# 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:

• To apply some distinct potentials to the shield and the strip and to calculate the charge that arises on the strip;

• To apply zero potential to the shield and to describe the strip as having constant but unknown potential and carrying the charge, and then to measure the potential that arises on the strip.

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 = 2·W / U2,

and when the charge is known:

C = q2 / 2·W.

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.