The Effective Permittivity of a Transmission Line in a Microstrip
Key Takeaways

Microstrip lines are one of the most popular choices of transmission lines for microwave and RF circuits.

The effective permittivity of a transmission line is required to characterize the behavior of microstrip line propagation.

The effective permittivity 𝜀e of a transmission line satisfies the following relationship:
1<𝜀_{e}<𝜀_{r}
Microstrip transmission lines are commonly used in microwave circuits
The continued growth of wireless communication has increased the demand for efficient microwave and RF circuits. The signal propagation in RF circuits must be fast, without much propagation delay or data loss. The propagation of signal on a transmission line in microwave and RF systems depends on characteristic impedance, which, in turn, is related to the permittivity and permeability of the medium containing the electromagnetic (EM) fields.
In the case of microstrip transmission lines, EM fields lie partly in the air and the dielectric between the strip conductor and the ground. In this article, we will discuss how to calculate the effective permittivity of a transmission line.
Microstrip Lines
Microstrip lines are a type of planar transmission lines that can accommodate both passive and active RF devices. The microstrip line is one of the most popular choices of transmission lines in microwave and RF circuits. They consist of a conductor fabricated on the dielectric substrate of permittivity ‘𝜀r’ with a grounded plane.
The dielectric material and the air above the microstrip makes it a transmission line with the inhomogenous dielectric medium. In microstrip lines, a portion of the EM field lines are confined to the dielectric medium and another portion is in the air above the dielectric substrate. This complicates the behavior of microstrip lines, and they fail to support the transverse electromagnetic (TEM) mode of propagation.
The Phase Velocity of TEM Fields
The phase velocity of the TEM fields vary in the dielectric medium and the air. The phase velocity of the TEM field in the dielectric region is given by the following equation:
In the air, the phase velocity is ‘c’, where c is the speed of light (3x10^{8} m/s). There is a phase mismatch in the dielectricair interface in a microstrip transmission line. To describe the characteristics of propagation in nonTEM mode, calculating the effective permittivity of the transmission line in a microstrip is required.
Calculating the Effective Permittivity of a Microstrip Transmission Line
Microstrip lines support quasiTEM mode due to the inhomogeneity in the dielectric medium affecting the EM field distribution. The effective permittivity of a transmission line is required to characterize the behavior of microstrip line propagation.
The effective permittivity 𝜀_{e} of a transmission line satisfies the following relationship:
1<𝜀_{e}<𝜀_{r}
The effective permittivity is influenced by substrate thickness ‘h’ and conductor width ‘w’, and is given by the equation:
Calculation Steps Based on QuasiTEM Approximation
Microstrip transmission lines can be approximated as quasiTEM mode, and by using the static solutions for the transverse electric field, the characteristics of a transmission line can be calculated. Here are the steps to calculate the effective permittivity of a microstrip transmission line:

Calculate the electrostatic potential everywhere in the dielectric. Let V_{o} be the calculated potential of the conductor strip and Q be the per unit length charge on the conductor. The line capacitance can be given by:

Repeat step 1, with 𝜀_{r}=1 for the capacitance of the line without dielectric (C_{air}).

TEM mode is satisfied in air medium:
The phase velocity of the TEM field in air,
L is independent of material property, so calculated L is the per unit length inductance of the transmission line with dielectric and air.

Characteristic Impedance,
Phase velocity,

Effective permittivity of the microstrip transmission line is given by:
From the last two equations, the effective permittivity of the transmission line can be written as:
The effective permittivity characterizes a transmission line as one with the homogenous dielectric medium of permittivity 𝜀e.
The effective permittivity of a transmission line is one of the main parameters in microwave and RF circuits. The advantages of microstrip lines have made them popular choices in RF circuits, therefore, there is always a need to find the effective permittivity to study the behavior of EM fields. The phase velocity of an EM field is dependent on the permittivity of the medium. The approximation of using quasiTEM mode is successful in microstrip lines, and it provides an approximate equivalent structure of the transmission line.
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