About Maximizing the Q of solenoid inductors that use ferrite rod cores, including charts of magnetic flux density and flux lines, with some actual Q and inductance measurements from simulations in FEMM
By Ben H. Tongue
Summary: Many factors interact to affect the Q of ferrite rod cored inductors. Part one of this Article identifies and comments upon some of them. A simplified model and an equivalent circuit is also discussed. The second part describes several ferrite cored inductors, along with measurements of inductance and Q. The third part displays graphs of flux density, flux lines, inductance and Q of several ferrite cored inductors. The fourth part shows how flux density changes when an air gap is inserted in the center of a ferrite rod. It also includes simulations showing the distribution of magnetic flux density and current density in the turns of the solenoid. The fifth part shows how Q varies when the ratio of the length of the solenoid to the core changes Also shown is a chart showing the change of Q when the conductor spacing is changed. The sixth part discusses important info about ferrite 61 and similar materials.
Part 1: Modeling ferrite cored inductors
Bulk factors that affect inductor Q:
A simplified equivalent circuit model for a ferrite cored inductor is shown in Fig. 1.
CLF=ferrite core loss-factor at a specific frequency [(30*10^-6) at 1 MHz for ferrite 61]
The simplified equivalent circuit shown in Fig. 1 provides a convenient way of think about the effect of placing a ferrite core in a solenoid. Ra and La represent the resistive loss and inductance of the air-cored solenoid without the core (we want to increase the resultant Q and inductance). Adding a core creates the effect of adding a parallel RL in series with the air coil. When a core is inserted into the air-cored solenoid, the series resistance of the solenoid in air is increased by the factor (FDFa-1) to account for the increased power loss in the copper wire caused by the increased flux density from the core
Magnetic flux density surrounding the solenoid turns conductor is not uniform along the length of the solenoid. It is greater at the ends than along its central part. Increasing the length/diameter of the solenoid reduces the percentage of total flux that penetrates the copper and thus reduces resistive copper losses (especially at the two ends of the winding). Increasing the ratio of the length of the ferrite rod to that of the solenoid further reduces the percentage of total flux that penetrates the copper, further reducing resistive losses.
The amount of electric field that penetrates the core is important, especially at the high end of the band . The Nickel/Zinc cores such as type 61 have a very high resistivity dielectric as well as a rather low dielectric constant (ε) that has a high dielectric loss tangent (tan δ): Losses caused by the high (tan δ) may be minimized by using construction methods that keep the parts of the solenoid that are at a high electrical potential spaced away from the core. For instance a coil former sleeve made of low loss, low dielectric constant material can be used to isolate the high impedance parts of the solenoid from the core.
I-squared-R resistive power loss in the conductor caused by the AC current flow: Increased series resistance of the solenoid reduces Q, especially at the low end of the band compared to the high end, since the inductive reactance is at a minimum there (if the resistance, as a function of frequency is constant). Proximity and skin effect losses increase the RF resistance of the conductor at the high end of the band more than at the low end. The use of litz wire reduces the loss across the band, but more so at the high end. If solid wire is used, spaced-turns winding will reduce the losses from proximity effects. An advantage of using Litz wire in a ferrite-rod inductor is that there seems to be little downside to Q from close spaced winding. This helps with obtaining a larger inductance with a smaller solenoid and ferrite core. The use of larger diameter wire to reduce one of these losses usually has the effect of requiring a larger solenoid and ferrite core in order to keep the inductance the same, requiring mind-numbing tradeoffs. Experimentation with 4" long by 1/2" diameter ferrite 61 rods and litz wire of 50/46, 125/46, 270/46 and 420/46 construction with an inductance 250 uH suggest that a winding length of about 1.5" of close-wound 125/46 litz wire is close to optimum, from the standpoint of Q. I've tried to use 660/46 litz with a 4"x1/2" ferrite 61 rod to attain a high Q inductance of about 250 uH. It never worked, probably because the length of the rod, being close to that of the solenoid, caused a high flux density condition to occur near the ends of the solenoid, creating extra copper loss. Lesson: For high Q, the rod should be longer than the solenoid, maybe three times as long.
Ferrite cores of the same specification often exhibit rather wide variations in their ferrite loss-factor (thus affecting the attainable Q when used as a core). They also vary, to a lesser degree, in initial permeability (µi). This affects the inductance. Generally, when selecting cores from a group having identical specifications, the ones with the least initial permeability will have the least hysteresis loss, especially at high frequencies. This provides a convenient way to select cores that will yield the highest Q coils, without actually measuring Q: Wind a solenoid on a thin walled, low loss form and measure its inductance after placing each core, in succession, centered in the coil. Generally the core providing the least inductance will provide the highest Q.
Comments: Consider the schematic in Fig. 1. La, Ra and Ra*(FDFa-1) define the inductance and Q of the air-cored solenoid (Before a ferrite core is inserted in an air-cored solenoid, FDF=1).
Summary: With no ferrite core present one has a low Q low inductance inductor. If one could construct a fully flux-coupled ferrite 61 core (LFEF~1.0), the Q at 1 MHz would be 1/ui*CLF=267. Highest Q occurs with an optimum value of LFEF, which also provides an intermediate value for real-world inductance. See Table 6, next to last entry.
In my experience with 1/2" diameter ferrite 61 rods aiming for 250 uH and using Litz wire, most of the time LFEF turns out to be greater than the value for maximum Q. An indication of this condition can be obtained by placing two extra cores, each co-axially aligned with the solenoid's core, one at each end of said core, to increase the LFEF. The Q is usually reduced even though the inductance is increased, showing that LFEF is too high for maximum Q. Proof of this can be attained by discarding the two extra cores and reducing the number of turns on the rod. Of course, inductance goes down, but Q will increase. To get the inductance back up and retain the higher Q, a solenoid and ferrite rod of larger diameter are required.
** Note the "no core" entry in Table 6 for inductor BB. The solenoid (with no core) has a Q of 88 (and an inductance of 17.6 uH).
Part 2: Measurements
Comparison of several conventionally wound Ferrite-cored solenoids having the same winding length and number of turns, but different diameters
Ferrite rod length=4", diameter=0.5", material=type 61, µi=125, ferrite loss factor (CLF)=30*10^-6, the "best ferrite core" was used, former=low loss thin wall tubing of various lengths, wire=125 /46 ga. litz, construction=conventional close wound solenoid of 58 turns having a length of about 1.625".
* Piece of polyethylene tubing having an OD of 0.625" and an ID of 0.50"
Comparison between a conventional and contra wound ferrite-rod cored solenoid using a "best" and a "worst" rod.
Article # 0, Part 12 for a mini-Article about the benefits of the contra-coil construction.
Ferrite rod length=4", Diameter=0.5", Material=type 61, µi=125, Ferrite loss factor (CLF)=30*10^-6, Former=polyethylene (not vinyl) tubing, ID=0.5", OD=0.625", length=5", Wire=125 strand/46 ga. litz, Construction=close-wound conventional solenoid of 58 turns having a length of about 1.625"
Ferrite rod length=4", Diameter=0.5", Material=type 61, µi=125, Ferrite loss factor (FEF)=30*10^-6, Former=polyethylene (not vinyl) tubing, ID=0.5", OD=0.625", length=5", Wire=125/46 ga. litz, Construction=close-wound contra wound solenoid of 58 turns and length of 1.625" (not wound as tightly as the conventional solenoid above).
The winding format for solenoids #1 and #2, below, are shown in Figs. 2 and 3. For clarity, the windings are shown as space wound, but the actual solenoids #1 and 2 close wound. Connections for the contra wound inductor shown in Fig. 3: For the series connection, join leads c and e. Lead d is hot and lead f is cold. For the parallel connection, join leads c and f. Join leads d and e. d/e is the hot and c/f is the cold connection.
The Q values given above were measured on an HP 4342A Q meter and corrected for the distributed capacity of the inductor (Co). The ferrite cores were purchased from CWS ByteMark in the third quarter of 2002. They may have changed vendors since then because some rods I purchased in the 3td quarter of 2004 resulted in lower Q coils than the values reported here. These rods also had two small, 180 degree apart, longitudinal flats along their entire length. CWS gracefully accepted a return of those rods and quickly refunded my money. The 'best' and 'worst' cores used in these measurements were from a group purchased from CWS ByteMark in the 3rd quarter of 2002.
Note the better high-band Q values recorded for the contra wound inductor. This is because the low Q distributed capacity from the dielectric of the ferrite (Co and Ro) is connected across an inductor having 1/4 the inductance (and reactance) value of the conventional wound solenoid. An observation: If the hot/cold connections to the contra wound coil in Fig. 3 are reversed, Q at 1710 kHz drops. This is because more loss from the low Q dielectric of the ferrite is coupled in to the stray capacitance.
Solid wire instead of litz?: Keep in mind that the work described here used close-wound 125/46 litz wire. If one duplicates 'Coil and Former B' in Table 2, except using 22 ga. solid copper wire (having the same diameter) as 125/46 litz, the Q values drop to about 1/6 of the values achieved with the litz wire. The cause is the large proximity effect resistive losses, as well as skin effect, in the solid wire. The proximity effect, but not the skin effect loss may be much reduced if the wires are space-wound. New trade-offs now must be considered: Same wire diameter, and therefore a longer solenoid, or a smaller wire diameter and the same overall length? If one wishes to use solid wire, it should probably be wound directly on the ferrite, not on a former. The overall Q will still be much less than when using litz, and the loss from the high (tan δ) dielectric of the ferrite will be pretty well swamped out because of the now higher losses from the skin and proximity effect losses. The Q values, using a close-wound solenoid of 22 ga. solid copper wire on a polyethylene former, as in 'Coil and Former' B in Table 2 are: 520 kHz: 130, 943 kHz: 141 and 1710 kHz: 150 when using the "best core". The Q drops only 3, 3, and 5 points respectively if the "worst core" is used.
Measurements to determine the (tan δ) of the dielectric of a 'medium core': Two adhesive copper foil coupons, 0.5"x1.75" were affixed to the 4", 0.5" diameter rod made of ferrite material 61 (3M sells rolls of thin copper foil with an adhesive on one side). The long dimension of each coupon was parallel to the axis of the rod with the two coupons set opposite to each other, 180 degrees apart. They formed a two plate capacitor having curved plates with the dielectric of the rod between them. The capacitance of this capacitor came out to be 6.5 pF. Measurements, using a Q meter and a high Q inductor were made that enabled calculation of the Q of this 6.5 pF capacitor. Q was 25 at 520 kHz, 35 at 943 kHz and 55 at 1710 kHz. Even though the distributed capacity of a ferrite rod inductor is only made up partially of this poor dielectric, it is, I believe, a previously unrecognized cause of the usual Q drop at the high end of the band. It is also, I believe, the cause of Q reduction in ferrite toroids when no gap is provided between the start and finish of the winding.
Part 3 - Flux density and flux line simulations, inductance and Q of several ferrite-cored inductors along with some measurements
David Meeker's "Finite Element Method Magnetics" program FEMM was used to generate Figs. 4-17. First a word about the displays: FEMM, as used here provides a 2-dimensional display of flux density (the colors) and flux lines (the black lines). Only half of the object being simulated is analyzed and displayed since only axisymmetric objects can be analyzed with the program. This saves simulation time, which can become very great. FEMM, at this time, cannot simulate using litz wire. That is why the following simulations and measurements use 22 ga. solid copper wire instead of the 125/46 litz used in Part 2.
Understanding the images: Visualize the axis of the ferrite rod as coincident with the y-axis of a conventional 3-D x, y and z coordinate graphing system with the center of the rod at the origin. The FEMM program discards everything to the left of the y, z plane that intersects the origin and displays a view of the other half. The images show field densities that exist in an x, y plane that intersects the origin. The vertical object at the left in each image is the ferrite rod. Its horizontal width is ½ that of the diameter of the actual round ferrite rod since the parts of the inductor to the left of a vertical y, z plane intersecting the origin have been discarded (see above). The vertical line of little circles to its right show the cross-sections of the turns of the solenoid wires. Fig. 4 is a plot of magnetic flux density and flux lines on an imaginary plane that cuts longitudinally through the center of ferrite rod inductor AA, shown mostly in purple. The outline of half the 4"x1/2" rod is shown at the left of the plot. If one measures, on the computer screen, the height and width of the rectangle, one can see that their ratio is 16. This is equal to the ratio of the 4" length of the rod to 1/2 of its 1/2" diameter. The large half-circle defines the area around the inductor that will be included in the simulation. It's made up mostly of air. The magnitude of the flux density can be seen from the colors on the display (see the chart). The range of flux density values for the display was purposely limited to about 20 to help supply flux density detail around the outer turns of the solenoid. That is why most all the core is colored purple (the flux density is above 4.000e-9 Tesla). Fig. 5 is a close-up simulation of the area near the upper turns of the solenoid. If one's browser has a zoom control, one can easily see how the flux density close to the surface of the wires of the end turns of the solenoid (even numbered Figs.) is greater than it is in the more central turns. High flux density in the copper equals high power loss (Q reduction).
Comment: Look at figs. 6 and 11 in Table 4. Inductors BB and EE are identical except for the length of the ferrite rod. It appears that about 10% of the end turns of solenoid BB are exposed to a flux density above 2.8e-9 Tesla (3 dB below the maximum plotted value of 4e-9 T). The corresponding percentage in solenoid EE about 50%. This shows that a high flux density around a greater percentage of turns results in lower Q. A parameter listing of the inductors is below Table 4. Note the Q values for inductors BB and EE.
Parameters of simulated inductors AA through DD, inductance and Q at 1 MHz:
Note that the Q difference between the "Best core" and the 'Worst core" is very small. This is because the main loss in this inductor is the high proximity loss in the solid close-spaced copper winding. The much lower ferrite core loss is swamped out and has little effect on Q, showing a Q ratio between the two of about 0.97. Compare these figures with those in Table 3 for a similar conventionally wound solenoid using close-spaced 125/46 litz wire. Proximity loss is greatly reduced in close-wound litz wire, compared to close-wound solid copper wire. The Q ratio here is about 0.75. Loss in the ferrite core swamps out the much lower proximity loss in the litz wire, and a much higher Q results.
Part 4 - The effect on impedance parameters of an air gap in the center of a ferrite rod inductor and on magnetic flux/current density in the wire cross-sections.
The images and text are based on FEMM simulations of inductor BB, shown in Figs. 6 and 7, the specs of which are shown below Table 4, but with the following difference: Instead of using one 4” long ferrite rod, two 2” co-axially oriented rods are used in each simulation with different air gaps between them (0.0000”, 0.0313”, 0.0625” and 0.1250”). The solenoid is centered on the gap. When the gap is 0.000” the result should be the same as if one solid 4” rod were used. The inductance and Q values for inductor BB are slightly different than the values in the new simulations, as stated below for a 0.0000” gap between two 2” rods. This is because the “meshing” parameter in the FEMM simulation was changed to reduce the time taken for the simulations.
The top half of a full image of a core/solenoid combination is a mirror image of the bottom half. In figures 12-17 advantage is taken of this characteristic by zooming in and not showing the entire rod so as to be able to get a larger image, thus supplying more detail.
The first group of four simulations have central gap widths of 0.0000”, 0.0313”, 0.0625” and 0.1250”. They are named GappedRodA0000, GappedRodB0313, Gapped RodC0625 and GappedRodD0125. They are intended to show magnetic flux density distribution in the ferrite and the air. The simulations are made at a frequency of 1 MHz with an AC current of 1 uA RMS in the solenoid. The actual magnetic flux density values can be estimated by comparing the color display to the color chart to the right of the images.
A second group of two simulations shows magnified views of two parameters of the GappedRodB0313 simulation. They are called GappedRodB0313_B (for showing flux density B in Teslas in the cross-sections of the individual wire turns) and GappedRodB0313_j (for showing the current density j in MA/m^2 in the cross-sections of the individual wire turns).
If one looks closely at the GappedRodB_B image, one can see how flux density is distributed in the wire cross-sections as a function of distance along the rod. As the textbooks say, very little flux exists in the interior of the wire. Where the external flux density is great, as it is at the ends of the rod and near the gap, the flux that penetrates the copper is confined near the outer periphery of the wire.
The distribution of current density in the turns as a function of position along the length of the rod is shown in image GappedRodB0313_j. This illustrates skin effect. Note the current density is not uniform in the wires because of proximity effect and the fact that the length of the solenoid is not very long, compared to the length of the rod.
Some specifications common to the inductors in Figs. 12 through 17: Ferrite rod length: two 2” long rods oriented coaxially, and spaced apart by 0.0000”, 0.0313”, 0.0625” or 0.1250”. Ferrite rod diameter=0.5”, Permeability of ferrite rod=125. Loss factor of ferrite at 1 MHz=30*10^-6, ID of solenoid=0.6263”. Number of turns=58. Wire: solid copper, OD=0.0263”. Length of solenoid=1.624”. Frequency at which the simulations are made=1 MHz.
It is interesting to see in Figs. 16 and 17 that the distribution of magnetic flux in the cross-section of the turns has the same shape as that of the current density.
Please note in the text accompanying Figs. 12-17 that the copper series loss component corresponds to the sum of Ra and Ra*(FDF-1) as shown in Fig. 1.
Results of the simulations:
Part 5 - Ferrite-rod inductor simulation experiments; all using centered solenoids 1.624" long and having 58 turns
The solenoids used in the simulations in Table 6 all use a conductor having a diameter of 0.0253". The only parameter varied is the core length. The simulations in Table 7 all use a 4" long core. The only parameter varied is the diameter of the conductor.
* Solenoid winding covers the full length of the core.
Table 6 shows that about 77% of the maximum Q is attained with a core about 2.4 times the length of the solenoid with the turns number, solenoid size, core length, etc used here. About 68% of the maximum inductance is attained. Note also that when the length of the core is shortened to approximately the length of the solenoid, Q drops precipitously. Resistive losses are mainly proximity effect losses. Hysteresis losses are magnetic losses in the ferrite core itself. Total losses are the sum of the two. There is a good lesson to be learned here: To maximize Q, do not cover the whole length of the core with the solenoid.
* Simulates winding the 58 turn solenoid directly on the 4" long ferrite core (solenoid ID=0.5013") instead of on a former having an ID of0.6263". Note that the the two simulations using a conductor diameter of 0.008995" show remarkably similar parameter values.
Table 7 shows the benefits of spaced winding when using solid wire. All the inductors in Table 7 use centered solenoids of 58 turns and a length of 1.624". The only variable is the diameter of the conductor, which controls the spacing of the turns (the winding pitch is held constant). The lesson here is that, when using solid copper wire, there can be a great Q benefit by space winding the solenoid and using an optimum size wire; in this case a Q of 431.9 vs 130.1 at 1 MHz, with solid wire. One can see that core losses change very little with the various conductor diameters (Hysteresis losses in ohms). Notice how, with a conductor diameter change from 0.02530 to 0.00006300", the AC copper loss decreases from 11.6 to 3.02 ohms overwhelming the increase in DC conductor resistance from 0.16 to 2.62 ohms.
See Table 3 for measured inductance and Q values of an inductor similar to inductor BB, but wound with 125/46 litz wire. Here the Q is even greater than in Table 7 because litz construction is less sensitive to proximity and skin effect losses than is solid wire.
Thanks must go to Brian Hawes for making me aware of the FEMM program and showing me how to use it.
Part 6: Perminvar ferrite, and what the term means
Normal nickel/zinc ferrites (NiZn), the types with less permeability as well as lower loss factors than manganese/zinc (MnZn) ferrites, are often used at RF because of their low loss at the higher frequencies. They do not suffer appreciably from permanent changes in permeability or loss factor from exposure to strong magnetic fields or mechanical shock such as grinding, or dropping on the floor.
Special nickel/zinc ferrites, called perminvar ferrites can achieve a considerably lower loss factor for the same permeability than normal nickel/zinc ferrites, and at higher frequencies. This result is achieved by adding a small amount of cobalt to the ferrite power before firing, but there is a catch. In order to actually achieve the lower loss factor, the ferrite core must be annealed by raising it to a temperature above its Curie temperature (the temperature at which it losses all its permeability), and then cooling it very slowly back down through the Curie temperature, and then to lower temperatures. This process usually takes about 24 hours. The Curie temperature of ferrite type 61 (a perminvar ferrite) is specified in the Fair-Rite catalog as being above 350 degrees C. The annealing process reduces the permeability somewhat, but reduces the loss factor substantially.
The low loss-factor property of the annealed perminvar ferrite can be easily degraded by mechanical shock, magnetic shock or just physical stress (as from a tight mounting clamp). The Fair-Rite catalog sheet for type 61 ferrite cautions "Strong magnetic fields or excessive mechanical stresses may result in irreversible changes in permeability and losses". Actually, the changes are reversible if one goes through the annealing process again. The MMG catalog, issue 1A, in writing about perminvar ferrites, adds: "Mechanical stresses such as grinding and ultrasonic cleaning increase the permeability and lower the Q, especially at the higher frequencies, although the changes in Q at the lower frequencies may be very small.
I suspect that there is now much less pressure on ferrite manufacturers to deliver a low loss product than in the past. Since time is money, maybe they now skimp on the annealing process. Several years ago I took some 4" x 0.5", mix 61 rods I had purchased from CWS ByteMark and had them re-annealed at the plant of a local ferrite manufacturer. The Q of a litz-wire coil using the re-annealed core, at 2.52 MHz, was increased by 12%. This indicates that the core was not originally properly annealed, or had been subjected to some mechanical or magnetic shock after being annealing by the manufacturer. I was informed, when I asked, that coil Q at high frequencies could be expected to increase by up to100% from the pre-annealed value. I chose the best of these re-annealed rods to be my "best ferrite core" rod in this Article. One source informed me that few ferrite manufacturers perform the annealing process anymore. Toroids made of type 61 material are still made here in the USA.
#29 Published: 10/07/2006; Revised: 01/07/08