No voltage induced in secondary coil of transformer with 5V 1.5 kHz voltage from Wien bridge oscillator in primary coil
The problem with your circuit is that the oscillator is not starting reliably when your load is your transformer. When the oscillator does not start, the output voltage is in the uV range.
One design goal of a Wien bridge oscillator is that it start reliably. Often there is another design goal which is that the amplifying element should not saturate. These competing design goals are often met by adding circuitry that stabilizes the amplitude of the oscillations.
When the oscillations are weak, the amplitude stabilization circuitry provides high positive feedback to increase the amplitude. When the oscillations are at the desired level, the positive feedback is reduced so that the overall closed loop gain is 1. If the oscillations are above the desired level, the positive feedback is reduced further, so that the oscillations dampen.
Components that are used for amplitude stabilization include small incandescent light bulbs, thermistors, back to back diodes, and JFETs.
Incandescent light bulbs provide amplitude stabilization through the varying resistance they have with temperature. If oscillation is weak, the filament is cold, and the resistance is low. If the oscillation is strong, the current heats the filament, which in turn increases its resistance.
One possible solution to your oscillator not starting when loaded with the transformer is to increase the value of R2 (or alternatively decrease the value of R1). You may double it, or even more. The down side is that this will surely cause the op amp to saturate if oscillations start. This will cause the waveform to deviate considerably from a pure sine wave. Saturation may also alter the frequency of oscillation somewhat. You may or may not care about these effects.
Another possible solution is to increase the number of turns on your transformer primary. This may not work without also increasing the value of R2 (or decreasing that of R1). When a transformer secondary only has a volt-meter across it, the transformer acts essentially like an inductor. The inductance seen at the primary is too low, which causes the op-amp to be heavily loaded, and this decreases the feedback. That in turn causes the oscillator not to start. Increasing the number of turns on your transformer will increase the primary inductance, and that is probably a good thing even if it is insufficient to start the oscillator.
Another solution is to use back to back diodes to achieve amplitude stabilization. I have provided a circuit here:



simulate this circuit &ndash; Schematic created using CircuitLab