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AS 91526 · 2024 paper · alternating current

Question 3(d) Merit to Excellence

Practises AC theory: RMS, reactance, impedance & resonance

Rachel connects a 70.0 mH inductor with a soft iron core and a 40.0 Ω bulb in series to a 110 V RMS, 60.0 Hz supply.

L70.0 mH
R40.0 Ω
V RMS110 V
f60.0 Hz

The iron core is reinserted and a capacitor is added to the circuit to achieve resonance.

Explain why the reactance of the capacitor needs to be 26.4 Ω. Begin by describing what happens at resonance.

The 2024 paper, page 7 ↗ Marking schedule, page 7 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

  1. 1 What is true of the circuit at resonance?

    The inductor’s and the capacitor’s reactances are equal, so the voltages across them are equal in size.

  2. 2 Why do those two voltages cancel?

    The inductor’s voltage is 90° ahead of the current and the capacitor’s is 90° behind, so they are 180° apart. On a phasor diagram they point opposite ways and add to zero.

  3. 3 What is left of the impedance, and what does that do to the current?

    Only the bulb’s resistance: Z = R = 40.0 Ω, the smallest Z can be. So the current is at its largest, 110 / 40.0 = 2.75 A.

  4. 4 Which reactance is already fixed, and what must the capacitor’s be?

    The inductor is unchanged (70.0 mH, with its core back in) and the supply is still 60.0 Hz, so XL is still 26.4 Ω. To cancel it, XC must be 26.4 Ω as well. (That takes C = 1/(2π × 60.0 × 26.4) ≈ 100 µF, though it is not asked.)
Answer At resonance XC = XL: the two voltages are 180° out of phase and cancel, so Z = R (its minimum) and the current is at its maximum. The inductor’s reactance at 60.0 Hz is 26.4 Ω, so the capacitor’s must be 26.4 Ω too.

What the marker wanted

  • Achieved One statement: the current is at its maximum, Z = R, Z is at its minimum, or the capacitor’s reactance cancels the inductor’s.
  • Merit Any two of those.
  • Excellence All of them: current at its maximum, Z = R (or its minimum), and the two reactances cancelling, with the reason: 180° out of phase, shown by an equation or a phasor diagram.

In our words, from NZQA’s 2024 marking schedule.

✕

Common mistake

Writing “XL = XC at resonance” and stopping. The schedule’s Excellence column also wants why they cancel and what that does to Z and the current, and NZQA’s 2024 report puts that whole explanation at Excellence.

How did it go?

Got it moves it further out. Review brings it back tomorrow.