Opposite phasing windings total inductance?

Hello,
In a transformer, will the inductances of two identical windings, wound in opposite directions (phase), and connected in series, cancel out?

I consider the attached saturable reactor, where the signal windings (load windings) are connected in series and are wound in opposite phases. This circuit works, because it is from a paper, but If the inductances of the load windings cancel out, how it should work at all?

No matter your circuit from paper, if you have a transformer with close wound primary and secundary (same number of turns), the inductance when put in series with "wrong" phase will be well below the inductance of the primary. the actual value depends on the coupling factor (k). High k gives low inductance when put in series.

The opposite: when put in "correct" phase, inductance is almost 4 times the value of the primary inductance (assuming Nsec = Nprim).

If you are familiar with mutual inductance: Ltot = Lprim+Lsec +or- 2*M where M = the mutual inductance

M = k*sqrt(Lprim*Lsec)

How can you say the load windings are connected in counter-phase ? However it seems to me a transductor, that is an electrically tunable inductor. In this case the primary and the secondary winding are constructed so they magnetics field are orthogonal one each other (as the picture suggest) so that there is no reciprocal interference (ideally k=0). There are may way to obtain this, depending on the geometry of the core. With respect to your picture I've expected the left side to be the RF winding while the right side the controlling winding.

The first conclusion is no really stringent "because it's from a paper?", but provided the paper is basically correct, the picture tells near to nothing about the working principle of a saturated inductor or transductor. Particularly the core geometry isn't shown.

I have seen two ways they do it. One way is to use two control windings (on the primary side) of opposite phase, but these have to be in two cores not one. The other way is using just one core and make the two opposite phase windings on the secondary side. They claim some advantages on this, like that the core cannot be saturated by currents in the secondary because they cancel out. But if they cancel out will the inductance cancel out as well? (coincident currents of opposite phase in the core).

Saturated inductors with a single magnetical path normally don't use "cancellation" methods.

The keyword RF suggests, that the additional magnetical path may be through air.
Show a picture of the core and coils geometry.

I do not have such a picture, but bor my tests I consider toroidal cores or solenoids (cylinder core).
I consider this document http://www.google.gr/url?sa=t&rct=j&...,d.ZGU&cad=rja