2023-05-10

Calculation of the saturation time of ferromagnetic cores through induction

In the previous part of this work, we presented the calculation of the saturation time of ferromagnetic cores,
using the coefficients of the Stoletov curve: \(k_{12}\, k_{23}\) or \(h_{12}\, h_{23}\).
A very simple calculation was shown for high-Q coils, and a slightly more complex one for medium-Q coils.
While manufacturers of ferromagnetic materials have not entered these coefficients into the passport data of their products,
finding them requires special laboratory research from the developer.
Therefore, in this part of the work, we will try to get around this moment and develop a methodology for calculating the saturation time of ferrites through one more passport characteristic - saturation induction.
Of course, such a calculation will be approximate, but its accuracy will be quite acceptable for the development of almost any device.

We will proceed from the formula for calculating the magnetic induction in the core:
\[ H = {B \over \mu_0 \mu} = {N\, I \over \ell} \tag{2.1}\]
Here: \(H\) is the magnetic field strength in the core,
\(B\) - magnetic field induction in the core,
\(\mu_0 \mu\) -- absolute and relative magnetic permeability, respectively,
\(N\) - the number of turns in the coil,
\(I\) - coil current,
\(\ell\) - the average length of the magnetic line of the core [1].

In the reference books of ferromagnetic materials, the parameter is given - saturation induction [2],
which is achieved at such a current through the core coil at which its induction no longer increases.
From here we can immediately correlate such a current and induction
\[ {B_s \over \mu_0 \mu} = {N\, I_s \over \ell} \tag{2.2}\]
where: \(B_s\) - saturation induction, \(I_s\) - saturation current.

Let's remember that the magnetic permeability depends on the current passing through the coil.
This pattern was presented in this work:
\[\mu(I) = \mu_{i} {1 + k_{12} I^2 \over 1 + k_{22} I^2 + k_{23} I^3} \tag {2.3}\]
Here: \(\mu_{i}\) is the initial permeability of ferrite, which is also given in reference books [2].
Substituting it into (2.2) we get:
\[ {B_s \over \mu_0 \mu_{i}} {1 + k_{22} I_s^2 + k_{23} I_s^3 \over 1 + k_{12} I_s^2} = {N\, I_s \over \ell} \tag{2.4}\]
The saturation current is large enough to be much greater than unity together with the coefficients.
Therefore, formula (2.4) can be simplified:
\[ {B_s \over \mu_0 \mu_{i}} {k_{23} I_s \over k_{12}} \approx {N\, I_s \over \ell} \tag{2.5}\]
As we can see, the currents are now canceling out, and the formula we need no longer contains it:
\[ {k_{12} \over k_{23}} {N \over \ell} \approx {B_s \over \mu_0 \mu_{i}} \tag{2.6}\]
Given practical enough precision, and a simple recalculation coefficients, we can now write:
\[ {h_{12} \over h_{23}} = {k_{12} \over k_{23}} {N \over \ell} = {B_s \over \mu_0 \mu_{i}} \tag{2.7}\]

Formula (2.7) allows you to find the ratio of the coefficients \(k_{12}/k_{23}\) or \(h_{12}/h_{23}\) through tabular data with sufficient accuracy.

Core Saturation Time

Now take formula (1.5) from previous part of this work
\[ \tau = \mu_0 \mu_i {h_{12} \over h_{23}} {S\, N \over U}\]
and rewrite it taking into account saturation induction and magnetic permeability from (2.7).
Thus, for a high-quality coil, which has no active resistance, the core saturation time is found as follows:
\[ \tau = {B_s S\, N \over U} \tag{2.8}\]
Recall that \(S\) is the cross-sectional area of the core, and \(U\) is the voltage applied to the coil.
The parameters \(B_s\) and \(S\) can be taken from reference data [2].
Interestingly, if the maximum possible magnetic flux flows through the coil, then this formula becomes quite simple:
\[ \tau = {\Psi_s \over U} \tag{2.9}\]
where \(\Psi_s\) is the flux linkage [3], or the total magnetic flux through all turns of the coil, provided that the induction in the coil is maximum.

For a medium-quality coil, where the active resistance \(R\) is significant, the core saturation time is found by one of the formulas:
\[ \tau = - {L_0 \over R} \ln \left(1 - {B_s \over Q_m U} \right)
\\
\tau = - Q_m S\, N \ln \left(1 - {B_s \over Q_m U} \right)
\tag{2.10}\]
where \(Q_m\) is the magnetic quality factor of the coil:
\[ Q_m = {\mu_0 \mu_i N \over R\, \ell} \]
Recall that here \(L_0\) is the initial inductance (at zero current), and \(\mu_0 \approx 1.257\cdot 10^{-6}\,(H/m) \).
Also, to ensure the necessary accuracy of the formula, the active resistance of the coil should not go beyond the following limits:
\[ R \leq {U \over 2} {\mu_0 \mu_i N \over B_s \ell} \tag{2.11}\]
or
\[ Q_m \geq {2 B_s \over U} \]
It is easy to see that with a sufficiently small active resistance of the coil, formula (2.10) turns into expression (2.8).

Calculation example

Let's take a CF139 core with an EE1306B magnetic core, where \(\mu_i = 1480, B_s = 0.49\, (T)\).
Based on the reference data on the shape of the magnetic circuit [2] and the methodology [1], we find
core cross-sectional area \(S = E\cdot F = 16.9\cdot 10^{-6}\, (m^2)\),
and the average length of the magnetic line of the core \(\ell = A + 2C + 2D - E = 28\cdot 10^{-3} \, (m)\).
We wind a coil with the number of turns \(N=10\) on the middle core of the core and apply voltage \(U = 10\, (V)\) to it.
Then, according to formula (2.8), the core saturation time will be equal to: \(\tau = 8.3\cdot 10^{-6}\, (s)\), or 8.3 µs.

If we want to take into account the resistance of the coil wire, which is, say, 1 Ohm, then the core saturation time must be recalculated using formula (2.10).
Then first we find the magnetic quality factor \(Q_m = 0.664\), after which we check it for condition (2.11),
and then we substitute it into (2.10), and we get the corrected core saturation time: \(\tau = 8.6\cdot 10^{-6}\, (s)\), or 8.6 µs.

Conclusions

In this work, approximate formulas were derived for calculating the saturation time of the core using reference data on ferromagnetic materials: saturation induction and initial magnetic permeability.
Their accuracy is sufficient for fast calculations with coils with high (formula 2.8) and medium (formula 2.10) quality factors.

Without taking into account the active resistance of the coil wire, the saturation time of the core is proportional to the saturation induction, the cross-sectional area of the core and the number of turns of the coil wound around it,
and inversely proportional to the voltage applied to the coil.
Accounting for this resistance leads to a small correction that takes into account the initial permeability and geometric parameters of the core.