Well what an interesting setup you have there !
That trace attached to your new post has no loops on end - what is it? Your setup with core not inserted perhaps? Or did you solve the mystery already?
The emf induced in the secondary coil is given to the power amplifier and then to the integrator.
(I assume your amplifier has high input impedance so there's not much current in sense coil compared to excitation coil.)
Since your excitation is sinewave, set your 'scope for ordinary time display and observe current & induced voltage as you slide the core into the solenoid.
Then take a look at input and output of "power amplifier" .
As core is inserted the induced signal gets much larger and we should look to see if we are overdriving that amplifier... if output sinewave peaks flatten out you are saturating the amplifier. You shouldn't be able to saturate an iron rod since so much of the magnetic path is through air.
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Think about what's going on inside that iron...
With an ideal inductor current and flux would be in phase and your test would show a straight line on your oscilloscope. That's what you'll get with air core and that's almost what the more recent trace looks like...
With real inductor you have hysteresis so expect a B-H loop.
Inside real iron you also have eddy currents which retard the flux because of Lenz's law.
Recall Gauss' h(dot)dl = i
enclosed counts ALL the current, not just what current you applied. If you use a voltage sensing coil to measure flux, even the current flowing in it counts.
Your 'scope trace on second post shows a small loop - if it's not your successful B-H loop, then perhaps it's just a little phase shift from eddy currents in your aluminum coil form or imperfect integration, or perhaps even an iron leg brace on bottom of work table underneath your solenoid. So take a look at table's underside..
Also note the aluminum coil form constitutes one shorted turn, if that turns out problematic maybe you can find some square plastic or use balsawood??
How do I conclude that my material has reached saturation?
i doubt you could saturate it but-
use 'scope to observe current and induced voltage vs time as you increase current . As you approach saturation, voltage will depart from a sine shape and i think get "spikey".
If you have a function generator and power amplifier that could drive current, , try this:
I've done it myself and it is a real eye-opener.
drive your setup with a triangle wave current instead of a sinewave and look at induced voltage.
A triangle wave current should give a square wave voltage and you'll see effect of "magnetic retardation" by eddy currents as rounded shoulders on your square wave.
You will see effect of frequency as dramatic rounding of the square wave's edges as frequency increases.
If i can find my old 'scope photos i'll photocopy and attempt to post them.