WOLFRAM|DEMONSTRATIONS PROJECT

Electrostatic Fields Using Conformal Mapping

A conformal mapping
f
produces a complex function of a complex variable,
w=f(z)
, so that the analytical function
f
maps the complex
z
plane into the complex
w
plane. This technique is useful for calculating two-dimensional electric fields: the curve in the
w
plane where either
Re[z]
or
Im[z]
is constant corresponds to either an equipotential line or electric flux. This Demonstration shows 10 examples of electrostatic fields often encountered in high voltage applications. The electric field is shown in the
u
-
v
plane (or the
w
plane, where
w=u+iv
). The electrodes correspond to either
x=
x
1
,
x
2
or
y=
y
1
,
y
2
, where
z=x+iy
(
x,y,u,v∈
). The 10 examples are:
• concentric circles:
f(z)=
-iz
e
• ellipses:
f(z)=sin(z)
• hyperbolas:
f(z)=sin(z)
• parabolas:
f(z)=
2
z
• bipolar circles:
f(z)=icot(z)
• Cassinian ovals:
f(z)=
-iz
e
+1
• elliptical pairs:
f(z)=sn(iz)
• blade to plate shape:
f(z)=cn(z)
• Maxwell curves:
(z+1+
z
e
)/π
• square edge using a function derived by the Schwarz–Christoffel method
Three options give slightly different boundary conditions or electrode potentials
x=
x
1
,
x
2
or
y=
y
1
,
y
2
. The parameters are shown on the right. The calculated electric fields are shown by color, normalized to the average field
E
ave
=|
x
1
-
x
2
|/g
or
|
y
1
-
y
2
|/g
, where
g
is the smallest distance between two electrodes. If you select option 1, the local field is high in the vicinity of sharp electrode edges. When selecting option 2 or 3, the values are reduced owing to blunted edge conditions. The white lines indicate the flux line and the dashed lines are the equipotential lines for constant
x
or constant
y
. Those two families of curves are orthogonal.