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Measurements on magnetic cores - a little theory
Here below a couple of methods that can be used to get some information about unknown magnetic cores material will be described:
- estimating the relative permeability
- estimating the relative permeability and permittivity (dielectric constant)
For the two methods above, a Qucs project with two schematics set up for doing all the needed calculations can be downloaded here. Brief instructions are in the schematics.
For some examples of ferrite materials characteristics determined using these methods see this page.
Before going into the details of the measurement methods, we need a little theory to describe the relationships between the measurements results and the material characteristics:
The permeability \( \mu \) of a material describes the ratio between the applied magnetic field strength \( H \) and the magnetic field density \( B \) inside the material: \( \mu = B/H \). Vacuum has also a magnetic permeability \( \mu_{\rm{0}} \) so often a material is also described using the relative permeability \( \mu_{\rm{r}} = \mu / \mu_{\rm{0}} \).
In general the (relative) permeability is a complex number, indicating that \( B \) may have a phase shift with respect to \( H \): \( \mu_{\rm{r}} = \mu_{\rm{r}}^{'} - j \mu_{\rm{r}}^{''} \).
Estimating the relative permeability
The impedance of an inductor with an air core is \( Z = j \omega L_{\rm{0}} \), while for the same inductor on a magnetic material \( Z_{\rm{F}} = j \omega \mu_{\rm{r}} L_{\rm{0}} \), so by taking the ratio of the two quantities the relative permeability can be determined:
\[ \mu_{\rm{r}} = \frac{Z_{\rm{F}}}{Z} \]\( Z_{\rm{F}} \) can be measured with a VNA after winding a few turns around the toroid under test, while \( Z \) can be computed from the number of turns and the toroid physical dimensions:
It can be shown that the inductance of an ideal air core toroidal inductor is
where the core constant \( C_{\rm{1}} \) is related only to the core dimensions
\[ C_{\rm{1}} = \frac{2 \pi}{h \ln(d_{\rm{o}} / d_{\rm{i}})} \]so knowing the toroidal inductor impedance and the core dimensions the complex relative permeability can be computed.
Estimating the relative permeability and permittivity - The coaxial line method
Determining the permeability and dielectric constant of materials is of course a problem that has been studied since a long time. Two classic papers describing a standard way of measuring these parameters are [1] [2]; in summary, the methods described there are based on filling a transmission line with the material under test and obtaining the material permeability and dielectric constant from the measured S-parameters or time-domain parameters of the line.
An extension of this method [5] is based on making two single-port reflection coefficient measurements, one with the line under test terminated by an open circuit and one with a short circuit termination, instead of a single two-port measurement. After some mathematical manipulations, the same basic parameters as in [1] and [2] are obtained.
The two main quantities that these methods need to then obtain the permeability and the dielectric constant of the material are:
- \( \Gamma_{\!\rm{inf}} \), the input reflection coefficient of an infinitely long transmission line homogeneously filled with the material under test
- the propagation factor \( P = e^{-\gamma L} = e^{-(\alpha + j \beta) L} \), where
\( \gamma \) propagation constant
\( \alpha \) attenuation constant
\( \beta \) phase constant
It can be shown that \[ \sqrt{\frac{\mu_r}{\epsilon_r}} = \frac{1 + \Gamma_{\!\rm{inf}}}{1 - \Gamma_{\!\rm{inf}}} = c_1 \] and \[ \mu_r \epsilon_r = -\left[ \frac{c}{\omega L} \ln\frac{1}{P} \right]^2 = c_2 \] so the permeability and dielectric constant can be computed as follows: \[ \mu_r = c_1 \sqrt{c_2} \] and \[ \epsilon_r = \frac{\sqrt{c_2}}{c_1} \]
When using the method described in [5], defining \( \Gamma_{\! \rm{o},\rm{s}} \) as the measured input reflection coefficient of the transmission line terminated with an open/short, respectively, it can be shown that
\[ \Gamma_{\!\rm{inf}} = Q \pm \sqrt{Q^2 -1} \, \text{, where } Q = \frac{1 + \Gamma_{\!\rm{s}} \, \Gamma_{\!\rm{o}}}{\Gamma_{\!\rm{s}} + \Gamma_{\!\rm{o}}} \] and \[ P^2 = \frac{\Gamma_{\!\rm{o}} - \Gamma_{\!\rm{inf}}}{1 - \Gamma_{\!\rm{o}} \, \Gamma_{\!\rm{inf}}} \] which allows to compute \(\epsilon_r\) and \(\mu_r\) using the equations shown above.There are some additional complications remaining, mainly due to the fact that \(P\) is a complex value and so the imaginary part of \(\ln(1/P)\) has infinite possible values, all differing by \(2 \pi \). Details on how to handle this are in the papers cited above; for our purposes, since the measurements were done starting at a rather low frequency, assuming the initial phase is close to zero and unwrapping it as the frequencies is increased works also quite well.
One important details is that all the reflection coefficients reference impedance is the impedance of the transmission line without the material under test, not the reference impedance (usually 50 Ω) of the measuring instrument.
References: