Calculations

Calculations

• Significand width of a number

  How many bits of a number this device keeps.

  bits = the first n where 1 + 2⁻ⁿ = 1

  Device number check accuracy.js Every app
  Exact

• Digits of a graphics float

  The number of significant digits stated for the graphics processor's float type.

  bits < 23: digits = max(1, ⌊(bits + 1) × log₁₀ 2⌋); bits ≥ 23: the fixed text "about 7"

  Device number check accuracy.js Every app
  64-bit

• Size and position text

  A dimension, port size or port position as text in millimeters.

  text = round(x × 10⁴) ÷ 10⁴

  Library configurator sizes library-configurator.js Every app
  64-bit

• Option value used for matching

  The text an option value is compared by when the page picks the variant that matches the selected options.

  numbers: round(v × 10⁴) ÷ 10⁴ as text; other values: trimmed text

  Library configurator sizes library-configurator.js Every app
  64-bit

• Mesh buffers

  The item's box or mesh as geometry for drawing, turned from z up to y up.

  Library 3D preview library-3d.js Every app
  32-bit, display only

• Camera position

  Where the camera sits as the view is turned and zoomed.

  position = target + dist × (sin φ sin θ, cos φ, sin φ cos θ)

  Library 3D preview library-3d.js Every app
  32-bit, display only

• Variant body buffers

  The chosen variant's box or mesh as geometry for drawing.

  Library 3D preview library-configurator.js Every app
  32-bit, display only

• Required and found values as text

  A limit or a measured value in words, such as ≤ 196.85 mm or 863.6–965.2 mm.

  text = round(x × 100) ÷ 100, then the unit

  Check report values problems.js Every app
  64-bit

• Built-in unit factors

  The factor from each built-in unit of mass, time, force, pressure, energy, power, current, voltage and volume to SI.

  lb = 0.45359237 kg; lbf = 4.4482216152605 N; psi = 6894.757293168 Pa; psf = 47.88025898 Pa

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Constants and fallback lengths

  pi, e, standard gravity g0, and the length units used when no unit table is given.

  g0 = 9.80665 m/s²; in = 0.0254 m; ft = 0.3048 m; yd = 0.9144 m

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Unit lookup

  The SI factor and dimension of a unit name.

  deg = π ÷ 180; rad = 1

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Dimension comparison

  Whether two values have the same dimensions, so they can be added, compared or converted.

  |aᵢ − bᵢ| < 1e-9 for each of the five exponents

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Power

  a ^ b for a value with units.

  value = aᵇ; dimensions × b

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Conversion with to

  M to kN m: checks the dimension and records the unit to show the answer in.

  shown = value ÷ factor of the target unit

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Shown amount

  The number and unit a row displays, and the amount passed to other modules.

  amount = value ÷ factor

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Square root and absolute value

  sqrt(a) and abs(a) for a value with units.

  √a halves the dimensions

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Rounding to a step

  round, floor and ceil, with an optional step such as round(x, 5 mm).

  round(a ÷ s) × s

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Minimum, maximum, sum and average

  min, max, sum and avg of values of one kind.

  sum = Σ aᵢ in order; avg = sum ÷ n

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Trigonometry, logarithms and exponent

  sin, cos, tan, their inverses, atan2, ln, log10, exp and pow.

  sin, cos, tan of radians; asin, acos need |x| ≤ 1; ln, log10 need x > 0

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Reading an app parameter

  @Name: the app's parameter as a value in SI.

  value = parameter value × factor of its unit

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Values written to driven parameters

  What each drive would set in the app: the row's answer in the parameter's unit.

  value = row value ÷ factor of the unit; change when |current − value| > 1e-9 × max(1, |value|)

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Time-weighted average

  The average of a value that changes at events: queue length, units busy, work in process.

  average = (Σ vᵢ × Δtᵢ) ÷ (t − t₀)

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Mean, deviation and percentiles of observations

  Waiting time, service time and time in system: mean, standard deviation, minimum, maximum, P50 and P95.

  m ← m + (x − m) ÷ n; s ← s + (x − m_old)(x − m_new); sd = √(s ÷ (n − 1)); P = sorted[⌊p × k⌋]

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Event time

  When a scheduled event happens.

  t_event = t_now + delay

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Conveyor slots and timing

  How many items fit on a belt, the gap between items and the travel time.

  slots = max(1, ⌊length ÷ pitch + 1e-9⌋); headway = pitch ÷ speed; travel = length ÷ speed

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Shift edges

  When a server goes on and off shift.

  x = t − ⌊t ÷ period⌋ × period; on shift when a ≤ x < b

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Server utilization

  The share of time a server's units were busy, blocked, down and idle.

  utilization = average busy units ÷ units; idle = max(0, 1 − busy − blocked − down)

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Throughput

  Finished entities per unit of time at a sink.

  throughput = count ÷ (t − t₀)

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Resource utilization

  The share of a shared resource's capacity that was in use.

  utilization = average in use ÷ average capacity

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Vehicle trip times

  How long a vehicle takes to reach the pick station and to carry the load to the drop station.

  to pick = distance ÷ speed + load time; carry = distance ÷ speed + unload time

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Student t factor

  The factor for a 95 % confidence range.

  df ≤ 30: table value; df ≤ 60: 2.0; df ≤ 120: 1.98; above: 1.96

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Replication mean and confidence half-width

  The mean of each result over independent runs and its 95 % range.

  mean = Σx ÷ n; sd = √(Σ(x − mean)² ÷ (n − 1)); half = t × sd ÷ √n

  Flow simulation (discrete events) engine.js Every app
  64-bit

• Uniform and normal draws

  The random numbers every arrival, service time and route choice is drawn from.

  u = (k + 0.5) ÷ 2³², k a 32-bit integer; z = √(−2 ln u) × cos(2π v)

  Flow simulation (discrete events) rng.js Every app
  64-bit

• Draw from a distribution

  One value from an exponential, uniform, triangular, normal, lognormal, Erlang, Weibull or empirical distribution.

  exp: −mean × ln u; tri: a + √(u (b − a)(c − a)) or b − √((1 − u)(b − a)(b − c)); weibull: scale × (−ln u)^(1 ÷ shape)

  Flow simulation (discrete events) rng.js Every app
  64-bit

• Mean of a distribution

  The mean used for load estimates and for routing to the least loaded target.

  tri: (a + m + b) ÷ 3; weibull: scale × Γ(1 + 1 ÷ shape)

  Flow simulation (discrete events) rng.js Every app
  64-bit

• Gamma function

  Γ(x), used for the mean of a Weibull distribution.

  Lanczos approximation, g = 7, 9 coefficients

  Flow simulation (discrete events) rng.js Every app
  64-bit

• Thermal voltage

  kT/q, the voltage scale of every diode and transistor junction.

  VT = 0.025852 V

  Circuit solver circuit.js Every app
  64-bit

• Value with an engineering suffix

  4.7k, 100n, 1meg: a component value read from text.

  value = number × factor of the suffix (t, g, meg, k, m, u, n, p, f); mil = 25.4e-6

  Circuit solver circuit.js Every app
  64-bit

• Source waveform value

  The value of a DC, sine or pulse source at a time.

  sin: off + amp × e^(−damp (t − delay)) × sin(2π f (t − delay) + phase); pulse: linear edges over tr and tf

  Circuit solver circuit.js Every app
  64-bit

• Diode current and conductance

  The current through a diode and its slope at a junction voltage.

  i = Is (e^(v ÷ (n VT)) − 1); g = Is e^(v ÷ (n VT)) ÷ (n VT)

  Circuit solver circuit.js Every app
  64-bit

• Transistor currents

  Collector and base current of a bipolar transistor.

  ic = If − Ir − Ir ÷ βr; ib = If ÷ βf + Ir ÷ βr; If = Is (e^(vbe ÷ VT) − 1)

  Circuit solver circuit.js Every app
  64-bit

• MOSFET level 1 current

  Drain current of a MOSFET in the triode and saturation regions.

  triode: id = kp (vov vds − vds² ÷ 2)(1 + λ vds); saturation: id = kp vov² (1 + λ vds) ÷ 2

  Circuit solver circuit.js Every app
  64-bit

• Switch conductance

  The conductance of a voltage-controlled switch.

  ln g = ln(1 ÷ roff) + a (ln(1 ÷ ron) − ln(1 ÷ roff)); a = 1 ÷ (1 + e^(−(v − vt) ÷ vh))

  Circuit solver circuit.js Every app
  64-bit

• Circuit equations

  The matrix and right side for one Newton step, at DC or at one time step.

  capacitor: geq = C ÷ h (backward Euler) or 2C ÷ h (trapezoid); inductor: r = L ÷ h or 2L ÷ h

  Circuit solver circuit.js Every app
  64-bit

• Newton iteration

  Repeats the linear solve until the voltages and currents stop changing.

  stop when |Δxᵢ| ≤ 1e-3 × max(|xᵢ new|, |xᵢ old|) + 1e-6 V (or 1e-12 A) for every unknown

  Circuit solver circuit.js Every app
  64-bit

• Operating point

  The DC voltages and currents of the circuit.

  direct Newton, then gmin stepping, then source stepping

  Circuit solver circuit.js Every app
  64-bit

• DC sweep

  The circuit solved for each value of a swept source.

  Circuit solver circuit.js Every app
  64-bit

• AC small-signal response

  Complex node voltages and branch currents at each frequency.

  capacitor: jωC; inductor: −jωL on its branch row; ω = 2π f

  Circuit solver circuit.js Every app
  64-bit

• Transient

  Voltages and currents over time.

  trapezoidal rule after a first backward-Euler step; output at t = k × tstep

  Circuit solver circuit.js Every app
  64-bit

• Junction voltage limiting

  Keeps a diode or transistor voltage from jumping too far in one iteration.

  v = v_old + vt ln(1 + (v_new − v_old) ÷ vt) when the step exceeds 2 vt above vcrit

  Circuit solver circuit.js Every app
  64-bit