Teaching guide for the module

Three-phase AC systems: Teacher's Guide

Teaching guide for the Three-Phase AC Systems simulator: explain in class three-phase systems, synchronous waveforms shifted by $120°$ , rotating phasors, star (Y) and delta (Δ) connections, neutral shift with unbalanced loads and the three-phase power triangle. Designed for industrial electrical engineering teachers.

Module: Three-Phase AC Systems · Four tabs: Waveforms · Star (Y) · Delta (Δ) · Power

Physical phenomenon

A three-phase system consists of three sinusoidal alternating voltages of equal amplitude and frequency, shifted by exactly 120° from each other. It is how electrical energy is generated, transmitted and distributed throughout the industrial world.

Each phase is described by:

$v_{k} (t) = V_{p e ak} \cdot sin (2 π f t - (k - 1) \cdot \frac{2 π}{3}), k = 1, 2, 3$

The three conductors are labelled L1 · L2 · L3 (IEC colours: red · yellow · blue) and in four-wire systems a fourth conductor, the neutral (N), is present.

The three loads can be connected in two configurations:

Star (Y): one terminal of each load to its line wire, the other to a common point (star centre, possibly tied to the neutral). $V_{L} = 3 \cdot V_{f}$ and $I_{L} = I_{f}$ hold.
Delta (Δ): the three loads in a closed series between the three lines. $V_{L} = V_{f}$ and $I_{L} = 3 \cdot I_{f}$ hold (at balanced load).

The total three-phase power is $P = 3 \cdot V_{L} \cdot I_{L} \cdot cos φ$ , independent of the connection type.

Key concepts

120° phase shift: constitutive condition of the system, not a special case.
Direct / inverse sequence: the order in which the three phases reach their peak (L1→L2→L3 or L1→L3→L2). Swapping two phase wires inverts the sequence: the principle of three-phase motor reversal.
Phase voltage $V_{f}$ (line-to-neutral) and line voltage $V_{L}$ (line-to-line). In Italy $V_{L} = 400 V$ and $V_{f} \approx 230 V$ .
Role of the neutral: keeps the phase voltages on the loads symmetric when imbalance between phases is significant. Its presence depends on the distribution system (TT/TN/IT) and on the type of utilities served.
Star-centre shift (Millman): without neutral and with unbalanced loads, the common point of the star shifts away from zero potential, generating overvoltages and undervoltages on the loads.
Three-phase power triangle: $P$ (active), $Q$ (reactive), $S$ (apparent), linked by $S = P^{2} + Q^{2}$ and $cos φ = P / S$ .
Constant instantaneous power: at balanced load and $cos φ = 1$ the sum of the three $p_{k} (t)$ is constant in time: three-phase motors run without pulsating torque, a structural advantage over single-phase.

How to use it in the classroom

Opening: Waveforms tab. Show the three sinusoids scrolling on the oscilloscope and the three phasors rotating while keeping a fixed $120°$ between them. Have students verbalise that they are looking at the same information in two different languages (temporal and geometric). Press SEQUENCE: INVERSE: the order of peaks and phasors flips. Explain that this is exactly what happens when you swap two phase wires of a three-phase motor, reversing its direction.

Development: Star (Y) tab. Verify the ratio $V_{L} / V_{f} = 3$ in the KPIs and link it to household sockets (230 V on a 400 V grid). Add imbalance with neutral ON: phase voltages stay symmetric, currents become different. Disconnect the neutral with imbalance present: the gold star-centre shift dot appears (complex Millman calculation) and phase voltages become unbalanced. This is the visual demonstration of why the neutral exists in residential cabinets.

Deeper exploration: Delta (Δ) tab. Show the triangular geometry and the animated flow distinguishing phase current $I_{f}$ (inside the side) from line current $I_{L}$ (on the outer wire). Compare with the Star tab at the same V_LINE, R, L, C: the line current in delta is $3$ times the one in star. This is the foundation of star-delta starting in industrial motors, where the starting torque drops to about $1/3$ of the rated delta torque.

Closing: Power tab. Animated power triangle. Move the slider $φ$ at constant $V_{L}$ and $I_{L}$ : $S$ stays fixed, $P$ and $Q$ swap along the hypotenuse. Introduce the power factor concept and the equivalent dual formula $P = 3 V_{f} I_{f} cos φ = 3 V_{L} I_{L} cos φ$ . Students see that three-phase power does not depend on the load connection.

Real-world examples

Domestic distribution. Household sockets draw a single phase from the neutral of a three-phase 400/230 V system, that is why apartment blocks are wired by spreading utilities across the three phases to balance the load at the substation.
Three-phase asynchronous motors. The heart of industrial automation: pumps, fans, compressors, conveyor belts. The rotating magnetic field produced by the three phase-shifted phases drags the rotor without brushes.
Star-delta starting. Industrial control cabinets with switching contactors: the motor starts in star (reduced winding voltage, contained inrush current) and switches to delta after a few seconds at running speed. Suitable for loads that start "unloaded" (fans, centrifugal pumps).
Energy transmission. High-voltage lines crossing the territory are three-phase: at the same transmitted power, a three-phase system requires less copper than the equivalent single-phase and maintains constant instantaneous power flow.
Industrial power factor correction. Capacitor banks in parallel with motors to bring $cos φ$ close to 1, eliminate reactive consumption penalties on the bill and free up capacity at the substation transformers.

Classroom discussion questions

Why do we have 230 V at home if the industrial grid is at 400 V?
If someone in a residential cabinet accidentally cuts the neutral wire, what happens to the single-phase loads on the three phases? And why might this not happen in an industrial cabinet with only balanced three-phase motors?
The same motor can be connected in star or in delta. Why does it draw less current in star? Why then do we want it to run in delta in the end?
At balanced load and unit power factor, the instantaneous sum of the three phase powers $p_{1} (t) + p_{2} (t) + p_{3} (t)$ is constant in time. What mechanical advantage does this bring to a three-phase motor compared to a single-phase one?
Two companies have the same contractual apparent power $S$ , but one has $cos φ = 0.95$ and the other $cos φ = 0.65$ . Which one pays more at the end of the year, and why?

Related modules

AC Behaviour (R, L, C): the underlying sinusoidal regime: the three-phase system is the composition of three AC waves shifted by $120°$ , with all the impedance and phasor theory already learned.
Power Factor & AC Power: the power factor $cos φ$ and correction apply identically in three-phase, and it is precisely in three-phase that correction is performed in industrial practice.