Magnitudes 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).

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.

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.

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.

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.

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 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).

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.

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?

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).

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.

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.

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.