ELECTENG 291 Fundamentals of Electrical Engineering
The Complete Exam Bible · S1 2027ELECTENG 291 Overview
- University of Auckland
- S1 2027
- 10 chapters
- Electrical Engineering
ELECTENG 291 Fundamentals of Electrical Engineering is the University of Auckland's Stage II circuit-analysis course — 15 points in the Faculty of Engineering (Electrical and Electronic Engineering), taught on the City Campus with three one-hour lectures a week plus a weekly tutorial and fortnightly labs.
- Assessed by Final Exam 40% · In-Person Tests 24% · Online Assignments 15% · Laboratories 12% · Tutorials 7% + 1 more
- Key terms Passive sign convention (PSC), Kirchhoff's laws (KVL & KCL), Linear circuit, Thévenin / Norton equivalent
- How to prepare Treat the three modules as one growing toolkit and keep the weekly rhythm the course is built around: attend the tutorial each week (it is worth 1% a…
- Most asked Where can I find past exam papers and practice for ELECTENG 291?
How ELECTENG 291 is assessed
| Component | Weight | Format |
|---|---|---|
| Final Exam | 40% | UoA S1 exam period; in-person, on paper (Mode C), formula page + restricted calculator; two equal free-form questions |
| In-Person Tests | 24% | Two 1-hour restricted in-person tests (12% each) |
| Online Assignments | 15% | Three online assignments (5% each) |
| Laboratories | 12% | Four graded labs (3% each); Lab 1 is a mandatory ungraded induction |
| Tutorials | 7% | Eleven weekly tutorials (1% each, Weeks 2-12), capped at 7% |
| Online Revision Assignment | 2% | A short online revision task |
What ELECTENG 291 covers
The whole course → one exam-ready map. ELECTENG 291 is assessed by a 40% final exam (in-person, on paper, two equal free-form questions) plus 60% coursework — two in-person tests (24%), three online assignments (15%), four graded labs (12%), weekly tutorials (7%) and a short online revision task (2%).
The three teaching modules — Circuit Analysis Methods, Transient Behaviours and AC Behaviours — map straight onto the chapters below; each chapter links to its free guide.
Circuit Fundamentals, Signals & Sources
Charge, current, voltage & power · the passive sign convention · independent & dependent sources (Module 1)Module 1 of University of Auckland ELECTENG 291 opens with the vocabulary every later chapter is written in: charge, current, voltage and power, the passive sign convention that fixes when power is absorbed or delivered, and the four ideal sources — independent and dependent voltage and current sources.
- Charge, current i [A] and voltage v [V]; current as the flow of charge, voltage as energy per unit charge
- Instantaneous power p = v·i [W] and the passive sign convention (PSC): current into the + terminal → power absorbed = +vi
- Absorbing vs delivering power: p > 0 absorbs, p < 0 delivers; conservation of power around a circuit (ΣP = 0)
- Ideal independent sources: the voltage source (fixes v) and the current source (fixes i)
- Dependent (controlled) sources: voltage- or current-controlled voltage/current sources, set by another circuit variable
- Signal description x(t) = X + x̃(t): a DC/average component X plus a time-varying (AC) component x̃(t)
Linearity & Superposition
Additivity & homogeneity · linear vs non-linear elements · the superposition principle (Module 1)Module 1.1 and 1.4 of University of Auckland ELECTENG 291 pin down what makes a circuit linear — it must be both homogeneous and additive — and then exploit that property with the superposition principle, analysing a circuit one independent source at a time.
- Linear iff BOTH homogeneous AND additive; otherwise non-linear
- Homogeneity (scaling): f{a·x(t)} = a·f{x(t)} for every constant a
- Additivity: f{x1(t) + x2(t)} = f{x1(t)} + f{x2(t)}
- Proof-by-counter-example: one violation of either property disproves linearity
- Worked cases: level-shifter y = x + K (non-linear for K ≠ 0), averaging/low-pass (linear), y = max(x) and the exponential diode (non-linear)
- The superposition principle: total response = sum of responses to each independent source acting alone
Resistive Circuits: Ohm's & Kirchhoff's Laws
Ohm's law · KVL & KCL · series/parallel · voltage & current dividers (Module 1)This chapter of Module 1 in University of Auckland ELECTENG 291 is the workhorse toolkit: Ohm's law (v = R i), Kirchhoff's voltage and current laws (KVL and KCL), combining resistors in series and parallel, and the voltage- and current-divider rules.
- Ohm's law v = R i (equivalently i = v/R, R = v/i) under the passive sign convention
- KVL: the sum of voltages around any closed loop is zero
- KCL: the sum of currents into any node is zero
- Resistors in series add: R_eq = R1 + R2 + …
- Resistors in parallel combine by reciprocals: 1/R_eq = 1/R1 + 1/R2 + … (two resistors: R_eq = R1R2/(R1+R2))
- Voltage divider: V_out = [R2/(R1 + R2)]·V_s across R2 in a series pair
Nodal & Mesh Analysis
The node-voltage method · the mesh-current method · choosing a method · supernodes/supermeshes (Module 1)This Module 1 chapter of University of Auckland ELECTENG 291 turns circuit-solving into a system: the node-voltage (nodal) method writes KCL at each essential node in terms of node voltages, and the mesh-current method writes KVL around each loop in terms of mesh currents.
- Node-voltage method: pick a reference node, write KCL at each essential node using node voltages and conductances, solve
- Mesh-current method: define loop currents, write KVL around each mesh, solve (the dual of nodal)
- Choosing a method: nodal suits many parallel branches / current sources; mesh suits series loops / voltage sources
- Supernode: a voltage source between two non-reference nodes — enclose both, write one KCL for the supernode plus a KVL constraint (e.g. V_A = V_B + 15 V)
- Supermesh: a current source shared by two meshes — combine them and add the current-source constraint
- Handling dependent sources: keep them active and add the controlling-variable relation as an extra equation
Thévenin & Norton Equivalents
Source transformation · finding V_th, R_th, I_N · maximum power transfer (Module 1)This Module 1 chapter of University of Auckland ELECTENG 291 shows how any linear two-terminal network collapses to a simple equivalent: the Thévenin form (an open-circuit voltage V_TH in series with R_TH) or the Norton form (a short-circuit current I_N in parallel with R_N), linked by R_TH = R_N = V_TH / I_N.
- Thévenin equivalent: open-circuit voltage V_TH in series with R_TH
- Norton equivalent: short-circuit current I_N in parallel with R_N
- The link R_TH = R_N = V_TH / I_N, and source transformation (V-source + series R ⇄ I-source + parallel R)
- Finding R_TH with only independent sources: zero them and find the equivalent resistance
- Finding R_TH with a dependent source present: apply a notional (test) source and take R_TH = V_test / I_test
- R_TH can be zero or negative with dependent sources — mathematically correct even when physically odd
Capacitors, Inductors & Energy Storage
The v-i relationships · stored energy · continuity of v_C and i_L · initial conditions (Module 2)Module 2 of University of Auckland ELECTENG 291 introduces the two energy-storage elements and the rules that make transients possible: the capacitor law i = C dv/dt and inductor law v = L di/dt (both under the passive sign convention), the energy stored (½Cv² and ½Li²), and the crucial continuity rules — a capacitor's voltage and an inductor's current cannot change instantly.
- Capacitor terminal law (PSC): i = C dv/dt — current is proportional to the rate of change of voltage
- Inductor terminal law (PSC): v = L di/dt — voltage is proportional to the rate of change of current
- DC steady state: a capacitor becomes an open circuit (i_C = 0) and an inductor becomes a short circuit (v_L = 0)
- Continuity: v_C(t) and i_L(t) are continuous — they cannot jump, so v_C(0⁺) = v_C(0⁻) and i_L(0⁺) = i_L(0⁻)
- Energy stored: capacitor ½C v_C², inductor ½L i_L² [J]
- Rate/state relations at a switching instant: dv_C/dt = i_C/C and di_L/dt = v_L/L
First-Order Transients: RC & RL Circuits
Natural & step response · the time constant τ = RC / τ = L/R · the general first-order solution (Module 2)Module 2 of University of Auckland ELECTENG 291 analyses circuits with a single (equivalent) energy-storage element — the RC and RL circuits — whose behaviour obeys a first-order ODE.
- First-order circuit = one equivalent energy-storage element → first-order ODE a1 dy/dt + a0 y = x(t)
- Time constant: τ = RC for an RC circuit, τ = L/R for an RL circuit — where R is the Thévenin resistance seen by the C or L
- The general first-order solution y(t) = y(∞) + [y(0⁺) − y(∞)]e^(−t/τ) (on the exam formula page)
- The three-step method: find y(0⁺) via continuity, find y(∞) at the new steady state, find τ
- Natural response (source removed, decay to zero) vs step response (settle to a new final value)
- Settling: the response is within about 1% of its final value after roughly 5τ
Second-Order Transients: RLC Circuits
The characteristic equation · overdamped/critically-damped/underdamped · natural frequency & damping (Module 2)This Module 2 chapter of University of Auckland ELECTENG 291 steps up to circuits with two energy-storage elements — the RLC circuit — governed by a second-order ODE.
- Second-order circuit = two energy-storage elements → second-order ODE and a characteristic (auxiliary) equation
- Undamped natural frequency ω0 = 1/√(LC) — the ideal loss-less LC oscillation rate [rad/s]
- Characteristic equation s² + 2αs + ω0² = 0 with roots s = −α ± √(α² − ω0²)
- Damping coefficient α (its form depends on the topology: series RLC α = R/2L, parallel RLC α = 1/2RC — confirm which against the Module 2.2 notes)
- Overdamped (α > ω0): two real distinct roots → sum of two decaying exponentials, no oscillation
- Critically damped (α = ω0): a repeated real root → the fastest non-oscillatory settling
AC Steady-State: Phasors & Impedance
Sinusoids, average & RMS · phasor representation · complex impedance Z = R + jX · admittance (Module 3)Module 3 of University of Auckland ELECTENG 291 analyses circuits driven by sinusoids without solving differential equations, by working in the frequency domain.
- Sinusoid (cosine convention): x(t) = A cos(ωt + φ), with ω = 2πf = 2π/T
- Average value X = (1/T)∫_T x dt and RMS value; for a sinusoid of amplitude A_p, X_rms = A_p/√2
- Phasor (peak convention): v(t) = V_p cos(ωt + θ_v) ↔ V = V_p∠θ_v
- Frequency-domain Ohm's law V = Z·I (bold = phasors)
- Element impedances: Z_R = R, Z_L = jωL (X_L = ωL), Z_C = 1/(jωC) = −j/(ωC) (X_C = −1/(ωC))
- General impedance Z = R + jX Ω, with |Z| = √(R² + X²) and arg Z = tan⁻¹(X/R)
AC Power & Phasor Circuit Analysis
Instantaneous, real, reactive & apparent power · power factor · the power triangle (Module 3)The final Module 3 chapter of University of Auckland ELECTENG 291 characterises power in AC circuits via the frequency domain: instantaneous, average (real), reactive and apparent power, the power factor and the power triangle, and analysing a complete AC circuit with phasors and impedances.
- Instantaneous power p(t) = v(t)·i(t) [W]
- Average (real) power P = ½ V_p I_p cos(θ_v − θ_i) = V_rms I_rms cos(θ_v − θ_i) [W]
- Reactive power Q = ½ V_p I_p sin(θ_v − θ_i) = V_rms I_rms sin(θ_v − θ_i) [VAR]
- Apparent power |S| = V_rms I_rms = √(P² + Q²) [VA] and the power triangle (P, Q, |S|)
- Power factor pf = cos(θ_v − θ_i) = P/|S|; leading (capacitive, θ < 0) vs lagging (inductive, θ > 0)
- Complex power S = ½ V I* = V_rms (I_rms)* with P = Re[S], Q = Im[S], pf angle = arg[S] (S is NOT a phasor)
It takes you from the Stage I foundations of ELECTENG 101 (the prerequisite) through the analysis, interpretation and evaluation of common DC and AC circuits, in three modules: Module 1 — Circuit Classifications & Analysis Methods (signals and sources, linearity, the load-line method, node-voltage and mesh analysis, superposition, and Thévenin/Norton equivalents); Module 2 — Transient Behaviours (capacitors and inductors, first-order RC/RL and second-order RLC circuits, and the Laplace transform for s-domain analysis); and Module 3 — AC Behaviours (sinusoids, phasors and complex impedance, and AC power characterisation via the frequency domain).
There is no prescribed textbook — the electronic lecture notes are released module by module on Canvas, with Alexander & Sadiku's Fundamentals of Electric Circuits recommended (not required) for extra reading. The grade is 40% a final written exam and 60% coursework.
The final examination is worth 40%, sat in-person in the Semester-1 exam period: it is on paper (Mode C — not Inspera), a printed booklet with a provided formula page and a restricted calculator, with two equal-weight free-form questions that you must justify verbally and mathematically. It is a must-sit hurdle — not sitting it gives a grade of DNS (Did Not Sit).
Confirm the exact date and materials rules on Canvas and the UoA exam timetable.
Worked example · free
AC load power characterisation — the signature Module 3 exam question
- +1Read the peak phasors off the cosine forms: V = 80∠0° V and I = 5∠−45° A. The power-factor angle is θ = θ_v − θ_i = 0° − (−45°) = +45°.
- +1Complex power (peak convention) S = ½ V I* = ½ · (80∠0°) · (5∠+45°) = ½ · 400∠45° = 200∠45° VA. (I* is the conjugate of the current phasor — flip the sign of its angle, so −45° becomes +45°.)
- +1(a) Average (real) power P = Re[S] = 200 cos(45°) = 200 · 0.7071 = 141.4 W.
- +1(b) Reactive power Q = Im[S] = 200 sin(45°) = 200 · 0.7071 = +141.4 VAR (positive).
- +1(c) P = +141.4 W > 0, so the load absorbs average power. (d) θ = +45° > 0 means the current lags the voltage, so the load is inductive (an inductive load has Q > 0 / a lagging power factor, pf = cos 45° = 0.707 lagging).
- +1(e) Impedance Z = V / I = (80∠0°)/(5∠−45°) = 16∠+45° Ω = 16(cos 45° + j sin 45°) = 11.31 + j11.31 Ω. The positive reactance (X = +11.31 Ω) confirms an inductive load, and P = |I_rms|²R = (5/√2)² · 11.31 = 12.5 · 11.31 = 141.4 W checks against part (a).
Key terms
- Passive sign convention (PSC)
- The bookkeeping rule the whole course uses: current is taken to enter the '+' voltage terminal of an element, so the power it absorbs is p = +vi. Flip the assumed current arrow and every KVL/KCL sign flips with it — a reversed arrow is a classic mark-losing slip.
- Kirchhoff's laws (KVL & KCL)
- KVL: the voltages around any closed loop sum to zero. KCL: the currents into any node sum to zero. Together with Ohm's law (v = R i) they are the backbone of every DC and AC circuit solve in ELECTENG 291.
- Linear circuit
- A circuit that is both homogeneous (scaling the input scales the output: f{a·x} = a·f{x}) and additive (f{x1+x2} = f{x1}+f{x2}). Only linear circuits obey superposition; a single counter-example to either property proves non-linearity.
- Thévenin / Norton equivalent
- Any linear two-terminal network reduces to a Thévenin equivalent (open-circuit voltage V_TH in series with R_TH) or a Norton equivalent (short-circuit current I_N in parallel with R_N), where R_TH = R_N = V_TH / I_N. With a dependent source present, R_TH can even be negative.
- Time constant (τ)
- The characteristic time of a first-order circuit: τ = RC for an RC circuit and τ = L/R for an RL circuit. The response settles to within about 1% of its final value after roughly 5τ, following y(t) = y(∞) + [y(0⁺) − y(∞)]e^(−t/τ).
- Complex impedance (Z = R + jX)
- The AC generalisation of resistance: Z = R + jX Ω, where R is resistance and X is reactance (X = ωL for an inductor, X = −1/(ωC) for a capacitor). Phasor Ohm's law is V = Z·I; |Z| = √(R²+X²) and arg Z = tan⁻¹(X/R).
ELECTENG 291 FAQ
Is ELECTENG 291 hard?
It is demanding but very learnable, and it is more cumulative than conceptually deep: Module 2 (transients) leans on Module 1 (circuit laws and Thévenin), and Module 3 (AC) reuses the same solving techniques in complex arithmetic — so falling behind is what makes it hard.
The pressure points students report are keeping the passive sign convention consistent, the AC-power sign of reactive power Q, and second-order (RLC) damping classification. The exam gives generous time (about 120 minutes of material with 180 minutes to write) and marks reward coherent, well-justified working — not speed — so steady weekly practice on tutorials and past tests generally makes it manageable.
Confirm assessment details on Canvas.
Can AI help me with ELECTENG 291?
Yes — as a study aid, not to sit assessments for you. Sia is an AI tutor that explains ELECTENG 291 the way it is actually taught and assessed at the University of Auckland: it will walk a KVL/KCL solve step by step, re-derive a Thévenin equivalent, check your phasor and AC-power arithmetic, and quiz you on fresh numbers, checking your reasoning as you go.
It does not do graded assessment for you — the University of Auckland's academic-integrity rules apply to the online assignments, tests and exam — so use it to understand the method and rehearse, and always confirm what is permitted on Canvas.
Where can I find past exam papers and practice for ELECTENG 291?
The course itself is the best source: past tests and exam material, the provided exam formula page, module supplementary problem sets and the weekly tutorials are released on Canvas, and the course notes that the four most recent past exam papers are a good indicator of difficulty (though not of this year's exact focus). Official exam solutions are not released.
This guide adds original worked examples that mirror the tests' anatomy with fresh numbers, solved line by line, and you can ask Sia to generate more practice in the same pattern for any chapter. Confirm the current materials on Canvas.
What are the exam and hurdle rules for ELECTENG 291?
The 40% final exam is a must-sit hurdle: you must attempt it to be eligible for a passing grade, and not sitting it gives a grade of DNS (Did Not Sit). Lab 1 is a mandatory ungraded induction that you must complete before you may attend the later graded labs. There is no separate numeric pass-mark or plussage hurdle stated — the course is graded on the 100% total across the exam and coursework.
Always confirm the current hurdle rules and dates on Canvas and the course outline.
Is the ELECTENG 291 exam open book, and what can I bring?
It is a restricted, on-paper exam (Mode C — not Inspera): a printed question/answer booklet with a provided formula page and a restricted calculator (no graphing, no full alphabet). You may bring one double-sided A4 sheet with anything you like written on it, plus your University ID and pens.
The provided formula page carries only the Laplace-transform pairs/properties and the first-order general-solution form — so the impedance definitions and the AC-power formulas (P, Q, |S|, power factor) are not given and belong on your own A4 sheet. Confirm the permitted materials for your sitting on Canvas.
Questions students ask in each ELECTENG 291 chapter
- What is the passive sign convention and why does it matter?
- It is the agreement that a reference current is drawn into the '+' voltage terminal of an element, so the power it absorbs is p = +vi. It matters because it makes every sign in the course consistent: the terminal laws v = Ri, i = C dv/dt and v = L di/dt are all written under it, and a single flipped current arrow reverses your KVL and KCL signs and can turn an absorbing element into a delivering one. Label the polarity and arrow first, every time. Chapter 01 · Circuit Fundamentals, Signals & Sources
- How do I prove a circuit is linear or non-linear?
- Test the two properties. For homogeneity, feed a·x and check the output equals a times the original output. For additivity, feed x1 + x2 and check the output equals f{x1} + f{x2}. If both hold in general, it is linear; if either fails for even one input, it is non-linear. Marks are awarded for setting up each property clearly and stating the concluding condition — not just the yes/no. Chapter 02 · Linearity & Superposition
- What is the difference between KVL and KCL?
- KVL is about loops: the voltages around any closed loop sum to zero (energy conservation). KCL is about nodes: the currents into any node sum to zero (charge conservation). You use KCL as the backbone of nodal analysis and KVL as the backbone of mesh analysis, but both, together with Ohm's law, appear in almost every solve. Chapter 03 · Resistive Circuits: Ohm's & Kirchhoff's Laws
- When should I use nodal versus mesh analysis?
- Count what you would have to solve. Nodal analysis needs one equation per essential node (minus the reference) and handles current sources and many parallel branches naturally. Mesh analysis needs one equation per independent loop and handles series voltage sources naturally, but it only applies cleanly to planar circuits. Pick whichever gives fewer unknowns; both are valid and should agree. Chapter 04 · Nodal & Mesh Analysis
- How do I find the Thévenin voltage and resistance?
- V_TH is the voltage across the open-circuited terminals. R_TH is the equivalent resistance seen from those terminals: with only independent sources, zero them (voltage sources → short, current sources → open) and reduce the resistor network; if there is a dependent source, instead apply a notional test source at the terminals and compute R_TH = V_test/I_test. A quick alternative is R_TH = V_TH/I_N using the short-circuit current. Chapter 05 · Thévenin & Norton Equivalents
- Why can't a capacitor voltage or inductor current change instantly?
- Because an instantaneous jump would require infinite power. The stored energy is ½Cv² for a capacitor and ½Li² for an inductor; changing v_C or i_L in zero time would change that energy in zero time, which is physically impossible. So v_C and i_L are continuous, and their pre-switch values carry straight across as the initial conditions for a transient. Chapter 06 · Capacitors, Inductors & Energy Storage
- What is the time constant and how do I find it?
- The time constant τ sets how fast a first-order circuit responds: τ = RC for an RC circuit and τ = L/R for an RL circuit. The R is the Thévenin resistance the capacitor or inductor 'sees' looking back into the rest of the circuit — so if there is more than one resistor, first reduce the network to its Thévenin equivalent at the storage element's terminals. After about 5τ the response is within roughly 1% of its final value. Chapter 07 · First-Order Transients: RC & RL Circuits
- How do I classify a second-order (RLC) response?
- Compare the damping coefficient α with the natural frequency ω0 = 1/√(LC), or look at the roots of the characteristic equation s² + 2αs + ω0² = 0. Two real distinct roots (α > ω0) → overdamped; a repeated root (α = ω0) → critically damped; complex roots (α < ω0) → underdamped, an oscillatory 'ringing' response at ω_d = √(ω0² − α²). Naming the case and writing the matching response form is the examinable skill. Chapter 08 · Second-Order Transients: RLC Circuits
- What is a phasor and why use one?
- A phasor is a complex number V = V_p∠θ_v that captures a sinusoid's amplitude and phase at a fixed frequency, so v(t) = V_p cos(ωt + θ_v). Because differentiation and integration of sinusoids become simple multiplications by jω in the phasor domain, AC circuits reduce to algebra: you solve them with the same series/parallel and nodal/mesh techniques as resistive circuits, just in complex numbers, then convert the answer back to a time-domain sinusoid. Chapter 09 · AC Steady-State: Phasors & Impedance
- What is the difference between real, reactive and apparent power?
- Real power P (watts) is the power actually dissipated or converted to work, P = V_rms I_rms cos θ. Reactive power Q (VAR) is the power that shuttles back and forth with inductors and capacitors without net consumption, Q = V_rms I_rms sin θ. Apparent power |S| (VA) is their combined magnitude, |S| = V_rms I_rms = √(P²+Q²) — the hypotenuse of the power triangle. The power factor cos θ = P/|S| ties them together. Chapter 10 · AC Power & Phasor Circuit Analysis
How to study for the exam
Treat the three modules as one growing toolkit and keep the weekly rhythm the course is built around: attend the tutorial each week (it is worth 1% a week, capped at 7%, and it is where the exam-style solving is rehearsed) and do the fortnightly labs — Lab 1 is a mandatory induction you must clear to attend the rest.
Drill circuit solves by hand from Module 1, always drawing the circuit and fixing the passive sign convention first: a flipped current arrow is the single most common way marks leak. Give the two in-person tests (12% each, restricted-book, free-form, show-all-working) full past-paper reps — the four most recent papers are a good difficulty guide.
Build your own double-sided A4 sheet as you go: the provided exam formula page carries only the Laplace pairs and the first-order solution, so the impedance definitions (Z_L = jωL, Z_C = 1/(jωC)) and every AC-power formula (P = ½V_pI_p cos θ, Q = ½V_pI_p sin θ, |S|, complex power S = ½V I*) must be on your sheet or in your head.
For the exam, budget time in proportion to marks across the two equal free-form questions, and remember the marking ethos: coherent, well-justified working with stated assumptions and correct units earns marks even when a final number slips. Sit the past tests in timed conditions, and always write the law you are using (KVL, KCL, continuity of v_C / i_L, V = Z·I) before the numbers.
Confirm the exam date and rules on Canvas and the UoA exam timetable.
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