Understanding Dimensional Resonance in High-Frequency Magnetic Cores

In the previous article in this series, we learned how eddy currents can degrade the frequency response of large, high-frequency magnetic cores. We also mentioned another phenomenon, dimensional resonance, that can limit the bandwidth and increase the losses of these cores. In this article, we’ll examine dimensional resonance just as we did eddy currents.

 

Propagation Velocity and Wavelength

Though both eddy currents and dimensional resonance alter the field distribution in large cores, they do so by completely different mechanisms. To understand the dimensional resonance effect, we first need to review some fundamental principles of electromagnetics.

From our college courses, we know that the propagation velocity of an electromagnetic wave inside a material is given by:

$$v~=~\frac{c}{\sqrt{\mu_r \epsilon_r}}$$

Equation 1.

 

where:

c is the speed of light in vacuum

μ_r is the relative permeability of the material

ε_r is the relative permittivity of the material.

The propagation velocity is the speed at which a wave travels through a medium. In a material having μ_r = 1 and ε_r = 1, for example, the propagation velocity is equal to the speed of light in vacuum. However, if a magnetic material has a permeability of μ_r = 100, then the wave velocity inside the material is one tenth what it would be in free space.

Noting that the wave travels a distance of one wavelength in one period, we can easily derive the following relationship between the propagation velocity, wavelength (λ), and frequency (f):

$$v~=~\frac{c}{\sqrt{\mu_r \epsilon_r}}~=~\lambda f$$

Equation 2.

 

The above equation shows that the wavelength is directly proportional to velocity at a given frequency. For example, if a magnetic material has a permeability of μ_r = 100, then the wave velocity and wavelength inside the material are both one tenth what they would be in vacuum. These short wavelengths can adversely affect high-frequency magnetic components, as we’ll soon see.

 

What is Dimensional Resonance?

The main advantage of ferrites is their low conductivity, which translates to smaller losses from eddy currents. However, the molecular structure of these materials can also lead to a high permittivity. For example, MnZn cores have a relative permittivity on the order of 10⁴. NiZn cores typically have a smaller relative permittivity, ranging from 100 to 10 over their usable frequency range.

Since MnZn cores can present high permeability and permittivity simultaneously within the frequency range of kHz through MHz, the propagation wavelength in these materials can be significantly shorter than in free space. For example, assume that μ_r = 1,000 and ε_r = 100,000 at f = 1 MHz for a certain ferrite material. Equation 2 shows that the wavelength reduces from about 300 m in free space to 3 cm inside the material.

The wavelength inside a magnetic core operating at high frequencies (or even low frequencies, if the core is large enough) may be comparable to the core’s cross-sectional dimensions. When this happens, different points of the core cross-section experience different magnetic field values. In other words, the field distribution isn’t uniform across the cross-section.

The cross-section of a core is perpendicular to the magnetic field. When one dimension of the core’s cross-section is equal to one half wavelength, the core will support a standing wave. This is known as dimensional resonance.

Figure 1 shows how dimensional resonance affects the field distribution across the cross-section of a high-permeability cylindrical core. The core is assumed to be lossless, with a diameter of 50 mm (a = 50) and a dielectric constant of 150,000.

 

Figure 1. Field distribution of a cylindrical core exhibiting dimensional resonance. Image used courtesy of G. R. Skutt

 

In these simulations, the core is exposed to a uniform field produced by a uniformly wound coil. The simulations show the field distribution when the sinusoidal current in the coil is at its peak.

In Figure 1(a), the frequency is too low for dimensional resonance to occur. The magnetic flux inside the core is uniform and in phase with the exciting current.

As the frequency is increased, however, the wavelength decreases in response. In Figure 1(b), it has decreased to the point where half a wavelength is equal to the core’s diameter (a = λ/2). It’s at this point that dimensional resonance occurs, producing a completely different field distribution than we saw in Figure 1(a). The field distribution continues to change when the frequency is increased even further, as we see in Figure 1(c) and Figure 1(d) (a = λ and a = 3λ/2, respectively).

 

Dimensional Resonance in MnZn and NiZn Materials

NiZn materials have far lower permeability and permittivity than MnZn materials. Because of this, an NiZn core will start exhibiting dimensional resonance at a much higher frequency than an MnZn core with the same cross-sectional dimensions. For NiZn cores used in typical RF applications, the dimensional resonance tends to be near 1 GHz.

Figure 2 gives some typical values for the resonance dimensions of MnZn and NiZn materials.

 

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Figure 2. Resonance dimensions of typical MnZn and NiZn materials. Image used courtesy of M. Kącki

 

For a given frequency of operation, the resonance dimension of the NiZn material is much larger than that of either MnZn core. To prevent the formation of standing waves in the material, the smallest cross-sectional dimension of the core should be smaller than half of the dimensional resonance wavelength.

 

The Combined Effect of Eddy Currents and Dimensional Resonance

The skin effect and dimensional resonance are two independent phenomena, so in theory they can be simulated and analyzed separately. However, because these two effects can occur simultaneously in a real-world magnetic core, it may not be feasible to experimentally isolate these two effects and evaluate them independently for a specific material. Figure 3 shows the overall field distribution when both of these effects occur together.

 

Figure 3. Field distribution due to the skin effect and dimensional resonance. Image used courtesy of G. R. Skutt

 

The core in Figure 3 has the same dimensions as the one we examined in Figure 1, and is simulated at the same four frequencies. Figure 1 examined only the effect of dimensional resonance—in this set of simulations, however, the conductivity of the material is set to a high value so that the skin effect is produced as well.

By comparing Figure 1(b) and Figure 3(b), we can observe how including the eddy currents raises the flux density near the surface of the core and reduces it at the center. This change in flux density, which is characteristic of the skin effect, is also apparent in the field distributions for the a = λ and a = 3λ/2 cases. It’s particularly striking in the a = 3λ/2 simulation.

 

How Different Combinations of Material Properties Affect Field Distribution

Now that we have a basic understanding of eddy currents and dimensional resonance in magnetic materials, let’s take a look at how our choice of material might affect the field distribution in the cross-section of a magnetic core. As an example, we’ll examine a lossless toroidal core with the following dimensions at a frequency of 500 kHz:

• An outer diameter of 90 mm.
• An inner diameter of 60 mm.
• A height of 20 mm.

The simplest case is a material with:

• Low conductivity (σ = 0.1 S/m).
• Low permittivity (ε_r = 1).
• High permeability (μ_r = 10,000).

The simulated field distribution for this core is shown in Figure 4.

 

Figure 4. Field distribution for μ_r = 10,000, ε_r = 1, and σ = 0.1 S/m. Image used courtesy of M. Kącki [PDF]

 

The flux distribution is almost uniform, though it gradually decreases between the inner radius of the core and the ring’s outer surface. This is consistent with the theoretical field distribution in an ideal toroid.

The second parameter combination of interest is:

• High conductivity (σ = 5 S/m).
• Low permittivity (ε_r = 1).
• High permeability (μ_r = 10,000).

The result is shown in Figure 5. As before, the simulation frequency is 500 kHz.

 

Figure 5. Field distribution for μ_r = 10,000, ε_r = 1, and σ = 5 S/m. Image used courtesy of M. Kącki [PDF]

 

The combination of high conductivity and high permeability leads to a small skin depth. The skin depth formula is reproduced below for your convenience:

$$\delta ~=~ \frac{1}{\sqrt{\pi f \mu \sigma}}$$

Equation 3.

 

A small skin depth means that the field is significantly attenuated at the center of the core. In this case, however, the low permittivity leads to a wavelength of 6 m. This wavelength is much greater than the core’s cross-sectional dimensions, which is why the above field distribution is unaffected by dimensional resonance. This situation is analogous to the skin effect that occurs in AC current distribution when one wavelength in the conductor is much larger than the skin depth.

Before we move on, it’s worth noting that the source for the above image mentions a permittivity of ε_r = 10,000 in the image’s caption. This is almost certainly a typo. The author clearly mentions in the text that a low permittivity is used for this simulation.

With that out of the way, let’s take a look at the third combination of properties:

• Low conductivity (σ = 0.1 S/m).
• High permittivity (ε_r = 50,000).
• High permeability (μ_r = 10,000).

Figure 6 shows the field distribution for this case.

 

Figure 6. Field distribution for μ_r = 10,000, ε_r = 50,000, and σ = 0.1 S/m. Image used courtesy of M. Kącki [PDF]

 

As we discussed earlier in the article, the combination of high permittivity and high permeability leads to a short wavelength. With these parameters, the wavelength is about 2.6 cm, which is close to the cross-sectional dimensions of the core. The simulation shows that the flux density reaches its maximum at the center of the core, as expected.

 

What If the Material Is Lossy?

In the above discussion, we considered only lossless materials, so μ_r and ε_r were assumed to be real values. For a lossy material, we need to use the complex permeability and permittivity values (\( \mu_r ~=~ \mu_s' ~-~ j \mu_s'' \) and \( \epsilon_r ~=~ \epsilon_p' ~-~ j \epsilon_p'' \)) in the wave equations. In this case, half the wavelength in the material is given by:

$$\frac{\lambda}{2} ~=~ \frac{\sqrt{2} \pi c}{\omega} ~\times~ \frac{1}{\sqrt{|\mu_r||\epsilon_r|~+~\mu_s'\epsilon_p' \big (1~-~\tan(\delta_m)\tan(\delta_d) \big )}}$$

Equation 4.

 

where the two loss tangent terms in the denominator are defined as:

$$\tan(\delta_m)~=~\frac{\mu_s''}{\mu_s'}$$

Equation 5.

 

and:

$$\tan(\delta_d)~=~\frac{\epsilon_p''}{\epsilon_p'}$$

Equation 6.

 

There are two different scenarios we need to consider for a lossy material. In one scenario, tan(δ_m)tan(δ_d) is much less than unity, meaning that the material isn’t significantly lossy. When that’s the case, Equation 4 shows that the wavelength is a relatively small value. Dimensional resonance can therefore occur.

In the other scenario, tan(δ_m)tan(δ_d) is much greater than unity. We’re therefore dealing with a significantly lossy material. Equation 4 now gives us a large value for the wavelength, which tells us that the core can’t support standing waves. In fact, the attenuation in this scenario is so large that the field is mostly concentrated in a shallow layer just underneath the core’s surface—the skin effect is dominant, in other words.

The skin depth for such a lossy material is given by:

$$\delta ~=~ \frac{\sqrt{2} c}{\omega} ~\times~ \frac{1}{\sqrt{|\mu_r||\epsilon_r|~-~\mu_s'\epsilon_p' \big (1~-~\tan(\delta_m)\tan(\delta_d) \big )}}$$

Equation 7.

 

In general, we need to consider both the wavelength of propagation and the skin depth of the material to determine whether the skin effect or dimensional resonance is dominant, or else if both effects are present simultaneously.

 

The Impact of Non-Uniform Fields

Non-uniform flux distribution can lead to localized magnetic saturation. When the flux distribution isn’t uniform, the total flux linking the turns of a winding isn’t simply proportional to the core cross-sectional area. In fact, dimensional resonance can cause equal amounts of magnetic flux in the positive and negative directions, leading to zero apparent permeability. Furthermore, non-uniform field distribution can cause substantially higher losses.

To reduce the effect of dimensional resonance, we must either:

1. Limit the frequency of operation.
2. Use a core with a smaller cross-section.

This concludes our discussion of dimensional resonance. I hope that this article, taken together with the previous one, has helped you to understand the origin and impacts of dimensional effects in high-frequency magnetic cores.