Colpitts oscillator frequency

The Colpitts oscillator is an LC oscillator whose tank uses a tapped capacitor pair (C1, C2) with a single inductor L. The feedback voltage is taken across one capacitor.

Circuit diagram (BJT, common emitter)

                 +VCC
                  |
        +---------+
        |         |
      [R1]      [RFC]
        |         |
        |         +----||---+---------+
        |         |    Cc2  |         |
        |         C        [C1]       |
  +-----+------B--|  Q1     |        [L]
  |     |         E        GND        |
  |   [R2]        |         |         |
  |     |       [RE]||[CE] [C2]       |
  |    GND        |         |         |
  |              GND        |         |
  |                         |         |
  +---------||--------------+---------+
            Cc1

The collector is coupled through Cc2 to the top of the tank. C1 runs from the top of the tank to ground, C2 from ground to the bottom of the tank, and L is across the whole C1-C2 series combination. The bottom of the tank is coupled back to the base through Cc1. The output is taken from the collector (or the tank).

Tank connections in a simple form:

 Collector o----+------+
                |      |
               [C1]    |
                |     [L]
 Emitter/GND o--+      |
                |      |
               [C2]    |
                |      |
 Base        o--+------+

C1 is between collector and ground, C2 between base and ground (emitter), and L is across the series combination. Bias: voltage divider R1, R2; emitter resistor RE with bypass capacitor CE; RF choke in the collector supply.

Working

The CE stage gives a 180° phase shift. The capacitor divider (with its junction at ground) gives another 180° between the collector and base voltages. So the total is 360° and the feedback is positive. The feedback fraction is β = C1/C2.

Derivation of frequency

Neglecting loading by the transistor (hie, hoe), the tank is L in parallel with the series combination of C1 and C2:

Ceq = C1 C2 / (C1 + C2)

At resonance the inductive and capacitive reactances are equal: ωL = 1/(ω Ceq)

f = 1 / (2π √(L Ceq)) = 1 / (2π √( L C1 C2 / (C1 + C2) ))

(A more exact derivation using the h-parameter model gives the same frequency when the transistor loading is negligible.)

Condition for oscillation: with Aβ ≥ 1 and β = C1/C2, the transistor gain must satisfy

hfe ≥ C2/C1

Example: L = 100 µH, C1 = 1 nF, C2 = 1 nF: Ceq = 0.5 nF; f = 1/(2π√(100×10⁻⁶ × 0.5×10⁻⁹)) = 712 kHz.