I am taking an analogue electronics course and have been given an assignment to create/develop a 64.5kHz Hartley oscillator using any 5V rail-to-rail operational amplifier. I am trying to design the circuit in simulation first before I build.
I will be using this circuit:
simulate this circuit – Schematic created using CircuitLab
The design equations we have been given for the Hartley oscillator are:
\$\omega_0 = \frac{1}{\sqrt{C_3(L1+L2)}}\$
\$K(\omega_0) = -\frac{L1}{L2}\$
\$A = \frac{L2}{L1} = \frac{R2}{R1}\$
I began by choosing \$L_1 = L_2 = 10\mu H\$ and then solving for \$C_3\$ with \$\omega_0 = 129000 \pi\$ to obtain \$C_3 = 3.04\times10^-7 F\$
then \$A = \frac{10\mu H}{10\mu H} = \frac{100k\Omega}{100k\Omega}\$
I've simulated in LTspice using the circuit below giving it a quick pulse to get the oscillation going:
The results:
From this I observe that the oscillation frequency is very close (64.3kHz) but the response dies out very quickly and does not sustain. My understanding of oscillators is that I want the negative feedback to be equal to positive feedback to fulfill Barkhausens criteria in which the magnitude of the total feedback should be 1 and the phase shift 0. I am obviously missing something important. Any advice on getting this circuit to have a sustained oscillation would be greatly appreciated.
edit:
Based on the suggestions below I added an extra resistor between the output and resonant circuit and adjusted my gain. Re-simulating gave these results in LTspice. I've added three screenshots below from the simulation. The first is over 10ms, the second I have zoomed in when the oscillation stabilizes and the third is a frequency domain plot showing an impulse at 64.5kHz: