20230504
Saturation time of ferromagnetic cores
This parameter is considered difficult to determine, because for its calculation, in the classical version, a lot of auxiliary data is needed.
The very same method for determining the saturation time of ferromagnetic cores is somewhat confused and not always clear; an example is given in [1].
For calculation, such techniques require, as a rule, the operating frequency of the steadystate process, which may not be known.
And often it is completely required to calculate the time of the transient process until the core is saturated, which these methods do not allow to do.
In this note, we will apply a simpler calculation based on the coefficients of the Stoletov curve,
which can be found using the methodology suggested in this paper.
Also, the note is intended to draw the attention of manufacturers of ferromagnetic materials to the fact that three more parameters should be indicated in the ferrite passport: \(h_{12}\, h_{22}\, h_{23}\).
These three coefficients make it possible to completely restore the Stoletov curve,
therefore, with their help, it is possible to determine the permeability of ferrite at any point of the magnetic field strength,
and also, quickly calculate and evaluate some important parameters of future devices based on such cores.
One of these parameters  ferrite saturation time  will be discussed in this work.
Note that all calculations will refer to closed cores, without a gap (Gapped).
Schematic diagram of checking the saturation time \(\tau\) is shown in Figure 1a.
It consists of a power source with a voltage \(U\), which is connected by a key \(SW\) to an inductor, the saturation time of the ferromagnetic core of which we find.
The coil, in general, has an active resistance \(R\).
Fig.1. a) Principal diagram of core saturation time test; b) highQ coil saturation plot; c) mediumQ coil saturation plot (blue) and nonsaturation coil plot (red)

Figures 1b and 1c show graphs of the current \(I\), which gradually increases until the core is saturated.
In Figure 1c, for comparison, a graph (red curve) is added for a coil that is operating without saturation.
In the figures \(t\) is the time.
Coefficients in Brief
They will constantly appear in further work, so we will start with them.
Three Interchangeable Coefficients
\[ k_{12},\, k_{22},\, k_{23} \tag{1.1}\]
And
\[ h_{12},\, h_{22},\, h_{23} \tag{1.2}\]
allow restore the Stoletov curve and determine the permeability of the ferrite at any point of the magnetic field strength.
The difference between them is the following.
The coefficients (1.1) are designed to calculate a specific coil with a core, for which the number of turns and inductance are known.
Coefficients (1.2) are more general and describe the magnetic properties of the core itself (without coil),
they must be indicated to the manufacturer in the ferrite passport.
The coefficients are converted into each other according to the following rule:
\[h_{12} = k_{12} \left({\ell \over N} \right)^2 \quad h_{22} = k_{22} \left({\ell \over N} \right)^2 \quad h_{23} = k_{23} \left({\ell \over N} \right)^3 \tag{1.3}\]
where: \(\ell\) is the average length of the magnetic line of the core [2],
and \(N\) is the number of turns of the coil.
High quality coil
In this part, we will consider the saturation time of the ferrite core for a coil that is expected to have a relatively high quality factor (Fig. 1b).
Then the formulas for calculating the core saturation time constant will be very simple.
They are derived from this work, from the denominator (7),
which must not be zero or become negative.
From here we get the core saturation time (or saturation time constant):
\[ \tau = {k_{12} \over k_{23}} {L_0 \over U} \tag{1.4}\]
where: \(L_0\) is the initial inductance of the coil (without current),
\(U\) is the voltage applied to the coil.
It is quite logical that the lower the inductance of the coil and the greater the voltage applied to it, the faster its core will saturate.
In reality, this time can be taken 510% less.
Formula (1.4) allows you to find a constant for a coil with a known inductance.
If we need a constant for the core itself (without a coil), then it can be found by the formula for recalculating the coefficients (1.3),
and formulas for calculating the inductance of coils with a closed core [3]:
\[ \tau = {h_{12} \over h_{23}} {L_0\, \ell \over U\, N} = \mu_0 \mu {h_{12} \over h_{23}} {S\, N \over U} \tag{1.5}\]
where: \(\mu_0 \mu\)  absolute and relative magnetic permeability, respectively [4],
and \(S\) is the crosssectional area of the core.
Interestingly, the saturation time constant does not depend on the length of the magnetic line \(\ell\).
For example, two toroidal coils with the same number of turns and core size can have the same saturation time, even if their core diameters are different.
For nonclosed magnetic cores, it is necessary to introduce additional coefficients into formulas (1.5) and (1.10), taking into account the uneven distribution of the magnetic field.
Average quality factor of a coil
In this case, we must at least partially take into account the active resistance of the coil winding \(R\) (Fig. 1c).
From a mathematical point of view, this can be done as follows: take the classical expression for the current in the LR circuit [5]
\[ I \approx {U \over R} \left( 1  e^{ \tau R / L_0} \right) \tag{1.6}\]
and substitute there the saturation current from the denominator of the formula (1.7) of this work
\[ I_s = {k_{12} \over k_{23}} \tag{1.7}\]
Deriving from this the saturation time, we get:
\[ \tau =  {L_0 \over R} \ln \left(1  {k_{12} \over k_{23}} {R \over U} \right) \tag{1.8}\]
Obviously, for small values of \(R\) formula (1.8) goes over into (1.4).
It is verified that the resulting formula works quite accurately if the following condition is met:
\[ R \leq {U \over 2} {k_{23} \over k_{12}} \tag{1.9}\]
This range is quite sufficient for most possible variants of real coils and cores.
Then, for a core with an assumed coil with an average quality factor, the saturation time will be calculated using one of the formulas:
\[ \tau =  {L_0 \over R} \ln \left(1  {h_{12} \over h_{23}} {R \over U} {\ell \over N} \right)
\\
\tau =  \mu_0 \mu {S\, N^2 \over \ell\, R} \ln \left(1  {h_{12} \over h_{23}} {R \over U} {\ell \over N} \right)
\tag{1.10}\]
As we see, here, in contrast to (1.5), it is necessary to take into account the average length of the magnetic line, but it is known both to the manufacturer of a specific ferromagnetic core and to the designer.
The active resistance range of the coil, for this formula to work correctly, must becan also be found from condition (1.9).
Conclusions
For a preliminary calculation of the core saturation time \(\tau\), you can take simple formulas (1.4) or (1.5), which assume the absence of active resistance in the coil.
As a rule, they do not differ from the more exact corresponding formulas (1.8) and (1.10) by more than 1.5 times.
Such simple calculations become possible if find coefficients for the Stoletov curve,
or indicate them in the passport of the core by the manufacturer of this ferrite.
Based on the first part of this work, in the second we will derive approximate formulas for some coefficients of the Stoletov curve, and core saturation time, based on reference data.
Materials used
 GOST R 586692019. 5.4 Determination of the time to saturation of current transformers by a graphical method using the characteristics of the magnetization of current transformers. [Site]
 Determination of magnetic circuit parameters.
 Wikipedia. Solenoid. Inductance.
 Wikipedia. Magnetic permeability.
 Wikipedia. LR chain.