Question

Find the operating frequency of a transistor Hartley oscillator for \(L_1 = 20\mu\text{H}\), \(L_2 = 40\mu\text{H}\) and mutual inductance between coils is \(2\mu\text{H}\) and \(C = 1\mu\text{F}\).

• A. \( \frac{62.5}{\pi} \text{ kHz} \)
• B. \( \frac{50.5}{\pi} \text{ kHz} \)
• C. \( 12.5\pi \text{ kHz} \)
• D. \( 50\pi \text{ kHz} \)

Hint:

Hartley oscillator resonant frequency: \(f_0 = 1 / (2\pi\sqrt{L_{eq}C})\).
Equivalent inductance for series aiding inductors with mutual inductance: \(L_{eq} = L_1 + L_2 + 2M\).
Ensure units are consistent (e.g., H, F, Hz). \(\mu = 10^{-6}\).

Answer 1

The Correct Option is A

The resonant frequency of a Hartley oscillator is given by: \[ f_0 = \frac{1}{2\pi\sqrt{L_{eq}C}} \] In a Hartley oscillator, the inductive part of the tank circuit consists of two inductors \(L_1\) and \(L_2\) in series, often with mutual inductance \(M\) between them. The equivalent inductance \(L_{eq}\) is given by \(L_{eq} = L_1 + L_2 + 2M\) (if the fields are aiding, which is typical for oscillator configuration). Given: \(L_1 = 20\mu\text{H} = 20 \times 10^{-6} \text{ H}\) \(L_2 = 40\mu\text{H} = 40 \times 10^{-6} \text{ H}\) \(M = 2\mu\text{H} = 2 \times 10^{-6} \text{ H}\) \(C = 1\mu\text{F} = 1 \times 10^{-6} \text{ F}\) Calculate \(L_{eq}\): \(L_{eq} = (20 + 40 + 2 \times 2) \times 10^{-6} \text{ H} = (60 + 4) \times 10^{-6} \text{ H} = 64 \times 10^{-6} \text{ H}\). Now calculate \(f_0\): \[ f_0 = \frac{1}{2\pi\sqrt{(64 \times 10^{-6})(1 \times 10^{-6})}} = \frac{1}{2\pi\sqrt{64 \times 10^{-12}}} \] \[ f_0 = \frac{1}{2\pi \times \sqrt{64} \times \sqrt{10^{-12}}} = \frac{1}{2\pi \times 8 \times 10^{-6}} \] \[ f_0 = \frac{1}{16\pi \times 10^{-6}} = \frac{10^6}{16\pi} \text{ Hz} \] To express in kHz and match options: \[ f_0 = \frac{1000 \times 10^3}{16\pi} \text{ Hz} = \frac{1000}{16\pi} \text{ kHz} \] \[ \frac{1000}{16} = \frac{500}{8} = \frac{250}{4} = \frac{125}{2} = 62.5 \] So, \[ f_0 = \frac{62.5}{\pi} \text{ kHz} \] This matches option (a). \[ \boxed{\frac{62.5}{\pi} \text{ kHz}} \]