Here it is shown the relationship between amplifier gain and the feedback fraction of Colpitts oscillator. This is useful in determining the condition for oscillation and choosing components values so as to setup the Colpitts oscillator for oscillation.
Colpitts oscillator are used to generate high frequency signals. The frequency is typically up to 500MHz. Colpitts oscillators can generate high frequency signal which op-amp cannot do. The op-amp frequency generation capability is dictated by its funity value.
Colpitts oscillator has two parts- the amplifier and the feedback network part. This is illustrated in the circuit diagram below.
The BJT amplifier is biased using voltage divider biasing technique(see the tutorial How to bias a BJT using voltage divider biasing). This sets up the quiescent operating point for the amplifier. The RFC(Radio Frequency Choke) is used to block the ac signal going to the supply. It has high inductive reactance at the oscillator frequency and thus appears open the the ac signal. The frequency of the Colpitts ocillator is given by,
\(f = \frac{1}{2*\pi \sqrt{L C}}\) --->(1)
where, \(C=\frac{C_1 C_2}{C_1+C_2}\) --->(2)
This can also be directly calculated using the online Colpitts oscillator calculator.
The condition for sustained oscillation of an oscillator is given by,
\(A_v B >1\) --->(3)
or,
\(A_v > \frac{1}{B}\) --->(4)
where \(A_v\) is the amplifier voltage gain and B is the feedback fraction.
Thus the voltage gain and the feedback fraction are inversely proportional. When the feedback is large, the voltage gain is small and vice versa.
The output voltage (\(V_{out}\)) appears across the capacitor C1 and the feedback voltage(\(V_f\)) appears across the capacitor C2.
It can derived that the feedback fraction(B) of the Colpitts oscillator is,
\(B=\frac{C1}{C2}\) --->(5)
Using this knowledge and from equation(4) we can say that the minimum voltage gain of the amplifier for oscillation is,
\(A_v > \frac{C2}{C1}\) --->(6)
The voltage gain of BJT amplifier is given by the following equation,
\(A_v = \frac{r_c}{r_{ace}}=\frac{ac \space collector \space resistance}{ac \space emitter \space diode \space resistance}\) ----->(7)
The ac collector resistance(\(r_c\)) is given by,
\(r_c = R_C || R_L = \frac{R_C R_L}{R_C+R_L}\) --->(8)The ac emitter diode resistance(\(r_{ace}\)) of a BJT derived by solid state physics is given by,
\(r_{ace} = \frac{25mV}{I_E}\) --->(9)
Using equation(8) and (9) in equation (7) we have,
\(A_v = \frac{R_C R_L I_E}{25mV(R_C+R_L)}\) --->(10)
The ac emitter resistance can also be determined using the h parameters(small signal characteristics) given by the equation below.
\(r_{ace} = \frac{h_{ie}}{h_{fe}}=\frac{h_{ie}}{\beta}=\frac{input \space impedance}{ac \space current \space gain}\) --->(11)
Now we can relate the voltage gain of the amplifier which is given by equation (10) and the feedback fraction through equation (6).
