Hartley Oscillator

Before the Colpitts circuit existed, Edwin Hartley patented an LC oscillator in 1915 that used a tapped inductor instead of two separate capacitors. The Hartley oscillator is still used today in AM radio transmitters and function generators because it is easy to tune with a single variable capacitor. A BC107 or 2N3904 transistor suits the low-to-mid frequency range well.

Core Concept

The Hartley oscillator differs from the Colpitts in that it uses a tapped inductor to produce the feedback signal. The inductor is split into two sections, L1 and L2, with a centre tap. The tank circuit consists of the total inductance (L1+L2) in parallel with a single tuning capacitor C.

The inductive voltage divider formed by L1 and L2 sets the feedback fraction. The voltage across L2 (from tap to ground) is fed back to the transistor base. Because both L1 and L2 carry the same current in a series path, the voltage divides in proportion to their inductance values. This makes the feedback fraction β = L2/(L1+L2).

The 2N3904 in common-emitter configuration gives a 180° phase shift. The inductive tap also produces a 180° phase shift due to the inversion between collector and emitter. Total loop phase is 360°, satisfying Barkhausen. The main advantage of the Hartley design is that changing a single capacitor C tunes the frequency without disturbing the feedback ratio, unlike Colpitts where changing either capacitor changes both frequency and feedback simultaneously.

Key Equations

Oscillation frequency:

f0 = 1 / (2π × sqrt((L1 + L2 + 2M) × C)) where L1 and L2 are the two inductance sections in henries, M is the mutual inductance between them (often zero if physically separated), and C is the tuning capacitor in farads.

If L1 and L2 are separate (no mutual coupling), M = 0, so:

f0 = 1 / (2π × sqrt(Ltotal × C)) where Ltotal = L1 + L2 in henries.

Feedback fraction:

β = L2 / (L1 + L2) (assuming M = 0). The Barkhausen gain condition requires the transistor gain Av ≥ L1/L2.

Given:
  L1 = 50 µH, L2 = 50 µH (equal, tapped at centre)
  C = 100 pF = 100 × 10^-12 F
  Mutual inductance M = 0 (physically separated coil sections)

Why this formula:
  Hartley frequency uses total inductance L1 + L2 with the tuning capacitor.

Formula:
  Ltotal = L1 + L2 = 50 + 50 = 100 µH = 100 × 10^-6 H
  f0 = 1 / (2π × sqrt(Ltotal × C))

Substitution:
  f0 = 1 / (2π × sqrt(100 × 10^-6 × 100 × 10^-12))
     = 1 / (2π × sqrt(10^-14))
     = 1 / (2π × 10^-7)

Calculation:
  f0 = 1 / (6.2832 × 10^-7)
     = 1.592 × 10^6 Hz

Feedback fraction:
  β = L2 / (L1 + L2) = 50 / 100 = 0.5
  Required transistor gain Av ≥ L1/L2 = 50/50 = 1 (easily satisfied)

Final Answer:
  f0 ≈ 1.59 MHz, feedback fraction β = 0.5

Exam Tip: GATE problems on Hartley oscillators often include mutual inductance M in the inductor. When the two coil sections are wound on the same core, the effective inductance is L1 + L2 + 2M, not L1 + L2. Students often forget to add 2M, causing an error in frequency calculation. Also note that the Barkhausen gain condition for Hartley is Av ≥ L1/L2, which is the inverse of the feedback fraction, similar to how the Colpitts condition is Av ≥ C1/C2.

Key Properties

• Tank circuit: total inductance (L1+L2+2M) in parallel with a single capacitor C. Changing C tunes frequency without affecting feedback.
• Feedback fraction β = L2/(L1+L2) for uncoupled sections. This determines what fraction of tank voltage reaches the transistor base.
• When L1 and L2 are on the same core, mutual inductance M adds to the effective tank inductance as Leq = L1+L2+2M.
• Barkhausen gain condition: transistor voltage gain Av must be at least L1/L2 for oscillation to be sustained.
• Typical frequency range is 100 kHz to 30 MHz. Above this, winding parasitic capacitance degrades performance.
• Hartley is easily tuned with a variable capacitor. This is why it was popular in early radio receivers where continuous tuning across the AM band was needed.
• Compared to Colpitts, Hartley has slightly lower frequency stability because inductors have more parasitic resistance (lower Q) than capacitors.

Quick Revision

• Hartley oscillator uses a tapped inductor (L1, L2) and one tuning capacitor C.
• f0 = 1 / (2π × sqrt((L1+L2+2M) × C)). Use 2M when inductors are coupled.
• Feedback fraction β = L2/(L1+L2). Gain condition: Av ≥ L1/L2.
• Loop phase: 180° from transistor (common emitter) + 180° from inductive tap = 360°.
• Advantage over Colpitts: single capacitor tunes frequency without changing β.
• Disadvantage: inductor winding has resistive loss, lowering Q compared to Colpitts capacitors.
• Practical range: 100 kHz to 30 MHz. Used in AM radio local oscillators.
• Exam trap: Forgetting to include 2M in the frequency formula when the two inductor sections share a magnetic core. This makes the calculated frequency higher than the actual frequency.