Ferrite in HF applications
Materials and properties
(published in Electron # 9, 2001)
In this article some properties will be
discussed about ferrite cores for inductors in HF applications. Related to
material properties, a few formulas will be derived that will have
interesting practical value when designing HF coils, transformers and baluns.
For a more fundamental discussion on these materials and properties, the book
by E.C. Snelling: "Soft ferrites", Butterworths Publishing,
As a background and to appreciate the derived formulas in this chapter please also refer to the introductory chapter in "Ferrites in HF applications".
Induction, permeability and flux density
When an electrical current is fed through a number of turns of electrical wire, an electro-magnetic field will be generated with a field strength of: H (A/m), which is related to the current strengths, the number of turns and the magnetic path length:
H = n . I / l (1)
H = magnetic field strength (A/m)
n = number of turns
I = electrical current (A)
l = magnetic path' length:
in case of a toroide: l = p . (D + d) / 2, with
D = outside diameter (m)
d = inside diameter (m)
This formula for the magnetic path length is an approximation that is fully adequate for 'run-of-the-mill' toroides in everyday applications. A more precise formula will take into account magnetic induction is increasing towards the inner diameter and will correct for this different path length accordingly.
The generated magnetic field will induce a magnetic induction: B in (ferrite) core material that may (and in case of ferrite will) be much larger than the initiating magnetic field:
B = m . H (2)
B = magnetic induction (Tesla, T, of V.s/m2)
m = permeability in H/m
Since permeability in ferrite materials is (much) larger than 1, almost all of the magnetic field will be inside the core material (low magnetic resistance) with a negligible amount outside (high magnetic resistance). Therefore just leading a wire through the center of a ferrite toroide already acts as a full turn.
Permeability is related to the type of core material and the magnetic field (current and number of turns); in alternating electrical fields also frequency is a parameter.
Let's look a bit closer at the relation between the magnetic induction: B and initiating magnetic field: H in figure 1. This figure is sub-divided into four quadrants, with positive values in the upper right hand quadrant and negative values in the lower left hand quadrant. Looking at the rising dashed line, we observe B to rise at rising H up to a certain level, after which this linear relation will flatten out and stay at a constant value at and after the induction saturation point, Bsat
From Bsat on, the magnetic induction does not change any more, so only permeability of free space is left: μ0 = 4 .p .10-7 H/m. Even some time before Bsat , the linear relationship between B and H is already lost and one may observe current distortions and hence distortion of the voltage across the inductor on this core. These distortions will produce harmonics we usually like to avoid in HF applications.
Energy and core loss
At a certain amount of magnetic field and induced magnetic induction, an amount of energy is stored in the inductor core. When still at a linear relationship, the energy density is equal to:
E = B . H / 2 (J / m3) (3)
Up to now we have been looking at the dashed line, starting at the origin. When the magnetic field is reduced from Bsat however, the induced field does not follow the dashed line any more but will follow the drawn line: a loop-type of figure will be followed from hereon. With the magnetic field H reduced to zero, a certain amount of induced field will remain inside the core (residual magnetism) , that may only be reduced to zero when the magnetic field H has been reversed and has reached a certain negative value. By further increasing the magnetic field, the induced field will increase as well (negatively), until saturation has been reached again, this time at the negative side. This behavior is repeated by reversing the magnetic field again. The specific loop form (hysteresis) strongly depend on the type of ferrite material and may vary from an almost perfect rectangle to an evenly almost perfect ellipsoid.
The reason for this behavior may be found at the microscopic material level, where small crystals reside. Inside these crystals magnetic domains exist (Weiss domains) with already aligned magnetic properties, this is known as ferrimagnetism. The external magnetic field H, will re-align these internal magnets, more so with increasing field strength. In this process, internal magnetic domain barriers have to be overcome, where energy will be lost. The shape of the hysteresis loop therefore has a profound relation with the amount of energy lost.
A better look to permeability
Looking at the 'standard' formula for inductance, we find the significance of permeability: m, as in:
L = nē .m .A / l (4)
L = inductance (Henry)
n = number of wire turns
A = core area (m2)
l = magnetic path length (m)
Permeability: m, may be subdivided into a general part, describing the 'space constant' m0 = 4 . p . 10 7 H/m, and the relative permeability: mr , describing specific core material, according to: m = m0 . mr.
For an air core, mr = 1, while for a some ferrite cores this specific permeability may go up to thousands and more. Therefore, a coil on a ferrite core may be have a very much higher inductance within the same volume than without this core. Vice versa, for the same inductance a coil on ferrite will have much less turns and so much less parasitic capacitance and therefore a higher application bandwidth. Especially with specific transmission-line transformers, that require as short a transmission line as possible, new applications become possible because of these ferrite materials. We will discuss these in one of the next chapters.
Maximum induction in the core
We have shown a relation between core induction and the electrical current in the inductor. This current will flow in relation to the voltage across the inductor: (UL) and its impedance (ZL), as in:
B = m .H
H = n .I / l
I = UL / ZL,
so we may write:
B = m .n .UL / ( l .ZL)
Voltage across the inductor is expressed as an effective value. For maximum inductance we need the maximum value of this (sinusoidal) voltage, that will be undistorted when no further saturated than about 20 % of the saturation inductance Bsat as specified by the manufacturer. We therefore may write:
Bmax = m .n .UL .Ö 2 / (l .ZL) = 0,2 .Bsat
and from this:
UL (inductie) = 0,14 .Bsat .l .ZL / (m .n)
From the formula we find that maximum voltage across the inductance is a (proportional) function of frequency. This is one of the reasons switch-mode power supplies operate at an elevated frequency since transformers may be much smaller, especially if high Bsat material is selected.
As we have seen in the inductance formula,
various parameters are related to the core form and type of material. To help
our calculations, many manufacturers make our life easy by presenting type
and form related values: m0, mr,
en A / l
in formula 4 in a single inductance
Attention: Some manufacturers prefer their
own definition that may lead to confusion. Especially some iron-powder
In our inductance calculations we now only
have to multiply
L = nē .
Table 1 (first chapter) presents a short impression of these factors as derived from toroide manufacturers specification: Ferroxcube, Siemens, Fair-Rite and Micrometals (Amidon supplier).
The shape factor F
The inductance factor is a very practical unit when calculating inductors and transformers. Results are reliable as long as application frequency is not too high, specifically not above 1 / 10 ferrimagnetic resonance frequency for that particular material. We will come back to this later.
At higher frequencies, we would like to know
the explicit coil shape factor to allow for losses to be brought into the
calculations. This shape factor is easily derived from the inductance factor
F = AL / mi = m0 . A / l (8)
This shape form factor F comes in handy.
As may be appreciated from formula 8, this form factor is related to the core area A and inversely related to the magnetic path length. This translates to higher inductance values on long tube-like coil formers as compared to more flattened toroides; hence the binocular and bead (tube) shapes we sometimes come across in HF applications.
Most of the above information may also be found in (manufacturers) data books. It should be noted that most manufacturers specify permeability to rather wide tolerances and +/- 25 % is no exception. Although smaller tolerances may be found as well, we should be aware that often permeability is rather sensitive to temperature variation, which sensitivity again to depend on the absolute temperature. This leads to property tolerances in the final application which should be taken into account when designing these components.
When designing at HF frequencies we usually
are forced to apply ferrite materials up to, or over ferrimagnetic resonance
frequencies. As we have seen, the inductance factor
Up to this moment we have been looking at inductance as a pure reactance. This may not be entirely true any more when moving to higher frequencies. Complex inductor impedance is usually described as a series circuit:
ZL = r + jwL, with "r" representing copper loss.
At higher frequencies Eddy-currents and hysteresis in the core material may no longer be neglected so we better incorporate these into our calculations. As may be appreciated from the impedance formula, reactance and loss come with a different phase relationship, which we may incorporate when changing specific permeability in formula 4 into:
mr = m - j.m (9)
m = pure inductivity
m = all core loss factors combined
Total complex impedance of our inductor on a ferrite core may now be described:
ZL = r + j.w.L = r + j.w.(n2 . m 0 .(m - j.m ) . A / l)
= r + w.n2.m 0 .m . A / l + j.w. n2 . m 0 .m . A / l (10)
and we once more find
an imaginary part: j .w . n ē. m . m 0 . A / l , (11)
and a real part: r + w . n ē. m . m 0 . A / l
At HF frequencies, copper loss "r" usually is (much) smaller than loss in the core material, so total inductor loss may be described as:
rF = w . n ē. m . m 0 . A / l (12)
We find that inductor loss "rF" is now also related to the operating frequency, next to the number of turns and the imaginary permeability, m .
Different frequency relationships
The loss factor: m is related to frequency, but to a different extend as the permeability factor: m'. Most manufacturers present these different dependencies in a useful graph as in figure 2. Unfortunately not all suppliers are presenting this type of information and one may wonder why some designers like to go along such trial and error road especially those designing for reproduction by others?
In figure 2 we find the frequency dependencies for m and m'. At the frequency where both are equal (here at about 5,5 MHz.) we find the ferrimagnetic resonance frequency, already mentioned before. At this frequency and even before this particular material may not be used in resonant circuits any more because of high loss. Up to and a little beyond this frequency the material may still be applied in (impedance) transformers and is still useful a long way beyond this resonant frequency when applied as a choke. In this last application, phase is not important as long as total impedance remains high, by whatever mechanism.
The inductance factor
Table 4: Enhancement factors for UL-power
Enhancement factors should be regarded with some care. In some applications the inductor / transformer is not free to radiate heat to the environment and some transformer manufacturers even apply molding raisin in antenna matching units with high isolation properties, trapping internally generated heat inside the cabinet. Therefore each specific application should be checked under worst case conditions before applying enhancement factors. In general it is prudent to measure internal temperatures first under controlled and worst case conditions before practically applying the component.
Also one should take care when applying impedance transformers in aerial systems. Although tuned antenna systems usually are design to operate around 50 Ohm, these easily may exhibit a much higher impedance when operated outside resonance. The antenna tuner at the transceiver side may match whatever impedance to the transceiver requirements, but the antenna transformer may be left to operate under a much different impedance regime (higher), hence much higher voltages than being designed for.
In this chapter we derived a formula for
maximum voltage across the inductor / transformer for maximum,
distortion-free operation. Next a different formula has been derived for the maximum
voltage related to internal power dissipation. It may be clear that at all
operating frequencies the lowest of these values should apply. It may be
instructive to find out how these maximum allowable values turn out in
practice. Therefore I calculated in table
Table 5: Maximum inductor voltages based on n=5, ∆T=28 K en Bmax = 0,2 Bsat.
Lower voltage applies
In table 5 we find that the maximum voltage
for this 5-turn inductor on a
When we need higher system power, we may
apply a bigger toroide e.g. as we may find in column 6, where a
Instead of this bigger core, one might also decide to apply more turns to enhance impedance, lowering internal power dissipation. Taking 6 i.s.o.5 turns, will enlarge inductor voltage to 76 V. and this translates to 114 W. in a 50 Ohm environment.
In column 9 we find 4C65 material to be more suitable for high voltages over much of the HF range, so higher system power up to around 20 MHz. This is a result of lower material loss and allows this 5 turns inductor to withstand around 100 V. between 4 and 20 MHz., to be applied in 200 W systems in a 50 Ohm environment. At frequencies above 20 MHz. not much difference exists any more for comparable core size between 4A11 and 4C65 material. This latter ferrite now also becomes lossy since we are approaching ferrimagnetic resonance for this material. In choke applications however both inductors will still do a very good job as may be seen in the impedance columns (2nd and 8th).
When comparing 3rd and 4th column, it may be noticed maximum inductance voltage is only important at lower frequencies (lower of the two voltages). For 4A11 material cross-over frequency is around 0,2 MHz., and for 4C65 MHz. below 1 MHz. Above these frequencies it is the internal power dissipation that will determine maximum allowable voltage across the inductor.
We noted before that transformer impedance should be at least four times system impedance to have negligible effect on signal. In a 50 Ohm system, the 5-turns inductor on 4A11 material will be 'invisible' at just over 1 MHz. with increasing impedance all the (frequency) way up.
For 4C65 material, the 5-turn inductor may be applied starting from 7 MHz. (8th column). We see the effect of a much lower initial permeability when compared to 4A11.
When designing an inductor / transformer on ferrite in wide frequency applications, it is good practice to determine maximum voltage for linearity as well for internal power dissipation. This is especially true when applying enhancement factors for different operation modes.
Bob J. van Donselaar