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.

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 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?

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.

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.

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 $400\ \text{V}$.

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 $230\ \text{V}$.

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.

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 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.

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.