Hartley oscillator

Hartley oscillator

This is under the category of tuned oscillator or resonant circuit oscillator
Operation
When $V_{cc}$ is applied ,the collector current begins to flow and the drop in collector voltage is coupled through capacitor C and $L_{2}$ .Thus the capacitor charges to its minimum voltage ,this voltage acts as initial excitation for tank circuit ,causing a current to flow in the LC circuit  .This current induces damped harmonic oscillation across $L_{1}$ and this circuit acts as input to base of transistor
This damped signal is amplified and appears at collector which is coupled as feedback to the tank circuit  C and $L_{1}$. The feedback voltage across $L_{2}$ is in phase with the input voltage across $L_{1}$ results in sustained oscillation
The feedback voltage is in phase shift with input voltage since 180° phase shift produced by the transistor and another 180° being provided by the tank circuit
Capacitor block ,dc component of the collector circuit ,but coupled c signal
As a results of this dc is out of tank circuit this energy loss due to tank circuit  is reduced and hence the oscillation is more stable

WRITING NODE EQUATION
    $\frac{V_π}{r_π}= \frac{V_π}{SL_1}+ (V_π-V_c)S_c=0$
    $g_mV_π+\frac{V_c}{R}= \frac{V_c}{SL_2}+ (V_c-V_π)S_c=0$
\end{array}\right]

= $\left[\begin{array}{cc}V_π\\V_c\\ \end{array}\right]=0$

$\left[\begin{array}{ccc}   (\frac{1}{r_π}+\frac{1}{SL_1}+S_c)&-S_c\\  (g_m-S_c)&\frac{1}{R}+\frac{1}{SL_2}+S_c\\ \end{array}\right]=0$
will continue.......