Calculations

Calculations

• Dense linear solve

  Solves the nodal equations of the pipe network and the load-flow Jacobian.

  Gaussian elimination with partial pivoting

  Pipe, duct and gas networks numeric.js Every app
  64-bit

• Bus admittance matrix

  The admittances that link every pair of buses.

  y = 1 ÷ (r + jx); Yff = (y + jb ÷ 2) ÷ |a|²; Yft = −y ÷ conj(a); Ytt = y + jb ÷ 2

  Load flow and short-circuit currents power.js Every app
  64-bit

• Power injected at each bus

  Real and reactive power from the voltages.

  Pᵢ = Σ Vᵢ Vⱼ (G cos δ + B sin δ); Qᵢ = Σ Vᵢ Vⱼ (G sin δ − B cos δ)

  Load flow and short-circuit currents power.js Every app
  64-bit

• Load flow iteration

  Corrects angles and voltages until the power balances.

  J Δx = (P_sched − P, Q_sched − Q); stop when max |mismatch| < 1e-9 pu

  Load flow and short-circuit currents power.js Every app
  64-bit

• Reactive limits of generator buses

  Switches a voltage-controlled bus to a fixed reactive power when it passes its limit.

  switch when Q_g > Qmax + 1e-9 or Q_g < Qmin − 1e-9

  Load flow and short-circuit currents power.js Every app
  64-bit

• Branch flows and losses

  Power at both ends of each branch and the loss in it.

  S_f = V_f conj(I_f); loss = P_f + P_t

  Load flow and short-circuit currents power.js Every app
  64-bit

• Per-unit base

  Ohms to per unit on a voltage and power base.

  Z_base = kV² ÷ MVA; z_pu = z ÷ Z_base

  Load flow and short-circuit currents power.js Every app
  64-bit

• Feeder voltage drop

  The three-phase voltage drop of a feeder.

  ΔV ≈ √3 I (R cos φ + X sin φ), φ = acos(pf)

  Load flow and short-circuit currents power.js Every app
  64-bit

• Sequence impedance matrices

  The impedance seen from every bus in the positive, negative and zero sequence.

  Z = Y⁻¹ for each sequence

  Load flow and short-circuit currents shortcircuit.js Every app
  64-bit

• Sequence to phase values

  Phase currents and voltages from the sequence components.

  I_a = I₀ + I₁ + I₂; I_b = I₀ + a² I₁ + a I₂; I_c = I₀ + a I₁ + a² I₂, a = −½ + j√3 ÷ 2

  Load flow and short-circuit currents shortcircuit.js Every app
  64-bit

• Fault current

  Initial and peak fault current, fault power, and the voltages and branch currents during the fault.

  3-phase: I₁ = c ÷ (Z₁ + Z_f); line to ground: I₁ = c ÷ (Z₁ + Z₂ + Z₀ + 3Z_f); i_p = κ √2 I_k, κ = 1.02 + 0.98 e^(−3R/X)

  Load flow and short-circuit currents shortcircuit.js Every app
  64-bit

• Complex division

  a ÷ b for the impedances and currents of the fault study.

  a ÷ b = a conj(b) ÷ (b_re² + b_im²)

  Load flow and short-circuit currents numeric.js Every app
  64-bit

• Complex matrix inverse

  Inverts a sequence admittance matrix.

  Gauss–Jordan elimination with pivoting by magnitude

  Load flow and short-circuit currents numeric.js Every app
  64-bit

• Curve constants

  The constants of the IEC and IEEE inverse-time curves and the cable factors.

  IEC normal inverse k = 0.14, α = 0.02; IEEE very inverse A = 19.61, B = 0.491, p = 2; cable k: copper PVC 115

  Protection curves and grading protection.js Every app
  64-bit

• Relay operating time

  How long a relay takes to trip at a current.

  IEC: t = TMS × k ÷ (M^α − 1); IEEE: t = TD ÷ divisor × (A ÷ (M^p − 1) + B), M = I ÷ pickup

  Protection curves and grading protection.js Every app
  64-bit

• Fuse or breaker curve from points

  The operating time of a device given as a table of current and time.

  ln t = ln t₀ + (ln I − ln I₀) ÷ (ln I₁ − ln I₀) × (ln t₁ − ln t₀)

  Protection curves and grading protection.js Every app
  64-bit

• Cable damage time

  How long a cable can carry a fault current.

  t = (k S ÷ I)²

  Protection curves and grading protection.js Every app
  64-bit

• Grading margin

  The time between a downstream and an upstream device at each fault current.

  margin = t_up − t_down; ok when margin ≥ the grading interval (0.3 s by default)

  Protection curves and grading protection.js Every app
  64-bit

• Pickup and cable checks

  Pickup against load and the smallest fault, and clearing time against the cable.

  pickup ≥ 1.05 × load; pickup ≤ smallest fault ÷ 1.5; clearing time ≤ (k S ÷ I)²

  Protection curves and grading protection.js Every app
  64-bit

• Coordinate and aperture format

  How a position and an aperture size are written in a Gerber file.

  X, Y = round(mm × 10⁶); size = 6 decimals

  Gerber and drill file numbers gerber.js Every app
  64-bit

• Pad as a flash or a region

  A pad, or its grown copy for the solder mask, in the file.

  axis-aligned: aperture (w + 2 × grow) by (h + 2 × grow); else a 4-corner region rotated by the pad angle

  Gerber and drill file numbers gerber.js Every app
  64-bit

• Drill sizes and positions

  The tool table and hole positions of the drill files.

  tool size = 3 decimals; X, Y = 4 decimals of a millimeter

  Gerber and drill file numbers gerber.js Every app
  64-bit

• Board size in the job file

  The board's width and height stated in the job file.

  size = extent of the outline, 4 decimals

  Gerber and drill file numbers gerber.js Every app
  64-bit

• Coordinate scale when reading

  The factor that turns the file's integers into coordinates.

  scale = 10^(−decimals of X)

  Gerber and drill file numbers gerber.js Every app
  64-bit

• Coordinates when reading

  Each X and Y of a Gerber file as a number.

  x = integer × scale

  Gerber and drill file numbers gerber.js Every app
  64-bit

• Drill file when reading

  Tool diameters and hole positions read back from a drill file.

  decimal text to the nearest 64-bit number

  Gerber and drill file numbers gerber.js Every app
  64-bit

• Real number when reading

  An 8-byte GDSII real as a number.

  value = sign × (mantissa ÷ 2⁵⁶) × 16^(exponent − 64)

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Real number when writing

  A number as an 8-byte GDSII real.

  mantissa = round(v × 2⁵⁶) after scaling v into [1/16, 1) by powers of 16

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  Exact

• Database unit

  User units per database unit and meters per database unit, read from the file.

  two 8-byte reals from the UNITS record

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Integers written to the stream

  Coordinates, widths, layers and array counts as the file stores them.

  coordinates and widths: 32-bit signed; layers and counts: 16-bit signed

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Placement transform

  A referenced cell's points reflected, magnified, rotated and moved.

  (x, y) → mag × (x, ±y), rotated by the angle in degrees, then translated

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Nested placements

  The combined transform of a cell placed inside a placed cell.

  mag = mag_parent × mag_child; angle = angle_parent ± angle_child; reflection toggles

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Array placement

  The position of each copy in a rows-by-columns array.

  position(i, j) = origin + i × (c − o) ÷ cols + j × (r − o) ÷ rows

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Path outline

  A path with a width turned into a polygon.

  offset = ± half width along the corner bisector ÷ max(0.2, cos of the half angle)

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Database units to user units

  Flattened coordinates in the file's user unit, usually micrometers.

  x_user = x_db × userPerDb

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Shapes and area per layer

  How many shapes each layer has and their total area.

  area = Σ |shoelace area|

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Width, spacing and area checks

  Shapes that are narrower, closer or smaller than a rule allows.

  area < minArea − 1e-12; 0 < distance < minSpace − 1e-12; facing-edge width < minWidth − 1e-12

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Narrowest width of a shape

  The smallest distance between two edges that face each other across the inside.

  distance from one edge's line to the other, for antiparallel edges that overlap

  GDSII layout: reals, flattening, areas and checks gds.js Every app
  64-bit

• Half-perimeter wire length

  The estimate of total connection length reported before and after placement.

  Σ over nets of (x_max − x_min) + (y_max − y_min)

  Board placement and routing place.js Every app
  64-bit

• Overlap and outline penalties

  The cost added when courtyards overlap or reach outside the board.

  cost = 50 × (overlap area + area outside the margin)

  Board placement and routing place.js Every app
  64-bit

• Annealing schedule

  How the search cools from large moves to small ones.

  T₀ = max(1, 0.15 (W + H)); T_end = 0.002 (W + H); T ← T × (T_end ÷ T₀)^(1 ÷ iterations)

  Board placement and routing place.js Every app
  64-bit

• Final part positions

  The positions stored on the board when placement ends.

  x, y = round(value × 1000) ÷ 1000

  Board placement and routing place.js Every app
  64-bit

• Routing grid

  The grid of points tracks may pass through.

  pitch = (track width + clearance) ÷ 2 by default; x = ox + i × pitch

  Board placement and routing route.js Every app
  64-bit

• Obstacle map

  Which grid points a track or a via of another net may not use.

  blocked when distance to copper < clearance + track width ÷ 2 + 0.25 × pitch

  Board placement and routing route.js Every app
  64-bit

• Path search

  The cheapest path on the grid from a pad to the copper already routed for its net.

  step cost 1, diagonal 1.4142, against the layer's direction × 1.4, via 14 per layer

  Board placement and routing route.js Every app
  64-bit

• Track points

  The corner points of a routed track.

  drop a point when |(b − a) × (c − b)| ≤ 1e-9

  Board placement and routing route.js Every app
  64-bit

• Probability of a basic event

  The probability of an event from a fixed value, a failure rate and time, a test interval, or repair data.

  failed by t: 1 − e^(−λt); tested every τ: 1 − (1 − e^(−λτ)) ÷ (λτ), or λτ ÷ 2 below 1e-9; repairable: MTTR ÷ (MTTF + MTTR)

  Fault trees and reliability block diagrams logic.js Every app
  64-bit

• At least k of n events

  The probability that k or more of a set of independent events occur.

  the distribution of the count, built one event at a time, summed from k up

  Fault trees and reliability block diagrams logic.js Every app
  64-bit

• AND, OR and voting gates

  The probability of a gate from its inputs when no event is repeated.

  AND: Π pᵢ; OR: 1 − Π (1 − pᵢ)

  Fault trees and reliability block diagrams logic.js Every app
  64-bit

• Top event with repeated events

  The probability of the top event when the same event appears in several places.

  P = p_x × P(tree | x occurs) + (1 − p_x) × P(tree | x does not)

  Fault trees and reliability block diagrams logic.js Every app
  64-bit