Exercises: Quantities of alternating current

Explain the difference between direct current and alternating current, and explain why long-distance electrical energy transmission is done using alternating current (transformers, stepping up voltage, and reducing losses).

Define each quantity of a sinusoidal wave in one sentence: peak value, RMS value, period, and frequency. State their International System (SI) units and the mathematical relationship between period and frequency.

The diagram shows the mains supply voltage, with its peak value and period indicated. Knowing that the RMS value of the mains is $230\ \text{V}$, calculate the peak value $V_{max} = V_{ef} \cdot \sqrt{2}$ and explain what the RMS value of an alternating voltage means physically.

Explain what the initial phase of a sinusoidal wave is and what it means for two waves of the same frequency to be "in phase" or "in antiphase". What happens when two identical waves in antiphase are added together?

The electrical grid in much of the Americas operates at $120\ \text{V}$ and $60\ \text{Hz}$, and the European grid at $230\ \text{V}$ and $50\ \text{Hz}$. Calculate the period of each grid and explain what precautions must be taken when connecting an American appliance in Europe (voltage and frequency).

The grid frequency is $50\ \text{Hz}$ because the first power stations were built to rotate at that speed and the entire synchronised grid inherited the value. Explain why an entire interconnected grid must operate at exactly the same frequency and what happens if a power station leads or lags in phase.

From a plot of voltage against time, state how the peak value and period are measured directly, and how the RMS value and frequency are deduced from them. What information cannot be determined by looking at an instantaneous value alone?

Distinguish between the mean value and the RMS value of a sinusoidal wave: what the mean value of a full cycle is, and why the mean value of half a cycle (or of a rectified wave) is used to characterise it. What value does a basic AC multimeter measure?

A sinusoidal wave has a peak value of $10\ \text{A}$ and a frequency of $50\ \text{Hz}$. a) Calculate its RMS value and its period. b) Write its mathematical expression $i(t) = I_{max} \cdot \sen(\omega t)$ after calculating the angular frequency $\omega = 2 \pi f$. c) State the value of the current at $t = 0$ and explain when it reaches its maximum for the first time.

A resistor of $100\ \Omega$ is connected to an RMS supply of $230\ \text{V}$. a) Calculate the RMS current passing through it and the average power dissipated ($P = V_{ef} \cdot I_{ef}$ in alternating current with a resistive load). b) Calculate the maximum voltage supported by the resistor and explain why knowing it matters when choosing its insulation specifications.

Explain what the impedance of an alternating current circuit is and how it differs from a pure resistance: which components (inductors and capacitors) introduce a phase shift between voltage and current, and in what unit impedance is measured.

A sinusoidal signal has a frequency of $60\ \text{Hz}$ and an RMS value of $120\ \text{V}$. a) Calculate the period, angular frequency $\omega$, and maximum value. b) Write the complete expression $v(t)$. c) Calculate the instantaneous value at $t = 1\ \text{ms}$ (note: $\omega t$ in radians).

A heater of $2\,000\ \text{W}$ is connected to an RMS mains supply of $230\ \text{V}$. a) Calculate the RMS current and the operating resistance of the heater. b) Calculate the peak current supported by the cable in each half-cycle. c) Explain why a $10\ \text{A}$ circuit breaker is suitable for this circuit.

A guitar string produces an 'A' note of $440\ \text{Hz}$. a) Calculate its period. b) The microphone converts it into an electrical signal of the same frequency and a peak value of $100\ \text{mV}$: write its expression $v(t)$. c) What frequency does the 'A' note an octave above have, and how does its period relate to the original one?

Two waves of the same frequency, wave 1 and wave 2, are out of phase by a quarter of a period, such that wave 1 reaches its maximum value before wave 2. a) Determine which one is leading and how many degrees and milliseconds of lead there are between them (the frequency is $50\text{ Hz}$). b) Explain what type of load (inductive or capacitive) could produce such a phase difference between voltage and current.

A full-wave rectifier bridge converts the sine wave into a sequence of single-sign 'humps'. Explain what the diode bridge does, what average value the rectified wave has ($\frac{2}{\pi}$ of the peak value), and why a filter capacitor is added afterwards to obtain nearly smooth direct current.

In a laboratory practical, you connect an oscilloscope probe with $10\text{X}$ attenuation to measure the secondary voltage of a transformer connected to the mains. A classmate claims that the transformer is faulty because they observe a sinusoidal wave of $6$ vertical divisions peak-to-peak on the screen at a scale of $2\ \text{V/div}$, and each cycle spans exactly $4$ horizontal divisions with the timebase at $5\ \text{ms/div}$. In addition, you notice that they forgot to activate the $\times 10$ factor in the oscilloscope menu.

1. a) Determine the actual values of peak voltage, RMS voltage, and frequency of the secondary signal, justifying the effect of the probe attenuation.
2. b) Refute your classmate's claim about the signal frequency using calculations and calculate the transformation ratio $m = V_1 / V_2$ knowing that the primary RMS mains voltage is $230\ \text{V}$.
3. c) If you change the timebase to $2\ \text{ms/div}$, predict how many horizontal divisions a half-period will now occupy and deduce what horizontal scale setting you should select to display a minimum of $4$ complete periods on a screen with $10$ horizontal divisions.

In a $50\ \text{Hz}$ alternating current circuit, the excitation voltage $v_1(t)$ is measured on channel 1 and the voltage drop $v_2(t)$ across a passive two-terminal component on channel 2 using a two-channel oscilloscope. Both signals are pure sinusoids of equal amplitude ($V_{\text{max}} = 12\ \text{V}$), but it is observed that $v_2(t)$ reaches its positive maximum value $5\ \text{ms}$ after $v_1(t)$.

1. a) Calculate the time delay and phase shift (in degrees and radians) of $v_2(t)$ with respect to $v_1(t)$. Justify whether it is a lead or a lag and reasonably deduce what type of ideal component (resistor, pure inductor, or pure capacitor) constitutes the dipole if $v_1(t)$ represents the voltage drop across a preceding resistor in series with said component.
2. b) If you switch the oscilloscope to X-Y mode assigning $v_1(t)$ to the horizontal axis and $v_2(t)$ to the vertical axis, predict and justify the geometric trajectory that will be drawn on the screen, calculating its characteristic dimensions.
3. c) An operator claims that if the mains frequency increases to $100\ \text{Hz}$ while keeping the amplitude and circuit components constant, the phase shift between both signals will double. Formally refute their claim by distinguishing between phase shift and time delay.

A self-consumption photovoltaic installation generates an active power of $5\ \text{kW}$ that it feeds to the main switchboard through a single-phase copper line (resistivity $\rho = 0{,}018\ \Omega \cdot \text{mm}^2/\text{m}$) $60\ \text{m}$ long ($120\ \text{m}$ of conductor wire in total) at an RMS mains voltage of $230\ \text{V}$.

1. a) Assuming that the inverter injects in phase with the voltage ($\cos \varphi = 1$), determine the minimum standard commercial cross-section of the conductors (to choose from $4$, $6$, $10$, or $16\ \text{mm}^2$) so that the relative voltage drop in the line does not exceed the limit of $1{,}5\ \%$ required by the REBT.
2. b) For the commercial cross-section chosen in the previous section, calculate the power dissipated by the Joule effect in the evacuation line and the transmission efficiency.
3. c) The utility company requires the inverter to operate with an inductive power factor of $\cos \varphi = 0{,}85$ while maintaining the $5\ \text{kW}$ injected. Predict quantitatively how the current through the cable, the new percentage voltage drop, and the power lost through the Joule effect change. Justify whether the chosen cross-section still complies with the regulatory criterion of $1{,}5\ \%$.

An inexpensive AC digital voltmeter internally measures the rectified average value of the signal ($V_{\text{med}}$) and multiplies it by a fixed constant $k = 1{,}11$ (form factor of a sinusoidal wave) to display a calibrated reading as an RMS value on the screen.

1. a) Analytically demonstrate that for a pure sinusoidal signal $v(t) = V_0 \operatorname{sen}(\omega t)$, the scale factor relating the RMS value ($V_{\text{ef}} = V_0 / \sqrt{2}$) to the rectified average value ($V_{\text{med}} = \frac{2}{\pi} V_0$) is indeed $k = \frac{\pi}{2\sqrt{2}} \approx 1{,}11$.
2. b) This instrument is used to measure an alternating voltage in a symmetrical square wave of amplitude $\pm 10\ \text{V}$. Calculate the true RMS value ($V_{\text{RMS}}$), the value the voltmeter will display on its screen, and the percentage relative error made by the reading.
3. c) If the output voltage of an unfiltered half-wave rectifier supplied with $230\ \text{V}$ RMS sinusoidal voltage is then measured, deduce whether the budget voltmeter will overestimate or underestimate the true thermal RMS value that a resistor connected to said output would dissipate, calculating the relative error numerically.

A power quality analyzer measures the current drawn by a set of computers and LED luminaires powered at $230\ \text{V}$ RMS ($50\ \text{Hz}$ sinusoidal). The Fourier spectral analysis of the phase current yields the following values:

• Fundamental component ($50\ \text{Hz}$): $I_{1,\text{ef}} = 20\ \text{A}$ with a lagging voltage-current phase shift $\varphi_1 = 30^\circ$.

• Third harmonic ($150\ \text{Hz}$): $I_{3,\text{ef}} = 8\ \text{A}$.

• Fifth harmonic ($250\ \text{Hz}$): $I_{5,\text{ef}} = 4\ \text{A}$.

(Higher-order harmonics are neglected).

1. a) Calculate the total RMS value of the current ($I_{\text{ef}}$) and the total harmonic distortion of current ($\text{THD}_I$).
2. b) Calculate the actual active power consumed and determine the total power factor of the installation ($FP = P / S$), comparing it quantitatively with the displacement power factor ($\cos \varphi_1$). Justify what causes the difference.
3. c) In a balanced three-phase distribution with these same loads, explain physically why the current through the neutral conductor does not cancel out and calculate the RMS current that would circulate through the neutral due exclusively to the third harmonic.

An ideal single-phase transformer of $230\ \text{V} / 24\ \text{V}$ at $50\ \text{Hz}$ has $N_1 = 1150$ turns in the primary winding and an internal ohmic resistance of the copper wire of $R_1 = 2\ \Omega$. In sinusoidal AC steady state, its no-load magnetizing reactance is $X_m = 460\ \Omega$.

1. a) From Faraday-Lenz's law of induction ($e = -N \frac{d\Phi}{dt}$), derive the relationship between the applied RMS voltage and the maximum magnetic flux $\Phi_{\text{max}}$ in the core, and calculate the value of $\Phi_{\text{max}}$ for the nominal AC voltage.
2. b) An operator mistakenly connects the primary winding to a $24\ \text{V}$ DC power source. Predict with reasoning the value of the voltage across the secondary terminals once the DC steady state is reached, and calculate the current that will circulate through the primary.
3. c) Justify, by means of a comparative calculation of the power dissipated by Joule heating in the primary, why the transformer would burn out in DC with only $24\ \text{V}$, whereas it withstands $230\ \text{V}$ in AC operating at no load.

In an industrial workshop, you want to verify the consumption of a single-phase machine connected to a $230\ \text{V}$ RMS, $50\ \text{Hz}$ line.

1. a) An operator clamps the clamp meter around the outer jacket containing both conductor wires (phase and neutral), and the clamp reads $0{,}00\ \text{A}$, even though the machine is running. Justify analytically using Ampère's law why this happens, and explain how the clamp must be placed to measure the load current.
2. b) After separating the conductors and placing the clamp around the phase wire, a reading of $16\ \text{A}$ is obtained. Using the analog voltage output of the clamp ($100\ \text{mV/A}$) and a voltage probe connected to an oscilloscope, it is observed that the voltage peak leads the current peak by $3{,}33\ \text{ms}$. Calculate the phase angle $\varphi$, the power factor, and the active power consumed by the machine.
3. c) Due to an insulation fault, a leakage current of $400\ \text{mA}$ is diverted to earth. If the clamp is placed again around the jacket that includes only phase and neutral (without the earth wire), predict with reasoning the value that the clamp will now register, and explain how a residual current device (RCD) uses this physical principle to protect users.

During the quality control of a $12\ \text{V}$ regulated linear DC power supply ($230\ \text{V}$ at $50\ \text{Hz}$ at the input), a pure resistive load of $4\ \Omega$ is connected to the output. The test bench yields the following readings:

• Input: RMS voltage of $230\ \text{V}$, RMS current of $0{,}35\ \text{A}$, and absorbed active power of $48\ \text{W}$.

• Output: average DC voltage of $11{,}8\ \text{V}$. When observing the output with the oscilloscope in AC coupling ($20\ \text{mV/div}$), a symmetrical triangular ripple of $3$ divisions peak-to-peak with a period of $10\ \text{ms}$ is observed. At no load, the DC voltage is $12{,}1\ \text{V}$.

1. a) Calculate the ripple frequency and identify the type of rectifier used. Determine the RMS ripple voltage ($V_{\text{riz,ef}} = \frac{V_{\text{riz,pp}}}{2\sqrt{3}}$), the percentage ripple factor at full load ($r = \frac{V_{\text{riz,ef}}}{V_{\text{CC}}} \cdot 100$), and the percentage load-to-no-load voltage regulation ($\frac{V_{\text{vacío}} - V_{\text{carga}}}{V_{\text{carga}}} \cdot 100$).
2. b) Calculate the overall energy efficiency of the power supply ($\eta$) and its input power factor ($FP$).
3. c) The design specifications require: ripple factor $r \le 0{,}3\ \%$, load regulation $\le 3\ \%$, and efficiency $\eta \ge 70\ \%$. Determine which criteria are met and which are not, and technically justify two modifications to the design (in the filter or in the regulator) to correct the detected shortcomings.