AS 91526 · 2023 paper · alternating current

Question 3(b) Merit to Excellence

Practises AC theory: RMS, reactance, impedance & resonance

Kate builds a series circuit with a signal generator at 200 Hz, an AC ammeter, a lamp of 15.0 Ω and a 1.20 µF capacitor.

f	200 Hz
R	15.0 Ω
C	1.20 µF

Kate raises the frequency from 200 Hz to 20 kHz, then to 200 kHz.

Give an in-depth explanation of what she observes at each frequency compared with part (a) — the impedance, the RMS current, and the brightness of the lamp.

The 2023 paper, page 8 ↗ Marking schedule, page 6 ↗

Work it out

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

1 How does a capacitor’s reactance change as the frequency rises?
X_C = 1/(2πfC) falls as f rises: a hundred times the frequency, a hundredth of the reactance. At 20 kHz X_C = 6.63 Ω; at 200 kHz, 0.663 Ω.
2 What does that do to the impedance at each frequency?
Z = √(15.0² + X_C²). At 20 kHz, Z = 16.4 Ω, down from 663 Ω. At 200 kHz, Z = 15.01 Ω: it cannot go below the lamp’s 15.0 Ω, so it barely changes from 20 kHz.
3 Take the signal generator’s voltage as fixed. What happens to the RMS current?
I = V / Z. At 20 kHz it is about 40 times what it was at 200 Hz (663.3 / 16.4). At 200 kHz it rises only about 9% more (16.4 / 15.01).
4 And the lamp?
Brightness follows the power, P = I²R. At 200 Hz the lamp is barely lit, if at all. At 20 kHz its power is about 1600 times bigger, so it glows brightly. At 200 kHz it gets about 19% more power again, which is hard to see.

Answer Taking the generator’s voltage as fixed: at 20 kHz, X_C drops to 6.63 Ω and Z to 16.4 Ω, so the current is about 40 times bigger and the lamp goes from barely lit to bright. At 200 kHz: X_C is 0.663 Ω, but Z can only fall to about R (15.01 Ω), so the current and brightness rise only a little more.

What the marker wanted

Achieved One idea: X_C or Z at either new frequency (6.63 Ω and 16.4 Ω; 0.663 Ω and 15.01 Ω), or that a higher frequency means a smaller reactance, a smaller impedance, more current or a brighter lamp.
Merit The current and the brightness both explained for one of the two frequencies, with one aspect missing.
Excellence A complete answer for both 20 kHz and 200 kHz. Numbers are not required if the reasoning is right and goes deep enough.

In our words, from NZQA’s 2023 marking schedule.

Common mistake

Explaining one frequency only. That is the Merit column; Excellence needs both, and NZQA’s 2023 report puts the whole chain (reactance, impedance, current and power) at Excellence.

How did it go?

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