Litz Wire Calculator: Strand Count, Diameter and AC Resistance

For copper, $\rho(T)=\rho_{20}[1+\alpha(T-20)]$ and $\delta_h=\sqrt{\rho/(\pi h f\mu_0)}$. Base resistivity follows Sullivan–Zhang (2014), attached p. 2; the temperature coefficient follows NBS Handbook 100 (1966), Table 3. Inputs are RMS currents. Straight length is total turns times mean turn length; litz adds strand take-up.

Whole strand-level proximity loss. Sullivan (1999), IEEE TPEL 14(2), Section II, Eq. 2, printed p. 284 (DOI 10.1109/63.750181), already includes the whole strand-level proximity loss under a uniform strand distribution. It does not require a separate internal-bundle term. Sullivan–Zhang (2014), attached pp. 3–5, expresses the same model as

Here $b_f$ is the core-window field breadth. The calculator also retains the small round-strand transport-skin increment, $F_s=1+d^4/(768\delta^4)$, from Peck (1992), printed p. 26, Eq. 13. At $d=\delta$ it gives 1.001302 versus the exact cylinder value 1.001301.

Optional conservative allowance, derived here. Disabled by default. For a uniformly populated circular bundle, averaging its self-field gives $\langle B^2\rangle=\mu_0^2 I^2/(8\pi^2 R^2)$. Integrating the fine-strand transverse-field loss produces the positive allowance

When enabled, $F_{\rm litz}=F_s+F_p+F_i$; otherwise it is $F_s+F_p$. This is the calculator's own conservative addition relative to the published model. Since Eq. 2 already includes whole strand-level proximity loss, this allowance may double-count part of it. It is neither a validated accuracy improvement nor a guaranteed upper bound. It does not calculate bundle-level circulating currents caused by imperfect transposition.

Diameter-dependent insulation. Sullivan (1999), Section III-F, Eqs. 3–4, printed p. 286, explains why coarse-wire minimum-build extrapolation is unsuitable for fine strands. We implement the fine-wire fit as

Single-build uses $a=1.12$, $\beta=0.97$; heavy-build uses $a=1.24$, $\beta=0.94$. These empirical fits are restricted to AWG 30–60; the calculator's 0.01 mm minimum is inside that range. Constant-enamel mode instead uses $d_o=d+2t$. The solid comparator always uses the entered constant enamel, as its diameter may exceed the fine-wire fit range. No standard's table is reproduced.

Packing factors and breadths. Let $b_b$ be usable bobbin breadth, $h$ available radial build, $k_w$ the entered whole-window packing and $k_b$ the entered unserved-bundle packing. The constraints are

Sullivan's Eq. 5 uses $b_b$, separately from Eq. 2's $b_f$. The paper's turn-packing factor $F'_p$ is relative to square packing: unity corresponds to circular cable area fraction $\pi/4$. Its Eq. 6 uses an effective litz-packing factor $F_{lp}$, including serving, filler and twist projection. Equating the two area descriptions gives

For unserved circular cable with no additional projection allowance, $F_{lp}=k_b$. For radial serving $s$ on finished diameter $D_L=\sqrt{F'_pb_bh/N}$, use $F_{lp}=k_b(1-2s/D_L)^2$. Further packing penalties also belong in $k_w$. Twist length is accounted separately by take-up. At full area use, bare-copper fill is $k_w(d/d_o)^2$, not $k_w$.

Optimization. For fixed diameter, harmonic loss is $A/n+Bn+C$. The optional allowance contributes the count-independent term $C$; its insulation model still affects the diameter comparison. Test adjacent first-bunch integers around $\sqrt{A/B}$ for every permitted outer-stage product, limited by area, bundle dimensions, strand count and first-bunch size. Compare all 121 diameters. This minimizes searched copper loss without a price model. Constant radial enamel, when anchored at a given diameter, overstates insulation for finer candidates and understates it for coarser candidates relative to a diameter-dependent build; it generally biases selection toward coarser strands. The RM5 optimization reference uses the paper's insulation fit and independently converted packing, without fitting any loss coefficient.

Derived simplified-design constant. Solving $F_p=F_{\rm target}-1$ gives

The user chooses the target. No economical-stranding table is copied. This guide excludes transport skin and the optional allowance.

Harmonics. $P=I_{\rm dc}^2R_{\rm dc}+\sum_{h=1,3,5}I_h^2R_{\rm ac}(hf)$, with $f_{\rm eff}=f\sqrt{\sum h^2I_h^2/\sum I_h^2}$ including DC in the denominator.

Solid round and foil. The round comparator uses cylindrical diffusion: $x=a/\delta$, $z=(1+j)x$, $S=zI_0(z)/(2I_1(z))$. Transport factor is $\Re(S)$; transverse-field shielding multiplies the fine-wire proximity term by $-4\Im(1/S)/x^2$. A continued fraction and large-argument expansion evaluate the Bessel ratio. This remains an averaged-field winding estimate.

Foil has copper thickness $t_f=A_{\rm cu}/b_b$. Dowell's whole-section factor is

For unequal breadths, $\eta=b_b/b_f$ and $u=t_f\sqrt{\eta}/\delta$. This is Dowell's resistance-preserving porosity transformation: spreading the foil to $b_f$ at unchanged thickness requires effective conductivity $\eta\sigma$, hence effective skin depth $\delta/\sqrt{\eta}$. See Mühlethaler (2012), Section 4.4.4, printed pp. 120–121. It approximates short foils; it does not resolve their edge fields. Small- and large-argument limits avoid cancellation and overflow.

• Copper, round individually insulated strands, constant copper temperature, equal strand currents and adequate transposition; negligible displacement current.
• Overall winding field is one-dimensional, parallel to layer faces and proportional to the entered winding current at every harmonic. Turns per MMF section represent identical series sections. Different winding-current waveforms require a resistance-matrix model.
• Core-window field breadth and usable bobbin breadth are separate inputs. Field breadth must be at least the physical breadth. No automatic coil-geometry conversion is inferred.
• Winding packing, unserved-bundle packing and take-up remain constant during the search. Whole-window packing must include serving, filler and twist projection. The reported bundle diameter is unserved.
• Strand insulation uses the selected constant-build or Sullivan (1999) single-/heavy-build power law. Those historical empirical fits are not supplier specifications. A constant build generally favors coarser candidates relative to a realistic diameter-dependent build.
• Sullivan Eq. 2 is already the whole strand-level proximity estimate. The optional positive bundle-field allowance is derived here, disabled by default, and can double-count loss already represented by Eq. 2.
• The solid comparison uses an equal-copper cylinder and an averaged winding field. Its insulation uses the constant enamel input. Foil spans the usable bobbin breadth, has one turn per layer and uses a Dowell porosity transformation for any breadth mismatch. Both comparators exclude litz twist take-up from their DC lengths.

The litz optimizer is restricted to $d\leq\delta$ at the highest nonzero entered harmonic. It finds the best of the sampled diameters and all permitted equal-group integer constructions within the count cap. It is not a continuous global optimum or a catalogue availability guarantee. Search-range errors apply only in optimization mode. An unavailable search produces an empty optimization sweep, while valid manual-wire results remain available.

The first-operation limit $n_{1,\max}=4(\delta/d)^2$ and subsequent 3-, 4- or 5-way stages follow Sullivan–Zhang (2014), attached pp. 3 and 5. The manual-mode frequency screen uses the entered first-bunch count. Bundle-level proximity currents require twist pitch short compared with the length exposed to a field, and radial transposition is needed to prevent bundle-level skin redistribution. These excess losses cannot be inferred from diameter and count alone. If a construction fails the screen, displayed losses remain ideal-litz estimates and actual losses can be higher. Unequal-group or unverified manual constructions need manufacturer confirmation. The suggested recipe is an admissible alternative for the evaluated count; the manual first-bunch entry is screened separately.

The geometric screen checks enamelled-circle area plus necessary round-envelope dimensions. Comparison utilization is the larger of the relevant area and dimension ratios; foil uses stack build divided by available build. A value no greater than one is necessary, not a fit guarantee. Finished diameter, serving, leads and bend radii require a winding layout. The foil fit check assumes the allocated build is available across the same breadth. The frequency sweep holds construction fixed, uses sinusoidal current at each point and omits the litz curve above the lower of the fine-strand and first-bunch screening frequencies. Both screens and the resulting curve limit are shown beside the chart.

Round-wire proximity estimates become uncertain when conductors are tightly packed and diameter exceeds skin depth. Foil assumes a 1-D field with negligible edge effects. No numerical precision displayed here represents a measured accuracy guarantee.

Not covered: air-gap fringing, isolated spiral/solenoid field geometry, arbitrary harmonics beyond those entered, unequal parallel paths, termination loss, core loss, dielectric loss, self-resonance, temperature rise or allowable current rating.

The diameter-dependent insulation fits are valid for AWG 30–60 only. The optimizer clips its upper diameter to AWG 30 when a fit is selected; manual use beyond that limit requires constant enamel from supplier data. The field-breadth input cannot be smaller than usable bobbin breadth. For short foil, the porosity transformation preserves DC resistance but does not replace a two-dimensional edge-field calculation. The optional bundle-field allowance is a sensitivity choice above the published whole-strand model, not a correction certified by Sullivan's paper.