In our ideal world, safety, quality and performance are paramount. However, the cost of the final component (which includes the ferrite) has in many cases, become the deciding factor. This article is written as an aide for the design engineer looking for alternative ferrite materials as a means to reduce cost.

**FERRITE APPLICATIONS**

The following are three major applications for soft ferrite:

1. Low Signal Level

2. Power

3. EMI

The required intrinsic material characteristics and core geometry are dictated by each specific application. The intrinsic characteristics controlling the performance of low signal level applications are permeability (particularly with temperature), low core loss, and good magnetic stability with time and temperature. Applications include high Q inductors, common mode inductors, wideband, matching and pulse transformers, antenna elements for radios and both active and passive transponders. For power applications the desirable characteristics are high flux density and low losses at the operating frequency and temperature. Applications include switchmode power supplies, magnetic amplifiers, dc-dc converters, power filters, ignition coils, and transformers for battery charging of electrical vehicles.

The intrinsic characteristic that most influences the performance of soft ferrite in suppression applications, is the complex permeability [1], which is directly proportional to the core’s impedance. There are three ways to use ferrites as suppressors of unwanted signals, conducted or radiated. The first, and least common, is as actual shields where ferrite is used to isolate a conductor, component or circuit, from an environment of radiated stray electromagnetic fields. In the second application, the ferrite is used with a capacitive element to create a low pass filter that is inductance – capacitance at low frequencies and dissipative at higher frequencies. The third, and most common use, is when the ferrite cores are used alone on component leads or in board level circuitry. In this application the ferrite core prevents any parasitic oscillations and/or attenuates unwanted signal pickup or transmission that might travel along component leads or interconnected wires, traces, or cables. In both the second and third application the ferrite core suppresses the conducted EMI by eliminating or greatly reducing the high frequency currents emanating from the EMI source. The introduction of the ferrite provides a sufficiently high frequency impedance that results in the suppression of the high frequency currents. Theoretically, the ideal ferrite would provide a high impedance at the EMI frequencies, and zero impedance at all other frequencies. In reality, ferrite suppresser cores provide a frequency dependent impedance. Low at frequencies below 1 MHz, and depending upon the ferrite material the maximum impedance can be obtained between 10 MHz to 500 MHz.

**COMPLEX PERMEABILITY**

As is consistent with electrical engineering principles in which alternating voltages and currents are denoted by complex parameters, so the permeability of a material can be represented as a complex parameter consisting of a real and an imaginary part. This is evidenced at high frequencies where the permeability separates into two components. The real component (μ’) represents the reactive portion, and is in phase [2] with the alternating magnetic field, whereas the imaginary component (μ”) represents the losses, and is out of phase with the alternating magnetic field. These may be expressed as series components (μ_{s}’ μ_{s}” ) or parallel components (μ_{p}’ μ_{p}”). The graphs in Figures 1, 2 and 3 show the series components of the complex initial permeability as a function of frequency for three ferrite materials. Material type 73 is a manganese zinc ferrite with an initial permeability of 2500. Material type 43 is a nickel zinc ferrite with an initial permeability of 850. Material type 61 is a nickel-zinc ferrite with an initial permeability of 125.

**Figure 1**

**Figure 2**

**Figure 3
**

Concentrating on Figure 3, the series components of 61 type material, we see that the real part of the permeability, μ_{s}’, remains constant with increasing frequency until a critical frequency is reached and then decreases rapidly. The losses, or μs”, rise, then peak as, μs’ falls. This decrease in μs’ is due to the onset of ferrimagnetic resonance. [3] It should be noted that the higher the permeability, the lower the frequency at which this occurs. This inverse relationship was first observed by Snoek and given the following formula:

eq. (1)

where:

ƒ_{res} = frequency at which μs” is maximum

γ = gyromagnetic ratio = 0.22 x 10^{6} A^{-1} m

μ_{i} = initial permeability

M_{sat} = 250-350 Am^{-1}

This same equation can be approximated by:

ƒ_{res} = B _{sat}/μ_{i} MHz eq. (2)

Since ferrite cores used in low signal level and power applications are concerned with magnetic parameters below this frequency, rarely does the ferrite manufacturer publish data for permeability and/or losses at higher frequencies. However, higher frequency data is essential when specifying ferrite cores used in the suppression of EMI.

**RELATIONSHIP BETWEEN COMPLEX PERMEABILITY AND IMPEDANCE**

The characteristic that is specified by most ferrite manufacturers for components used in EMI suppression is the impedance. The impedance is easily measured on readily available commercial analyzers with direct digital readouts. Unfortunately, the impedance is usually specified at particular frequencies and is the scalar quantity representing the magnitude of the complex impedance vector. Although this information is valuable, it is often not sufficient, especially when modeling the ferrite’s circuit performance. In order to accomplish this, the impedance value and

phase angle for the components, or the complex permeability for the specific material must be available.

But even before beginning to model the performance of a ferrite component in a circuit, the designer should know the following:

- The frequency of unwanted signals.
- The source of the EMI (radiated/conducted).
- Operating conditions (environment).
- If high resistivity is required because of multiple turns, conductor pins in a connector filter plate or position in the circuit.
- The circuit impedance, source and load.
- How much attenuation is required.
- The allowable space on the board.
- The design engineer can then compare materials at the relevant frequencies for the complex permeability, heeding effects of temperature and field strength. After this, core geometry can be selected, from which the inductive reactance and series resistance can be defined.

**The Equations**

The impedance of a ferrite core in terms of permeability is given by:

Z = jωμL_{o} eq. (3)

and

μ = μ’ – jμ” = (μ_{s}’^{2} +(jμ”_{s})^{2})^{1/2} eq. (4)

where

μ’= real part of the complex permeability

μ”= imaginary part of the complex permeability

j = unit imaginary vector

L_{o}= the air core inductance

therefore

Z = jωL_{o} (μ’ −jμ”) eq. (5)

The impedance of the core is also considered to be a series combination of inductive reactance (XL) and the loss resistance (Rs), both of which are frequency dependent. A loss free core would have an impedance that would be given by the reactance:

X = jωL_{s} eq. (6)

A core that has magnetic losses may be represented as having an impedance:

Z = R_{s} + jωL_{s} eq.(7)

where:

R_{s} = total series resistance =R_{m} + R_{e}

R_{m} = Equivalent series resistance due to the magnetic losses

R_{e} = equivalent series resistance for copper losses

At low frequencies the impedance of the component is primarily the inductive reactance. As frequency increases, the inductance decreases at the same time the losses increase and the total impedance increases. Figure 4 is a typical curve of X_{L}, R_{s} and Z vs. frequency for our a medium permeability material.

**Figure 4**

Knowing that the magnetic quality factor

Q = μ’/μ” = ωL_{s}/R_{s} eq. (8)

then the inductive reactance is made directly proportional to the real part of the complex permeability by L_{o}, the air core inductance:

jωL_{s} = j ωL_{o}μ_{s}’

The loss resistance is also made directly proportional to the imaginary part of the complex permeability by the same constant:

Rs= ωLoμs”

Substituting into equation (7) for impedance:

Ζ = ωL_{o}μ_{s}”+ jωL_{o} μ_{s}’

and factoring:

Z = jωL_{o} ( µ_{s}’ – jµ_{s}’’) eq. (9)

In equation 9, the core material is given by µ_{s}’ and µ_{s}’’, and the core geometry is given by L_{o}. Thus, knowing the complex permeability for different ferrites a comparison can be made to obtain the most suitable material at the desired frequency or frequency range. After the optimum material is chosen, the best size component can be selected. The vector representation for both the complex permeability and impedance is found in Figure 5.

**Figure 5**

If the manufacturer supplies graphs of complex permeability vs frequency for the ferrite materials recommended for suppression applications, then a comparison of core shapes and core materials for optimizing impedance is straight forward. Unfortunately this information is rarely made available. However, most manufacturers supply curves of initial permeability and losses vs. frequency. From this data a comparison of materials for optimizing core impedance can be obtained.

**Material Selection
**

**Example**Referring to Figure 6, Initial Permeability & Loss Factor [4] vs. Frequency for Fair-Rite 73 material, supposing a designer wants to guarantee maximum impedance between 100 and 900 kHz. 73 material is chosen. For purposes of modeling, the designer also needs to know the reactive and resistive portions of the impedance vector at 100 kHz (105 Hz) and 900 kHz. This information can be derived from the graphs as follows:

At 100kHz μ_{s}’=μ_{i} = 2500 and (Tan δ/μ_{i}) = 7 x 10^{-6} since Tan δ = μ_{s}”/μ_{s}’ then μ_{s}” = (Tan δ/μ_{i}) x (μ_{i})2 = 43.8

Calculating for the complex permeability:

μ = μ’ − jμ” = (μ_{s} ’ 2 +( jμ”_{s})^{2})^{1/2} = 2500.38

It should be noted that, as expected μ” adds very little to the total permeability vector at this low frequency. The impedance of the core is primarily inductive.

However at 900 kHz μ_{s}” has become a significant contributor

μ_{s}’= 2100, μ_{s}”=1014 μ = 2332

**Figure 6**

Core Selection

The designer knows that the core must accept a #22 wire, and fit into a space of 10 mm by 5 mm. The inside diameter will be specified as .8 mm. Solving for the estimated impedance and its components, first a bead with an outside diameter of 10 mm and a height of 5mm is chosen:

at 100 kHz

since Z= ωL_{o} μ and Toroidal L_{o}= .0461 N^{2} log_{10} (OD/ID) Ht 10^{-8} (H)

then

Z= ωL_{o} (2500.38) = (6.28 x 10^{5}) x .0461 x log_{10} (10/.8) x 5 x (2500.38) x 10^{-8} = 3.97 ohms

where

R_{s} = L_{o} ω μ_{s}” = .069 ohms

X_{L} = L_{o} ωμ_{s}’= 3.97 ohms

at 900 kHz

Z = 33.3 ohms, R_{s} =14.48 ohms, X_{L} =30.0 ohms

Then a bead with an outside diameter of 5 mm and a length of 10 mm is selected:

at 100 kHz

Z= ωL_{o} (2500.38) = (6.28 x 10^{5}) x .0461 x log_{10} (5/.8) x 10 x (2500.38) x 10^{-8}= 5.76 ohms

where

R_{s}= L_{o} ω μ_{s}” = .100 ohms

X_{L} = L_{o} ω μ_{s}’ = 5.76 ohms

at 900 kHz

Z= 48.1 ohms, R_{s} = 20.9 ohms, X_{L} = 43.3 ohms

In this instance, as in most, maximum impedance is achieved by using a smaller OD with the longer length. If the ID were larger, for instance 4 mm, the reverse whould have been true.

This same approach can be used if graphs of Impedance per unit Lo and phase angle vs. frequency are supplied. Figures 9, 10 and 11 are representative of such curves for the same three materials used throughout this article.

**Example**

The designer wants to guarantee maximum impedance for the frequency range of 25 MHz to 100 MHz. The available board space is again 10mm by 5mm and the core must accept a #22 awg wire. Referring to Figure 7 Impedance per unit L_{o} for three ferrite materials, or Figure 8, complex permeability for the same three materials, an 850 μ_{i} material is chosen. [5] Using the graph of Figure 9, Z/L_{o} for the medium permeability material , at 25 MHz, is 350 x 10^{8} ohm/H. Solving for the estimated impedance:

Z= 350 10^{8} x .0461 x log_{10} (5/.8) x 10 x 10^{-8}

Z=128.4 ohm Φ = 30 degrees

X_{L} = Z sin Φ = 126.8 ohms

R_{s} = Z cos Φ = 19.81 ohms

and at 100 MHz:

Z= 179.8 ohms Φ= 0

X_{L}= 0 ohms Rs= 179.8 ohms

**Figure 7**

**Figure 8**

**Figure 9**

The same approach may be used for different materials, dimensions, and frequencies.

The previous discussion assumed that the core of choice was cylindrical. If the ferrite core being used is for flat ribbon, or bundled cable, or a multi-hole plate, the calculation for the L_{o} becomes more difficult, and fairly accurate figures for the cores path length and effective area must be obtained in order to calculate the air core inductance. This can be done by mathematically sectioning the core and summing the calculated path length and magnetic area for each section. In all cases though, an increase, or decrease in impedance will be directly proportional to an increase or decrease in the height/length of the ferrite core. [6]

Figure 10

Figure 10

**Figure 11**

**RELATIONSHIP BETWEEN IMPEDANCE AND ATTENUATION**

As stated most manufacturers are specifying cores for EMI applications in terms of impedance, but often the end user needs to know the attenuation. The relationship that exists between these two parameters is:

Attenuation = 20 log_{10} ((Z_{s} +Z_{sc} + Z_{L}) / (Z_{s} + Z_{L})) dB

where

Z_{s} = Source impedance

Z_{sc} = Suppressor core impedance

Z_{L} = Load impedance

The relationship is dependent on the impedance of the source generating the noise and the impedance of the load receiving it. These values are usually complex numbers that can be infinite in scope and not easily obtained by the designer. Selecting a value of one ohm for both the load and the source impedance, as may be the case when the source is a switch mode power supply and the load many low impedance circuits, simplifies the equation and allows comparison of ferrite cores in terms of attenuation.

Under these conditions, the equation reduces to:

Attenuation = 20 log_{10} (Z_{sc}/2) dB

The graph in Figure 12 is a family of curves that show the relationship between the shield bead impedance and the attenuation for a number of commonly used values of the load plus the generator impedance.

**Figure 12**

Figure 13 is the equivalent circuit of an interference source with an internal impedance of Z_{s}, generating an interference signal through the series impedance of the suppressor core Z_{sc} and the load impedance Z_{L}.

**Figure 13**

**THE ENVIRONMENT**

**Temperature**

As stated, ferrite’s magnetic parameters can be effected by temperature and field strengths.

Figures 14 and 15 are graphs of impedance vs. temperature for the same three ferrite materials. The most stable of these materials is the 61 material, with a decrease in impedance of 8% at 100º C and 100 MHz. This is compared to a 25% drop in impedance for the 43 material at the same frequency and temperature. These curves, when supplied, may be used to adjust the specified room temperature impedance if desired attenuation is to be at elevated temperatures.

Figure 14

Figure 14

**Figure 15**

**Field Strength**

As in the case of temperature, dc and 50 or 60 Hz power current will also effect the same intrinsic ferrite characteristics which in turn with result in lowering of the impedance of the core. Figures 16, 17 and 18 are typical of curves that illustrate the effect of biases on impedance a ferrite material. The curve depicts the degradation of impedance as a function of field strength for a specific material as a function of frequency. It should be noted that as frequency increases the effect of biases diminishes.

**Figure 16**

**Figure 17**

**Figure 18**

**NEW MATERIALS**

Since the compilation of this data, Fair-Rite Products has introduced two new materials. Our 44 which is a nickel zinc, medium permeability material and our 31 which is a manganese zinc high permeability material.

Figure 19 is a plot of impedance vs. frequency for the same size bead in 31, 73, 44 and 43 materials. 44 material is an improved 43 material with higher dc resistivity, 109 ohm cm, better thermal shock characteristics, temperature stability and higher curie temperature (T_{c}). When compared to our 43 material, 44 material has slightly higher impedance vs. frequency characteristics. Still material 31 exhibits higher impedance than either 43 or 44 throughout the measured frequency range. Designed to alleviate the problem of dimensional resonance that effects low frequency suppression performance of the larger manganese-zinc cores, 31 has found successful applications as cable connector suppressor cores and large toroidal cores. Figures 20 is a curve of impedance vs frequency for 43, 31, and 73 material for a Fair-Rite core with an OD of .562”, ID of .250 and a HT of 1.125. When comparing Figure 19 to Figure 20 it should be noted that for the smaller core in Figure 19, for frequencies up to 25 MHz, 73 material is the optimum suppression material. However as the core cross section increases, the maximum frequency decreases . As can be shown by the data in Figure 20 where the highest frequency where 73 is optimum is 8 MHz. Also noteworthy is that 31 material is superior from 8 MHz to 300 MHz. However, being a manganese zinc ferrite, 31 material has a much lower volume resisitivity of 102 ohm-cm and exhibits greater changes in impedance with extreme temperature variation.

**Figure 19**

**Figure 20**

**Figure 21**

**Glossary**

**Air Core Inductance – L _{o} (H)**

The inductance that would be measured if the core had unity permeability and the flux distribution remained unaltered.

General formula L

_{o}= 4π N

^{2}10

^{-9}(H)

C1

Toroidal L

_{o}= .0461 N

^{2}log

_{10}(OD/ID)

Ht 10

^{-8}(H)

Dimensions in mm’s

**Attenuation – A (dB)**

The decrease in signal magnitude in transmission from one point to another. It is a scalar ratio of the input magnitude to the output magnitude in decibels.

**Core Constant – C ^{1} (cm^{-1})**

The summation of the magnetic path lengths of each section of a magnetic circuit divided by the corresponding magnetic area of the same section.

**Core Constant – C ^{2} (cm^{-3})**

The summation of the magnetic path lengths of each section of a magnetic circuit divided by the square of the corresponding magnetic area of the same section.

**Effective Dimensions of a Magnetic Circuit **

Area Ae (cm^{2}), Path Length le (cm) and Volume Ve (cm^{3})

For a magnetic core of given geometry, the magnetic path length, the cross-sectional area and the volume that a hypothetical toroidal core of the same material properties should possess to be the magnetic equivalent to the given core.

**Field Strength – H (oersted)**

The parameter characterizing the amplitude of the field strength.

H = .4 π NI/le (oersted)

**Flux Density – B (gauss)**

The corresponding parameter for the induced magnetic field in a area perpendicular to the flux path.

**Impedance – Z (ohm)**

The impedance of a ferrite may be expressed in terms of its complex permeability.

Z = jωL_{s} + R_{s} = jωL_{o}(μ_{s}’- jμ_{s}”) (ohm)

**Loss Tangent – tan δ**

The loss tangent of the ferrite is equal to the reciprocal of the Q of the circuit.

**Loss Factor – tan δ/μ _{i}**

The phase displacement between the fundamental components of the flux density and the field strength divided by the initial permeability.

**Phase Angle – Φ**

The phase shift between the applied voltage and current in a inductive device.

**Permeability – μ**

The permeability obtained from the ratio of the flux density and the applied alternating field strength is…..

Amplitude Permeability, μa – when stated values of flux density, are greater than that used for initial permeability.

Effective Permeability, μe – when a magnetic circuit is constructed with an airgap or airgaps, and then the permeability is that of a hypothetical homogeneous material which would provide the same reluctance.

Incremental Permeability, μΔ – when a static field is superimposed.

Initial Permeability, μi – when the flux density is kept below10 gauss.

**References**

- Soft Ferrites, Properties and Applications, 2nd Edition, 1988, Snelling E. C., Butterworth, Boston MA.
- Magnetic Materials and their Applications, 1974, Heck, C. Crane, Russak and Company, New York, NY.
- “‘Prayer Beads’ Solve Many of Your EMI Problems,” Parker C., Tolen, B., Parker, R., EMC Technology, Vol. 4, No. 2, 1985.
- Fair-Rite Products Soft Ferrite Catalog, 16th Edition, 2008.

**Notes**

- See Glossary for definition of other permeability terms
- In phase is when the maxima and minima of the magnetic field, H, and those of the introduction B, coincide. Out of phase is when the maima and minimum is displaced by 90°
- Ferromagnetic Resonance is also called spin precession resonance
- The loss factor, which is the normalization of loss tangent per unit if permeability, is a material property describing the loss characteristics per unit of permeability.
- This remains true so long as the increase in height/length does not cause the core to be in dimensional resonance.

Carole Parker has a Bachelor’s Degree in electrical engineering and A Master’s Degree in business administration from Rensselaer Polytechnic Institute. She has been employed by Fair-Rite Products for over 35 years. Starting as an application engineer in 1974, she was promoted to Director of Sales and Marketing in 1993, and became President of the Company in 1996. |