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capacitance

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capacitance   { CAPACITANCE.PDE

See discussion in Help section "Electromagnetic Applications | Electrostatics".

}

TITLE 'Capacitance per Unit Length of 2D Geometry'

{ 17 Nov 2000 by John Trenholme }

SELECT

errlim 1e-4

thermal_colors on

plotintegrate off

VARIABLES

v

DEFINITIONS

mm = 0.001           ! meters per millimeter

Lx = 300 * mm         ! enclosing box dimensions

Ly = 150 * mm

b = 0.7               ! radius of conductor / radius of entire cable

x0 = 0.25 * Lx       ! position and size of cable raised to fixed potential

y0 = 0.5 * Ly

r0 = 15 * mm

x1 = 0.9 * Lx

y1 = 0.3 * Ly

r1 = r0

epsr                 ! relative permittivity of any particular region

epsd = 3             ! relative permittivity of cable dielectric

eps0 = 8.854e-12     ! permittivity of free space

eps = epsr * eps0

v0 = 1               ! fixed potential of the cable

energyDensity = dot( eps * grad( v), grad( v))/2   ! field energy density

EQUATIONS

div( eps * grad( v)) = 0

BOUNDARIES

region 1 'inside'  epsr = 1

start 'outer'  ( 0, 0) value( v) = 0

line to ( Lx, 0) line to ( Lx, Ly) line to ( 0, Ly) line to close

region 2 'diel0'  epsr = epsd

start 'dieb0'  ( x0 + r0, y0)

arc ( center = x0, y0) angle = 360

region 3 'cond0'  epsr = 1

start 'conb0'  ( x0 + b * r0, y0) value( v) = v0

arc ( center = x0, y0) angle = 360

region 4 'diel1'  epsr = epsd

start 'dieb1'  ( x1 + r1, y1)

arc ( center = x1, y1) angle = 360

region 5 'cond1'  epsr = 1000         ! fake metallic conductor

start 'conb1'  ( x1 + b * r1, y1)

arc ( center = x1, y1) angle = 360

PLOTS

grid(x,y) zoom(x0-1.1*r0, y0-1.1*r0, 2.2*r0)   ! 3-term zoom generates a square window }

contour( v) as 'Potential'

contour( v) as 'Potential Near Driven Conductor'

zoom(x0-1.1*r0, y0-1.1*r0, 2.2*r0)

contour( v) as 'Potential Near Floating Conductor'

zoom(x1-1.1*r1, y1-1.1*r1, 2.2*r1)

elevation( v) as 'Potential from Wall to Driven Conductor' from ( 0,y0) to ( x0, y0)

elevation( v) as 'Potential from Driven to Floating Conductor' from ( x0, y0) to ( x1, y1)

contour( energyDensity) as 'Field Energy Density'  png(3072,2)

contour( energyDensity) as 'Field Energy Density Near Floating Conductor'

zoom(x1-1.2*r1, y1-1.2*r1, 2.4*r1)

elevation( energyDensity) from ( x1 - 2 * r1, y1) to ( x1 + 2 * r1, y1)

as 'Field Energy Density Near Floating Conductor'

contour( epsr) paint on "inside" as 'Definition of Inside'

SUMMARY png(3072,2)

report sintegral( normal( eps * grad( v)), 'conb0', 'diel0') as 'Driven charge'

report sintegral( normal( eps * grad( v)), 'outer', 'inside') as 'Outer charge'

report sintegral( normal( eps * grad( v)), 'conb1', 'diel1') as 'Floating charge'

report sintegral( normal( eps * grad( v)), 'conb0', 'diel0') / v0 as 'Capacitance (f/m)'

report integral( energyDensity, 'inside') as 'Energy (J/m)'

report 2 * integral( energyDensity, 'inside') / v0^2 as 'Capacitance (f/m)'

report 2 * integral( energyDensity, 'inside') / ( v0 * sintegral( normal( eps * grad( v)), 'conb0', 'diel0'))

as 'cap_by_energy / cap_by_charge'

END