Reducing equivalent self inductance of an inductor
You are correct that you can reduce the flux in the core by having half of the turns go in one direction and half of the turns go in the opposite direction. Can you reduce it to 0? Well that depends upon what you mean.
Because the wires with current flowing one way are physically distant from the wires flowing in the opposite direction, each will produce flux. At a distance those fluxes will &quot;cancel&quot; each other out. However, there will be some flux near each wire.
If the stars in heaven are aligned perfectly, none of this &quot;leakage&quot; flux will make a loop around the window of the core. For all practical purposes, you could say that the flux in the core is 0. Of course, there is bound to be be some imbalances. The spacing between wires will vary at least microscopically from one place to another, so the flux around the core window may not be mathematically 0, even if it is 0 for practical purposes.
Such a coil, by the way, would not be useful as an inductor, because an inductor works by generating flux!
Edit:
&quot;how can a series Inductance reduces (L=L/2-L/2) because (L=L1+L2) series connection are additive&quot;
One can add the inductance values of two inductors wired in series, only if the inductors are magnetically isolated. When inductors coils are coupled magnetically, they form a transformer.
If two &quot;identical&quot; coils are wound on a core, and wired so that they generated flux in the same direction, the inductance would not be increased by a factor of 2, but by a factor of 4! (Inductance generally is proportional to the number of turns squared). Magnetic coupling matters in problems like these.