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Hartley oscillator frequency

Short answer

The tank uses tapped inductance (L1, L2) and a capacitor C. Frequency f = 1/[2π√((L1+L2+2M)C)]. Feedback from the tap to base, with common‑emitter phase shift, gives positive feedback. Gain condition hfe ≥ L1/L2.

Draw the circuit diagram of the Hartley Oscillator and write down the expression for its frequency of oscillation.

The Hartley oscillator is an LC oscillator in which the tank circuit has two inductors (a tapped coil) L1 and L2 and one capacitor C. The feedback signal is taken from the tap between the inductors.

Circuit diagram (BJT, common emitter)

 Collector o---+------+
               |      |
             [ L1 ]   |
               |     [ C ]
 Ground (tap) o|      |
               |      |
             [ L2 ]   |
               |      |
 Base      o---+------+

The tank has C across the series combination of L1 and L2; the junction of L1 and L2 is at ac ground. The transistor is a common-emitter stage with voltage-divider bias, an emitter resistor with bypass capacitor, an RF choke in the collector supply and coupling capacitors.

  • L1 is between the collector and the tap; L2 is between the tap and the base. The voltage across L2 is fed back to the base.
  • Because the tap is at ac ground, the collector and base voltages are 180° apart. This 180°, with the 180° shift of the common-emitter stage, gives 360° (positive feedback).

Frequency of oscillation

Total inductance with mutual inductance M between the coils: Leq = L1 + L2 + 2M

f = 1 / (2π √(Leq C)) = 1 / (2π √((L1 + L2 + 2M) C))

For no mutual coupling: f = 1/(2π √((L1 + L2) C)).

Condition

Gain: hfe ≥ L1/L2 (for a BJT amplifier with negligible loading), that is, the loop gain at least unity (Aβ ≥ 1), where β = L2/L1.

Hartley oscillators are used for RF generation, and frequency is varied easily by a variable capacitor.

✓ Answer checked by Claude Opus 5.5 (high effort)

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