Elec1: Microstrip transmission line

A shielded microstrip transmission line consists of a substrate, a microstrip, and a shield.

Problem Type:
Plane-parallel problem of electrostatics.

Geometry:
The transmission line is directed along z-axis, its cross section is shown on the sketch. The rectangle ABCD is a section of the shield, the line EF represents a conductor strip.

Given:
Relative permittivity of air ε = 1;
Relative permittivity of substrate ε = 10.

Problem:
Determine the capacitance of a transmission line.

Solution:
There are several different approaches to calculate the capacitance of the line:

Both these approaches make use of the equation for capacitance:

C = q / U.

Other possible approaches are based on calculation of stored energy of electric field. When the voltage is known:

C = 2W / U2,

and when the charge is known:

C = q2 / 2W.

Experiment with this example shows that energy-based approaches give little bit less accuracy than approaches based on charge and voltage only. The first approach needs to get the charge as a value of integral along some contour, and the second one uses only a local value of potential, this approach is the simplest and in many cases the most reliable.

Results:

Theoretical result:

C = 178.1 pF/m.

Approach 1:

C = 177.83 pF/m (99.8%).

Approach 2:

C = 178.47 pF/m (100.2%).

Approach 3:

C = 177.33 pF/m (99.6%).

Approach 4:

C = 179.61 pF/m (100.8%).

The first and third approaches are illustrated in the Elec1_1.pbm problem in the Examples folder, and the Elec1_2.pbm explains the second and the fourth approaches.