Teaching guide for the module

Behaviour of R, L, C in alternating current: Teacher's Guide

Teaching guide for the AC Behaviour (R, L, C) simulator: explain in class the phase shift between voltage and current in reactive components, impedance, rotating phasors and resonance in a series RLC circuit. Basic and Advanced modes for technical secondary schools, electrical and electronics specialisations.

Module: AC Behaviour (R, L, C) · Basic / Advanced mode · Oscilloscope + phasor diagram

Physical phenomenon

In sinusoidal alternating current, voltage and current are no longer linked by simple proportionality as in DC. Each passive component introduces a characteristic phase shift between $v (t)$ and $i (t)$ :

Pure resistor (R): $i (t)$ in phase with $v (t)$ , $φ = 0°$ .
Pure inductor (L): $i (t)$ lags $v (t)$ by $90°$ .
Pure capacitor (C): $i (t)$ leads $v (t)$ by $90°$ .

The impedance $Z$ is introduced as the generalisation of resistance to the sinusoidal regime:

$Z_{R} = R, Z_{L} = j ω L, Z_{C} = \frac{1}{j ω C}$

where $ω = 2 π f$ is the angular frequency. The modulus $∣ Z ∣$ sets the amplitude ratio, the argument $φ$ sets the phase shift.

In a series RLC circuit, total impedance is $Z = R + j (ω L - 1/ ω C)$ , and resonance occurs when $ω L = 1/ ω C$ , i.e. $f_{0} = \frac{1}{2 π L C}$ : the reactive part vanishes, $Z = R$ , and current peaks.

Phasors are rotating vectors representing sinusoidal quantities: length is the peak value, angle is the initial phase. They are the standard language of AC electrical engineering.

Key concepts

Phase shift $φ$ : the electrical "signature" of each component in AC.
Reactance: the "non-dissipative" opposition to current flow, frequency-dependent ( $X_{L} = ω L$ , $X_{C} = 1/ ω C$ ).
Impedance $Z$ : generalises resistance, is a complex number (modulus + phase).
RMS value: for a pure sinusoid $V_{R M S} = V_{p e ak} / 2$ . It is the quantity read on a multimeter.
RLC resonance: the frequency at which the circuit behaves as purely resistive.
Phasor vs sinusoid: two languages for the same reality: the sinusoid is the "time projection" of the rotating phasor.

How to use it in the classroom

Opening: Basic mode. Select the R component: V and I are overlapping sinusoids, in phase. Switch to L: current "trails" by a quarter period. Then C: current "leads". Have students verbalise what they see before writing $φ = \pm 90°$ on the board. This is the moment when the three rules sink in intuitively.

Development: switching to Advanced. In Advanced mode, phasors appear next to the oscilloscope. Press the PHASORS tab in the visualisation panel. Show that the current phasor forms an angle of $- 90°$ (inductive) or $+ 90°$ (capacitive) with the voltage phasor. Explain that phasor and sinusoid represent the same quantity in two languages: temporal (sinusoid) and geometric (phasor).

Deeper exploration: frequency dependence. Vary the frequency: with an inductor the current amplitude decreases as $f$ rises (because $X_{L}$ grows). With a capacitor the opposite happens. This is a direct visual demonstration leading naturally to the filter concept, you can preview the Filters module here.

Closing: RLC resonance. Select the series RLC circuit and vary the frequency until the V and I phasors are aligned: this is resonance. Amplitudes peak. Have students compute $f_{0} = 1/ (2 π L C)$ from the chosen values and verify the match.

Real-world examples

Electrical distribution. Power grids run in AC (50 Hz in Europe) to allow voltage transformation via transformers, impossible in DC.
Asynchronous motors. They work by exploiting the phase shift of sinusoidal currents to generate a rotating magnetic field. The motor's inductive behaviour is the main cause of low power factor in industrial loads.
Radio tuning. A parallel RLC circuit is "tuned" to the desired frequency by adjusting $L$ or $C$ , the principle of an analogue radio tuner.
Switching power supplies. They use inductors and capacitors at switching frequencies of hundreds of kHz to transfer energy efficiently.
Capacitive compensation. Capacitor banks are installed in industrial cabinets to compensate the inductive phase shift of motors (a natural bridge to the Power Factor module).

Classroom discussion questions

With a single resistor, V and I are in phase. What does "in phase" mean graphically, and why is it a notable property in AC?
What happens to an inductor's reactance if I double the frequency? And to a capacitor's?
At RLC series resonance, why is current at its maximum despite L and C being present, which "block" current?
A multimeter shows 230 V on a household socket, but the peak voltage is about 325 V. Why?
Why does the power grid use AC and not DC, even though our devices run on DC?

Related modules

Ohm's Law & Power Management: Ohm's Law in DC is the conceptual foundation of impedance: in AC the "resistance" becomes a complex number.
Capacitor Charge & Discharge: the same RC circuit, here in sinusoidal regime, there in transient regime.
Filters: direct application of $Z$ 's frequency dependence to select signal bands (low-pass, high-pass, band-pass).
Power Factor & AC Power: the phase shift $φ$ introduced here translates into active, reactive and apparent powers.
Three-Phase AC Systems: the extension of the sinusoidal regime to three phases shifted by $120°$ , the foundation of industrial distribution.