Resonant transformer - sign of the voltage reinduced in the primary winding

htg

Consider a biresonant transformer with a small coupling coefficient k << 1.
Let the voltage applied to the primary LC circuit be U1(t)=U*sin(2*Pi*f*t), then also the current I1 will be sinusoidal. The voltage induced in the secondary winding and the current in the secondary LC circuit will be a positive multiple of -cos(2*Pi*f*t). Then the voltage reinduced in the primary winding will be a positive multiple of -sin(2*Pi*f*t), so it will tend to decrease the current in the primary winding. As far as I know, drawing current from the secondary winding INCREASES the current through the primary winding.
What is the explanation?

NascentOxygen

See what you think of this:

Apply a voltage sin(ωt) and a current sin(ωt) flows, setting up a magnetic flux sin(ωt)
The voltage induced in the secondary by that flux is dɸ/dt, viz., cos(ωt)
and causing a secondary current of cos(ωt)
This secondary current produces a flux of cos(ωt), and the derivative of that, dɸ/dt, inducing a component in the primary of –sin(ωt)

The increased difference between the source emf and the coil´s emf causes increased primary current.

(I leave you to prepend an amplitude co-efficients to each time-varying term.)