Abstract
The novel technique of generalizing complex-to-complex impedance transformers (CCITs) with miniaturization is introduced in this paper. The generalized CCITs are designed based on conjugately characteristic-impedance transmission lines (CCITLs) along with the Meta-Smith charts (MSCs), resulting in convenient design equations. For illustration, a prototype of generalized CCITs is designed, simulated, and implemented with an asymmetric compact microstrip resonant cell (ACMRC), which is one of the CCITLs. The measurement results confirm that the proposed technique offers approximately 28.5% shorter in physical length compared to the CCIT designed using standard transmission lines.
1. Introduction
One of the fundamental radio-frequency modules of modern electronic devices is an impedance matching network (IMN) [1–3]. The simplest matching transformer is in the form of resistance-to-resistance transformers, i.e., quarter-wave transformers (QWTs). But the most general form is the complex impedance-matching transformer. Complex-to-complex impedance transformers (CCITs) are appeared in several papers [4–7], which can be implemented using transmission lines (TLs). In practice, TLs can be designed to be either symmetric, such as coplanar waveguide TLs and compact microstrip resonant cells (CMRCs) [8], or asymmetric such as asymmetric CMRCs (ACMRCs).
This paper proposes an alternative mathematical model of a generalized CCIT based on conjugate characteristic-impedance transmission lines (CCITLs) to achieve significant size reduction. Note that CCITLs have been introduced since 2004 [9–11]. In [9], the basic theory of CCITLs and the Meta-Smith charts (MSCs) were proposed, including their potential applications. In [10], quarter-wave-like transformers (QWLTs) based on CCITLs were successfully applied for miniaturized resistance-to-resistance transformers. Recently, a novel reactance-to-reactance transformer (RRT) based on CCITLs and MSCs has been proposed for size reduction [11].
In this paper, QWLTs and RRTs based on CCITLs are generalized for CCITs. It is found that the newly proposed CCIT can be designed using the MSCs and implemented using an ACMRC, where the CST Microwave Studio [12] is employed for simulations. After comprehensive review [13–18], the CCITs can be implemented using six approaches as follows: only one TL section, one TL and stubs, two TLs and stubs, three TLs and stubs, several TLs, and coupled TLs. As pointed out in [18], these methods cannot treat all possible complex termination impedances due to the lack of an available systematic design method, including unnecessarily complicated designs. Among the six approaches, the CCITs implemented with a single TL on each transformer are recommended due to their useful, simple, reliable, diverse applications and the possible implementation of all other conventional designs [18]. Thus, this paper is considered only one TL section implemented with an ACMRC.
The originality and benefit of the proposed CCIT technique are to generalize CCITs implemented with a single standard TL to overcome some limitations. When both standard TL and CCITL can be employed to implement CCITs using only one TL section for specific input and output impedances, it can be shown that the CCIT implemented using the CCITL can offer shorter in both physical and electrical lengths compared to those of the CCIT implemented using the standard TL. In addition, the former can provide a wider range of input and output impedances for complex impedance matching compared to that of the single standard TL method.
The rest of this paper is organized as follows. Section 2 illustrates the theory of conventional CCITs and proposed CCITs. Both simulated and measured results are demonstrated in Section 3, including discussions. Finally, conclusions are presented in Section 4.
2. Theoretical Background
2.1. CCITS Implemented Using a Standard TL
A CCIT implemented using a standard TL is shown in Figure 1(a). A standard TL of length , possessing the characteristic impedance (), propagation constant (), and electrical length (), is employed to match between a complex output impedance () and a complex input impedance (), where . In Figure 1(a), the output () and input () reflection coefficients can be expressed as [11].
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The complex impedance matching problem can be easily designed with the Smith chart, as shown in Figure 2. Figures 2(a) and 2(b) illustrate the Smith chart for the cases of and , respectively. Note that and in Figure 2 are the normalized and with respect to , respectively. In addition, and in Figure 2 are the angles associated with and in the complex reflection coefficient plane, respectively.
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In the case of , , and are related to the phase relationship as [11] where Equations (1) and (2) are employed to obtain Equation (3). Similarly, and are related to the phase relationship in the case of as [11]
It is obvious that Equations (3) and (4) are identical to each other. After rearranging Equation (3), of the CCIT can be found as
Note that is independent of θ. In Equation (5), exists if the following condition is satisfied:
However, it is implied from Equation (6) that this CCIT cannot match an arbitrary to an arbitrary in all cases. Once is known, of the CCIT can be obtained as where and are the normalized and with respect to , respectively.
2.2. Generalized CCITS Implemented Using a CCITL
The CCIT implemented using a CCITL is shown in Figure 1(b) to match between a complex output impedance ZL to a complex input impedance , where , and are the characteristic impedances, propagation constant, and electrical length of the CCITL, respectively. Note that can be expressed as [11] where is the absolute value of and ϕ is the argument of Note that ϕ is set in the range of . In Figure 1(b), and represent the load and input reflection coefficients for the CCITL, respectively, which can be expressed as [9]
In [9], the MSCs are employed to design the stub tuners and periodic TL structures. In this work, the MSCs are applied to design CCITs implemented using CCITLs. From the MSC viewpoint, there are 4 cases to consider as follows [11]:
Case 1: and (see Figure 3(a))
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Case 2: and °
Case 3: and
Case 4: and
In Figure 3(a), and are the normalized and with respect to , respectively. Note that and are real and imaginary parts of , respectively. Similar to the cases of the Smith chart in Section 2.1, and in all cases of MSCs above are also related via the phase relationship. For case 1, Equations (8)–(10) can be readily rearranged as [11].
After rearranging Equation (11) by considering only the magnitude for both sides of Equation (11), can be determined straightforwardly as
Note that depends on ϕ but is independent of . In addition, when (corresponding to the standard TL method), Equation (12) is reduced to Equation (5) as expected. Once is known via Equation (12), of the CCIT implemented using a CCITL can be expressed as where
Thus, Equations (12) and (13) are the design equations for the CCIT implemented using a CCITL. Note that Equation (12) requires that the term inside the square root, and must be nonnegative; i.e., the following conditions are satisfied:
It should be pointed out that, when , Equations (15) and (16) are reduced to Equation (6) as expected. Note that Equations (15) and (16) depend on ϕ, providing another degree of freedom and possibly leading to a wider range of and for complex impedance matching, compared to that of the standard TL method. For CCITs implemented using only one TL section, there is a forbidden region in the impedance domain, defined as an area where one TL cannot transform a complex impedance into another one [18–20]. Thus, the CCIT implemented using a CCITL can possibly reduce the forbidden region due to the reason explained above. However, similar to the CCIT in Section 2.1, this CCIT may not match an arbitrary to an arbitrary in all cases.
For cases 2 to 4, it is found that the CCIT design equations are still Equations (12) and (13). Note that the standard TL and the Smith Chart can be considered the special case of CCITLs and MSCs when [9]. As expected, it can be shown that Equations (12) and (13) are reduced to Equations (5) and (7), respectively.
3. Results and Discussions
For illustration, the CCIT implemented using a CCITL is designed to match a complex output impedance to a complex input impedance . In this case, ϕ is assumed to be -9.02°. Using Equation (12), is found to be 49.63 Ω. Figure 3(b) illustrates the MSC design for normalized and , corresponding to case 4. It is found from the MSC when that , , and , which are almost identical to the design parameters from Equations (9) and (10) ( and ) and Equation (13) () as expected.
The schematic diagrams of the CCITs are shown in Figures 4(a) and 4(b) for the CCIT implemented using a standard TL and a CCITL based on an ACMRC, respectively. For this load , the resistor of 60.4 Ω and capacitor of 1 pF are connected in series to obtain at the operating frequency of 2 GHz. For the CCIT implemented using a standard TL (called the 1st CCIT), Equations (5) and (7) are employed to determine and in Figure 4(a), respectively. It is found that and . For the CCIT implemented using a CCITL (called the 2nd CCIT), Equations (12) and (13) are employed to determine and in Figure 4(b), respectively. For a given , it is found that and . To implement both CCITs, the CST Microwave Studio is employed to simulate desired microstrip structures on the FR-4 substrate (dielectric constant of 4.3, loss tangent of 0.02, and substrate thickness of 1.6 mm), as illustrated in Figure 5, where their dimension is given in mm. It is clearly seen that the CCITL approach offers approximately 100 MHz wider bandwidth, as shown in Figure 6, with a 20% shorter in physical length compared to that of the standard TL approach. Note that the input reflection coefficient in Figure 6 is defined as where is the input impedance of the considered CCIT and is the desired input impedance of the CCIT design at the operating frequency of 2 GHz, which is equal to 15.9-j23.3 Ω in this case.
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Finally, the CCITs are integrated with the shunt stub tuning circuit operating at 2 GHz, as shown in Figure 7, for both standard TL and CCITL implementations. The CST Microwave Studio will also be used to appropriately adjust the formerly designed microstrip structures of both CCIT implementations (called the modified 1st and 2nd CCITs) due to coupling effects after integrating the CCITs into the shunt stub tuning circuit. The optimized designs are presented with dimensions in Figures 8(a) and 8(b) for the CCITs using a standard TL and a CCITL implemented by an ACMRC, respectively. In addition, the insertions in Figures 8(a) and 8(b) illustrate CCIT prototypes implemented using a standard TL and a CCITL implemented by an ACMRC, respectively. Note that the proposed design offers approximately 28% shorter in physical length compared to that of the CCIT implemented using a standard TL. Furthermore, the comparison between the CCITs implemented using a standard TL and a CCITL based on an ACMRC in terms of electrical length, physical length, and bandwidth is summarized in Table 1 for clarity.
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Figure 9 also illustrates the plots of the magnitude of the input reflection coefficient () for both simulation and measurement of both CCITs implemented using a standard TL and a CCITL based on an ACMRC. It is clearly seen that the measurement results offer better at -27.5 dB and -34.3 dB compared to the simulation results at -22 dB and -25 dB of the CCIT implemented using the standard TL and the CCITL, respectively. However, the measurement results also show a slightly shifted operating frequency from 2 GHz to 2.1 GHz and 2.2 GHz for the CCITs implemented using the standard TL and the CCITL, respectively.
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In addition, Table 2 illustrates the comparison between the CCITs implemented using a standard TL in [20] and a CCITL using for and , where five values of are considered and the TL parameters are given for both the standard TL and the CCITL for each . From Table 2, it is found that, for specific and , the CCIT implemented using the CCITL can offer significantly shorter electrical lengths compared to those of the CCIT implemented using the standard TL.
4. Conclusions
The generalized CCITs implemented using a CCITL based on an ACMRC are proposed in this paper, where they are systematically designed using the MSCs, resulting in convenient design equations. It is shown that the proposed method can miniaturize both the physical and electrical sizes of desired CCITs. In addition, the simulated and measured results confirm that the CCIT implemented using the CCITL can indeed offer shorter in both physical and electrical lengths, compared to those of the CCIT implemented using the standard TL. Furthermore, the CCIT implemented using a CCITL can possibly reduce the forbidden region, compared to that of the CCIT implemented using a single standard TL due to an extra degree-of-freedom parameter ϕ of the CCITL model.
Data Availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was funded by the King Mongkut’s University of Technology North Bangkok, contract no. KMUTNB-65-KNOW-12.