Index Inductance L-reactance L-impedance L-Quality winding factor max. voltage max. power measuring

Some practical formulas Inductance: L = n^2 . mo . mr . A / l (H) with: n = number of turns mo = absolute permeability (4. p .10^-7) mr = relative permeability (air = 1) For inductors with magnetic core materials, mr will split-up starting around ca 100 kHz into m’ - (j) m”, both a function of frequency in different ways, with m” to represent material loss. Refer to manufacturer for details. A = effective magnetic winding area (m^2) l = effective magnetic path length (roughly inductor length in case of air coils) (m) Note: This (theoretical) formula will also hold for inductors without magnetic core when l => 3 .d, and turns are in a single layer. For different shapes, more practical formulas are available. Reactance: XL = w . L (Ohm), with: w = 2. p . f (Hz) Impedance: ZL = rs + j XL (Ohm), or in absolute terms: ZL = Ö(rs ^2+ XL^2) Series resistance rs is representing all ('Ohmic') loss, including copper loss (wire, DC), skin-effect (wire, AC) and core loss Quality factor: Q = XL / rs At operating-frequencies above 1/10 times the ferri-magnetic resonance frequency (see manufacturer; m’ = m”), core loss is the dominant parameter, i.e. n^2. w . m”. A / l > rs. Therefore Q = m’ / m” for inductors on core. Note: Some manufacturers prefer to specify: loss = 1 / Q = tand = m”/ mi, with mi the initial permeability in stead of the frequency-dependent: m’! This will present incomparable numbers with real Q < 10. Winding factor: AL = mo . mr . A / l (nH/n^2), Accordingly: L = n^2 . AL (as long as foperation < fr ./5 , with fr = ferri-magnetic resonance frequency, where m’ = m”) Note: Some manufacturers prefer to specify AL in mH/ 100 turn, especially at low permeability materials to make numbers look more like those of high permeability materials. Divide by 10^4 to obtain standard definition. Maximum inductor voltage (core dissipation): Umax = Ö (Pmax . (Q/6+1/Q) . XL) with: Pmax = maximum power in core material for temperature rise = 28 K At 'standard' 36 mm. ferrite toroid (effective volume = 8,6 cm^3), Pmax = 4 Watt (freely radiating thermal resistance is ca 7 K / Watt). Scale different core size to root of volumes. Maximum inductor voltage for linear application: Umax = 0.89 . Bsat . n . f . A , with: Bsat = saturation induction as presented by manufacturer. For ferrite materials, lower of maximum voltage for dissipation or maximum voltage for linear application will determine application. Above a few 100 kHz., usually core dissipation is the determining factor. To illustrate maximum voltage behavior, the graph is showing maximum values for a one-turn inductor at 36 mm., toroid of 4C65 material. Red-curve for maximum voltage for a core temperature rise of 28 K, blue-curve for maximum voltage for linear behavior and green curve for total impedance. Example: At 1 MHz. maximum voltage for linear application (blue curve): 31 Volt. At a two turns inductor, this would be: 62 V., at 3 turns: 93 V. etc. Same applies to maximum voltage at 1 MHz. for a core dissipation of 4 Watt (temperature rise of 28 K, red-curve) is: 12 Volt, at two turns; 24 V. From 150 kHz. and above maximum voltage for core dissipation is determining factor for maximum system power (lower of the two curves). The green curve is showing total impedance ZL, scaling however with n^2. At 10 MHz. total impedance is 11,4 Ohm, at two turns this will be (4 x) 45,6 Ohm and at three turns (9x) 102,6 Ohm etc. Quick measurements Measuring initial permeability: Cc = 270 pF, Ck = 0,1 mF (good quality, low tolerance, no decoupling) Put 10 turns at the unknown core material. Change frequency of LF generator until maximum reading at the voltmeter. The (resonance) frequency ‘fr‘ will be found between 1 and 150 kHz. for almost all ferrite toroids. At ‘fr‘ : mi = 2 . 10^9. l / (fr^2 . A), with: l = magnetic path length (ca 0.9 x circumference at toroids) A = area of one turn Measuring quality factor: Diagram as above; Cc = 2,7 pF Generator is HF-type and to be set at operating frequency of inductor. Change Ck until maximum voltage at meter. Q = (f+ + f-) / 2(f+ - f-) = m’/m” , with: f+ = frequency above resonance when voltage is down by 0,7 x maximum. f- = frequency below resonance when voltage is down by 0,7 x maximum. When Ck < 50 . Cc, latter should be changed for a smaller type to avoid circuit damping by the generator or meter. More information: Soft Ferrites and Accessories, Ferroxcube Data Handbook; Soft ferrites, E.C. Snelling, Butterworths Publishing, Stoneham; Transmission line transformers, J. Sevick, ARRL; Ferrieten in HF applications Bob van Donselaar, mailto:on9cvd@amsat.org