Resources Numerical accuracy
Calculations
Every calculation under our calculation policies: what it works out, its formula and the precision it runs at. Open one to see its code as it is in the released app.
Calculations
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Thermal voltage
kT/q, the voltage scale of every diode and transistor junction.
VT = 0.025852 V
Circuit solver circuit.js Every app64-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 app64-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 app64-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 app64-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 app64-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 app64-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 app64-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 app64-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 app64-bit -
Operating point
The DC voltages and currents of the circuit.
direct Newton, then gmin stepping, then source stepping
Circuit solver circuit.js Every app64-bit -
DC sweep
The circuit solved for each value of a swept source.
Circuit solver circuit.js Every app64-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 app64-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 app64-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 app64-bit -
Sweep and frequency lists
The source values of a DC sweep and the frequencies of an AC analysis.
dc: start, start + step, …; ac dec or oct: f = fstart × (fstop ÷ fstart)^(k ÷ N); lin: fstart + (fstop − fstart) k ÷ (points − 1)
Circuit solver circuit.js Every app64-bit -
Magnitude and phase
The size and angle of a complex result.
|z| = hypot(re, im); phase = atan2(im, re) × 180 ÷ π
Circuit solver circuit.js Every app64-bit -
Real linear solve
Solves the circuit equations of one Newton step.
LU with partial pivoting, then back substitution
Circuit solver linalg.js Every app64-bit -
Complex linear solve
Solves the AC equations and the antenna's impedance matrix.
LU with partial pivoting on separate real and imaginary arrays
Circuit solver linalg.js Every app64-bit -
Magnetic and electric constants
μ0 and ε0 used by every field result.
μ0 = 4π × 10⁻⁷ H/m; ε0 = 8.8541878128 × 10⁻¹² F/m
Magnetic and electric field solvers fields.js Every app64-bit -
Field of straight wire segments
The magnetic flux density at a point from current-carrying segments.
B = μ0 I ÷ (4π d) × (z_b ÷ √(z_b² + d²) − z_a ÷ √(z_a² + d²))
Magnetic and electric field solvers fields.js Every app64-bit -
Loop as straight segments
A circular loop turned into chords for the segment formulas.
n chords with their ends on the circle, 128 by default
Magnetic and electric field solvers fields.js Every app64-bit -
Mutual inductance of two loops
The Neumann integral over two loops given as segments.
M = μ0 ÷ 4π × Σ Σ (dl₁ · dl₂) ÷ R
Magnetic and electric field solvers fields.js Every app64-bit -
Complete elliptic integrals
K(k) and E(k), used for coaxial loops.
K = π ÷ (2a) at the limit of the arithmetic–geometric mean; E = K (1 − Σ 2ⁿ⁻¹ cₙ²)
Magnetic and electric field solvers fields.js Every app64-bit -
Mutual inductance of coaxial loops
Maxwell's formula for two circular loops on one axis.
M = μ0 √(r₁ r₂) ((2 ÷ k − k) K − (2 ÷ k) E); k² = 4 r₁ r₂ ÷ ((r₁ + r₂)² + d²)
Magnetic and electric field solvers fields.js Every app64-bit -
2D finite-element solve
The potential at every mesh node and its gradient in every triangle.
∇·(k ∇u) = −s on linear triangles; element matrix k (bᵢ · bⱼ) A
Magnetic and electric field solvers fields.js Every app64-bit -
Magnetic field, energy and saturating iron
Flux density per triangle, stored energy per meter, and the iteration for a B–H curve.
B = (∂A/∂y, −∂A/∂x); W = Σ ½ ν B² A
Magnetic and electric field solvers fields.js Every app64-bit -
Electric field and energy
Field per triangle and stored energy per meter.
E = −∇V; W = Σ ½ ε E² A
Magnetic and electric field solvers fields.js Every app64-bit -
Capacitance from energy
Capacitance per meter from the stored energy.
C = 2W ÷ ΔV²
Magnetic and electric field solvers fields.js Every app64-bit -
Force from the Maxwell stress
The force per meter on a body in the solved field.
F = Σ ((B · n) B − ½ B² n) × edge length ÷ μ
Magnetic and electric field solvers fields.js Every app64-bit -
Conjugate-gradient solve
Solves the finite-element equations.
stop when ‖r‖₂ ≤ tol × ‖b‖₂
Magnetic and electric field solvers linalg.js Every app64-bit -
Constants
The speed of light, μ0, ε0 and the impedance of free space.
c = 299792458 m/s; μ0 = 4π × 10⁻⁷; ε0 = 1 ÷ (μ0 c²); η0 = μ0 c
Wire antenna (method of moments) wire.js Every app64-bit -
Gauss–Legendre points
The sample points and weights of the integrals along segments.
2, 3 and 4 points on [0, 1]
Wire antenna (method of moments) wire.js Every app64-bit -
Segments and node merging
How wires are cut into segments and joined at shared points.
segments per leg = ⌈L ÷ maxSeg⌉, maxSeg = λ ÷ 20 by default; merge distance = 1e-6 × max(largest coordinate, λ ÷ 1000)
Wire antenna (method of moments) wire.js Every app64-bit -
Kernel integrals over a segment pair
The coupling between two segments.
∫∫ cos(kR) ÷ (4πR), R = √(|r − r′|² + a²)
Wire antenna (method of moments) wire.js Every app64-bit -
Impedance matrix
The matrix that links the currents to the source voltage.
Z_mn = jωμ (u · u′) ∫∫ f_m f_n G + (1 ÷ jωε) ∫∫ f_m′ f_n′ G
Wire antenna (method of moments) wire.js Every app64-bit -
Input impedance
The impedance seen at the feed point.
Z_in = V ÷ I at the feed
Wire antenna (method of moments) wire.js Every app64-bit -
Radiation intensity
Power per solid angle in a direction.
U = η0 k² ÷ (32π²) × (|N_θ|² + |N_φ|²)
Wire antenna (method of moments) wire.js Every app64-bit -
Radiated power and directivity
Total radiated power, the strongest direction and the directivity.
P = Σ U sin θ Δθ Δφ; D = 4π U_max ÷ P
Wire antenna (method of moments) wire.js Every app64-bit -
Rotation about an axis
The rotation matrix for an angle about an axis.
R = cos θ I + sin θ [a]× + (1 − cos θ) a aᵀ
Rotations, transforms and small linear algebra (motion) math.js Every app64-bit -
Roll, pitch and yaw from a matrix
The three angles of a rotation matrix.
pitch = asin(−R₃₁); roll = atan2(R₃₂, R₃₃); yaw = atan2(R₂₁, R₁₁)
Rotations, transforms and small linear algebra (motion) math.js Every app64-bit -
Rotation vector from a matrix
Axis times angle of a rotation, used for orientation error.
θ = acos((tr R − 1) ÷ 2); v = θ ÷ (2 sin θ) × (R₃₂ − R₂₃, R₁₃ − R₃₁, R₂₁ − R₁₂)
Rotations, transforms and small linear algebra (motion) math.js Every app64-bit -
Rigid transforms
Product, inverse and application of a rotation with a translation.
A·B = (R_A R_B, p_A + R_A p_B); A⁻¹ = (R_Aᵀ, −R_Aᵀ p_A)
Rotations, transforms and small linear algebra (motion) math.js Every app64-bit -
Dense linear solve
Solves the small systems of inverse kinematics and linkages.
Gaussian elimination with partial pivoting
Rotations, transforms and small linear algebra (motion) math.js Every app64-bit -
Positive-definite test
Whether a symmetric matrix, such as a mass matrix, is positive definite.
Cholesky factorization succeeds
Rotations, transforms and small linear algebra (motion) math.js Every app64-bit -
Singular values
The singular values of a Jacobian, used to detect singular poses and to count degrees of freedom.
σᵢ = √(eigenvalues of J Jᵀ or Jᵀ J)
Rotations, transforms and small linear algebra (motion) math.js Every app64-bit -
Pose error
How far a link is from its target in position and orientation.
e = (p_target − p, rotation vector of R_target Rᵀ)
Inverse kinematics and joint trajectories ik.js Every app64-bit -
Inverse kinematics
The joint values that bring a link to a target position, orientation or both.
Δq = Jᵀ (J Jᵀ + λ² I)⁻¹ e
Inverse kinematics and joint trajectories ik.js Every app64-bit -
Planar two-link arm, closed form
Both elbow solutions of a two-link arm for a tip position.
cos q₂ = (x² + y² − l₁² − l₂²) ÷ (2 l₁ l₂); q₁ = atan2(y, x) − atan2(l₂ sin q₂, l₁ + l₂ cos q₂)
Inverse kinematics and joint trajectories ik.js Every app64-bit -
Trapezoidal velocity profile
Position, speed and acceleration of a rest-to-rest move with a speed and an acceleration limit.
t_a = v ÷ a; T = 2 t_a + (d − v² ÷ a) ÷ v; short move: t_a = √(d ÷ a), T = 2 t_a
Inverse kinematics and joint trajectories traj.js Every app64-bit -
Cubic profile
A move with zero speed at both ends.
q = q₀ + Δ (3τ² − 2τ³)
Inverse kinematics and joint trajectories traj.js Every app64-bit
A calculation is added only after its source was read. Each statement can be checked in the code shown on its policy's page; the app repositories are public. When a release changes a calculation, its page says so until it has been read again.