The microplasma system that uses a stripline resonator is a promising plasma system because, capitalizing on the resonance property of the plasma source, it can readily generate plasma under atmospheric conditions without the assistance of an additional matching network. However, precise determination of the resonance frequency before device manufacturing is very important in practice. Therefore, the evolution trends of the resonance frequency of the discharge have to be investigated, based on source impedance analysis, to get an insight into the discharge matching condition. In this paper, a means of determining the discharge source impedance, called conformal mapping, and its application to the microwave parallel stripline resonator discharge source is presented and discussed.
In many rf systems, some of the output power of the rf generator does not reach the load because of power loss in the line and matcher. When a plasma chamber is used as the load, the power loss in the chamber is non-intuitive because plasma is a non-linear dielectric material. Thus, to accurately measure the power loss in the chamber, an rf sensor can be connected to the chamber. Microplasmas have recently received much attention for development and application. For a combination of the potentials of low temperature plasmas, which have the advantage of being in the micro scale, the discharge creates a highly reactive environment that contains charged particles, excited species, radicals, and photons; this is applicable to various fields, including bio-medical applications (treatment of living tissues, tissue sterilization, and blood coagulation), dental treatment, displays, radiation sources, microchemical analysis systems, gas analyzers, and photodetectors. Microplasmas can be generated over wide pressure range from a few mTorr up to a few atmospheres. Typically, operation of microplasma systems at atmospheric pressure is more favored as its size can be reduced by eliminating the micro-pump and it only requires low power, making its integration into microsystems and portable devices possible [1].
Recently, the microwave parallel stripline resonator discharge source has been widely investigated due to its ability to use the source resonance to generate plasma even under difficult conditions such as that at atmospheric pressure, without assistance from an additional matching box. However, precise determination of the resonance frequency before device manufacturing is very important in practice. Therefore the evolution trends of the resonance frequency of the discharge have to be investigated, based on source impedance analysis, to get an insight into the discharge matching condition. In this paper, a means of determining the discharge source impedance, called conformal mapping, and its application to the microwave parallel stripline resonator (MPSR) discharge source is presented and discussed.
In this section, we the first explain our motivation for this work before we beginning calculations. Figure 1 is an example of the S11 parameter of the MPSR, obtained using commercial CST microwave studio software (CST-Computer Simulation Technology Company). As shown in Fig. 1, the parameter spectrum has a sharp minimum value at approximately 0.74 MHz, reflecting strong resonance in the device. A simple calculation based on Fig. 1 shows that the 3 dB quality factor (
One can apply an electric circuit model to estimate the resonant frequency and net impedance of an MPSR and thereby determine the key parameters that control the performance of the device. This results in the formulation of closed-form expressions that are useful for designing an MPSR and analyzing the designs. However, MPSR geometry is not like that of a stripline or other well-known geometries. Therefore, the device parameters must be determined carefully. There are various methods to do this, such as finite difference methods, Green’s functions, and conformal mapping. In this paper, the characteristic impedance, capacitance, and inductance of the design are calculated using the conformal mapping method. The data obtained after calculation, simulation with a commercial program, and conducting an experiment based on an MPSR device are presented for comparison.
Conformal mapping is an important means of solving a wide range of physical problems, such as those in fluid flow, aerodynamics, thermomechanics, electrostatics, and elasticity [2–4], because it helps to reduce complicated mathematical problems in complex geometry and 2D symmetry to simpler ones. Particularly, in electrostatics and transmission lines, conformal mapping has been used to estimate the potential, electric field, and capacitance [5–10]. The difference between conformal mapping and normal mapping is that the mapping function used in conformal mapping is the analytic and nonzero version of its derivative on the mapped region. Therefore, these transformations have following properties.
First, a harmonic function satisfying the Laplace’s equation is transformed into a harmonic function.
Second, with conformal mapping, the Dirichlet and Neumann boundary conditions remain unchanged in the transformed region. Third, the conformal function preserves the capacitances of the corresponding conductors. The second property of conformal mapping implies that, instead of solving a boundary value problem directly on the original plane, it can be transformed into a simpler problem on the mapped plane. The third property is the key to estimating capacitances where a two-dimensional region with an abnormal distribution of conductors or boundaries is transformed into a region where capacitance is known, for example, in parallel plates. A widely used conformal transformation for capacitance calculation is the Schwarz-Christoffel mapping [2,11].
The Schwartz-Christoffel transformation transforms the real axis of the original map (on the
Or
where
Previous studies have shown that the TEM wave mode is favored on the stripline and that the electric field along a transmission line satisfies the Laplace’s equation [1]. This implies that the electric field around a stripline has the properties of an electrostatic field. Therefore, conformal mapping can be used to determine the resonator capacitance.
Figure 3 shows the sample of electric field vectors distribution along two branches of the resonator from a side view (a), and front view (b). The tangential component of the electric field is zero at the symmetry plane ff′, called virtue ground. This is equivalent to placing an electric wall at ff′; in other words, the potential will be zero at the electric boundary. Therefore, each impedance that connects between two branches of the resonator can be considered as two halves connected to the virtual ground. Figure 3(c) shows the equivalent capacitance network for the resonator operated in an odd resonant mode. In this circuit,
The sequence for calculating mutual capacitance between two legs of the resonator in a rectangle is shown in Fig. 4 using the (a) original structure, (b) intermediate structure, upper half-plane, (c) intermediate structure with scaled axis, and (d) parallel plane capacitor. First, we use the Schwarz–Christoffel transformation, given in
The mapping equation from the
Because the angle at points
where
where
Constants
substituting the boundary conditions
Solving the equations in system 10, we obtained
and the position of
To obtain
and the solution is
With
In our work
The transformation from the upper-half plane
Now,
By substituting conditions
where
By substituting the boundary conditions of
where
The half-side capacitance
However, as mentioned above, the strip width of the resonator is not infinitely wide; therefore, the contribution of the fringe field in the plane normal to the strip line toward the capacitance must be considered. To estimate the edge effect in this case, we consider the capacitance per unit length between two finite parallel planes as shown in Fig. 5. A finite parallel plane capacitor shown in Fig. 5(a) with fringe field at two ends can be transformed to an infinite parallel plane capacitor with uniform electric field by applying conformal mapping method. Therefore, the capacitance obtained from the geometry in Fig. 5(d) includes the contribution of the edge effect of the parallel capacitor, as shown in Fig. 5(a). The contribution of the fringe field to the capacitance for this case can be determined from the different values of the capacitances of the infinite parallel capacitor and finite parallel capacitor.
Similarly, the capacitance of parallel capacitors is given as
where
We can determine the contribution of the capacitance per unit length of the stripline direction caused by the fringe field, as shown in Fig. 6, as
Applying
and the half-side capacitance
The discharge gap capacitance of the resonance can be given as
where
where
The left-half component
The sequence for calculating
where
The fringe effect (Fig. 9) is
where
The effective dielectric constant at the discharge gap is given as
where
Figure 10 shows the schematic of the strip to the ground capacitance in two dimensions perpendicular to the stripline direction and the sequential mapping of the half of the strip to ground capacitance by Schwarz–Christoffel mapping method. Similarly, the mapping function of the rectangle in the
and the mapping function from the upper half of the
where
The capacitance between one leg of the resonator and ground, as shown in Fig. 11, is
From the capacitance equivalence circuit for the resonator of MPSR operation in the odd-resonance mode shown in Fig. 3, the capacitance of one leg of the resonator can be given as
The characteristic impedance of each branch, as shown in Fig. 12, is
The formation of the discharge gap shifts the resonant frequency of this device to a lower frequency. There are two main reasons for this effect. First is the lengthening of the two legs of the resonator owing to the contribution of two strip segments at the two ends of the resonator. The amount of shift in resonant frequency in this case is
where
where
The second reason is the shifting of resonance frequency due to the contribution of the capacitance at the discharge gap
Thus, the resonant frequency of MPSR can be given as
The result of the dependency of the resonant frequency on the gap width for a fixed thickness of the stripline (100
In this paper, the capacitance and characteristic impedance of MPSR line were analyzed based on the conformal mapping method, and the evolution trend of the resonance frequency was investigated to understand the discharge matching condition. The calculation results are in good agreement with the results of the commercial CST microwave studio software. This paper contributes toward the understanding of the operation principle and optimization of MPSR.
This work was supported by the Chungnam National University (CNU).