RC and LCR circuits, oscillations and resonance
Answers are for parents and teachers.
1
A series RC circuit has R = 20 Ω and XC = 15 Ω on a 100 V supply. Find Z, I, VR, VC, tan θ (θ from the 3-4-5 triangle), and say whether the voltage leads or lags.
Z = √(20² + 15²) = 25 Ω. I = 100 ÷ 25 = 4 A. VR = 80 V and VC = 60 V (√(80² + 60²) = 100 V ✓). tan θ = 15 ÷ 20 = 0.75, so θ = 37°, and the voltage lags the current.
2
A series LCR circuit has R = 10 Ω, XL = 40 Ω and XC = 16 Ω on 130 V. Find X, Z, I, the power used and tan θ, and say whether it is inductive or capacitive.
X = 40 − 16 = 24 Ω. Z = √(10² + 24²) = 26 Ω. I = 130 ÷ 26 = 5 A. P = 5² × 10 = 250 W. tan θ = 24 ÷ 10 = 2.4 (θ about 67°). XL > XC, so it is inductive and the voltage leads.
3
An oscillating circuit has L = 2 mH and C = 8 µF. Find its natural frequency (π = 3.14).
L = 2 × 10⁻³ H, C = 8 × 10⁻⁶ F, LC = 1.6 × 10⁻⁸ s², √(LC) = 1.265 × 10⁻⁴ s. f = 1 ÷ (2 × 3.14 × 1.265 × 10⁻⁴) = 1259 Hz, about 1.26 kHz.
4
A radio coil has L = 100 µH and its variable capacitor goes from 50 pF to 200 pF. Find the range of frequencies it can be tuned to (π = 3.14).
C = 50 pF: LC = 10⁻⁴ × 5 × 10⁻¹¹ = 5 × 10⁻¹⁵, √(LC) = 7.07 × 10⁻⁸ s, f = 1 ÷ (2 × 3.14 × 7.07 × 10⁻⁸) = 2.25 × 10⁶ Hz = 2.25 MHz. C = 200 pF: LC = 2 × 10⁻¹⁴, √(LC) = 1.414 × 10⁻⁷ s, f = 1.13 MHz. The range is about 1.13 MHz to 2.25 MHz (C four times larger, f half as large).
5
Spot the error. A student has R = 8 Ω, XL = 35 Ω and XC = 20 Ω in series and writes Z = 8 + 35 + 20 = 63 Ω. Find the mistake and the correct Z.
R, XL and XC are not added as plain numbers: XL and XC point in opposite ways, so X = 35 − 20 = 15 Ω, and R is at right angles to X. Z = √(8² + 15²) = 17 Ω.