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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.

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Here below is a simplified version.

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

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We will use Thevenin circuit to analyse C2 first.

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

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

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

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

Similarly, we will use Thevenin circuit to analyse C1.

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

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

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

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

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.

This is the book link Click here for book info

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

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  • \$\begingroup\$ What was the question? \$\endgroup\$ Dec 10, 2022 at 8:29
  • \$\begingroup\$ The voltage across capacitors cannot be replaced with independent sources. Why did you do that? \$\endgroup\$
    – sarthak
    Dec 10, 2022 at 9:44
  • \$\begingroup\$ @sarthak Laplace Transform Analysis \$\endgroup\$
    – kile
    Dec 10, 2022 at 9:57
  • \$\begingroup\$ @TimWilliams How do you get the formula of the book? And how do you analyse the circuit? \$\endgroup\$
    – kile
    Dec 10, 2022 at 9:58
  • \$\begingroup\$ Writing equations and solving. I should not transforming schematic. \$\endgroup\$
    – Antonio51
    Dec 10, 2022 at 16:18

1 Answer 1

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Searched for a working circuit ...
Here are the schematic and behavior.

Starting the oscillator ... (weird !)

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Steady state oscillator ... Don't know if it stops after some time (simulation longgg).

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Here is a Maple sheet for calculating ... frequency and plot of the collector voltage.

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Zeroes and poles are listed.
The oscillator frequency oscillation is confirmed by microcap v12 simulation.

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