Toroid calculator: turns from the AL value, and the flux check

The turns a ferrite or powder toroid needs for a target inductance, from the core's AL value; the inductance a given winding gives; and the AL of an unmarked core from one measurement. Beside the answer it carries the AL tolerance into the inductance, checks the peak flux density against a limit for a sine, square or current drive, and works out the wire the winding takes and its resistance. The defaults are TDK's worked resonant-circuit inductor, 640 µH on AL = 100 nH, which the note winds with 80 turns, and the page's tests hold the calculation to TDK's and the Unitrode handbook's printed figures.

Fig 1 — The winding, drawn one stroke per turn up to 48, and L = AL·N² against turns. 640 µH on AL = 100 nH needs 80.00 turns, rounded up to 80, which give 640 µH. The band is AL ±10 %: 576 µH to 704 µH at 80 turns. Peak flux density 1.13 mT, 0.38 % of the 300 mT limit.

Calculate

Inductance factor AL (nH)

Target inductance L (µH)

AL tolerance (± %)

Core cross-section Ae (mm²)

Drive

Winding voltage, RMS U (V)

Frequency f (kHz)

Flux density limit B (mT)

Outside diameter OD (mm)

Inside diameter ID (mm)

Core height HT (mm)

Wire diameter, insulated d (mm)

Lead allowance (mm)

Wire resistance (Ω/m)

Turns N = (L/AL)^½, rounded up: 80 (exact 80.00)
Inductance at 80 turns, L = AL·N²: 640 µH
L with AL at ±10 % (derived: L ∝ AL): 576 µH to 704 µH
Turns that reach 640 µH even at the lowest AL (derived): 85
Peak flux density B̂ = U/(4.44·f·N·A_e), sine (derived): 1.13 mT · 0.38 % of limit
Length of one turn, (OD − ID) + 2·HT + π·d (derived): 31.2 mm
Turns one layer holds, π(ID − d)/d (derived): 125
Wire for 80 turns and the leads · DC resistance: 2.69 m · 1.20 Ω

How this is calculated

Standard: TDK (EPCOS), Ferrites and accessories, Application notes, May 2017 (section 1.3, pp. 3–4; Symbols and terms, p. 33; Cautions and warnings, p. 36); TI SLUP132, Unitrode Magnetics Design Handbook (Dixon, 2001), pp. 2-5, 5-5, 5-7 to 5-10

$A_L = \frac{L}{N^2}, \qquad L = A_L N^2, \qquad N = \sqrt{\frac{L}{A_L}}$: TDK's symbol table (p. 33): AL "Inductance factor; AL = L/N2", in nH. SLUP132 Eq 3c (p. 5-7), L = N²AL nanohenrys, AL "expressed in milliHenrys/1000 turns², or nanoHenrys/turn²". TDK p. 4: "the equation N = (L/AL)1/2 yields 80 turns" for 640 µH on 100 nH. The calculator rounds N up to a whole turn and reports the L that gives.
$L_{min,max} = A_L\,(1 \mp t)\,N^2$: The AL tolerance t carried into the inductance. At a fixed number of turns L is proportional to AL, so the band is the same percentage. Derived; the tolerance itself comes from the core datasheet.
$L = \mu_0 \mu_e N^2 \frac{A_e}{l_e} \quad\Rightarrow\quad A_L = \frac{\mu_0 \mu_e A_e}{l_e}$: SLUP132 Eq 3b (p. 5-7), in SI with dimensions in metres; µe is the effective permeability of the core with its gap, discrete or distributed. Eq 7 (p. 5-9) solves it for µe "(or calculate the inductance factor, AL…)". µ0 = 4π × 10⁻⁷ H/m.
$\Delta B = \frac{1}{N A_e}\int E\,dt$: Faraday's law as SLUP132 writes it (p. 2-5), "where ∫Edt = applied Volt-seconds, N = turns, and Ae = core cross-section area"; "The total flux swing, ΔB, is twice the peak flux swing referred to in the core loss curves".
$\hat B_{sine} = \frac{U}{4.44\, f N A_e}, \qquad \hat B_{square} = \frac{U}{4 f N A_e}$: Derived from the line above. A half cycle of a sine of RMS value U, √2·U·sin ωt, integrates to 2√2·U/ω; halving the swing gives the peak, and 2π/√2 = 4.443. A symmetric ±U square wave holds U for 1/(2f) per half cycle. U is the RMS voltage in both, which for the square wave is U itself.
$\hat B = \frac{L\,\hat I}{N A_e} = \frac{A_L N \hat I}{A_e}$: SLUP132 Eq 5 (p. 5-8), N = L·ΔI/(ΔB·Ae), solved for the flux with the swing taken from zero current: the peak flux a peak current sets while L holds. Derived. The handbook's example: 2.2 µH, 10 A of ripple, 0.046 T and Ae = 0.97 cm² give 4.93 turns, rounded to 5 (p. 5-10).
$l_N = (OD - ID) + 2\,HT + \pi d, \qquad N_{layer} = \left\lfloor \frac{\pi\,(ID - d)}{d} \right\rfloor$: Derived geometry: the perimeter of a rectangular core section, plus π·d because the wire's centre sits d/2 off the core all the way round; and the turns that fit side by side around the inside of the hole. The wire for the coil is then N·lN plus the leads, and its resistance that length times the wire's Ω/m, TDK's method on p. 4.

Assumptions

AL is the datasheet value, measured at a low flux density. It holds while the core is well below saturation and at frequencies where the material's permeability is still its low-frequency value.
The flux check is linear: it takes L, and so AL, as constant up to the peak. A powder core loses inductance gradually as the flux rises, so the figure is the flux at the datasheet AL, not the inductance at that flux.
The sine and square forms assume no DC across the winding. The current form takes the swing from zero: for a choke, enter DC plus half the ripple.
The turn length is for one tight layer on a sharp-edged rectangular section. Rounded edges shorten it a little; a second layer lengthens it.
Ae is the datasheet's effective cross-section. The default is the illustrative core's geometric section.

What sets the turns: the AL value

A toroid's inductance rises with the square of its turns, and the number that links the two is the core's inductance factor. TDK's ferrite application notes define it in their table of symbols: A_L, "Inductance factor; A_L = L/N2", in nanohenrys (p. 33). The Unitrode Magnetics Design Handbook, reissued by TI as SLUP132, gives the same quantity "expressed in milliHenrys/1000 turns², or nanoHenrys/turn²", writes it as L = N²A_LnanoHenrys (Eq 3c, p. 5-7), and says what it is for: it "provides a convenient method for calculating inductance for an existing gapped core with a given number of turns". A datasheet AL of 100 nH means one turn gives 100 nH, ten turns 100 times that, and 80 turns 6400 times it.

So the calculation in each direction is one line. For a target inductance, N = √(L/A_L), rounded up to a whole turn; for a winding, L = A_LN²; for an unmarked core, wind a known number of turns, measure L, and A_L = L/N². Behind the AL is the core's geometry and its effective permeability. SLUP132's Eq 3b, L = µ₀µ_eN²A_e/l_e, says A_L = µ₀µ_eA_e/l_e: the cross-section A_e over the magnetic path length l_e, times a permeability that the handbook says "typically ranges from 10 (for a large gap) to 300 (for a small gap)" for gapped and powder cores, and which runs to thousands for an ungapped ferrite: "The permeability of power ferrite materials is in the range of 1500 to 3000" (p. 2-2). That is why AL values span three orders of magnitude, and why the same inductance can need five turns or five hundred.

Two more things decide whether the turns the formula gives are the turns to wind. The AL has a tolerance, and L carries it straight through: at a fixed N, L is proportional to AL, so ±10 % on AL is ±10 % on L. And the winding has to carry its current or voltage without driving the core into saturation, which AL says nothing about. The calculator works out both, and the wire the winding takes.

Turns for a target inductance: a table

Whole turns, rounded up, for four inductances on four AL values. The AL values are round numbers chosen to span the range from a low-permeability powder core to an ungapped ferrite, not specific parts.

L	A_L 25 nH	A_L 100 nH	A_L 400 nH	A_L 2000 nH
10 µH	20	10	5	3
100 µH	64	32	16	8
1.0 mH	200	100	50	23
10 mH	633	317	159	71

Read across a row and the square root shows: sixteen times the AL needs a quarter of the turns. Read down a column and a hundred times the inductance needs ten times the turns. The bottom-left corner is a winding that will not fit on most cores, the top-right one that is too coarse to trim: at 10 µH on 2000 nH, one turn more raises L by 78 %, against 1.0 % for one more turn at 1 mH on 25 nH. Where few turns are needed, a lower-AL core gives finer control.

Worked example: TDK's 640 µH resonant-circuit inductor

TDK's note works a complete design in section 1.3 (pp. 3–4): "A SIFERRIT pot core inductor is required with an inductance of L = 640 µH and a minimum quality factor Q = 400 (tan δL = 1/Q = 2.5 · 10-3) for a frequency of 500 kHz. The temperature coefficient αe of this inductor should be 100 · 10-6/K in the temperature range +5 to +55 °C." The core is a pot core rather than a toroid, but the AL arithmetic is the same for any closed core, and the note works every step of it. The calculator's defaults are its numbers.

The temperature coefficient sets the AL. The M33 ferrite has a relative temperature coefficient α_F of "about 1,6 · 10-6/K", and a gapped core's coefficient is α_F times its effective permeability, so the wanted 100 · 10⁻⁶/K needs µ_e = α_e/α_F = 62.5, the note's 62.5. "With pot core P 18 × 11 (B65651): µe = 47.9 for AL = 100 nH." Then the turns: "For an AL value of 100 nH and an inductance of 640 µH the equation N = (L/AL)1/2 yields 80 turns." The square root is 80.00, exactly 80, so no rounding is needed.

The wire comes next. The litz has an overall diameter of 0.367 mm and 0.444 Ω/m, and "The length of an average turn lN on the above former is 35.6 mm. The length of litz wire necessary for the coil is therefore 80 · 35.6 mm = 2848 mm plus say 2 · 10 cm for the connections, giving a total length of 3.04 m." The calculator's 2848 mm agrees; the sum with the connections is 3.048 m, which the note gives as 3.04 m. "The average resistivity of this wire is 0.444 Ω/m; the total DC resistance is thus 3.04 m · 0,444 Ω/m ≈ 1,35 Ω." From 3.04 m it is 1.350 Ω and from 3.048 m 1.353 Ω; both round to 1.35 Ω. Last, the check: α_e = µ_e·α_F = 47.9 × 1.6 · 10⁻⁶/K = 76.6 · 10⁻⁶/K, the note's 76.6, and "Actual measurement yielded 90 · 10-6/K."

On the calculator's illustrative toroid, 25 mm outside, 15 mm inside and 10 mm high, the same 80 turns of the same litz need 31.2 mm per turn, 2.69 m with the leads, and 1.20 Ω. One layer around the 15 mm hole holds 125 turns, so 80 fit in one. With the AL at ±10 %, the 80 turns give 576 µH to 704 µH; to be sure of 640 µH at the low end, 85 turns. At 10 V RMS and 500 kHz the peak flux density is 1.13 mT, and it would take about 2.7 kV across the winding to reach 0.3 T: a resonant-circuit inductor is limited by its losses and its Q long before saturation.

Peak flux density: the saturation check

AL gives the inductance; it does not say how much the core can carry. That is the flux density, and SLUP132 gives the relation on p. 2-5: "peak-to-peak flux swing, ΔB, is calculated from Faraday's Law, where ∫Edt = applied Volt-seconds, N = turns, and A_e = core cross-section area", ΔB = ∫E dt/(N·A_e). The same page warns which number a core-loss curve wants: "The total flux swing, ΔB, is twice the peak flux swing referred to in the core loss curves as “Flux Density”." The calculator works in the peak value, B̂, which is what a saturation limit is quoted as.

For a voltage across the winding the integral is over half a cycle. A sine of RMS value U gives B̂ = U/(4.44·f·N·A_e), the 4.44 being 2π/√2 = 4.443; a symmetric ±U square wave gives B̂ = U/(4·f·N·A_e). Both are derived here from SLUP132's integral rather than printed in it. At the same RMS voltage the square wave drives -10 % more flux, because it spends its whole half cycle at full voltage. Frequency is in the denominator, so the same winding at 50 Hz instead of 500 kHz carries ten thousand times the flux: 11.3 T for the default's 10 V on paper, far past where any ferrite saturates, which is why a mains transformer needs so many more turns than a switching one.

For a choke the drive is a current, and SLUP132's turns equation for an inductor (Eq 5, p. 5-8), N = L·ΔI/(ΔB·A_e), solved for the flux gives B̂ = L·Î/(N·A_e) with the swing taken from zero. The handbook's buck output inductor (p. 5-10) uses it the other way round: 2.2 µH and 10 A of ripple at a swing of 0.046 T on an ETD34 of A_e = 0.97 cm² gives "N = 2.2 · 10/(.046 · 0.97) × 10⁻² = 4.93 → 5 Turns". The calculator's form gives 4.93, rounded to 5, and at the 65 A short-circuit peak the five turns reach 0.295 T, under the 0.3 T the design was set to. Those 2.2 µH on five turns are an AL of 88 nH.

The limit is the designer's. SLUP132 suggests a "conservative saturation limit, B_MAX (perhaps 3000Gauss (0.3Tesla) for power ferrite)" (p. 5-7), the calculator's default. A saturation figure is not the only ceiling: "For acceptable losses, flux density swing ΔB must be restricted to much less than B_SAT" (p. 2-5), and at switching frequencies core loss usually sets the swing.

Ferrite or powder toroid for a choke

The current form exposes the trap in winding a choke on a high-AL ferrite toroid. Substitute L = A_LN² and the flux is B̂ = A_L·N·Î/A_e: for a given core, the higher the AL, the more flux each ampere makes, and adding turns makes it worse, not better. Take 1 mH carrying 1 A peak on the illustrative 25/15/10 core with A_e = 50 mm². On an AL of 2000 nH, an ungapped ferrite (µ_e about 2000, taking l_e as the mean circumference, π(OD + ID)/2, an approximation), it needs 23 turns and reaches 0.92 T: 3.1 times a 0.3 T limit, which this winding reaches at about 0.33 A. On an AL of 100 nH (µ_e about 100, a powder core's range) it needs 100 turns and reaches 0.20 T. Both figures are illustrative; the ratio between them, which for the same inductance is the square root of the AL ratio before rounding the turns, is not.

SLUP132 describes what a distributed gap buys. Powder cores "store their energy in a non-magnetic gap that is distributed throughout the entire core", with "effective permeabilities in the range of ≈15 to ≈200" (p. 2-2), and "Toroidal powdered metal cores, with windings distributed uniformly around the entire core, can be used in any inductor or flyback transformer application. Stray magnetic flux and EMI propagation is very low" (p. 5-5). The alternative of cutting a gap in a ferrite ring is ruled out on the same page: "a gapped ferrite toroidal core is a very bad choice. Windings distributed around the toroid will not conform to discrete gaps, resulting in large stray fields, radiated EMI, and inductance values that cannot be calculated."

The price is a soft knee. "Rounding of the B-H characteristic … causes incremental inductance to decrease substantially as the DC operating point is raised. Typically, the inductance may be halved at an operating flux density of 0.4Tesla (4000 Gauss), only half way to saturation" (p. 2-2). For a powder core, the flux the calculator reports is the flux at the datasheet AL; the inductance at that flux has to come from the manufacturer's DC-bias curve.

Wire length per turn from the core dimensions

TDK's example takes the length of an average turn, l_N, from the coil-former data sheet, and notes that it "always refers to the fully wound former. If the former is not fully wound, the length of an average turn must be corrected according to the extent of the winding" (p. 4). A toroid has no former, so the calculator derives the turn from the core: the wire wraps the rectangular section, whose perimeter is (OD − ID) + 2·HT, and its centre runs d/2 outside the core, which adds π·d. On the illustrative core that is 30 mm of perimeter plus 1.15 mm for the 0.367 mm litz. Real cores have rounded edges, which shorten the turn slightly.

The hole limits the turns, not the outside. Side by side around the inside, the wire centres lie on a circle of diameter ID − d, so one layer holds π(ID − d)/d turns. Past that the winding goes to a second layer, each turn of which is longer than the first layer's, and the calculator's wire length and resistance become underestimates; it says so when the turns do not fit. The resistance is then length times the wire's Ω/m, exactly as TDK works it.

Where the AL model stops being valid

AL is a small-signal figure. SLUP132, on powder cores: "be aware that the permeability quoted by the manufacturer may not apply at the conditions of the application" (p. 5-8). Near saturation, and on a powder core well before it, L falls below AL·N².
Mechanical stress lowers it. TDK's cautions: "The higher the stresses are in the core, the lower is the value for the initial permeability. Thus the embedding medium should have the greatest possible elasticity" (p. 36). Potting a toroid in a hard compound can move the inductance.
The winding's capacitance takes over at high frequency. On ring cores for interference suppression, TDK notes that "The maximum of the impedance curve shifts to lower frequencies as the number of turns increases; this is due to the capacitive effect of the turns" (p. 11). More turns for more inductance lower the frequency up to which the part is an inductor at all.
Temperature moves it. A gapped core's coefficient is µ_etimes the material's, and "For effective permeabilities µe <80, however, due to the influence of the winding an additional temperature coefficient of approx. (10 to 30) · 10-6/K must be included in the calculation" (TDK, p. 4); the example measured 90 · 10⁻⁶/K against a calculated 76.6.
NiZn ferrite can be changed by overload: "The magnetic properties of NiZn-materials can change irreversible in high magnetic fields" (TDK, p. 36). An AL measured after a saturating event may not be the datasheet's.
The flux forms assume a clean drive. Any DC across a winding driven with AC walks the flux: SLUP132 notes that "The smallest dc voltage component in an applied ac waveform will slowly but inevitably “walk” the flux into saturation" (p. 4-6).

Common toroid mistakes

Rounding the turns down. SLUP132 on rounding N (p. 5-8): "If N is rounded down to a smaller integer, the core may saturate, or, if the core is loss limited, core loss will be greater than planned." The calculator rounds up and shows the inductance the whole turns give.
Designing to the nominal AL. With a tolerance on AL, the turns that give the target at nominal give less at the low end. If the target is a minimum, wind the turns the calculator gives for the lowest AL.
Winding a DC choke on an ungapped high-AL ferrite. Its AL gives the inductance in few turns, and the same AL makes the flux per ampere large; a fraction of an ampere can saturate it. Check the current drive before choosing the core.
Using the sine form for a square wave. 4.44 belongs to a sine; a square wave of the same RMS voltage needs the 4, and carries more flux.
Confusing peak and peak-to-peak flux. Core-loss curves are read at the peak, "use ΔB/2 to enter the core loss curves" (SLUP132 p. 2-5); a saturation limit is also a peak.
Trusting an AL measured on one or two turns. With so few turns, anything outside the core, the leads and the meter's fixture, is a larger share of what the meter reads. More turns, and a drive level and frequency close to the application's, give an AL that applies.