Exercises: Three-phase systems and phasors

The diagram shows the three voltage waves of a three-phase system. a) Describe what they have in common and how they differ. b) Measure on the graph the phase shift between two consecutive waves, indicated by the question mark, and express it in degrees and as a fraction of a period. c) Reason why all three never equal zero simultaneously at any given instant.

Explain three advantages of a three-phase system over a single-phase system: the instantaneous power delivered is constant (does not pulse), transmitting the same power requires less copper, and three-phase motors are self-starting. Why do you think all energy generation and transmission are three-phase?

Define phase voltage and line voltage in a three-phase system: between which points each is measured and what values they have in the European grid ($230\ \text{V}$ and $400\ \text{V}$). State which voltage a domestic socket actually uses.

Describe the star and delta connections of a three-phase load: how the three components are joined in each (a common junction versus a closed loop) and in which of the two an accessible neutral point exists. Sketch both diagrams freehand.

You can do it on paper and send a photo with your answer.

Numerically verify the relationship $U_L = \sqrt{3} \cdot U_F$ using European grid values: start from $U_F = 230\ \text{V}$ and calculate $U_L$ (take $\sqrt{3} = 1{,}732$). Comment on the small difference from the nominal value of $400\ \text{V}$.

The diagram shows the phasor diagram (Fresnel diagram) of the three voltages of a balanced three-phase system. a) Explain what each phasor represents (magnitude and angle) and in which direction the assembly rotates. b) State the angle marked with the question mark between two consecutive phasors. c) Reason what the sum of the three phasors is equal to at any given instant.

Explain the function of the neutral conductor in a three-phase network: what current it carries when the loads across the three phases are equal and what happens when they are different. Reason what would happen to the voltages supplied to homes if the neutral wire were severed while the phases were unbalanced.

A distribution line has four wires: the three phases (R, S, T) and the neutral (N). State the voltage between any two phases and between a phase and the neutral, and explain how a single-phase home is connected to this network and why the homes in a neighbourhood are distributed across the three phases.

Three identical impedances of $23\ \Omega$ are connected in star to a three-phase supply with a line voltage of $400\ \text{V}$. a) Calculate the voltage across each impedance. b) Calculate the phase current and the line current. c) Explain why line current equals phase current in a star connection.

Three identical impedances, of $23\ \Omega$ each (pure resistances), are connected in delta to a three-phase network with a line voltage of $400\ \text{V}$.

1. a) Calculate the voltage across each impedance and its phase current.

1. b) Calculate the line current, $I_L = \sqrt{3} \cdot I_F$.

1. c) Compare the power consumed by the same three impedances connected in delta and in star to that same $400\ \text{V}$ network, and obtain the ratio between them ($P_\triangle / P_Y$).

A balanced three-phase load connected to $400\ \text{V}$ draws a line current of $16\ \text{A}$ with a power factor of $0{,}9$. a) Calculate the active power using $P = \sqrt{3} \cdot U_L \cdot I_L \cdot \cos \varphi$ (take $\sqrt{3} = 1{,}732$). b) Calculate the energy consumed over $8\ \text{h}$ of operation. c) Explain what the power factor means physically in this installation.

A three-phase motor with a useful power output of $5\ \text{kW}$ operates at $400\ \text{V}$ with $\cos \varphi = 0{,}85$. a) Calculate the line current it draws (rearrange the three-phase power formula). b) If the efficiency of the motor were $88\,\%$, calculate the electrical power absorbed and the actual current taking this into account. c) Comment on why the current increases when considering efficiency.

A three-phase workshop operating at $400\ \text{V}$ has a line current of $32\ \text{A}$ and a power factor of $0{,}75$. a) Calculate the apparent power, $S = \sqrt{3} \cdot U_L \cdot I_L$. b) Calculate the active power. c) Calculate the reactive power by applying Pythagoras' theorem to the power triangle, $Q = \sqrt{S^2 - P^2}$, and comment on whether the installation would pay a reactive power penalty.

Three different loads, $23\ \Omega$, $46\ \Omega$, and $92\ \Omega$, are connected in star to a three-phase supply of $400\ \text{V}$ with neutral. a) Calculate the current in each phase. b) Explain what voltage each load experiences despite being different, and what role the neutral plays in this. c) Reason why the neutral conductor carries current in this case but not in the balanced case.

The diagram shows three impedances connected to a three-phase supply with neutral. a) Identify the type of connection. b) Calculate the line voltage marked with the question mark, knowing that each impedance experiences $230\ \text{V}$. c) Reason between which two points on the diagram you would connect a single-phase appliance rated at $230\ \text{V}$.

Three houses in a residential area are supplied from the same three-phase line, each connected to a different phase. a) Explain why the distribution company distributes the houses across the three phases instead of connecting them all to the same phase. b) Reason what happens to the current imbalance when one of the houses has everything switched off. c) Comment on what measure is taken in large buildings to improve balance (distribution of circuits across phases in each dwelling).

In a workshop connected to a balanced three-phase network of $400\ \text{V}$ and $50\ \text{Hz}$, a $3\ \text{kW}$ motor with inductive $\cos \varphi = 0{,}8$ and a purely resistive $2\ \text{kW}$ furnace operate simultaneously. An operator measures the current in each general power supply line with a clamp meter, expecting to obtain the value resulting from directly adding the currents calculated for each machine separately.

1. a) Calculate the nominal line current drawn by each load separately and the scalar sum of both currents.
2. b) Determine, using the active and reactive power balance (Boucherot's method), the actual line current that the clamp meter will register and the overall power factor of the workshop.
3. c) Justify phasorially the discrepancy between both results and explain whether the arithmetic sum always overestimates or underestimates the total line current for loads with different power factors.

A company must transmit an active power of $10\ \text{kW}$ with $\cos \varphi = 1$ over a distance $L$ from the main switchboard to a work module. Two cabling alternatives are evaluated using copper conductors of identical length and operating at the same admissible current density ($j = I/S$):

• Single-phase option at $230\ \text{V}$ (two conductors: phase and neutral).

• Three-phase option at $400\ \text{V}$ (three balanced phase conductors).

1. a) Calculate the line current required in each of the two options.
2. b) Derive the relationship between the required conductor cross-sectional area in single-phase ($S_M$) and in three-phase ($S_T$).
3. c) Calculate the percentage of total copper volume saved by choosing the three-phase option over the single-phase option, adding up all the wires in each line.
4. d) Proving the expression for Joule losses in the line ($P_p = \sum R \cdot I^2$), analyze whether the total energy losses during transmission are greater, smaller, or equal in the three-phase option while maintaining the same current density.

In a building connected to a three-phase mains supply with neutral ($400\ \text{V}$ line-to-line and $230\ \text{V}$ line-to-neutral), an accidental break occurs in the neutral conductor at the service entrance. At that instant, the load connected to phase R corresponds to a $2300\ \text{W}$ resistive heater ($R_R = 23\ \Omega$), while on phase S only an electronic device in standby mode is on, drawing $23\ \text{W}$, equivalent to a resistance $R_S = 2300\ \Omega$. Phase T is unloaded at that moment.

1. a) Draw the resulting equivalent circuit between phases R and S after the neutral break and calculate the actual RMS voltage that each of the two appliances will withstand.
2. b) Predict the behavior of both loads by calculating the power each will dissipate in this situation compared to its rated power.
3. c) Justify why a conventional miniature circuit breaker does not trip in response to this fault and what specific protection device must be installed to safeguard the installation.

One phase of an industrial warehouse connected to $230\ \text{V}$, $50\ \text{Hz}$ draws an inductive current $I_L = 20\ \text{A}$ with a lagging power factor $\cos \varphi = 0{,}8$. To reduce the current at the service entrance without altering the operating conditions of the machinery, a capacitor drawing a purely capacitive current $I_C = 8\ \text{A}$ is connected in parallel with this phase.

1. a) Draw the phasor diagram (Fresnel diagram) to scale taking the phase voltage as the reference phase ($V = 230 \angle 0^\circ\ \text{V}$), showing the phasors for voltage, inductive current $I_L$, capacitive current $I_C$, and the resulting total current $I_{\text{tot}}$.
2. b) Calculate analytically the total current drawn from the grid and the new power factor of the phase after connecting the capacitor.
3. c) Justify physically, based on the energy exchange between magnetic and electric fields, why the capacitor decreases the total line current without reducing the active power delivered to the loads.

After replacing the distribution transformer of a pumping station, a centrifugal pump driven by a three-phase induction motor is connected to the $400\ \text{V}$, $50\ \text{Hz}$ grid. Upon putting it into service, the pumped flow rate is nearly zero and abnormal mechanical cavitation noise is detected. The maintenance team suspects a phase sequence reversal of the lines R, S, and T.

1. a) Justify mathematically or through the time evolution of phasors how the phase sequence generates the rotating magnetic field in the stator and on which parameter its direction of rotation depends.
2. b) Design the necessary procedure in the motor's terminal box to reverse its rotation and demonstrate with phasors why this procedure reverses the direction of the rotating field.
3. c) Describe the operating principle of a phase sequence indicator and explain what preliminary check must be performed before starting critical rotating machinery that cannot tolerate reverse rotation.

Design the three-phase main power supply line ($400\ \text{V}$, $50\ \text{Hz}$) in copper (resistivity $\rho = 0{,}018\ \Omega \cdot \text{mm}^2/\text{m}$) to supply a workshop whose total installed power is $15\ \text{kW}$ with an inductive $\cos \varphi = 0{,}8$, located at a distance of $60\ \text{m}$ from the distribution board. The technical specifications require that the voltage drop does not exceed $1{,}5\ \%$ of the line-to-line voltage and that the conductor operates with a maximum current density of $4\ \text{A/mm}^2$.

The standard commercial conductor cross-sections available are: $4\ \text{mm}^2$, $6\ \text{mm}^2$, $10\ \text{mm}^2$, and $16\ \text{mm}^2$.

1. a) Calculate the line current that will flow through the installation at full load.
2. b) Calculate the minimum cross-section required according to the thermal current density criterion and according to the maximum allowable voltage drop criterion.
3. c) Select the final standard cross-section, justifying your decision.
4. d) Choose the commercial rating of the four-pole miniature circuit breaker ($25\ \text{A}$, $32\ \text{A}$, or $40\ \text{A}$) so that it ensures the protection of the circuit complying with $I_{\text{line}} \le I_n \le I_{\text{admissible}}$ for the chosen conductor.

In a single-phase supply ($v(t) = V_{\text{max}} \sin(\omega t)$ and $i(t) = I_{\text{max}} \sin(\omega t)$ for a resistive load), the instantaneous power pulses at twice the grid frequency and drops to zero twice in each cycle. In contrast, in a balanced three-phase system with three identical resistors $R$ connected in wye (star), the phase voltages are phase-shifted by $120^\circ$ from each other.

1. a) Write the analytical expressions for the instantaneous power absorbed by each phase ($p_1(t)$, $p_2(t)$, $p_3(t)$) as a function of time and the RMS phase voltage $V_F$.
2. b) Prove analytically that the total instantaneous power $p_{\text{total}}(t) = p_1(t) + p_2(t) + p_3(t)$ is constant and independent of time (rely on the trigonometric identity $\sin^2 x = \frac{1 - \cos(2x)}{2}$ and the phasor sum of three components phase-shifted by $120^\circ$).
3. c) From this result, analyze and predict the mechanical advantages that a three-phase motor has over a single-phase motor of the same power in terms of instantaneous torque, vibrations, and bearing service life.

A worker needs to connect a single-phase heater rated for $230\ \text{V}$ with a nominal power of $2300\ \text{W}$ to a construction site distribution board. Upon checking that the cable only provides three phases without a neutral wire at $400\ \text{V}$, they state: "Since it's a robust resistor, I'll connect it between any two phases; it will heat up a bit faster, but it won't be damaged."

1. a) Quantitatively refute the worker's statement by calculating the resistance of the heater and the actual power it will dissipate when connected to $400\ \text{V}$. Determine the percentage of excess power that the equipment will experience.
2. b) Reasonably predict the thermal consequences and direct electrical hazards that this maneuver will cause to the heating element and the installation.
3. c) Using a phasor diagram of the three phase voltages shifted by $120^\circ$, deduce the geometric proof of the relationship $V_L = \sqrt{3} \cdot V_F$ and justify why the neutral is essential to obtain the designed $230\ \text{V}$.

Topic Challenge. How would you optimize the electrical consumption of a small local industrial facility to avoid penalties? Design a technical proposal for reactive power compensation using LibreOffice Calc or GeoGebra.

1. Data collection: Check the actual system frequency ($f \approx 50\text{ Hz}$) on the public platform of Red Eléctrica de España (REE's RedOS website). Assume a small industrial unit with a balanced three-phase supply ($V_L = 400\text{ V}$, $V_F = 230\text{ V}$) feeding a set of motors with $P = 18\text{ kW}$ and an inductive power factor $\cos\varphi = 0,68$.

2. Phasor and power analysis: Calculate the RMS values of line and phase currents, as well as the reactive power $Q$ (kvar) and apparent power $S$ (kVA). Draw the qualitative phasor diagram of voltages and currents.

3. Reactive compensation: Determine the capacitive reactive power $Q_C$ and the capacitance $C$ per phase required by connecting a delta-connected capacitor bank to raise the power factor to $\cos\varphi' = 0,95$.

4. Technical proposal: Write a brief report with a comparative table of currents and powers (before and after), justifying the reduction in losses.

Evaluation criteria:

• Precision in the calculation of three-phase quantities ($V$, $I$) and powers ($P$, $Q$, $S$, $\cos\varphi$).

• Rigor in drawing the phasor diagram and sizing the capacitance $C$.

• Clarity and technical justification in the final comparative proposal.