Your assumption that the core flux cancels out to zero is not valid because you don't have perfect coupling between L1 and L2. If you had, then the inductance for the series-additive connected inductor should measure 120 (not 90) uH, because inductance is proportional to the square of the number of turns, and you're essentially doubling the number of turns. Likewise, you would measure no inductance for the series-opposing connection. Note that even if there was no coupling you would still measure 60 uH for the series-additive pair, so it seems there is a fair amount of leakage flux.
In the opposing flux configuration, the effect on the circuit is similar to replacing the coupled inductor with two separate inductors having much lower inductance values. As a result, the current ripple is higher, and the associated core losses are higher.
In a transformer, increasing the magnetic path length reduces flux and permeability and increases reluctance of the core?
True.
The magnetic path length (MPL) is analogous to the length of a conductor for electrical current - increasing the MPL will increase its reluctance (magnetic 'resistance'). For a given magneto-motive force (MMF) in ampere-turns the flux produced will depend upon the value of reluctance in ampere-turns per Weber and so an increase in reluctance will produce a decrease in flux. (See Hopkinson's Law)
Permeability is analogous to 'conductivity' and depends upon the material as well as the path length and cross-sectional area. Increasing reluctance will decrease permeability.
[In electrical terms if the resistance is increased the conductance is decreased]
Why does the equation of relative permeability of a gapped core show that permeability is increased as you increase the MPL?
False
The reason an 'air gap' is used is to prevent the core from saturating. {n.b. the term 'air gap' also refers to other materials such as nylon that is used to fill the gap.} In other words you are increasing the reluctance of the circuit and as we have already seen this decreases the permeability.
As you don't show the equation you're referring to I'll just go through the basic theory.
The gap reluctance, Rg, is given by:
Rg = Lg / (μ0 * Ae)
where Lg is gap length, μ0 is the permeability of a vacuum (very similar to value for air) and Ae is the cross sectional area.
The core reluctance, Rc, is
Rc = Lc / (μ0 * μr * Ac)
Where Lc is the magnetic core length, μr is the relative permittivity of the core material (>>μ0) and Ac is the area of the core.
The total reluctance of the magnetic circuit, Rt, is
Rt = Lc / (μ0 * μr * Ac) + Lg / ( μ0 * Ae)
Generally Ae = Ac = A (maintains cross sectional area across gap) so this becomes
Rt = (1 / ( μ0 * A)) * ( (Lc / μr) + Lg)
If μr is very large (as in most cases of magnetic core material) it is the length of the airgap, Lg, that dominates the Reluctance value. Reluctance INCREASES with air gap length (and MPL) and if you increase Reluctance you DECREASE the effective Permeability.
Best Answer
You need to make sure that the ferrite can be permanently magnetized. This is sometimes referred to as "hard" ferrite — one that has a nearly-square B-H curve.
Most of the modern ferrites used in EMC work are "soft" ferrites, designed mainly for their loss characteristics, and they do not retain any significant magnetization.