Electrical Engineering Level 6: Engineering Mathematics I Explained

Engineering Mathematics I

Vocational Context: The Math of Diagnostics

At Level 6, you are not just designing new systems; you are responsible for fixing the expensive, complex problems that others cannot solve.

When a multi-million-pound production line trips intermittently, or a high-voltage transformer overheats without overload, you cannot “guess.” You must apply Mathematical Root Cause Analysis (RCA).

This protocol outlines how Senior Engineers use:

  • Calculus to tune control loops (PID) and analyze transient instability.
  • Algebra to model fault flows and network balance.
  • Trigonometry to analyze harmonic distortion and power quality.

Calculus in Automation: PID Control Loops

The Concept:

In modern industrial automation, the Proportional-Integral-Derivative (PID) controller is the brain of the operation. It uses calculus to maintain stability (e.g., keeping a furnace at exactly 1000°C).

Vocational Application:

  • Proportional: Algebraic error correction.
  • Integral: Calculus used to look at the past error accumulation to eliminate steady-state offset.
  • Derivative: Calculus used to look at the rate of change to predict future errors and dampen overshoot.

The Engineer’s Job:

If a system is oscillating (hunting) or sluggish, the engineer must “tune” these parameters. This requires a conceptual understanding of how changing the integral time constant affects the system’s stability over time.

Algebra in Protection: Symmetrical Components

The Concept:

When analyzing unbalanced three-phase faults (e.g., a Single Line-to-Ground fault), standard Ohm’s law fails. Engineers use Symmetrical Components—a linear algebra method that breaks unbalanced phasors into three balanced sets (Positive, Negative, and Zero sequence).

Vocational Application:

  • Fault Modeling: Using algebraic matrices to predict current flow during a specific fault type.
  • Relay Settings: Modern digital protection relays calculate these sequences mathematically. The engineer must set the logic thresholds (algebraic inequalities) to ensure the relay trips for a fault but stays closed for a heavy motor start.

Trigonometry in Power Quality: Harmonic Analysis

The Concept:

Non-linear loads (Variable Speed Drives, LED lighting) draw current in pulses, not smooth sine waves. This creates Harmonics—frequencies that are integer multiples of the fundamental 50Hz.

Vocational Application:

  • Fourier Analysis: Engineers use trigonometric principles (Fast Fourier Transform – FFT) to break down a distorted wave into its component sine waves.
  • The Risk: High harmonic frequencies cause “Skin Effect” (increasing resistance) and overheat neutral conductors.
  • The Fix: The engineer calculates the Total Harmonic Distortion (THD) and sizes active harmonic filters to inject an opposing waveform (trigonometric cancellation).

Regulatory Framework: Compliance through Analysis

Your mathematical analysis is the proof required for legal compliance.

  • Electricity at Work Regulations 1989 (Reg 5): Requires that electrical systems are not overloaded. Harmonic currents can overload cables even if the RMS current looks normal. Mathematical analysis proves you have addressed this invisible risk.
  • BS 7671 (Appendix 11): Provides guidance on Harmonic distortion. Compliance requires verifying that THD levels do not exceed equipment ratings.

Comprehensive Learner Task: Major Failure Investigation

Task Brief

You have performed the mathematical analysis and identified the root causes of a critical system failure. Now, you must document your findings to justify the necessary rectifications.

You are required to produce a formal Mathematical Diagnostic Report that translates complex mathematical diagnostics into clear engineering solutions.
Report Requirements (Evidence Generation) You must compile a technical report consisting of the following three diagnostic sections.

Part 1: Harmonic Investigation (Trigonometry Application)

  • The Analysis: Provide a written analysis of the waveform distortion caused by VSDs.
  • The Math: Explain how Fourier analysis (Trigonometry) helps identify specific harmonic frequencies. Explain why the “Summation” of these harmonics is causing the transformer to overheat (Eddy currents) despite the low RMS current reading.

Part 2: System Stability Check (Calculus Application)

  • The Analysis: Evaluate the PID control loop behavior of the robots.
  • The Math: Discuss the Rate of Change (Derivative) settings. Explain mathematically how an incorrect derivative value is causing the system to react aggressively to minor fluctuations, leading to instability and PLC crashes.

Part 3: Network Verification (Algebra Application)

  • The Analysis: Present an evaluation of the Neutral Conductor sizing.
  • The Math: Use Algebraic Logic to demonstrate why “Triplen Harmonics” (Zero Sequence) are adding up in the neutral conductor instead of cancelling out. Prove algebraically that the current neutral sizing is insufficient.

Part 4: Regulatory & Solution Statement

  • The Conclusion: Verify that the current operation violates the Electricity at Work Regulations 1989 (due to thermal risk) and confirm that your proposed solution restores compliance with BS 7671.

Submission Guidelines / Evidence for Portfolio

To achieve the credits for this unit, you must upload the following specific evidence to your learner portal:

  • Evidence Type: “Mathematical modelling reports”