Calculations

Calculations

• Permeability of free space

  The constant μ0 used in every hand formula.

  μ0 = 4 × 10⁻⁷ × π

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

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• Round conductor as a polygon

  The corners of the polygon that stands for a round wire.

  pᵢ = (cx + r cos(2πi ÷ n), cy + r sin(2πi ÷ n)), n = 48

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

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• C-core gap field

  The flux density in the air gap of a C-core electromagnet by the magnetic circuit formula.

  B = μ0 × NI ÷ ((g + l ÷ μr) × 10⁻³), l = 2(w − tk) + 2(h − tk) − g

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

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• Magnet in a closed iron frame

  The flux density in a permanent magnet set in one leg of a closed frame.

  B = Br × (lm ÷ μrm) ÷ (lm ÷ μrm + L ÷ μr), L = 2(w − tk) + 2(h − tk) − lm

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

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• Field round one wire

  The flux density at two radii from the center of a round conductor.

  B = μ0 × I ÷ (2π × r × 10⁻³), r = 2a

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

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• Force between two wires

  The force per meter between two parallel conductors.

  F = μ0 × |I₁ × I₂| ÷ (2π × d × 10⁻³)

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

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• Template sizes within their ranges

  Clamps each size a person types to the template's allowed range.

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

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• Frame of the study's plane

  The origin and axes of the plane the result is drawn in.

  x = (helper × n) ÷ |helper × n|; y = n × x

  Magnetics add-on: the result drawn as display bodies addons/magnetics/lib/bodies.js

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• Triangle cut to the crop rectangle

  Cuts a field triangle to the rectangle that is shown.

  Magnetics add-on: the result drawn as display bodies addons/magnetics/lib/bodies.js

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• Field color bands

  The color band each triangle falls in.

  band = min(9, floor(|B| ÷ top × 10))

  Magnetics add-on: the result drawn as display bodies addons/magnetics/lib/bodies.js

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• Defining constants

  The pound, standard gravity, inch and foot, and the pound-force, gallon and British thermal unit built from them.

  lbf = 0.45359237 × 9.80665; gal = 231 × 0.0254³; BTU = 1055.05585262 J

  Engineering units: factors, dimensions and conversion engineer/eng-units.js

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• Unit table

  Each unit's factor to SI and its dimension.

  Engineering units: factors, dimensions and conversion engineer/eng-units.js

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• SI prefixes

  A prefixed unit such as kPa or mm from its base unit.

  factor = prefix × unit factor

  Engineering units: factors, dimensions and conversion engineer/eng-units.js

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• Combining units

  The factor and dimension of a product or quotient of two units.

  f = f_a × f_bᵉ; dimension = d_a + e × d_b

  Engineering units: factors, dimensions and conversion engineer/eng-units.js

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• Reading a unit text

  Turns a text such as W/(m^2 K) into one factor and one dimension.

  Engineering units: factors, dimensions and conversion engineer/eng-units.js

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• Value to SI

  A value in a given unit as its SI value.

  SI = v × f

  Engineering units: factors, dimensions and conversion engineer/eng-units.js

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• Value from SI

  An SI value in a given unit.

  v = SI ÷ f

  Engineering units: factors, dimensions and conversion engineer/eng-units.js

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• Unit to unit

  A value converted from one unit to another of the same kind.

  result = v × f_from ÷ f_to

  Engineering units: factors, dimensions and conversion engineer/eng-units.js

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• Temperature points

  Celsius and Fahrenheit readings to and from kelvin.

  K = °C + 273.15; K = (°F − 32) × 5 ÷ 9 + 273.15

  Engineering units: factors, dimensions and conversion engineer/eng-units.js

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• Power and logarithm used

  Which power and logarithm functions the curves are evaluated with.

  Material packs: property curves by temperature engineer/material-pack.js

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• One curve segment

  The fitted curve's value at a temperature.

  y = 10^(Σ cᵢ Lⁱ), L = log10 T; or y = c₀ + T(c₁ + T(c₂ + T(c₃ + T c₄)))

  Material packs: property curves by temperature engineer/material-pack.js

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• Property at a temperature

  The property value with its unit, range, fit error and source.

  value = curve(T) × scale

  Material packs: property curves by temperature engineer/material-pack.js

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• Series coefficients

  The factors 1/n! of the sine and cosine series.

  cₙ = ±1 ÷ n!

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Angle reduction

  Brings an angle into [−π, π].

  n = floor(x ÷ 2π + ½); r = (x − n × 2π_hi) − n × 2π_lo

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Sine series

  The sine of a reduced angle.

  sin r = r(c₀ + c₁r² + … + c₁₂r²⁴)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Cosine series

  The cosine of a reduced angle.

  cos r = c₀ + c₁r² + … + c₁₂r²⁴

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Sine

  sin x for any finite x.

  sin r = sin(π − r) for r > π/2; sin r = sin(−π − r) for r < −π/2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Cosine

  cos x for any finite x.

  cos r = −cos(π − |r|) for |r| > π/2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Exponential

  The exponential eˣ of a number.

  k = floor(x ÷ ln 2 + ½); r = (x − k × ln2_hi) − k × ln2_lo; eˣ = 2ᵏ × Σ rⁿ ÷ n! (n = 0 … 18)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Natural logarithm

  The natural logarithm ln x of a number.

  x = m × 2ᵉ, m in [√2/2, √2]; ln m = 2(s + s³/3 + … + s²⁹/29), s = (m − 1) ÷ (m + 1); ln x = e × ln2_hi + (ln m + e × ln2_lo)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Base-2 logarithm

  The base-2 logarithm log₂ x of a number.

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Base-10 logarithm

  The base-10 logarithm log₁₀ x of a number.

  log₁₀ x = ln x ÷ 2.302585092994046

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Whole-number power

  xⁿ for a whole n ≥ 0.

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Power

  A number x raised to a power y.

  xʸ = exp(y × ln x) for a fractional y

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Arctangent

  The arctangent atan x of a number.

  atan z = 2·atan(z ÷ (1 + √(1 + z²))), twice; then 4(z − z³/3 + … + z⁴¹/41)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Two-argument arctangent

  The angle of the point (x, y).

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Arcsine

  The arcsine asin x of a number in [−1, 1].

  asin x = atan2(x, √((1 − x)(1 + x)))

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Arccosine

  The arccosine acos x of a number in [−1, 1].

  acos x = 2·atan2(√(1 − x), √(1 + x))

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Tangent

  The tangent tan x of an angle.

  tan x = sin x ÷ cos x

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Hyperbolic sine

  The hyperbolic sine sinh x of a number.

  sinh x = (eˣ − e⁻ˣ) ÷ 2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Hyperbolic cosine

  The hyperbolic cosine cosh x of a number.

  cosh x = (eˣ + 1 ÷ eˣ) ÷ 2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Hyperbolic tangent

  The hyperbolic tangent tanh x of a number.

  tanh x = (e²ˣ − 1) ÷ (e²ˣ + 1)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Cube root

  The cube root ∛x of a number.

  r₀ = exp(ln|x| ÷ 3); r ← r − (r³ − |x|) ÷ (3r²), twice

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Sphere

  Signed distance to a sphere.

  d = √((x − cx)² + (y − cy)² + (z − cz)²) − r

  Field shapes: signed distance functions field/field.js

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• Box with rounded edges

  Signed distance to a box, optionally with rounded edges and corners.

  q = |p − c| − (half − round); d = |max(q, 0)| + min(max(q_x, q_y, q_z), 0) − round

  Field shapes: signed distance functions field/field.js

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• Cylinder

  Signed distance to a cylinder between two points.

  t = (p − m)·u; d_r = |p − m − t·u| − r; d_t = |t| − L ÷ 2; d = |max(d_r, d_t, 0)| + min(max(d_r, d_t), 0)

  Field shapes: signed distance functions field/field.js

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• Torus

  Signed distance to a torus about the z axis.

  d = √((√(x² + y²) − R)² + z²) − r

  Field shapes: signed distance functions field/field.js

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• Half space

  Signed distance to a plane.

  d = (n·p − d₀) ÷ |n|

  Field shapes: signed distance functions field/field.js

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• Solid of revolution

  Signed distance to the solid under a profile r(x) turned about the x axis.

  ρ = √(y² + z²); d = least distance from (x, ρ) to the profile's segments; inside when ρ ≤ r(x)

  Field shapes: signed distance functions field/field.js

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• Union, intersection and subtraction

  Combines shapes by union, intersection or subtraction.

  union = min(a, b); intersection = max(a, b); subtraction = max(a, −b)

  Field shapes: signed distance functions field/field.js

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