With given magnetic field H, one can obtain the magnetic flux density from the relation
B = µ * H, with µ = µ0 * µr
µ0 : absolute magentic permablility, 4pi×10−7 Vs/(Am)
µr : relative magnetic permeability
The resulting total flux Φ in the core is then
Φ = B * A,
with A the core's cross section area.
If now H changes with time, then also B does and will induce a voltage U2
in the secondary windings of n turns:
U2 = -d(Φ/dt)*n
For fast changing input current (input frequency is high compared to the low pass time constanat R2*C1)
the voltage at the capacitor C1 is small, thereby nearly all the voltage appear over R2
I2 = U2/R2
I2 is stored by capacitor C1 resulting in voltage Uy, which is then the integral of
I2 over time.
Uy = integral ( U2/(R2*C1)) dt
= 1 / (R2*C1) integral (U2) dt
= 1 / (R2*C1) integral (-d(Φ/dt)*n) dt
= 1 / (R2*C1) integral (-d(B*A/dt)*n) dt
= n*A / (R2*C1) B
= n*A / (R2*C1) * µ0 * µr *H
and with H = n * I1 = n * Ux / R1
Uy = n² *A / (R2*C1) * µ0 * µr * Ux / R1
Then relative permeability is given by
µr = ( Uy* R1 ) / ( n² *A / (R2*C1) * µ0 * Ux)
As one can see,
µr ~ Uy / Ux
so the slope of trace on the scope is proportional to µr