Understanding the Effect of Eddy Currents on the High-Frequency Behavior of Magnetic Cores
The formation of eddy currents can cause problems for magnetic cores at high frequencies. To mitigate these undesirable effects, we first need to understand them.
In previous articles, we used the concept of permeability to characterize the interaction of a material’s atomic magnetic moments with an externally applied field. Since it’s a macroscopic representation of atomic behavior, we might expect the permeability to be independent of the material’s physical dimensions.
That expectation would be incorrect. Due to dimensional resonance and eddy currents, a material's magnetic properties for an AC excitation can vary with its dimensions. These phenomena are particularly important in high-power, high-frequency applications of magnetic materials.
By understanding these effects, we can minimize the number of design iterations needed to choose the right core material and dimensions for a magnetic component. This article explores the effect of eddy currents on the high-frequency behavior of magnetic materials. The next article in this series will examine dimensional resonance.
Permeability as a Function of Material Dimensions
We know that the real and imaginary parts of permeability change with frequency. Figure 1 shows the datasheet frequency response of Ferroxcube's 3E10 material.
Figure 1. Permeability of the 3E10 material vs. frequency. Image used courtesy of Ferroxcube
Though the plot above doesn’t reference it, the frequency response of an MnZn core is dependent on core size. The 3E10 material is MnZn-based. Figure 2 shows how its frequency response changes with its physical dimensions.
Figure 2. Permeability of the 3E10 material vs. frequency for different core dimensions. Image used courtesy of M. Kącki [PDF]
In the above figure, the designations T6, T29, T50, and T80 represent different-sized toroidal cores used in the experiment. The numbers indicate the cores’ outer diameters in millimeters. T6, for example, has an outer diameter of 6 mm.
A visual inspection reveals that the T29 core is the largest core that performs identically to what’s reported on the datasheet. The frequency response rolls off at a lower frequency for the T50 core.
The datasheet shows the frequency performance of small-sized cores, but high-power applications use large cores to handle the required power levels. When operated at high frequencies, the loss density of a large core can be significantly higher than the manufacturer’s specified value. Eddy currents make the flux distribution across the cross-section of the core uneven, potentially leading to degraded permeability performance.
The Basics of Eddy Currents and the Skin Effect
In a previous article series, we went into great detail on the relationship of eddy currents and the skin effect in current-carrying conductors. Some of the basic concepts are relevant here—let’s review them before we dive in too far. First, Faraday’s law and Lenz’s law:
- Faraday’s law states that a changing magnetic field induces a voltage, and hence a current, in a conductive wire.
- Lenz’s law states that this induced current generates a magnetic field in the opposite direction of the original magnetic flux.
When a bulk piece of conductive material is placed in a changing magnetic field, the flux produces circulating currents. Figure 3 illustrates how these currents, known as eddy currents, form.
Figure 3. Eddy currents are produced when flux passing through a conductive material changes. Image used courtesy of Sciencefacts
When AC current flows through a wire, the time-varying magnetic field from the current produces eddy currents in the wire (Figure 4).
Figure 4. Eddy currents in a current-carrying conductor. Image used courtesy of Steve Arar
If we compare the direction of the eddy current with that of the main current, it’s apparent that these two currents have the same direction near the surface of the conductor. However, they have opposite directions near the center of the conductor.
As a result, the overall current isn’t evenly distributed across the cross-section of the wire. Instead, the AC current mostly tends to flow through a shallow layer just underneath the surface of the conductor. This is known as the skin depth.
A similar phenomenon occurs when a conductive material is used as the core of an inductor, as Figure 5 illustrates. Though we don’t intend to pass a current through the core, unwanted eddy currents are produced by exposure to a changing magnetic flux.
Figure 5. Eddy currents induced in an electrically conductive core. Image (adapted) used courtesy of Frenetic
In this figure:
- Ba is the magnetic flux from the applied field.
- Be is the induced magnetic flux.
- i(t) is the time-varying input current.
- ie is the eddy current.
Ba is produced by the time-varying input current, meaning that it is itself time-varying. Faraday’s law therefore applies, and Ba induces eddy currents (ie). In accordance with Lenz’s law, ie induces a new magnetic flux (Be) that opposes Ba.
Field Distribution in a Cylindrical Core
Eddy currents can render the flux distribution across the cross-section of a core non-uniform, just like they do to the AC current distribution in a wire. In fact, we use the same equation to assess the flux distribution in a magnetic core as we do to find the skin depth (δ) for AC current distribution:
$$\delta ~=~ \frac{1}{\sqrt{\pi f \mu \sigma}}$$
Equation 1.
where:
f is the frequency of operation
µ is the magnetic permeability of the conductor, measured in H/m
σ is the conductivity, measured in S/m.
Note that the skin depth is independent of the material’s permittivity. In the next article, we’ll see that the permittivity of the materials plays a key role in creating dimensional resonance—the other phenomenon that can degrade the high-frequency performance of the core.
The AC current distribution in a round wire depends on the ratio of the conductor’s radius to its skin depth at the frequency of interest. If the radius is much larger than the skin depth, most of the current flows through the skin depth of the wire. When the skin depth and the radius of the conductor are comparable, however, the whole cross-sectional area of the conductor is almost equally effective in carrying AC current.
Similarly, the field distribution in a cylindrical core depends on the ratio of the core radius to the skin depth. Figure 6 illustrates the field distribution for a cylindrical core at four different frequencies. The core has a diameter of 50 mm (a = 50 mm).
Figure 6. The effect of skin depth on field distribution. Image used courtesy of G. R. Skutt
In these simulations, a uniformly wound coil is used to produce a uniform excitation field that is applied to the core. The simulations correspond to the point in time when the sinusoidal current in the coil is at its peak.
At f = 60 kHz, the skin effect is negligible. The magnetic flux in the core is uniform and in phase with the exciting current. As we go to higher and higher frequencies, the ratio of the core diameter to the skin depth increases, making the skin effect more and more pronounced. At f = 520 kHz, most of the flux is concentrated in a shallow region below the surface of the core.
Skin Depth of Different Magnetic Materials
To determine whether a core is large enough to support significant eddy currents, we need to consider its material properties as well as the frequency of operation. Table 1 compares the skin depths of some commonly used magnetic materials with that of copper. The parameters that influence skin depth—permeability, conductivity, and frequency—are tabulated as well.
Table 1. Skin depth for some typical parameter values. Data used courtesy of G. R. Skutt
| Material | Permeability (μr) | Conductivity (σ in S/m) | Skin Depth: f = 60 Hz |
Skin Depth: f = 100 kHz |
Skin Depth: f = 1 MHz |
| Copper | 1 | 5.8 × 107 | 8.5 mm | 0.21 mm | 0.07 mm |
| Silicon Steel | 5 × 104 | 2 × 106 | 0.21 mm | 0.005 mm | 0.002 mm |
| MnZn Ferrite | 3,000 | 0.5 | 1.7 m | 41.1 mm | 13 mm |
| NiZn Ferrite | 100 | 0.01 | 65 m | 1.6 m | 0.5 m |
Copper has a high conductivity (σ = 5.8 × 107) but a permeability close to that of free space (μ = μrμ0 = 1 × 4π × 10-7 H/m), leading to a small skin depth of about 70 μm at 1 MHz. Silicon steel has both high permeability and high conductivity, producing an even smaller skin depth of about 2 μm at 1 MHz. This is why iron-core devices need to combat the skin effect by using finely laminated core structures.
Ferrites have high permeability but relatively low conductivity, reducing the effect of eddy currents. This is the main advantage ferrites have over other magnetic materials. They also have a relatively large skin depth.
For example, an NiZn ferrite core has a skin depth of 0.5 m at 1 MHz. The MnZn ferrite material, on the other hand, has a high-frequency skin depth comparable to the dimensions of a typical core. The difference is due to the MnZn material having a much greater conductivity than the NiZn material. However, even that “greater” conductivity is much lower than the conductivity of silicon steel or copper.
Reducing the Skin Effect
Non-uniform flux distribution is undesirable—it can create localized magnetic saturation, reduce permeability, and substantially increase losses. The skin effect may put constraints on the maximum core dimensions that we can use in a high-frequency core. Laminated structures can be used to break the conduction path through the core, reducing the effect of eddy currents, but the practical minimum lamination thickness can limit the achievable performance.
Even though there are a limited number of solutions to combat the skin effect, it’s important to understand this phenomenon. It helps us to know how magnetic cores perform at high frequencies and how we can minimize, or at least recognize, these effects in our designs. In the next article, we’ll explore dimensional resonance and how it interacts with the skin effect to produce the overall field distribution in a high-frequency magnetic core.
Featured image used courtesy of Adobe Stock
