.PowerSystems.AC3ph.Impedances

Information

Contains lumped impedance models and can also be regarded as a collection of basic formulas. Shunts are part of a separate package.

General relations.

  r = R / R_base                  resistance
  x = 2*pi*f_nom*L/R_base         reactance
  g = G / G_base                  conductance
  b = (2*pi*f_nom*C) / G_base     susceptance
  G_base = 1/R_base

A) Symmetric systems.

The reactance-matrix in abc-representation is

          [x_s, x_m, x_m
  x_abc =  x_m, x_s, x_m
           x_m, x_m, x_s]

and corresponds to the following diagonal matrix in dq0-representation

          [x, 0, 0
  x_dq0 =  0, x, 0
           0, 0, x0]

with the relations

  x   = x_s - x_m           stray reactance (dq-components)
  x0  = x_s + 2*x_m         zero-reactance (o-component)
  x_s =  (2*x + x0)/3       self reactance single conductor
  x_m = -(x - x0)/3         mutual reactance

Coupling.

  -x_s/2 <  x_m <  x_s
  uncoupled limit:          x_m = 0,               x0 = x
  fully positive coupled:   x_m = x_s,             x0 = 3*x_s
  fully negative coupled:   x_m = -x_s/2,          x0 = 0
  'practical' value:        x_m = -x_s*(2/13),     x0 = (3/5)*x

The corresponding resistance matrix is

                  [r, 0, 0
  r_abc = r_dq0 =  0, r, 0
                   0, 0, r]

The susceptance matrices in abc- and in dq0-representation are

          [ b_pg + 2b_pp, -b_pp,         -b_pp
  b_abc =  -b_pp,          b_pg + 2b_pp, -b_pp
           -b_pp,         -b_pp,          b_pg + 2b_pp]
          [ b_pg + 3*b_pp, 0,             0
  b_dq0 =   0,             b_pg + 3*b_pp, 0
            0,             0,             b_pg]

where _pg denotes 'phase-to-ground' and _pp 'phase-to-phase'.

The corresponding conduction matrices are (in analogy to susceptance)

          [ g_pg + 2g_pp, -g_pp,         -g_pp
  g_abc =  -g_pp,          g_pg + 2g_pp, -g_pp
           -g_pp,         -g_pp,          g_pg + 2g_pp]
          [ g_pg + 3*g_pp, 0,             0
  g_dq0 =   0,             g_pg + 3*g_pp, 0
            0,             0,             g_pg]

B) Non symmetric systems.

  x_abc is an arbitrary symmetric matrix with positive diagonal elements

  r_abc is an arbitrary diagonal matrix with positive elements

  b_abc (phase-to-ground) is an arbitrary diagonal matrix with positive elements

  b_abc (phase-to-phase) is of the form

          [b_pp[2] + b_pp[3], -b_pp[3],           -b_pp[2]
  b_abc = -b_pp[3],            b_pp[3] + b_pp[1], -b_pp[1]
          -b_pp[2],           -b_pp[1],            b_pp[1] + b_pp[2]]

  g_abc(phase-to-ground) is an arbitrary diagonal matrix with positive elements

  g_abc(phase-to-phase) is of the form

          [g_pp[2] + g_pp[3], -g_pp[3],           -g_pp[2]
  g_abc = -g_pp[3],            g_pp[3] + g_pp[1], -g_pp[1]
          -g_pp[2],           -g_pp[1],            g_pp[1] + g_pp[2]]

where

  index 1 denotes pair 2-3
  index 2 denotes pair 3-1
  index 3 denotes pair 1-2

The corresponding dq0-matrices are all obtained by Park-transformation P

  x_dq0 = P*x_abc*transpose(P)
  r_dq0 = P*r_abc*transpose(P)
  b_dq0 = P*b_abc*transpose(P)
  g_dq0 = P*g_abc*transpose(P)

Contents

Name	Description
Resistor	Resistor, 3-phase dq0
Conductor	Conductor, 3-phase dq0
Inductor	Inductor with series resistor, 3-phase dq0
Capacitor	Capacitor with parallel conductor, 3-phase dq0
Impedance	Impedance (inductive) with series resistor, 3-phase dq0
Admittance	Admittance (capacitive) with parallel conductor, 3-phase dq0
ResistorNonSym	Resistor non symmetric, 3-phase dq0.
InductorNonSym	Inductor with series resistor non symmetric, 3-phase dq0.
CapacitorNonSym	Capacitor with parallel conductor non symmetric, 3-phase dq0.
Varistor	Varistor, 3-phase dq0
Partials	Partial models