# Using Complex Permeability to Characterize Magnetic Core Losses

## In this article, we use the concept of magnetic field intensity to help explain how complex permeability models the losses of a magnetic core.

Permeability, which relates the magnetic field inside a material to an external field, is a key parameter for the ferromagnetic materials used in electrical components. At very low frequencies, a real-valued permeability can describe the material’s magnetization. At higher frequencies, however, magnetic material manufacturers use a complex value for permeability. The imaginary part of this complex permeability works to account for core losses—but how?

In this article, we’ll dive into the theory of complex permeability. We’ll begin our discussion by examining the concept of magnetic field intensity, which will be crucial to understanding some of the math later on. Note that when we refer to “magnetic materials” in this article, we specifically mean *ferromagnetic* materials.

### Magnetic Field Intensity

Previous articles in this series stated that for a material placed in a uniform magnetic field (*B*_{0}), the total magnetic field inside the material is given by:

$$\overrightarrow{B}~=~\mu_{r}\overrightarrow{B_{0}}$$

**Equation 1.**

**Equation 1.**

where *μ _{r}* is the material’s relative permeability.

When analyzing the effect of magnetic fields on a material, we need to constantly distinguish between *B*_{0} and *B*. To make this distinction clearer, we’ll define another field quantity—the magnetic field intensity, symbolized by *H*.

The magnetic field intensity is defined as the externally applied magnetic field divided by the permeability of free space (*μ*_{0}):

$$\overrightarrow{H} ~=~ \frac{\overrightarrow{B_0}}{\mu_0}$$

**Equation 2.**

**Equation 2.**

Using this new field quantity, Equation 1 can be rewritten as:

$$\overrightarrow{B} ~=~ \mu_0 \mu_r \overrightarrow{H}$$

**Equation 3.**

**Equation 3.**

Introducing a new quantity might seem redundant at first glance, but it’s actually a handy way to clarify which field we’re referring to. We use specific names for the *H* and *B* fields to highlight the following differences between them:

*H*is the magnetic field intensity (or strength).*B*is the magnetic flux density, or sometimes the magnetic induction.

Let’s look at an example.

### Example: Magnetic Field Intensity of a Solenoid

Consider the solenoid in Figure 1*.*

**Figure 1.** An example solenoid. Image used courtesy of Steve Arar

**Figure 1.**An example solenoid. Image used courtesy of Steve Arar

We want to answer two questions:

- What is the magnetic flux density (
*B*) and the magnetic field intensity (*H*) of this solenoid when a core isn’t used? - How does the flux density change when we insert a core with relative permeability
*μ*?_{r}

Assuming that the turns are closely spaced and the field inside the coil is uniform, we can apply Ampere’s law to find the field inside the coil. Without going through the steps in detail, the end result for an air-core solenoid is:

$$\overrightarrow{B} ~=~ \mu_0 \frac{N}{l} I ~=~ \mu_0 n I$$

**Equation 4.**

**Equation 4.**

where:

*N *is the total number of turns

*l* is the length of the solenoid

*I* is the current

*n* is the number of turns per unit length.

Because no magnetic core is being used, the magnetic field described in Equation 4 arises from the current through the coil without any contribution from core magnetization (*B* = *B*_{0}). Dividing this value by *μ*_{0} gives us the magnetic field intensity:

$$\overrightarrow{H} ~=~ \frac{\overrightarrow{B}}{\mu_0} ~=~ \frac{\mu_0nI}{\mu_0} ~=~ nI$$

**Equation 5.**

**Equation 5.**

If we insert a core, the magnetic flux density becomes:

$$\overrightarrow{B}~=~\mu_{0} \mu_{r} \overrightarrow{H}~=~\mu_{0} \mu_{r} n I$$

**Equation 6.**

**Equation 6.**

Due to the presence of the core, *B* now includes two magnetic field components:

- The magnetic field created by the electric current.
- The magnetic field created by the magnetization of the core material.

The magnetic field intensity, which itself is produced by electric currents, can therefore be thought of as the driving force that produces the magnetic flux density. Permeability quantifies the ease with which *H* gives rise to *B*.

### How Does a Magnetic Core Change Inductance?

Next, let’s see how the inductance of an air-core solenoid changes when we insert a magnetic core. The inductance of a circuit (*L*) is defined as the total magnetic flux passing through the circuit per unit current flowing through the circuit. For an air-core solenoid, we have:

$$L~=~\frac{N \Phi}{I}~=~\frac{NBA}{I}$$

**Equation 7.**

**Equation 7.**

where:

\(\Phi \) is the magnetic flux passing through each turn

*I* is the current flowing through the coil

*A* is the cross-sectional area of the solenoid.

With no magnetic core, we have \(\overrightarrow{B} ~=~ \mu_{0} \overrightarrow{H}\). Inserting a core increases the magnetic flux density by a factor of *μ _{r}*.

For instance, let’s say that the core material's permeability is 500 times greater than the permeability of free space. Using this core increases the field inside the coil by a factor of 500 for a given current. Equation 7 shows that the inductance of the coil also increases by the same factor.

Based on the above, if the inductance of an air-core solenoid is *L*_{0}, the inductance of the same solenoid with a magnetic core would be:

$$L ~=~ \mu_r L_0$$

**Equation 8.**

**Equation 8.**

Now that we know how a magnetic core changes the inductance of the circuit, we can use Equation 8 to see how it changes the impedance. Since an ideal air-core solenoid acts as an inductor with inductance *L*_{0}, its impedance is:

$$Z_0 ~=~ jL_0 \omega$$

**Equation 9.**

**Equation 9.**

When the core is inserted, the inductance—and, consequently, the impedance—is multiplied by the relative permeability, resulting in:

$$Z ~=~ j\mu_r L_0 \omega$$

**Equation 10.**

**Equation 10.**

### Accounting for the Core Losses: Complex Permeability

So far, we’ve assumed that the core is lossless. That’s why Equation 10 produces a purely inductive impedance, which we know dissipates zero average power. In reality, some of the input energy will be lost in the core as heat. How do we model these core losses?

Noting that resistors are the electrical components that represent losses, we need a way to include an additional resistive term in the impedance equation above. The only core property present in the equation is the permeability, so that’s the parameter we’ll modify. As you may have already guessed, we need to define the permeability as a complex value to account for the core losses. The equation for complex permeability is:

$$\mu_r ~=~ \mu_r' ~-~ j \mu_r''$$

**Equation 11.**

**Equation 11.**

By substituting complex permeability into Equation 10, we obtain:

$$\begin{eqnarray}Z ~&=& ~jL_0 \omega \big (\mu_r' ~-~ j \mu_r'' \big ) \\&=& ~jL_0 \omega \mu_r' ~+~ L_0 \omega \mu_r''\end{eqnarray}$$

**Equation 12.**

**Equation 12.**

There are now two different terms in our impedance equation:

- An inductive term that originates from the real part of the permeability (
*μ*′). This term shows that the core increases the flux through the coil and hence its inductance._{r} - A resistive term that originates from the imaginary part of the permeability (
*μ*″). This term is associated with the loss in the material._{r}

Equation 12 leads to the equivalent circuit model for the inductor shown in Figure 2, which consists of an ideal inductor in series with a resistor.

**Figure 2. **Equivalent circuit for series representation of core losses. Image used courtesy of Steve Arar

**Figure 2.**Equivalent circuit for series representation of core losses. Image used courtesy of Steve Arar

If the series circuit model in Figure 2 is converted to its parallel equivalent, the parallel resistance will likewise model the core losses.

### Magnetization Out of Phase With Applied Field

We defined complex permeability in Equation 11. Let’s see what that equation really means.

We know that permeability describes the relationship between the external field applied to a magnetic material and the field produced inside of it. The real part of the permeability in Equation 11 corresponds to the material magnetization that occurs in phase with the external field. This is the behavior that we expect from an ideal, lossless core.

On the other hand, the imaginary part of the permeability shows that some magnetization of the material occurs 90 degrees out of phase with the applied field. This phase shift leads to an induced voltage across the inductor that’s in phase with the current flowing through the circuit, producing a resistive term in the overall impedance equation.

Figure 3 shows the phase relationship between *B* and *H* fields for two different cores. Figure 3(a) corresponds to a lossless core, Figure 3(b) to a lossy one.

**Figure 3.** The phase relationship between the B* and *H* fields. Image used courtesy of Steve Arar*

**Figure 3.**The phase relationship between the

In the lossless core, \(\overrightarrow{B}\) is in phase with \(\overrightarrow{H}\). In the lossy core, however, the total field inside the material lags the magnetic field intensity. The phase lag of \(\overrightarrow{B}\) with respect to \(\overrightarrow{H}\) is called the loss angle. As we see in the above figure, it’s symbolized by δ_{m}.

The tangent of the loss angle is given by:

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

**Equation 13.**

**Equation 13.**

The loss tangent is also the ratio of a coil’s equivalent series resistance (neglecting copper resistance) to its reactance. This ratio is equal to the reciprocal of the inductance’s quality factor (*Q*), giving us:

$$\tan(\delta_m) ~=~ \frac{R}{\omega L}~=~\frac{1}{Q}$$

**Equation 14.**

**Equation 14.**

### Evaluating the Material’s Frequency Behavior

Frequency affects the permeability of magnetic materials. To evaluate the material’s performance properly, we need to know the values of *μ _{r}*′ and

*μ*″ at the frequency range of interest. Examining these values helps us verify that the material has enough real permeability at the desired frequency without incurring significant core loss.

_{r}The datasheet of a magnetic material usually illustrates the complex permeability as a function of frequency. For example, Figure 4 shows the permeability plots for 61 Material, an NiZn ferrite material created by Fair-Rite.

**Figure 4.** Permeability of Fair-Rite’s 61 Material vs. frequency. Image used courtesy of Fair-Rite

**Figure 4.**Permeability of Fair-Rite’s 61 Material vs. frequency. Image used courtesy of Fair-Rite

The real permeability of this material is nearly constant at low frequencies (below about 10 MHz). The imaginary permeability is likewise almost negligible. At low frequencies, we therefore have an effectively lossless core with high permeability. The material can be used over this frequency range to increase magnetic coupling in a transformer or to realize-high value inductors.

As we go to higher frequencies, the real permeability rises slightly and then falls rather rapidly. The imaginary permeability—or, equivalently, the loss—also rises to a maximum and then decreases rapidly. The frequency at which the imaginary permeability reaches its maximum is known as the *ferroresonance frequency*.

61 Material is one of the highest-frequency ferrite materials available. As you can see in Figure 4, its real permeability falls below 1 at around 600 MHz. This is why high-frequency transformers can’t rely on magnetic materials to increase the coupling between the primary and secondary coils.

### Wrapping Up

In this article, we learned how the imaginary part of complex permeability captures core losses at high frequencies. It’s worth noting that the concept of complex permeability is mainly used in signal applications, where the core is operated at field levels much lower than saturation.

At high field values, using the imaginary permeability to model losses can make the losses seem smaller than they really are. That’s why we usually don’t use the complex permeability in power electronics, where the nonlinear frequency behavior of the materials is more pronounced. We’ll explore the high-frequency, high-power behavior of magnetic materials in future articles.

*Featured image used courtesy of Adobe Stock*

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