Substrate Dielectric Extraction Using a Band Stop Resonator Sensor
The microstrip line is one of the most popular guided wave structures that is commonly used for designing microwave components. Typically, microstrip lines are planar and simple to fabricate, requires a smaller footprint, is easier to design, and can readily be integrated with other components on a board or wafer. In this case, a microstrip based T-Line resonator with an open ended stub is used for realizing a band stop filter. It is well know that the resonators are commonly used in the design of many RF or μ-wave circuits such as filters and oscillators. A crucial advantage of these resonators is that the resonance frequency is based on the substrate properties on which these are fabricated. In reality, we can determine the effective dielectric constant of the substrate using the resonance frequency and few other design parameters from the T-line resonator.
In this work, two different workflows are presented (i) a simple procedure to design the T-line resonator on a High Frequency laminate and (ii) the inverse problem of using the resonance frequency and the design parameters to extract the real part of the substrate dielectric constant.
(i) A Open Stub Bandpass Filter
This is most commonly referred to as a T-line resonator that consists of an open-circuit length of a transmission line in parallel to the 50 ohm microstrip line. The circuit will behave as a parallel resonant one when the length of the stub is λ/2 or multiples of λ/2. As an example, a T-line resonator with resonance frequency of 5 GHz is simulated on a high frequency laminate of standard thickness (1.52 mm) and having the dielectric properties of 3+ j0. First, using the in-built transmission line calculator of Feko, the width of the 50 Ohms microstrip line for the chosen substrate at the chosen resonance frequency (5 GHz) is calculated and it is found to be 3.82 mm. Using planar Green's function approach, the T-line resonator is modelled and the schematic as well as the simulation results are shown in Fig 1 and 2 respectively.
Fig 1. T-Line resonator.
Fig 2. Simulated S-parameters for the open stub bandpass filter.
Based on the chosen dimensions, a band stop filter is achieved operating at 5 GHz.
(ii) The inverse extraction routine:
For the inverse problem, one should have the measured/simulated results of the T-line resonator built on an unknown substrate and one should have the knowledge of the dimensions of the open stub (length and width). Once these two entities are known and the order of resonance is determined, the real part of the effective dielectric constant of the substrate can be extracted using a simple formula as shown below. In this case, if Fig 2 is considered as the measured results with a resonance at 5 GHz generated using the line length and width to be 3.82 mm and 8.4 mm for the open ended stub, then the effective dielectric constant can be determined by taking into account the length and width of the open-ended stub and a correction factor for the T-junction. Ignoring the open-end effects, an approximate formula for calculating the effective dielectric constant is given as follows
where,
n is the order of resonance
c is the speed of light
fr is the resonance frequency
l is the length of the open stub
w is the width of the feedlines, and
teff is the correction factor for the T-junction.
Based on the above formula, the effective dielectric constant for the substrate on which the T-line is built is estimated to be 2.89 (~3). This is closer to the actual real part of the dielectric constant (3) used in the simulation and the slight deviation may be due to ignoring the open ended effects. The inverse loop showcases the accuracy of the formula used to determine the unknown dielectric constant of the substrate. This technique is useful for characterizing unknown dielectric substrates and if the substrate absorbs certain materials. this technique can work to identify the absorbed substance's dielectric properties thereby paving way for realization of a good substrate base sensing technique.