How it works
An LC oscillator sustains oscillation by satisfying the Barkhausen criterion: loop gain |Aβ| = 1 and total phase shift around the loop = 0° (or 360°). The LC tank circuit provides the frequency-selective positive feedback. In a Colpitts oscillator, two series capacitors C1 and C2 form a capacitive voltage divider; the transistor amplifier compensates for tank circuit losses. Resonant frequency fr = 1/(2π√(L × Ceq)) where Ceq = C1C2/(C1+C2). Feedback fraction β = C1/C2, so voltage gain must satisfy A = C2/C1. In a Hartley oscillator, two inductors L1 and L2 replace the capacitors for the tap: fr = 1/(2π√(Leq × C)) where Leq = L1 + L2 + 2M (M is mutual inductance). Hartley is used at lower frequencies; Colpitts offers better frequency stability at RF.
Key points to remember
Barkhausen criterion: Aβ = 1 at exactly the oscillation frequency and loop phase = 0°. Colpitts fr = 1/(2π√(L·Ceq)), Ceq = C1C2/(C1+C2). Hartley fr = 1/(2π√(Leq·C)), Leq = L1+L2+2M. For stable oscillation, startup requires |Aβ| slightly > 1; amplitude-limiting (by transistor saturation or AGC) brings it back to 1 in steady state. Crystal oscillators replace the LC tank with a quartz crystal to get frequency stability of 10 ppm or better, compared to ±1% for LC. Clapp oscillator (Colpitts with series capacitor in the inductor branch) further improves stability by reducing the effect of transistor parasitic capacitances on fr.
Exam tip
The examiner always asks you to derive the frequency of oscillation for Colpitts and Hartley oscillators — write the formula, identify Ceq or Leq from the circuit, and state the Barkhausen criterion before calculating fr.