**Colpitts Oscillator using Op Amp:**

The Colpitts Oscillator using Op Amp circuit show in Fig. 16-4 is similar to the op-amp phase shift oscillator, except that an LC network is used to produce the necessary phase shift in the feedback voltage. In this case, the LC network acts as a filter that passes the oscillating frequency and blocks all other frequencies. The filter circuit resonates at the required oscillating frequency.

where X_{CT} is the total capacitance in parallel with the inductor. This gives the resonance frequency (and oscillating frequency) as,

Capacitors C_{1} and C_{2} are connected in series across L_{1}; so,

Consideration of the LC network shows that its attenuation (from the amplifier output to input) is due to the voltage divider effect of L and C_{1.} This gives,

It can be shown that the required 180° phase shift occurs when

and this gives,

As in the case of all oscillator circuits, the loop gain must be a minimum of one to ensure oscillation. Therefore,

or,

When deriving the above equations, it was assumed that the inductor coil resistance is very much smaller than the inductor impedance; that is, that the coil Q factor (ω L/R) is large. This must be taken into considered when selecting an inductor. It was also assumed that the amplifier input resistance is much greater than the impedance of C_{1} at the oscillating frequency. Because of the inductor resistance and the amplifier input resistance, and because of stray capacitance effects when the oscillator operates at a high-frequency, the amplifier voltage gain usually has to be substantially larger than C_{1}/C_{2}.

**Circuit Design:**

Colpitts oscillator design can commence with selection of the smallest capacitor (C_{2}) much larger than stray capacitance, or with selection of a convenient value of L. To keep the amplifier input voltage to a fairly low level, the feedback network is often designed to attenuate the output voltage by a factor of 10. This requires that C_{1}/C_{2} ≈ 10. (It should be recalled that large A_{CL} values require larger op-amp bandwidths.) Also, X_{C2} should be much larger than the amplifier output impedance. Using the desired oscillating frequency, L can be calculated from Eq. 16-4. Amplifier input resistor R_{1} must be large enough to avoid overloading the feedback network, (R_{1} ≫ X_{C1}). Resistor R_{2} is determined from A_{CL} and R_{1}.

**BJT Colpitts Oscillator:**

A Colpitts oscillator using a single BJT amplifier is shown in Fig. 16-6(a). This is the basic circuit, and its similarity to the op-amp Colpitts oscillator is fairly obvious. A more complex version of the circuit is shown in Fig. 16-6(b).

Components Q_{1}, R_{1}, R_{2}, R_{E} and C_{E} in (b) are unchanged from (a), but collector resistor R_{C} is replaced with inductor L_{1}. A radio frequency choke (RFC) is included in series with V_{CC} and L_{1}. This allows dc collector current (I_{C}) to pass, but offers a very high impedance at the oscillating frequency, so that the top of L_{1} is ac isolated from V_{CC} and ground. The output of the LC network (L_{1}, C_{1}, C_{2}) is coupled to via C_{c} to the amplifier input. The circuit output voltage (v_{o}) is derived from a secondary winding (L_{2}) coupled to L_{1}. As in the case of the BJT phase shift oscillator, the transistor current gain is important. Circuit analysis gives Eq. 16-4 for frequency, and for current gain,