Index

B/H curve

Permeability

A_L factor

Q factor

Umax induct.

Thermal R

Umax power

Udissp vs pwr

Ferrite in HF applications

Materials and properties

(published in Electron # 9, 2001)

Introduction

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, Stoneham and the Ferroxcube Data Handbook: "Soft Ferrites and Accessories" is especially recommended.

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

Magnetic field

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)

of which:

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.

Magnetic induction

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)

of which:

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.

B-H curve

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, B_sat

From B_sat 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 B_sat , 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 B_sat 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 = n² .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: m_r , describing specific core material, according to: m = m_{0 .} m_r.

For an air core, m_r = 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: (U_L) and its impedance (Z_L), as in:

B = m .H

H = n .I / l

I = U_L / Z_L,

so we may write:

B = m .n .U_L / ( l .Z_L)

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 B_sat as specified by the manufacturer. We therefore may write:

B_max = m .n .U_L .Ö 2 / (l .Z_L) = 0,2 .B_sat

and from this:

U_{L (inductie)} = 0,14 .B_sat .l .Z_L / (m .n)

Since:

Z_L = 2 .p .f .L, and also:

L = n².m .A / l

the formula for the maximum allowable voltage across an inductor on a ferrite core for linear behavior:

U_{L (inductie)} = 0,89 .B_sat .f .n .A (5)

We will find this formula again at various places in this and other chapters.

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 B_sat material is selected.

Inductance factor A_L

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: m_0, m_r, en A / l in formula 4 in a single inductance factor: A_L, expressed in nH/n² (nano-Henry per turn squared):

A_L = m_{0 .} m_{r .} A / l (nH/n²) (6)

Attention: Some manufacturers prefer their own definition that may lead to confusion. Especially some iron-powder suppliers prefer A_L: micro-Henry per 100 windingen (μH/100 turns) as this will produce bigger numbers by a factor of 10! Better recalculate to the mainstream definition as in (6) when in a design process.

In our inductance calculations we now only have to multiply A_L by the number of turns squared, to directly find coil inductance:

L = n² . A_L (nH) (7)

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 when dividing A_L by the initial permeability: m_i, usually also specified by the manufacturer.

F = A_L / m_i = 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.

Inductance tolerance

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.

Complex permeability

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 A_L has been determined for low frequencies only so we better have a good look again at parameters outside this area.

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:

Z_L = 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:

m_r = m^’ - j.m^” (9)

with:

m^’ = pure inductivity

m^” = all core loss factors combined

Total complex impedance of our inductor on a ferrite core may now be described:

Z_L = 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 ². m^’ . m ₀ . A / l , (11)

and a real part: r + w . n ². 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:

r_F = w . n ². m^” . m ₀ . A / l (12)

We find that inductor loss "r_F" 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?

Figure 2: Complex permeability related to frequency

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 A_L at higher frequencies

The inductance factor is very practical when calculating impedances at low frequencies and when the inductor may be regarded as lossless. At higher frequencies core loss has to be incorporated, but since the out-of-phase relationship will have to be handled as a complex quantitie. We therefore calculate total inductor impedance:

__________ __________________________________

|Z_t| = \/ r_F² + (X_L)² = \/ (w.n ².m^”. m ₀ .A / l)² + (w.n ².m^’.m ₀ .A / l )²

__________

= w . n ². m ₀ . (A / l) . \/ (m^” )² + (m^’)² (13)

The part under the 'root' we call ‘m_C‘ and we may use this number directly when calculating choke impedance.

Analogue to A_L we determine different 'inductance' factors after finding the shape factor 'F' (formula 8):

A_RF(f) = F . m”_(f), for calculating inductor loss,

A_L(f) = F . m' _(f) , for calculating inductance, and

A_Z(f) = F . m_C(f) , for total coil impedance (14)

Above factors have been calculated as a design aid in table 2 in the previous chapter for a number of popular ferrite toroides for the frequency range 1,5 - 50 MHz. In this table, italic values are extrapolated from the graphs and should therefore be used with some care, more so for every next extrapolation step.

Quality factor Q

For inductors in resonant circuits, inductor quality is important as this is determining selectivity and power loss. This quality factor is specified as:

Q = w . L / r (15)

We will take a closer look at the Q factor in conjunction with ferrite materials, with inductance and loss factors as in formula 11. We may now write:

Q = (w . n². m^’ . m₀ . A / l ) / (w . n². m^”. m₀ . A / l ) = m^’ / m^” (16)

Formula 16 is showing that for most ferrite materials, inductor quality is almost entirely determined by the ferrite material properties. At real high quality factor (Q > 100) as with some ferrites or at very low frequencies copper loss should be taken in account as well.

Using manufacturers specifications we may now directly determine ferrite type for resonant application at a certain (HF) frequency from the inductance (m^’) and loss figures (m^”).

Let's look at table 2 again, with this little tool as a pointer. We will find that hardly any ferrite type qualifies as a core for high quality resonant circuits at HF frequencies, except 4C65 material for up to just under 10 MHz. This is showing that ferrites are not the material of choice for high quality resonant circuits at HF. Usually carbonyl powder iron cores are the better choice here, also because of better basic temperature coefficients.

Power loss in ferrite cores

Before we have determined maximum voltage across an inductor on a ferrite core for linear application. Above we also found material loss mechanisms that will absorb part of the signal power. Especially in high power application this lost power will make the core heat up and this is when we should be cautious not to approach high (Curie) temperatures, where ferrite material looses all permeability and only free-space inductance will determine inductance. Therefore we should take a closer look at this heat mechanism and governing factors.

Thermal resistance

Since most ferrite materials exhibit a positive temperature coefficient for permeability, inductance will rise with temperature which usually is a positive factor in transformer and choke applications. Relation between internal power dissipation and generated temperature is:

ΔT = P * R_th

We did some test to determine R_th. in toroide core shapes. From these tests we found the well known 36 mm. toroide to exhibit a thermal resistance of 7 K/W. For a temperature rise of 28 K, this core should not dissipate more than 4 Watt, provided this core may freely exchange heat with the environment. This temperature rise of 28 K is usually sufficient to keep almost all ferrite materials below Curie temperature, up to an environmental temperature of 65 C., which is usually quite satisfactory.

Thermal resistance in ferrites is related a materials constant and the amount of material involved. From more tests and factory information it was found that this R_th is related to the square root of material volume as in:

with 'a' a scaling factor.

For a toroide, volume may be expressed as in:

V = p . h .(D² – d²) / 4

with 'h', 'D' and 'd' as in formula 1.

For the specific 36 mm. toroide in most of our thermal tests, we found V = 10,32 cm³. We may now generalize our formula for the temperature rise of ferrite materials:

P = ∆T / Rth = ∆T * a *√ (V) (17)

and, after entering the values from the thermal tests:

4 = 28 * a * √ (10,32)

out of which: a = 0,044.

We may now determine maximum allowable power dissipation for any toroide shape and size and allowable temperature rise. As an example we would like to know maximum power dissipation in a 55,8 mm ferrite toroide (V = 29,9 cm³) at an allowable temperature rise of 40 K.:

P_max = ∆T * 0,044 * √ V = 40 * 0,044 * √ (29,9) = 9,5 watt.

For the maximum temperature rise as in above derivation the following conditions are relevant:

- When an antenna transformer is heating up on a sunny day, core temperature may easily go up to over 60 °C even without any additional power applied. An additional temperature rise of 30 K because of internal power dissipation may then bring total core temperature close to boiling water, when most other (plastic) materials already are giving in (isolation material, transmission-line coating / internal support materials).

- We have been looking at Curie temperatures as an upper limit. With 4C65 ('61') type of ferrite this point is reached at 350 °C, but 4A11 ('43') type already is limited at 125 °C. With the latter material when applied in the above example with an antenna transformer on a hot summer day, not much margin is left.

- A ferrite toroidal shape is often applied as a core material to transformers and chokes. Wire materials are usually coated with insulating and support materials also to ensure electrical characteristics (characteristic impedance). Since these wires and 'lines' are applied with some mechanical tension, deformation due to temperature rise may take place long before these materials are giving in, causing electrical characteristics to change beyond a desirable level.

For above reasons, maximum temperature rise due to internal power dissipation should be limited to 30 - 40 K. at all times.

Power dissipation

After calculating loss resistance as in formula 12, we now also have a means of determining total power dissipated in these losses. Inductor current will follow from:

I = U_L / Z_C,

with total internal power dissipation in the impedance series circuit:

P = ( U_L/ Z_C )² . r_F (18)

where

Z_C = w . n² . m₀ . m_c . A / l (formula 13, using m_c i.s.o. the root)

As we have seen, internal power dissipation is limited to a maximum value P_max. We now may derive a maximum value for the voltage across the inductor, limited by internal power dissipation:

U_{L(dissipation)} = SQRT (P_max (Q + 1/Q) X_L) (19)

In our formula development we have been calculating with small signal parameters. At large drive the relation between the filed strength H and the magnetic flux density B is no longer linear. A larger part of the hysteresis loop is opening up making for additional material loss. These additional losses are apparent when measuring the quality parameter, Q. In a number of temperature runs at various frequencies, this Q-loss and additional heat development in the core material may be catered for by slipping this Q-loss into formula 19. The final formula than becomes:

U_{L(dissipation)} = SQRT (P_max (Q/6 + 1/Q) X_L) (19a)

In many applications, voltage across the inductor is easily derived from other system quantities, e.g. total system power in a particular system impedance, usually 50 Ohm in power applications. The maximum voltage across the inductor / transformer on a ferrite core is a practical tool for determining 'fitness' of this component for such applications. We will find this formula again at various places in this and other chapters.

Addition factors to internal power limits

All above calculations apply to continuous power dissipation in a ferrite core. In radio-ham applications this is not very often the case. Usually we are listening for much longer periods than we transmit, although exceptions have been spotted. When limiting our transmissions to 5 minutes maximum and listening for the same period of time, maximum internal power dissipation of our ferrite core materials may easily be enhanced by a factor of 2.

When operating in SSB mode, a large margin is noticeable between effective and peak power; a factor of 5 and more may easily be measured, depending on type of speech, signal quality and type of speech-processing. Also when operating in CW mode an enhancement factor of 3 is applicable. Operating in frequency and phase modulation, carrier is maximum during the entire transmission period and switched of during listening.

Enhancement factors may also be taken into account for the voltage across the inductor / transformer according to formula 19, when the square root is taken of above factors (presented for power).

modulation type	enhancement factor
continuous carrier	1
FM, 50 % Tx	1,4
CW, idem	2,4
SSB with processor, idem	2,4
SSB, idem	3,2

Table 4: Enhancement factors for U_L-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.

Maximum induction or dissipation?

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 5 a choke with five turns on two different materials (4A11 and 4C65) at two toroide sizes. Although these calculations show high precision, it should be noted that inductance factor are specified with a tolerance of 25 %.

		4A11 material					4C65 material
	36 mm. toroide			55 mm. toroide			36 mm. toroide
f	Zc	UL	UL	Zc	UL	UL	Zc	UL	UL
MHz	W	(dissip.)	(induction)	W	(dissip.)	(induction)	W	(dissip.)	(induction)
0.2	35	37	34	43	45	63
0.5	98	39	86	121	47	158
1	215	37	171	265	45	315	32	61	197
1.5	346	36	257	426	43	473	47	86	296
4	1087	51	685	1338	59	1260	126	105	789
7	1116	57	1199	1374	68	2206	221	122	1381
10	1336	63	1713	1645	76	3151	328	127	1973
15	1580	71	2570	1945	85	4726	567	113	2959
20	1708	74	3427	2101	89	6302	808	91	3946
30	1953	80	5140	2405	99	9453	1184	76	5919
40	2079	82	6853	2560	99	12603	1543	58	7892
50	2210	87	8567	2710	105	15754	1969	66	9865

Table 5: Maximum inductor voltages based on n=5, ∆T=28 K en B_max = 0,2 B_sat.

Lower voltage applies

In table 5 we find that the maximum voltage for this 5-turn inductor on a 36 mm., 4A11 toroide at 10 MHz. is limited to 63 V., which translates to around 80 Watt in a 50 Ohm environment.

When we need higher system power, we may apply a bigger toroide e.g. as we may find in column 6, where a 58 mm. 4A11 toroide (allowing maximum internal dissipation of 6,8 Watt) will allow 76 V. at 10 MHz. across the inductor, which is translated into around 116 Watt in a 50 Ohm environment.

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, on9cvd@veron.nl