Let's distinguish between Input Current and Output Current. The Input Current does whatever it wants, and as long as you stay on the linear portion of the saturation curve, the Output Current and the flux will be proportional to it.
Once the Input Current moves the operating point into the saturated (nonlinear) region, the Output Current and the flux are no longer proportional and your instrument (meter, relay, etc.) does not accurately reflect the actual current. Some relays take advantage of this but that's another topic.
If you severely overload the CT the flux will become relatively fixed with no relation to the Input Current.
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...which is why CTs must be correctly selected in the first case so that the original poster's postulate is a non-starter. GA.
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The picture below is the current and voltage when mains voltage is applied to a "plugtop" transformer 240:24 ratio 0.2 amp rating at 24V (got by oscilloscope). The 24 volt winding current is zero (no-load). These transfo are cheap and have high saturation and losses. The principals of saturation apply to Voltage transformers and current transformers the same - electrons and magnetic fields do not care what a transformer does.
The blue trace is mains voltage 244V rms 50 Hz. The red is current, 61 mA rms. The trace is one cycle, 50 ms, so one horizontal division is 2ms.
One could make a current transformer [CT] from this unit by having a single turn, or several turns, in place of the 24 volt winding - it would be called a "wound primary CT" to distinguish it from CTs in which the primary is a straight piece of busbar.
The important point is that busbar type CTs are normally check saturation tested by applying a high voltage to the secondary winding, the iron core reacts to amperes x turns - it does not know, for example, any difference between 100 amps in one turn primary and 1 amp in 100 turns secondary.
Back to the picture, the effects of saturation are clearly seen, note....
The current is NOT sinusoidal, it is low for 2 divisions horizontal, then has a peaked rush of current over 3 divisions.
The current lags the voltage by nearly 1/4 cycle or 90 degrees - the lagging inductive current effect.
Note the current as it rises upwards from zero on the right. The voltage is high, it is 75 to 100% of peak value from 1.3 to 3 divisions from the middle, but the current is just increasing steadily.
Remember that the instantaneous back-emf in an inductor is given by number of turns x rate of change of flux linking the turns and that - after allowing for resistance drops - it must equal the supply voltage.
Over the time noted in 3, the back emf is not changing much so the magnetic flux (rate of increase x time unit) must be increasing nearly evenly with time. There is not much current, so saturation is not happening, flux is increasing in proportion to current - permeability μ is nearly constant.
Near the end of the peak of voltage, a sharp increase in current occurs. This is because saturation is occurring, the flux is not increasing steadily with magnetizing current now, it takes more and more current increase to get the increase of flux in each millisecond needed to give the necessary back-emf.
Over the last two time divisions, even though the voltage is falling, the current increases steeply, as the "flattening top" of the "DC" magnetic saturation curve takes effect.
What happens in extreme saturation is that current only flows in short spikes near primary voltage zero. Because the spike is very short, flux is changing very rapidly [from negative saturation to positive saturation] - this induces very high voltage in the CT winding and its load.
Consequently, CT windings and their secondary wires and meters have to be well insulated and robust to survive high peak currents and voltages.
Finally, I note that CT often get saturation in normal operation - many loads, say incandescent lamps, motors, capacitors and transformers have inrush currents of 10 times steady load or more when switched on.
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