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AS 91526 · 2025 paper · AC inductors and capacitors

Question 3(c) Merit to Excellence

Practises AC theory: RMS, reactance, impedance & resonance

The 0.600 H coil (an ideal inductor plus a 2.40 Ω resistor) is connected in series with a 68.0 Ω resistor to a variable-frequency AC supply set to 25.0 Hz and 9.00 V RMS.

L0.600 H
R(coil)2.40 Ω
R68.0 Ω
V RMS9.00 V
f25.0 Hz

The nail is removed and a 1.20 × 10⁻⁴ F capacitor is added in series with the coil and the resistor. As the frequency is reduced, the voltage across the 68.0 Ω resistor rises to a maximum at one particular frequency.

Explain why it reaches a maximum at that one frequency. No calculations required.

The 2025 paper, page 8 ↗ Marking schedule, page 6 ↗

Work it out

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

  1. 1 As the frequency falls, what happens to XL and to XC?

    XL = 2πfL falls with frequency and XC = 1/(2πfC) rises. At 25.0 Hz, XL ≈ 94 Ω is bigger than XC ≈ 53 Ω, so at some lower frequency they are equal.

  2. 2 What is special about the frequency where they are equal?

    That is resonance. The voltages across the inductor and the capacitor are the same size and opposite in phase, so they cancel.

  3. 3 With the reactances cancelled, what is left of the impedance?

    Only resistance: Z = R, the smallest it can be. The current is in phase with the supply and at its maximum, and the whole supply voltage sits across the circuit’s resistance, the 68.0 Ω plus the coil’s 2.40 Ω. So Z = 70.4 Ω, I = 9.00 / 70.4 = 0.128 A, and the 68.0 Ω resistor gets 0.128 × 68.0 ≈ 8.69 V of the 9.00 V: slightly less than all of it, because the coil’s resistance takes the rest.

  4. 4 Link that back to the 68.0 Ω resistor.

    VR = IR, so the biggest current gives the biggest voltage across the resistor. Either side of that frequency one reactance outweighs the other, Z rises and the current falls.
Answer At one frequency XL = XC: resonance. The inductor and capacitor voltages cancel, Z = R is at its minimum, the current is at its maximum, and so VR = IR is at its maximum. (It is about 18.8 Hz, though no calculation is asked for.)

What the marker wanted

  • Achieved One idea: resonance, XL = XC, VL = VC, VR = VS, Z = R, Z at its minimum, or I at its maximum.
  • Merit Names resonance and gives most of the chain: equal reactances (or voltages), Z = R and at its minimum, the current at its maximum.
  • Excellence Explains resonance and gives all of it: XL and XC equal and opposite so they cancel, VR = VS, Z = R at its minimum, and I at its maximum.

In our words, from NZQA’s 2025 marking schedule.

✕

Common mistake

Naming resonance, or writing XL = XC, and stopping there. On its own that is one Achieved idea; the grade comes from explaining why equal reactances make the current, and so VR, as large as it can be.

How did it go?

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