Colpitts Oscillator

RF signal generators in laboratory equipment and the local oscillator in AM radio receivers both use a Colpitts oscillator to produce stable sine waves. It uses two capacitors and one inductor in a pi-network to generate frequencies from a few kilohertz up to several hundred megahertz. The BC547 transistor is commonly used for implementations up to 100 MHz.

Core Concept

The Colpitts oscillator uses an LC tank circuit for frequency selection. The tank has one inductor L and two capacitors C1 and C2 in series forming the capacitive leg. Energy oscillates between the magnetic field in L and the electric field in C1 and C2. The resonant frequency is set by L and the series combination of C1 and C2.

The capacitive voltage divider formed by C1 and C2 also serves as the feedback network. The voltage across C2 is fed back to the base of the transistor. The ratio C1/(C1+C2) determines the feedback fraction β. This elegant dual use of C1 and C2 for both frequency determination and feedback is what makes the Colpitts topology compact and popular.

The transistor provides the amplification needed to satisfy the Barkhausen magnitude condition |Aβ| = 1. The BC547 in common-emitter configuration gives a phase shift of 180°. The capacitive voltage divider also introduces a 180° phase shift, so the total around the loop is 360°, satisfying the Barkhausen phase condition.

Key Equations

Oscillation frequency:

f0 = 1 / (2π × sqrt(L × Ceq)) where Ceq = C1×C2/(C1+C2) is the series equivalent capacitance of C1 and C2.

Feedback fraction:

β = C1 / (C1 + C2) assuming equal impedances. This is the fraction of output voltage fed back to the transistor base.

Barkhausen magnitude condition:

gm × RL ≥ C1 / C2 where gm is the transistor transconductance in A/V and RL is the effective collector load resistance in Ω.

Given:
  L = 10 µH = 10 × 10^-6 H
  C1 = 100 pF = 100 × 10^-12 F
  C2 = 100 pF = 100 × 10^-12 F

Why this formula:
  Colpitts frequency is determined by L and Ceq = series combination of C1 and C2.

Formula:
  Ceq = (C1 × C2) / (C1 + C2)
  f0 = 1 / (2π × sqrt(L × Ceq))

Substitution:
  Ceq = (100 × 10^-12 × 100 × 10^-12) / (100 × 10^-12 + 100 × 10^-12)
      = (10^-20) / (200 × 10^-12)
      = 50 × 10^-12 F = 50 pF

  f0 = 1 / (2π × sqrt(10 × 10^-6 × 50 × 10^-12))
     = 1 / (2π × sqrt(5 × 10^-16))
     = 1 / (2π × 2.236 × 10^-8)
     = 1 / (1.405 × 10^-7)

Calculation:
  f0 = 7.12 × 10^6 Hz

Final Answer:
  Oscillation frequency f0 ≈ 7.12 MHz

Exam Tip: The most common Colpitts mistake in GATE is using C1 in parallel instead of in series for the equivalent capacitance. C1 and C2 are in series in the tank circuit, so Ceq = C1C2/(C1+C2), not C1+C2. Also remember: feedback fraction β = C1/(C1+C2), not C2/C1. If C1 = C2, then Ceq = C/2 and β = 1/2. The Barkhausen gain condition gm·RL ≥ C1/C2 is also tested.

Key Properties

• Colpitts uses two capacitors and one inductor. The series equivalent capacitance Ceq = C1C2/(C1+C2) sets the resonant frequency with L.
• Feedback fraction β = C1/(C1+C2). Increasing C1 relative to C2 increases feedback but also changes the oscillation frequency.
• The BC547 transistor in common-emitter gives 180° phase shift. The capacitive divider adds another 180°, giving 360° total loop phase.
• Oscillation frequency stability is better than RC oscillators because the LC tank has a high Q factor, typically 50 to 200.
• Practical Colpitts circuits operate from a few kilohertz to several hundred megahertz. Above 100 MHz, parasitic capacitances become significant.
• The radio frequency choke (RFC) in the collector supply line provides DC bias while presenting high impedance to the RF signal.
• If C1 = C2 = C, then Ceq = C/2 and f0 = 1/(2π × sqrt(L × C/2)), which is sqrt(2) higher than with one capacitor C alone.

Quick Revision

• Tank circuit: L in parallel with series combination of C1 and C2.
• Ceq = C1C2/(C1+C2) for the frequency formula.
• f0 = 1 / (2π × sqrt(L × Ceq)).
• Feedback fraction β = C1/(C1+C2).
• Total loop phase = 180° (transistor) + 180° (cap divider) = 360°.
• Gain condition: gm × RL ≥ C1/C2 for sustained oscillation.
• Higher Q of LC tank gives better frequency stability than RC oscillators.
• Exam trap: Using Ceq = C1 + C2 (parallel formula) instead of C1C2/(C1+C2) (series formula). C1 and C2 are always in series in the tank arm of a Colpitts oscillator.