# Colpitts oscillator with a series inductance π-network

In a Colpitts oscillator, the feedback occurs via a series inductance π-network. An, based on a common-emitter BJT amplifier, is shown in the circuit below.

Here below is a simplified version.

This circuit is equivalent to the following with voltages inside the circuits.

We will use Thevenin circuit to analyse C2 first.

$$\frac{1} {R} + \frac{1} {1/(s C_2) + s L} = \frac{1} {R_{p_{2}}}$$

$$R_{p_2} = \frac{R\,\left(1+C_{2}\,L\,s^2\right)}{1+C_{2}\,R\,s+C_{2}\,L\,s^2}$$

$$I_2(s) = \frac{V_0} {s} \frac{R_{p_2}} {\bigl(R_{p_2} + 1 / (s C_1) \bigl)} \frac{1} {1/(s C_2) + s L}$$

$$I_2(s) = \frac{C_{1}\,C_{2}\,R\,V_{0}\,s}{1+C_{2}\,R\,s+C_{1}\,R\,s+C_{2}\,L\,s^2+C_{1}\,C_{2}\,L\,R\,s^3}$$

Similarly, we will use Thevenin circuit to analyse C1.

$$\frac{1} {R} + \frac{1} {1/(s C_2) + s L} = \frac{1} {R_{p_{2}}}$$

$$R_{p_2} = \frac{R\,\left(1+C_{2}\,L\,s^2\right)}{1+C_{2}\,R\,s+C_{2}\,L\,s^2}$$

$$I_2(s) = \frac{V_0} {s} \frac{R_{p_2}} {\bigl(R_{p_2} + 1 / (s C_1) \bigl)} \frac{1} {1/(s C_2) + s L}$$

$$I_2(s) = \frac{C_{1}\,C_{2}\,R\,V_{0}\,s}{1+C_{2}\,R\,s+C_{1}\,R\,s+C_{2}\,L\,s^2+C_{1}\,C_{2}\,L\,R\,s^3}$$

According to Thevenin equivalent circuit analysis, we know that

$$I_1(s) + I_2(s) = g_m V_{\pi}$$

$$\Rightarrow \frac{C_{2}\,\left(V_{\pi }+C_{1}\,R\,V_{\pi }\,s+C_{1}\,R\,V_{0}\,s\right)}{1+C_{2}\,R\,s+C_{1}\,R\,s+C_{2}\,L\,s^2+C_{1}\,C_{2}\,L\,R\,s^3} = g_m V_{\pi}$$

However, the book gave me a total different answer.

Here was what the book said. The book An Introduction to Radio Frequency Engineering is written by Christopher Coleman.

$$s C_2 + g_m + (\frac{1} {R} + s C_1) (1 + s^2 C_2 L) = 0$$

• What was the question? Dec 10, 2022 at 8:29
• The voltage across capacitors cannot be replaced with independent sources. Why did you do that? Dec 10, 2022 at 9:44
• @sarthak Laplace Transform Analysis
– kile
Dec 10, 2022 at 9:57
• @TimWilliams How do you get the formula of the book? And how do you analyse the circuit?
– kile
Dec 10, 2022 at 9:58
• Writing equations and solving. I should not transforming schematic. Dec 10, 2022 at 16:18