Losses in microstrip lines
The attenuation constant of the dominant microstrip mode depends on geometric factors, electrical properties of the substrate and conductors, and on the frequency. For a nonmagnetic dielectric substrate, two types of losses occur in the dominant microstrip mode: (1) dielectric loss in the substrate and (2) ohmic skin loss in the strip conductor and the ground plane. The sum of these two losses may be expressed as losses per unit length in terms of an attenuation factor a. From ordinary transmission-line theory, the power carried by a wave traveling in the positive z direction is given by :
The attenuation constant a can be expressed as
where αd is the dielectric attenuation constant and αc is the ohmic attenuation constant.
The gradient of power in the z direction in Eq. (2) can be further expressed in terms of the power loss per unit length dissipated by the resistance and the power loss per unit length in the dielectric. That is,
where σ is the conductivity of the dielectric substrate board.
Substitution of Eq. (3) into Eq. (2) results in :
When the conductivity of a dielectric cannot be neglected, the electric and magnetic fields in the dielectric are no longer in time phase. In that case the dielectric attenuation constant, as expressed is given by :
where σ is the conductivity of the dielectric substrate board in Ω/cm. This dielectric constant can be expressed in terms of dielectric loss tangent as :
Then the dielectric attenuation constant is expressed by :