Saturation and hysteresis are both fundamental characteristics of magnetic core materials. They make the B-H curve nonlinear and multivalued, complicating the design of magnetic components. They also cause distortion and power loss.
We discussed hysteresis in two previous articles. In this article, we’ll learn about magnetic core saturation and how it relates to nonlinear behavior. We’ll then explore several different permeability definitions that allow us to characterize various aspects of a nonlinear B-H curve.
What is Magnetic Core Saturation?
When a ferromagnetic material is exposed to a magnetic field, its magnetic domains align with the external field, producing a strong field inside the material. A sufficiently large external field will eventually make all of the material’s magnetic domains align with it. Beyond this point, which is illustrated in Figure 1(b), no more alignment is possible. The material is then said to be saturated.
Figure 1. Ferromagnetic domains in (a) the absence of an external field and (b) in the presence of a strong external field. Image used courtesy of R. Nisticò
Magnetic core saturation can have a detrimental impact on the performance and efficiency of magnetic devices. Since the magnetic domains of the material can’t align any further, the magnetic flux of a saturated core is almost constant. A winding wrapped around the core therefore behaves almost like a short circuit, leading to distortion, overheating, and damage to the device.
The magnetic flux of a saturated core also becomes almost independent of the winding current, limiting the core’s ability to play its role as a medium for transferring or storing energy. In most applications, magnetic devices must be carefully designed to prevent core saturation.
Saturation and the B-H Curve
As you may recall from our earlier discussion of complex permeability, we use two different field quantities to analyze the material’s response to an external magnetic field:
- B, the flux density. This is defined as the total magnetic field inside the material divided by the material’s cross-sectional area.
- H, the field intensity. This is defined as the external magnetic field divided by the permeability of free space (μ0).
In general, we use the following expression to describe the relationship between the B and H fields:
$$B~=~ mu_0 mu_r H$$
Equation 1.
where μr is the material’s relative permeability. A material’s B-H curve represents this relationship by placing B on the vertical axis of a graph and H on the horizontal axis.
For a non-magnetic material, the B-H curve is linear. A single permeability value is therefore obtained for arbitrary values of H. For example, Figure 2 shows the B-H curve for an air-core inductor. The relative permeability of air is μr = 1, so the slope of the line is equal to μ0.
Figure 2. The B-H curve for an air-core inductor. Image used courtesy of Steve Arar
By contrast, Figure 3 depicts the relationship between B and H in a ferromagnetic material.
Figure 3. The B-H curve of a ferromagnetic material. Image used courtesy of Steve Arar
As you can see, the B-H characteristic of a ferromagnetic material isn’t a straight line. Instead, μr depends on H. Increasing the H field makes the flux density go up until it hits its maximum value (Bsat). Similarly, if we apply a large current in the opposite direction, the material saturates at –Bsat.
The Effect of Temperature on Saturation Flux Density
A ferromagnetic material’s permeability and saturation flux density change with temperature. For example, Figure 4 shows the typical hysteresis loops of Fair-Rite’s 61 Material at 25 °C and 100 °C.
Figure 4. Typical hysteresis loops of Fair-Rite’s 61 Material. Image used courtesy of Fair-Rite
As the temperature rises from 25 °C to 100 °C, the saturation flux density drops from around 2500 gauss to 2200 gauss. This isn’t terribly extreme: for some ferrites, a temperature rise from 20 °C to 90 °C can halve the value of Bsat. The design of a magnetic device must therefore account for temperature variations to avoid core saturation at its highest operating temperature.
The Bmax Magnetization Curve
Because core losses increase with the amplitude of the excitation signal, the maximum magnetic flux density of a design may be determined by either core saturation or core losses. To separate the core’s nonlinearity from its hysteresis loss, we use the model in Figure 5.
Figure 5. Hysteresis loops for different H field values (orange loops) and the Bmax magnetization curve (the green curve). Image used courtesy of Steve Arar
Under steady-state conditions, a range of hysteresis loops are obtained for different values of H. These are the orange loops in the figure. Connecting the tips of all the hysteresis loops produces the Bmax magnetization curve, which is shown in green. The Bmax magnetization curve gives the maximum flux density (Bmax) for different maximum field intensity (Hmax) values. It represents a core model that ignores the hysteresis loss.
As the shape of the Bmax magnetization curve demonstrates, the B-H characteristic is highly nonlinear. To more easily characterize different aspects of this curve, we define permeability in a number of different ways. These include:
- Amplitude permeability (µa).
- Initial permeability (µi).
- Incremental permeability (μΔ).
- Differential permeability (µd).
We’ll discuss all four of these definitions over the rest of this article.
Amplitude Permeability and Initial Permeability
In alternating magnetization, we usually care most about the peak values of the B and H fields. That being the case, we want to use the amplitude permeability, which is the slope of the line that goes from the origin to the tip of the hysteresis loop for a given value of H. The amplitude permeability, which is sometimes also called the large-signal permeability, is denoted by µa.
The peak B and H values correspond to the tips of the hysteresis loops at different H values. In other words, they’re points on the Bmax magnetization curve we examined above. We can therefore obtain µa by dividing B by H at any point on the magnetization curve. In mathematical language:
$$mu_a ~=~ frac{1}{mu_0} ~times~ frac{B}{H}$$
Equation 2.
When H is very low—close to zero—we refer to the amplitude permeability as the initial permeability (µi):
$$mu_i ~=~ frac{1}{mu_0} ~times~ lim_{Delta H rightarrow 0} frac{Delta B}{Delta H}$$
Equation 3.
Initial permeability is a key performance metric when assessing soft magnet materials. It’s especially important in telecommunications applications, where very low drive levels are involved. According to IEC 60401-3, the value of µi for a material is defined using a closed magnetic circuit—for example, a closed, ring-shaped, cylindrical coil—for f ≤ 10 kHz and B < 0.25 mT at a temperature of T = 25 °C.
Figure 6 illustrates the concepts of initial permeability and amplitude permeability.
Figure 6. Illustration of the initial permeability (µi) and amplitude permeability (µa) for a nonlinear B-H curve. Image used courtesy of Steve Arar
The slope of the purple line, which connects the origin to Point A, is Point A’s amplitude permeability. The material’s amplitude permeability continues to increase until it reaches its maximum at Point B. Beyond that point, µa decreases as the core approaches saturation. This is typical behavior for a magnetic material. The slope of the magenta line, which gives the ratio of B to H at very low excitation levels, is the initial permeability.
As an aside, it’s often important to know the position of the maximum permeability and the course of µa versus B. Figure 7, which is once again taken from the datasheet of Fair-Rite’s 61 Material, shows the material’s amplitude permeability for different excitation levels.
Figure 7. The amplitude permeability of Fair-Rite’s 61 Material for different excitation levels. Image used courtesy of Fair-Rite
With that, let’s examine another two permeability definitions.
Incremental Permeability and Differential Permeability
In many inductor applications, the core is excited by an AC signal riding on a DC component. When that’s the case, the AC component produces minor B-H loops on the Bmax magnetization curve. These B-H loops are drawn in cyan in Figure 8.
Figure 8. Minor B-H loops in a ferromagnetic material. Image used courtesy of Steve Arar
Each loop in the above figure corresponds to a different DC value of the H field (HDC). The incremental permeability (μΔ) at a given value of HDC is equal to:
$$mu_{Delta}~=~frac{1}{mu_{0}}~times~frac{Delta B}{Delta H}$$
Equation 4.
In other words, the incremental permeability is the slope of the minor loop at a given field intensity value. Figure 9 shows how the incremental permeability of a ferromagnetic material changes with the field intensity.
Figure 9. Typical incremental permeability versus the H field. Image used courtesy of Steve Arar
At low excitation levels, the incremental permeability is equal to the initial permeability. It increases with the value of H until it reaches a maximum. As we continue to increase the excitation level, the core saturates and the incremental permeability approaches that of free space.
By way of example, Figure 10 shows the incremental permeability of 61 Material. As you can see, it follows the same pattern as Figure 9.
Figure 10. The incremental permeability of Fair-Rite’s 61 Material for different excitation levels. Image used courtesy of Fair-Rite
Finally, the differential permeability (µd) is defined as the slope of the Bmax magnetization curve at a given operating point. The slopes of the two green curves in Figure 11 illustrate incremental and differential permeability.
Figure 11. Incremental and differential permeability definitions. Image used courtesy of Omron
It’s worth mentioning that some resources, such as the book “High-Frequency Magnetic Components” by M. Kazimierczuk, define incremental permeability the same way that we defined differential permeability. Figure 12 provides another example of a resource that defines incremental permeability in this way. This figure comes from the online help files for QuickField, a finite element analysis software package.
Figure 12. Some references don’t make a distinction between incremental and differential permeabilities. Image used courtesy of QuickField
Wrapping Up
The B-H characteristic of a ferromagnetic material is highly nonlinear. In this article, we examined four definitions of permeability that can help us describe this nonlinear behavior. We also learned about core saturation, a major concern when designing magnetic components. In the next article, we’ll discuss how to avoid saturation in inductors and transformers.
As a final note, please be aware that the above permeability definitions capture the behavior of magnetic materials only at low frequencies. At these frequencies, the B and H fields are in phase and the permeability is a real value. As we increase the frequency, these two vectors are no longer in phase.
To correctly describe the material’s behavior over a wide frequency range, we use the concept of complex permeability. There are also a few other low-frequency permeability definitions, such as effective and apparent permeability, that we didn’t cover in this article due to space constraints.
