# Permeability concept in Inductors

- Posted by Tomáš Zedníček
- On March 4, 2021
- 0

## Permeability μ

### Permeability describes an important effect in ferromagnetic materials. If a ferromagnetic material is placed in a magnetic field, it is observed that the magnetic flux becomes concentrated in this material. Analogous to electric resistance, the ferromagnetic material presents a good conductor for the field lines. Permeability may therefore be described as a magnetic conducting or penetrating property.

*Figure 1: Ferromagnetic material in a magnetic field*

The factor by which the induction (B) changes through the introduction of the material is called the **relative permeability (μ _{r})**.

The equation for relative permeability is extended for the space filled by the material:

*Figure: EQ-1_17*

The induction in the core material (B_{F}) in our example on page 19 (a constant permeability of μ_{r} = 800 is assumed) is given by:

*Figure: EQ-1_18*

The relative permeability of the material is however not constant but strongly non-linear. The permeability of a material is essentially dependent on:

- The magnetic field strength H (dependent on operating conditions → hysteresis curve)
- The frequency f (frequency dependent complex permeability)
- The temperature T (→ temperature drift, → Curie temperature)
- The material used

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**Typical permeabilities (μ _{r})**:

- Iron powder cores, superflux cores 50 … 150
- Manganese-zinc cores 300 … 20000
- Nickel-zinc cores 40 … 1500

**Complex permeability**

*Figure: Frequency dependence of permeability and impedance*

*Figure: EQ-1_20*

The introduction of complex permeability allows separation into an ideal (zero loss) inductive component and a frequency dependent resistive component which represents the losses of the core material. This treatment can be applied to** all** core materials and clearly differentiates between inductors and EMC ferrites.

The inductive component is represented by (μ^{I}) and the resistive component by (μ^{II}). The following applies to transformation on the impedance level:

*Figure: EQ-1_21*

with L_{0} = inductance of an air coil of the same construction and field distribution, without core material (μ_{r} = 1).

For the series impedance (Z):

*Figure: Equivalent impedance circuit diagram*

with L_{0} = inductance of an air coil of the same construction and field distribution, without core material (μ_{r} = 1).

*Figure: EQ-1_22*

Multiplying out and dividing into real and imaginary parts provides the following relationship:

- Loss component R
_{S}= ωL_{0}μS^{II} - Inductance component X
_{LS}= ωL_{0}μS^{I}

*Figure: Loss angle (δ)*

For the loss angle (δ), tan δ is given by:

*Figure : EQ-1_23*

A large angle (δ) means a high core loss; the phase relationship between voltage and current at the inductor is less than 90°.

Furthermore:

μ_{S}^{I} = μ_{i}

μ_{S}^{II} = μ_{i} · tan δ

(μ_{i} = initial permeability)

Similarly, inductance and resistance can also be presented as a parallel equivalent circuit; the following relationships apply:

*Figure : EQ-1_24*

These frequency dependent components can be measured with the aid of impedance analysis and represented in an associated graph:

*Figure : Impedance curve for SMD ferrite 742 792 034*

Observations from the above measurement graph:

- The inductance is stable in a certain frequency range, to show strong frequency dependence above approx. 10 MHz.

Above 100 MHz the inductance falls sharply, down to zero at approx. 250 MHz. - The loss component (R) grows continuously with frequency and reaches the same value as the X component at the so-called ferromagnetic resonance frequency. The

resistance value rises until the high MHz range and dominates over the impedance (Z).

The component shown here – a SMD ferrite – serves the user as a broadband absorber or filter component, as a result of its broadband loss resistance (R).

**Comparing core materials**

Core materials are only used effectively in the construction of inductors within a limited frequency range, as a result of the frequency dependent loss components. Core losses rise sharply above a typical frequency limit. The core material may then be used then as a filter component.

This relationship and also the limits of core materials are illustrated in the following graph:

*Figure : Inductive parts of impedance and their frequency dependence for various core materials*

*Figure: Resistive parts of impedance and their frequency dependence for various core mater*

**Observations:**

- Iron powder materials (Fe): May be used as pure inductance up to approx. 400 kHz; the R loss component dominates thereafter up to approx. 10 MHz (also beyond depending on the core material). The core is no longer effective in the frequency range above approx. 20 MHz.
- Manganese-zinc cores are inductive up to frequencies around 20 MHz – 30 MHz, typically with losses rising above 10 MHz. The core material is no longer effective in the frequency range above –approx. 80 MHz.
- Nickel-zinc cores are inductive up to frequencies around 60 MHz, above this, the core material shows losses up to frequencies of 1 GHz and more.

This quantitative comparison illustrates why nickel-zinc ferrites have become predominant in the EMC field. The core material can perform an effective filter function in the frequency range of greatest interest.

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