \times X_C = \frac{1}{2\pi f}$
    2. Divide both sides by $X_C$: $C = \frac{1}{2\pi f X_C}$
  • To solve for Supply Frequency ($f$): $f = \frac{1}{2\pi C X_C}$

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Scientific Notation and Engineering Notation

Electrical quantities range from minute currents (such as $0.000005\text{ A}$ leakage current) to massive power ratings (such as $25,000,000\text{ W}$ in power stations). Writing these values with long strings of zeros invites errors.

Standard Form (Scientific Notation)

In standard form, any positive number is written as: $A \times 10^n$ Where $1 \le A < 10$ and $n$ is an integer (positive or negative).

• Example: $45,000\text{ W} = 4.5 \times 10^4\text{ W}$
• Example: $0.00018\text{ A} = 1.8 \times 10^{-4}\text{ A}$

Engineering Notation

Engineering notation modifies scientific notation so that the exponent $n$ is always an integer multiple of 3 (e.g., $10^3, 10^6, 10^{-3}, 10^{-6}, 10^{-9}$). This matches standard SI metric prefixes directly:

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Triangles, Trigonometry and Statistics

Unit 202 LO1 also lists triangles and trigonometry and statistics. Both appear as short, practical questions.

Pythagoras' Theorem

In a right-angled triangle the longest side (the hypotenuse, $c$) is related to the other two sides by: $c^2 = a^2 + b^2 \implies c = \sqrt{a^2 + b^2}$ Example: a cable tray rises 3 m vertically over a horizontal distance of 4 m. The length of tray needed is $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$ m.

The same right-angled triangle appears in a.c. theory, where impedance is the hypotenuse of resistance and reactance ($Z = \sqrt{R^2 + X^2}$), and in the power triangle.

Sine, Cosine and Tangent (SOH CAH TOA)

For an angle $\theta$ in a right-angled triangle: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Check that your calculator is in degree (D) mode, not radians, before using these keys.

Statistics

Statistics questions usually ask you to summarise a set of readings, such as a batch of test results or monthly energy use.

Data is often presented as bar charts, pie charts, line graphs or tally charts. In a pie chart each sector angle is the item's share of 360°: a circuit using 25% of a building's energy is drawn as a $0.25 \times 360^\circ = 90^\circ$ sector.

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Common Algebraic Mistakes in Electrical Exams

1. Denominator Isolation Errors: When transposing $X_C = \frac{1}{2\pi f C}$ to solve for $C$, students often mistakenly write $C = 2\pi f X_C$. The product $2\pi f$ remains in the denominator: $C = \frac{1}{2\pi f X_C}$.
2. Calculator Bracket Omission: Calculating $\frac{230}{2\pi \times 50 \times 0.05}$ without enclosing the denominator in brackets: entering 230 / 2 * π * 50 * 0.05 produces $\frac{230}{2} \times \pi \times 50 \times 0.05 \approx 903.2$, rather than the correct value $\frac{230}{(2 \times \pi \times 50 \times 0.05)} \approx 14.64$.
3. Forgetting Powers and Roots: In $P = I^2 R$, writing $I = \frac{P}{R}$ instead of $I = \sqrt{\frac{P}{R}}$.
4. Sign Inversion in Multi-Term Sums: In KVL equations ($V_T = V_1 + V_2 + V_3$), transposing for $V_2$ gives $V_2 = V_T - V_1 - V_3$, not $V_2 = V_T + V_1 - V_3$.