Patent Publication Number: US-9853340-B2

Title: Coupled slow-wave transmission lines

Description:
RELATED APPLICATIONS 
     The present application claims the benefit of U.S. Provisional Patent Application No. 62/074,457, filed Nov. 3, 2014, the disclosure of which is incorporated herein by reference in its entirety. 
     The present application claims priority to and is a continuation-in-part of U.S. patent application Ser. No. 14/921,218, filed Oct. 23, 2015, entitled “SLOW-WAVE TRANSMISSION LINE FORMED IN A MULTI-LAYER SUBSTRATE,” which claims priority to U.S. Provisional Patent Application No. 62/074,457, filed Nov. 3, 2014, the disclosures of which are incorporated herein by reference in their entireties. 
    
    
     FIELD OF THE DISCLOSURE 
     The present disclosure relates to transmission lines, and specifically to transmission lines configured to transmit slow-wave signals. 
     BACKGROUND 
     Mobile computing devices, such as mobile phones and computer tablets, continue to employ designs focused on decreasing size requirements. The trend toward miniaturization of mobile computing devices requires the use of smaller internal components. Tunable filters are one such internal component that affect the overall size of a mobile computing device. One way to construct a tunable filter is through the use of transmission lines. Notably, tunable filters require slower wave signals, and thus, transmission lines used to construct tunable filters should be designed to transmit wave signals at compatible speeds. Three factors that affect the speed at which transmission lines transmit wave signals are size, permittivity (∈), and permeability (μ). 
       FIG. 1  illustrates an exemplary transmission line  10  disposed along a ground plane  12 . The transmission line  10  is separated from the ground plane  12  by a distance (D), wherein, as a non-limiting example, the distance (D) may include a dielectric layer (not shown). Further, the transmission line  10  is employed using a low cost, low permittivity (∈ low ) material. The speed at which a wave signal is transmitted (the velocity factor (Vf) (not shown)) by the transmission line  10  is inversely proportional to the square root of the relative permittivity (Vf=1/√∈(r)). Thus, the ∈ low  material causes the transmission line  10  to have a higher Vf as compared to transmission lines constructed using a higher permittivity material. To delay a transmitted wave signal in light of the higher Vf, the transmission line  10  is designed with a longer length (L long ) so as to require a transmitted wave signal to travel a further distance. Additionally, the transmission line  10  is designed with a wider width (W wide ) to reduce loss. Therefore, to transmit a wave signal at a speed that is compatible with a tunable filter while achieving low loss, the transmission line  10  requires a larger area to overcome the higher Vf associated with the ∈ low  material. However, the larger area of the transmission line  10  may not be desirable for tunable filters implemented in mobile computing devices with limited area requirements. 
     To transmit a wave signal at a speed that is compatible with a tunable filter while requiring less area than the transmission line  10 , a transmission line may be constructed using a high permittivity ∈ high  material. In this manner,  FIG. 2  illustrates an exemplary transmission line  14  employed using a high cost, ∈ high  material disposed along a ground plane  16 . Notably, the transmission line  14  is separated from the ground plane  16  by a distance (D). The ∈ high  material causes the transmission line  14  to have a lower Vf as compared to transmission lines constructed using a ∈ low  material, such as the transmission line  10 . Because the transmission line  14  has a lower Vf, a transmitted wave signal does not need to be delayed by employing a longer length (L long ), allowing the transmission line  14  to be designed with a shorter length (L short ). However, the transmission line  14  is also designed with narrower width (W narrow ), which causes increased loss. Thus, although the transmission line  14  consumes less area than the transmission line  10 , the transmission line  14  incurs greater loss and requires a higher cost material. 
     Therefore, it would be advantageous to employ a transmission line designed to transmit wave signals at speeds compatible with tunable filters while achieving reduced area, costs, and loss. 
     SUMMARY 
     The present disclosure relates to coupled slow-wave transmission lines. In this regard, a transmission line structure is provided. The transmission line structure includes a first undulating signal path formed from first loop structures. The transmission line structure also includes a second undulating signal path formed from second loop structures. The second undulating signal path is disposed alongside of the first undulating signal path. Further, a first ground structure is disposed above or below either one or both of the first undulating signal path and the second undulating signal path. In this manner, based on factors such as, but not limited to, geometry of the first and second undulating signal paths and the distance between the first and second undulating signal paths, the first and second undulating signal paths may magnetically couple to one another. Such coupling may allow the transmission line structure to be used in a filter structure. 
     According to one embodiment, a transmission line structure is disclosed. The transmission line structure comprises a first undulating signal path comprising first loop structures. The transmission line structure further comprises a second undulating signal path comprising second loop structures and disposed alongside of the first undulating signal path. The transmission line structure further comprises a first ground structure disposed above or below at least one of the first undulating signal path and the second undulating signal path. 
     Those skilled in the art will appreciate the scope of the disclosure and realize additional aspects thereof after reading the following detailed description in association with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings incorporated in and forming a part of this specification illustrate several aspects of the disclosure, and together with the description serve to explain the principles of the disclosure; 
         FIG. 1  is a diagram of an exemplary transmission line; 
         FIG. 2  is a diagram of an exemplary transmission line with a shorter length and narrower width; 
         FIG. 3  is a cross-sectional diagram of an exemplary multi-layer laminate printed circuit board (PCB); 
         FIGS. 4A-4C  are diagrams of exemplary slow-wave transmission lines with an undulating signal path; 
         FIG. 5A  is a cross-sectional diagram of an exemplary slow-wave transmission line with an undulating signal path; 
         FIG. 5B  is a cross-sectional diagram of the slow-wave transmission line with the undulating signal path in  FIG. 5A  disposed in a multi-layer laminate PCB; 
         FIG. 6A  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a T-shaped pattern; 
         FIG. 6B  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a P-shaped pattern; 
         FIG. 7A  is a cross-sectional diagram of the slow-wave transmission line disposed in the T-shaped pattern in  FIG. 6A ; 
         FIG. 7B  is a cross-sectional diagram of the slow-wave transmission line disposed in the P-shaped pattern in  FIG. 6B ; 
         FIG. 8A  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a U-shaped pattern and employs I-shaped ground bars; 
         FIG. 8B  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a T-shaped pattern and employs I-shaped ground bars; 
         FIG. 8C  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a P-shaped pattern and employs I-shaped ground bars; 
         FIG. 8D  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a T-shaped pattern and employs T-shaped ground bars; 
         FIG. 8E  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a P-shaped pattern and employs L-shaped ground bars; 
         FIG. 9A  is a cross-sectional diagram of the slow-wave transmission line disposed in the T-shaped pattern that employs the T-shaped ground bars; 
         FIG. 9B  is a cross-sectional diagram of the slow-wave transmission line disposed in the P-shaped pattern that employs the L-shaped ground bars; 
         FIG. 10A  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a double-L-shaped pattern; 
         FIG. 10B  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a double-T-shaped pattern; 
         FIG. 10C  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a polygonal-shaped pattern; 
         FIG. 10D  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line is disposed in a rounded pattern; 
         FIG. 11  is a diagram of an exemplary slow-wave transmission line employing a shield structure along an undulating signal path; 
         FIG. 12  is a diagram of an exemplary double-folded slow-wave transmission line with an undulating signal path; 
         FIG. 13  is a diagram of an exemplary slow-wave transmission line with an undulating signal path employed as a discrete device mounted on a PCB; 
         FIG. 14A  is a diagram of an exemplary solenoid-type slow-wave transmission line with an undulating signal path disposed around a ground structure; 
         FIG. 14B  is a diagram of an exemplary solenoid-type slow-wave transmission line with an undulating signal path disposed between a first and second ground structure; 
         FIG. 15A  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line includes insulator layers formed from a material having a permittivity greater than a certain value; 
         FIG. 15B  is a diagram of an exemplary slow-wave transmission line with an undulating signal path, wherein the slow-wave transmission line includes insulator layers formed from a material having a permeability greater than a certain value; 
         FIG. 16A  is a diagram of an exemplary slow-wave transmission line with an undulating signal path formed using integrated circuit (IC) and laminate processes; 
         FIG. 16B  is a diagram of an exemplary slow-wave transmission line with an undulating signal path formed using IC and laminate processes; 
         FIG. 17  is a diagram of an exemplary slow-wave transmission line illustrating exemplary magnetic fields induced by an exemplary current flow; 
         FIG. 18A  is a top-level diagram of a transmission line structure that includes a first undulating signal path a distance from and aligned with a second undulating signal path; 
         FIG. 18B  is a top-level diagram of a transmission line structure that includes a first undulating signal path another distance from and aligned with a second undulating signal path; 
         FIG. 19A  is a top-level diagram of a transmission line structure that includes a first undulating signal path a distance from and not aligned with a second undulating signal path; 
         FIG. 19B  is a top-level diagram of a transmission line structure that includes a first undulating signal path another distance from and not aligned with a second undulating signal path; 
         FIG. 20A  is a top-level diagram of a transmission line structure that includes a first undulating signal path aligned with a second undulating signal path, wherein a wall structure is disposed between the first and second undulating signal paths and perpendicular to a first ground structure; 
         FIG. 20B  is a cross-sectional diagram of the transmission line structure in  FIG. 20A ; 
         FIG. 21A  is a top-level diagram of a transmission line structure that includes a first undulating signal path aligned with a second undulating signal path, wherein another wall structure is disposed between the first and second undulating signal paths and perpendicular to a first ground structure; 
         FIG. 21B  is a cross-sectional diagram of the transmission line structure in  FIG. 21A ; 
         FIG. 22A  is a diagram of a transmission line structure that includes a first undulating signal path and a second undulating signal path magnetically coupled by a floating loop structure; 
         FIG. 22B  is a diagram of a transmission line structure that includes a first undulating signal path and a second undulating signal path magnetically coupled by another floating loop structure; 
         FIG. 23A  is a diagram of a transmission line structure that includes a first undulating signal path and a second undulating signal path magnetically coupled by a floating loop structure controlled by a switch; 
         FIG. 23B  is a diagram of a transmission line structure that includes a first undulating signal path and a second undulating signal path magnetically coupled by another floating loop structure controlled by a switch; 
         FIG. 24  is a diagram of a transmission line structure that includes a first undulating signal path and a second undulating signal path magnetically coupled by a floating ring structure; and 
         FIG. 25  is a diagram of a transmission line structure that includes a first undulating signal path and a second undulating signal path magnetically coupled by first and second plate structures. 
     
    
    
     DETAILED DESCRIPTION 
     The embodiments set forth below represent the necessary information to enable those skilled in the art to practice the disclosure and illustrate the best mode of practicing the disclosure. Upon reading the following description in light of the accompanying drawings, those skilled in the art will understand the concepts of the disclosure and will recognize applications of these concepts not particularly addressed herein. It should be understood that these concepts and applications fall within the scope of the disclosure and the accompanying claims. 
     It will be understood that, although the terms first, second, etc. may be used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first element could be termed a second element, and, similarly, a second element could be termed a first element, without departing from the scope of the present disclosure. As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items. 
     Relative terms such as “below” or “above,” or “upper” or “lower,” or “horizontal” or “vertical” may be used herein to describe a relationship of one element, layer, or region to another element, layer, or region as illustrated in the Figures. It will be understood that these terms and those discussed above are intended to encompass different orientations of the device in addition to the orientation depicted in the Figures. 
     The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the disclosure. As used herein, the singular forms “a,” “an,” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises,” “comprising,” “includes,” and/or “including” when used herein specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. 
     Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs. It will be further understood that terms used herein should be interpreted as having a meaning that is consistent with their meaning in the context of this specification and the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein. 
     The present disclosure relates to coupled slow-wave transmission lines. In this regard, a transmission line structure is provided. The transmission line structure includes a first undulating signal path formed from first loop structures. The transmission line structure also includes a second undulating signal path formed from second loop structures. Notably, the second undulating signal path is disposed alongside of the first undulating signal path. Further, a first ground structure is disposed above or below either one or both of the first undulating signal path and the second undulating signal path. In this manner, based on factors such as, but not limited to, geometry of the first and second undulating signal paths and the distance between the first and second undulating signal paths, the first and second undulating signal paths may magnetically couple to one another. Such coupling may allow the transmission line structure to be used in a filter structure. 
     Before discussing details of the slow-wave transmission line for transmitting slow-wave signals beginning in  FIG. 4A , details of a multi-layer laminate printed circuit board (PCB) are first discussed.  FIG. 3  illustrates an exemplary multi-layer laminate PCB  18  employing metal layers M 1 -M 5  alternating with dielectric layers D 1 -D 4 . Each of the metal layers M 1 -M 5  is constructed of a conductive material. Further, each dielectric layer D 1 -D 4  is constructed of a substrate material having a particular dielectric value. To form the multi-layer laminate PCB  18 , vias (not shown) used to electrically connect corresponding metal layers M 1 -M 5  are drilled in corresponding dielectric layers D 1 -D 4  and clad or plated with a conductive material. Additionally, the metal layers M 1 -M 5  are disposed in an alternating manner with the dielectric layers D 1 -D 4 , wherein circuit traces are etched into each metal layer M 1 -M 5 , or alternatively, circuit traces are metal-plated and have dielectric material pressed onto the metal. The metal and dielectric layers M 1 -M 5 , D 1 -D 4  are connected using a lamination process to form the multi-layer laminate PCB  18 . In this manner, the multi-layer laminate PCB  18  may support circuits designed to be fabricated in a multi-layer substrate. 
       FIG. 4A  illustrates an exemplary slow-wave transmission line  22  with an undulating signal path  24  formed in a multi-layer substrate. The undulating signal path  24  in the slow-wave transmission line  22  employs loop structures  26 ( 1 ),  26 ( 2 ). The loop structure  26 ( 1 ) includes via structures  28 ( 1 ),  28 ( 2 ) connected by an intra-loop trace  30 ( 1 ). Similarly, the loop structure  26 ( 2 ) includes via structures  28 ( 3 ),  28 ( 4 ) connected by an intra-loop trace  30 ( 2 ). The undulating signal path  24  further includes an inter-loop trace  32  that connects the two loop structures  26 ( 1 ),  26 ( 2 ). Constructing the slow-wave transmission line  22  with the undulating signal path  24  in this manner increases the distance that a slow-wave signal must travel through the slow-wave transmission line  22  as compared to a transmission line employing a straight, non-undulating signal path having a similar length. Requiring the slow-wave signal to travel an increased distance delays the slow-wave signal so as to be more compatible with speeds required by tunable filters without incurring an increase in area. 
     Additionally, constructing the slow-wave transmission line  22  as described above causes each loop structure  26 ( 1 ),  26 ( 2 ) to form a corresponding loop inductance  34 ( 1 ),  34 ( 2 ). The loop inductance  34 ( 1 ) is formed between the via structures  28 ( 1 ),  28 ( 2 ) and the intra-loop trace  30 ( 1 ), while the loop inductance  34 ( 2 ) is formed between the via structures  28 ( 3 ),  28 ( 4 ) and the intra-loop trace  30 ( 2 ). Further, the slow-wave transmission line  22  includes a first ground structure  36  disposed along the undulating signal path  24 , thus forming a first distributed capacitance  38  between the undulating signal path  24  and the first ground structure  36 . Although the first ground structure  36  is substantially planar in this embodiment, other embodiments may employ the first ground structure  36  in alternative shapes. 
     Resonance generated by an inductance-capacitance (LC) network formed by the loop inductances  34 ( 1 ),  34 ( 2 ), and the first distributed capacitance  38  increases the effective dielectric constant (i.e., increases the relative permittivity ∈(r)) of the slow-wave transmission line  22 . Such an increase in relative permittivity ∈(r) reduces the corresponding velocity factor (Vf) (Vf=1/√∈(r)), thus reducing the speed of the slow-wave signal. Therefore, the slow-wave transmission line  22  is designed to transmit slow-wave signals at speeds compatible with tunable filters by forcing the slow-wave signal to travel a further distance as described above, as well as by slowing down the slow-wave signal using the LC network. 
     Notably, the slow-wave transmission line  22  may achieve the described delay and speed reduction of the slow-wave signal even when employing a low cost, low permittivity (∈ low ) material having a high velocity factor (Vf). Thus, the slow-wave transmission line  22  may be designed to achieve such benefits while avoiding increased cost associated with high cost, high permittivity (∈ high ) material. 
     Additionally, in this embodiment, the loop structure  26 ( 1 ) is constructed so that the via structure  28 ( 1 ) is disposed within a lateral pitch (L P ) of the via structure  28 ( 2 ), wherein the lateral pitch (L P ) is less than a height (H) of each via structure  28 ( 1 ),  28 ( 2 ). The loop structure  26 ( 2 ) is similarly constructed so that the via structure  28 ( 3 ) is disposed within the lateral pitch (L P ) of the via structure  28 ( 4 ), wherein the lateral pitch (L P ) is less than a height (H) of each via structure  28 ( 3 ),  28 ( 4 ). Notably, each via structure  28 ( 1 )- 28 ( 4 ) has a corresponding width (W) and depth (DPT). Constructing the loop structures  26 ( 1 ),  26 ( 2 ) in this manner increases the corresponding loop inductances  34 ( 1 ),  34 ( 2 ), thus allowing the LC network in the slow-wave transmission line  22  to further reduce the speed at which the slow-wave signal is transmitted. 
     While the slow-wave transmission line  22  in  FIG. 4A  is designed to delay and reduce the speed of a slow-wave signal as previously described, alternative embodiments that achieve reduced loss may be employed. In this manner,  FIG. 4B  illustrates an exemplary slow-wave transmission line  22 ′ with an undulating signal path  24 ′. The slow-wave transmission line  22 ′ includes certain common components with the slow-wave transmission line  22  in  FIG. 4A . Such common components that have an associated number “X” in  FIG. 4A  are denoted by a number “X′” in  FIG. 4B , and thus will not be re-described herein. 
     The slow-wave transmission line  22 ′ includes a loop structure  26 ′( 1 ) constructed with via structures  28 ′( 1 ),  28 ′( 2 ) connected by an intra-loop trace  30 ′( 1 ). Similarly, the slow-wave transmission line  22 ′ includes a loop structure  26 ′( 2 ) constructed with via structures  28 ′( 3 ),  28 ′( 4 ) connected by an intra-loop trace  30 ′( 2 ). Notably, the via structures  28 ′( 1 )- 28 ′( 4 ) are elongated via structures, wherein a width (W′) of each via structure  28 ′( 1 )- 28 ′( 4 ) is approximately equal to at least twice a depth (DPT′) of each via structure  28 ′( 1 )- 28 ′( 4 ), as opposed to the width (W) that is approximately equal to the depth (DPT) of each via structure  28 ( 1 )- 28 ( 4 ) in  FIG. 4A . Because the via structures  28 ′( 1 )- 28 ′( 4 ) employ a width (W′) approximately equal to at least twice the depth (DPT′), the corresponding intra-loop traces  28 ′( 1 ),  28 ′( 2 ) have a substantially similar width (W′). Further, a resistance (R) of a conductive material is inversely proportional to area (A), and thus, the larger width (W′) of the via structures  28 ′( 1 )- 28 ′( 4 ) and the intra-loop traces  28 ′( 1 ),  28 ′( 2 ) reduces the resistance (R) of the slow-wave transmission line  22 ′ as compared to that of the slow-wave transmission line  22  in  FIG. 4A . In this manner, the lower resistance (R) reduces the loss experienced by a slow-wave signal transmitted through the slow-wave transmission line  22 ′. 
     Similarly,  FIG. 4C  illustrates an exemplary slow-wave transmission line  22 ″ with an undulating signal path  24 ″. The slow-wave transmission line  22 ″ also includes certain common components with the slow-wave transmission line  22  in  FIG. 4A . Such common components that have an associated number “X” in  FIG. 4A  are denoted by a number “X″” in  FIG. 4C , and thus will not be re-described herein. In this manner, via structures  28 ″( 1 )- 28 ″( 4 ) are elongated via structures, wherein a width (W″) of each via structure  28 ″( 1 )- 28 ″( 4 ) is approximately equal to at least five times a depth (DPT″) of each via structure  28 ″( 1 )- 28 ″( 4 ). Further, because the via structures  28 ″( 1 )- 28 ″( 4 ) employ a width (W″) approximately equal to at least five times the depth (DPT″), corresponding intra-loop traces  30 ″( 1 ),  30 ″( 2 ) have a substantially similar width (W″). Thus, the larger width (W″) of the via structures  28 ″( 1 )- 28 ″( 4 ) and the intra-loop traces  30 ″( 1 ),  30 ″( 2 ) reduces the resistance (R), and hence, the loss, of the slow-wave transmission line  22 ″ as compared to that of the slow-wave transmission lines  22 ,  22 ′ in  FIGS. 4A, 4B , respectively. 
       FIG. 5A  illustrates a cross-sectional diagram of an exemplary slow-wave transmission line  40  similar to the slow wave transmission lines  22 ,  22 ′, and  22 ″ of  FIGS. 4A-4C . The slow-wave transmission line  40  includes a first ground structure  42  disposed in a first metal layer (M 1 ). A first dielectric layer (D 1 ) is disposed above the first ground structure  42 . Additionally, an undulating signal path  44  is included above the D 1  layer. In this manner, the undulating signal path  44  includes loop structures  46 ( 1 ),  46 ( 2 ). The loop structure  46 ( 1 ) includes an intra-loop trace  48 ( 1 ) disposed in a fourth metal layer (M 4 ) that connects via structures  50 ( 1 ),  50 ( 2 ). The via structure  50 ( 1 ) employs an inter-via trace  52 ( 1 ) disposed in a third metal layer (M 3 ) that connects vias  54 ( 1 ),  54 ( 2 ) disposed in a second and third dielectric layer (D 2 , D 3 ), respectively. The via structure  50 ( 2 ) employs an inter-via trace  52 ( 2 ) disposed in M 3  that connects vias  54 ( 3 ),  54 ( 4 ) disposed in D 2 , D 3 , respectively. The loop structure  46 ( 2 ) includes an intra-loop trace  48 ( 2 ) disposed in M 4  that connects via structures  50 ( 3 ),  50 ( 4 ). The via structure  50 ( 3 ) employs an inter-via trace  52 ( 3 ) disposed in M 3  that connects vias  54 ( 5 ),  54 ( 6 ) disposed in D 3 , D 2 , respectively. The via structure  50 ( 4 ) includes an inter-via trace  52 ( 4 ) disposed in M 3  that connects vias  54 ( 7 ),  54 ( 8 ) disposed in D 3 , D 2 , respectively. 
     Further, the slow-wave transmission line  40  includes an intra-loop trace  56  disposed in a second metal layer (M 2 ) that connects the loop structures  46 ( 1 ),  46 ( 2 ). Segment traces  58 ,  60  disposed in M 2  are connected to the vias  54 ( 1 ),  54 ( 8 ), respectively, to complete the undulating signal path  44 . Notably, this embodiment includes a second ground structure  62  disposed in a fifth metal layer (M 5 ) above a fourth dielectric layer (D 4 ) along the undulating signal path  44  opposite of the first ground structure  42 . As described in further detail below, the second ground structure  62  forms a second distributive capacitance (not shown) between the undulating signal path  44  and the second ground structure  62 . 
       FIG. 5B  illustrates the slow-wave transmission line  40  of  FIG. 5A  disposed in an exemplary multi-layer laminate PCB  64  similar to the multi-layer laminate PCB  18  of  FIG. 3 . Notably, the slow-wave transmission line  40  is disposed in a U-shaped pattern, wherein the loop structures  46 ( 1 ),  46 ( 2 ) are disposed adjacent to one another and each loop structure  46 ( 1 ),  46 ( 2 ) is employed with a substantially equal size and U-shape. Further, in addition to loop inductances  66 ( 1 ),  66 ( 2 ) formed within the loop structures  46 ( 1 ),  46 ( 2 ), respectively, a loop inductance  66 ( 3 ) is formed between the loop structures  46 ( 1 ),  46 ( 2 ). Because the loop structures  46 ( 1 ),  46 ( 2 ) are disposed adjacent to one another and are substantially the same size, the loop inductance  66 ( 3 ) is substantially equal to each of the loop inductances  66 ( 1 ),  66 ( 2 ). 
     Further, a first distributive capacitance  68  is formed between the first ground structure  42  and the undulating signal path  44 , and a second distributive capacitance  70  is formed between the second ground structure  62  and the undulating signal path  44 . Intra-loop capacitances  72 ( 1 ),  72 ( 2 ) are formed between the via structures  50 ( 1 ),  50 ( 2 ) and  50 ( 3 ),  50 ( 4 ), respectively, and an inter-loop capacitance  74  is formed between the via structure  50 ( 2 ) and the via structure  50 ( 3 ). Thus, the first and second distributive capacitances  68 ,  70 , the intra-loop capacitances  72 ( 1 ),  72 ( 2 ), and the inter-loop capacitance  74  combine with the loop inductances  66 ( 1 ),  66 ( 2 ), and  66 ( 3 ) to form an LC network. In this manner, the slow-wave transmission line  40  is designed to transmit slow-wave signals at speeds compatible with tunable filters by forcing the slow-wave signal to travel a further distance, as well as by slowing down the slow-wave signal using the LC network. 
     In addition to the U-shaped slow-wave transmission line  40  in  FIGS. 5A, 5B , other embodiments may employ slow-wave transmission lines in alternative shapes and achieve similar functionality. In this manner,  FIG. 6A  illustrates an exemplary slow-wave transmission line  76  disposed in a T-shaped pattern. The slow-wave transmission line  76  includes an undulating signal path  78  that includes loop structures  80 ( 1 ),  80 ( 2 ) connected by an inter-loop trace  82 . Notably, the T-shaped pattern is also formed between the loop structures  80 ( 1 ),  80 ( 2 ). Because the loop structure  80 ( 1 ) is disposed in the T-shaped pattern, the loop structure  80 ( 1 ) includes four via structures  84 ( 1 )- 84 ( 4 ) and three intra-loop traces  86 ( 1 )- 86 ( 3 ). In this manner, the via structure  84 ( 1 ) is connected to the via structures  84 ( 2 ),  84 ( 3 ) by the intra-loop traces  86 ( 1 ),  86 ( 2 ), respectively. Further, the via structure  84 ( 2 ) is connected to the via structure  84 ( 4 ) by the intra-loop trace  86 ( 3 ). Similarly, the loop structure  80 ( 2 ) includes four via structures  84 ( 5 )- 84 ( 8 ) and three intra-loop traces  86 ( 4 )- 86 ( 6 ). The via structure  84 ( 5 ) is connected to the via structures  84 ( 6 ),  84 ( 7 ) by the intra-loop traces  86 ( 4 ),  86 ( 5 ), respectively. Further, the via structure  84 ( 6 ) is connected to the via structure  84 ( 8 ) by the intra-loop trace  86 ( 6 ). First and second ground structures  88 ,  90  are also included in the slow-wave transmission line  76 . 
     Further,  FIG. 6B  illustrates an exemplary slow-wave transmission line  92  disposed in a P-shaped pattern. The slow-wave transmission line  92  includes an undulating signal path  94  having loop structures  96 ( 1 ),  96 ( 2 ) connected by an inter-loop trace  98 . Notably, the P-shaped pattern is also formed between the loop structures  96 ( 1 ),  96 ( 2 ). Because the loop structure  96 ( 1 ) is disposed in the P-shaped pattern, the loop structure  96 ( 1 ) includes three via structures  100 ( 1 )- 100 ( 3 ) and two intra-loop traces  102 ( 1 ),  102 ( 2 ). In this manner, the via structure  100 ( 1 ) is connected to the via structure  100 ( 2 ) by the intra-loop trace  102 ( 1 ). Further, the via structure  100 ( 2 ) is connected to the via structure  100 ( 3 ) by the intra-loop trace  102 ( 2 ). Similarly, the loop structure  96 ( 2 ) includes three via structures  100 ( 4 )- 100 ( 6 ) and two intra-loop traces  102 ( 3 ),  102 ( 4 ). The via structure  100 ( 4 ) is connected to the via structure  100 ( 5 ) by the intra-loop trace  102 ( 3 ). Further, the via structure  100 ( 5 ) is connected to the via structure  100 ( 6 ) by the intra-loop trace  102 ( 4 ). First and second ground structures  104 ,  106  are also included in the slow-wave transmission line  92 . As described in detail below, the T-shaped slow-wave transmission line  76  and the P-shaped slow-wave transmission line  92  are configured to transmit slow-wave signals with similar advantages as those provided by the U-shaped slow-wave transmission line  40  in  FIGS. 5A and 5B . 
       FIG. 7A  is a cross-sectional diagram of the slow-wave transmission line  76  disposed in the T-shaped pattern in  FIG. 6A . The slow-wave transmission line  76  is disposed in a multi-layer substrate similar to the slow-wave transmission line  40  in  FIG. 5B . Thus, the via structures  84 ( 1 )- 84 ( 4 ) in the loop structure  80 ( 1 ) and the via structures  84 ( 5 )- 84 ( 8 ) in the loop structure  80 ( 2 ) are constructed using vias and intra-via segments as described with reference to the slow-wave transmission line  40 , and thus will not be re-described herein. 
     Additionally, loop inductances  108 ( 1 ),  108 ( 2 ) are formed within the loop structures  80 ( 1 ),  80 ( 2 ), respectively. A loop inductance  108 ( 3 ) is also formed between the loop structures  80 ( 1 ),  80 ( 2 ). A first distributed capacitance  110  is formed between the first ground structure  88  and the undulating signal path  78 . A second distributed capacitance  112  is formed between the second ground structure  90  and the undulating signal path  78 . Further, intra-loop capacitances  114 ( 1 ),  114 ( 2 ) are formed between the via structures  84 ( 3 ),  84 ( 4 ) and  84 ( 7 ),  84 ( 8 ), respectively. An inter-loop capacitance  116  is formed between the loop structures  80 ( 1 ),  80 ( 2 ). Thus, the loop inductances  108 ( 1 )- 108 ( 3 ), the first and second distributed capacitances  110 ,  112 , the intra-loop capacitances  114 ( 1 )- 114 ( 2 ), and the inter-loop capacitance  116  combine to form an LC network. In this manner, the slow-wave transmission line  76  is designed to transmit slow-wave signals at speeds compatible with tunable filters by forcing the slow-wave signal to travel a further distance, as well as by slowing down the slow-wave signal using the LC network. 
       FIG. 7B  is a cross-sectional diagram of the slow-wave transmission line  92  disposed in the P-shaped pattern in  FIG. 6B . The slow-wave transmission line  92  is disposed in a multi-layer substrate similar to the slow-wave transmission line  40  in  FIG. 5B . Thus, the via structures  100 ( 1 )- 100 ( 3 ) in the loop structure  96 ( 1 ) and the via structures  100 ( 4 )- 100 ( 6 ) in the loop structure  96 ( 2 ) are constructed using vias and intra-via segments as described with reference to the slow-wave transmission line  40 , and thus will not be re-described herein. 
     Additionally, loop inductances  118 ( 1 ),  118 ( 2 ) are formed within the loop structures  96 ( 1 ),  96 ( 2 ), respectively. A loop inductance  118 ( 3 ) is also formed between the loop structures  96 ( 1 ),  96 ( 2 ). A first distributed capacitance  120  is formed between the first ground structure  104  and the undulating signal path  94 . A second distributed capacitance  122  is formed between the second ground structure  106  and the undulating signal path  94 . Further, intra-loop capacitances  124 ( 1 ),  124 ( 2 ) are formed between the via structures  100 ( 1 ),  100 ( 3 ) and  100 ( 4 ),  100 ( 6 ), respectively. An inter-loop capacitance  126  is formed between the loop structures  96 ( 1 ),  96 ( 2 ). Thus, the loop inductances  118 ( 1 )- 118 ( 3 ), the first and second distributed capacitances  120 ,  122 , the intra-loop capacitances  124 ( 1 )- 124 ( 2 ), and the inter-loop capacitance  126  combine to form an LC network. In this manner, the slow-wave transmission line  92  is designed to transmit slow-wave signals at speeds compatible with tunable filters by forcing the slow-wave signal to travel a further distance, as well as by slowing down the slow-wave signal using the LC network. 
     Notably, impedance can vary within a slow-wave transmission line due to its structure. Thus, it may be desirable to better control the impedance within a slow-wave transmission line. In this manner, ground bars connected to corresponding ground structures may be disposed within and between loop structures of a slow-wave transmission line to help regulate the impedance throughout the structure. 
       FIG. 8A  illustrates an exemplary U-shaped slow-wave transmission line  40 ′ that employs certain common components with the slow-wave transmission line  40  in  FIGS. 5A, 5B . Such common components that have an associated number “X” in  FIGS. 5A, 5B  are denoted by a number “X′” in  FIG. 8A , and thus will not be re-described herein. In this manner, the slow-wave transmission line  40 ′ includes loop structures  46 ′( 1 ),  46 ′( 2 ), as well as first and second ground structures  42 ′,  62 ′. Further, the slow-wave transmission line  40 ′ also employs I-shaped first ground bars  128 ( 1 ),  128 ( 2 ) connected to the first ground structure  42 ′ and disposed within the loop structures  46 ′( 1 ),  46 ′( 2 ), respectively. The slow-wave transmission line  40 ′ also includes an I-shaped second ground bar  130  connected to the second ground structure  62 ′ and disposed between the loop structures  46 ′( 1 ),  46 ′( 2 ). By disposing the I-shaped first ground bars  128 ( 1 ),  128 ( 2 ) and the I-shaped second ground bar  130  in this manner, the impedance through the slow-wave transmission line  40 ′ is more regulated. 
     Further,  FIG. 8B  illustrates an exemplary T-shaped slow-wave transmission line  76 ′ that employs certain common components with the slow-wave transmission line  76  in  FIG. 6A . Such common components that have an associated number “X” in  FIG. 6A  are denoted by a number “X′” in  FIG. 8B , and thus will not be re-described herein. In this manner, the slow-wave transmission line  76 ′ includes loop structures  80 ′( 1 ),  80 ′( 2 ), as well as first and second ground structures  88 ′,  90 ′. Further, the slow-wave transmission line  76 ′ also employs I-shaped first ground bars  132 ( 1 ),  132 ( 2 ) connected to the first ground structure  88 ′ and disposed within the loop structures  80 ′( 1 ),  80 ′( 2 ), respectively. The slow-wave transmission line  76 ′ also includes an I-shaped second ground bar  134  connected to the second ground structure  90 ′ and disposed between the loop structures  80 ′( 1 ),  80 ′( 2 ). By disposing the I-shaped first ground bars  132 ( 1 ),  132 ( 2 ) and the I-shaped second ground bar  134  in this manner, the impedance through the slow-wave transmission line  76 ′ is more regulated. 
     Further,  FIG. 8C  illustrates an exemplary P-shaped slow-wave transmission line  92 ′ that employs certain common components with the slow-wave transmission line  92  in  FIG. 6B . Such common components that have an associated number “X” in  FIG. 6B  are denoted by a number “X′” in  FIG. 8C , and thus will not be re-described herein. In this manner, the slow-wave transmission line  92 ′ includes loop structures  96 ′( 1 ),  96 ′( 2 ), as well as first and second ground structures  104 ′,  106 ′. Further, the slow-wave transmission line  92 ′ also employs I-shaped first ground bars  136 ( 1 ),  136 ( 2 ) connected to the first ground structure  104 ′ and disposed within the loop structures  96 ′( 1 ),  96 ′( 2 ), respectively. The slow-wave transmission line  92 ′ also includes an I-shaped second ground bar  138  connected to the second ground structure  90 ′ and disposed between the loop structures  96 ′( 1 ),  96 ′( 2 ). By disposing the 1-shaped first ground bars  136 ( 1 ),  136 ( 2 ) and the 1-shaped second ground bar  138  in this manner, the impedance through the slow-wave transmission line  92 ′ is more regulated. 
     Notably, although the ground bars described in  FIGS. 8A-8C  are I-shaped ground bars, other embodiments may achieve similar function when employing ground bars with alternative shapes. In this manner,  FIG. 8D  illustrates an exemplary T-shaped slow-wave transmission line  76 ″ that employs certain common components with the slow-wave transmission line  76 ′ in  FIG. 8B . Such common components that have an associated number “X′” in  FIG. 8B  are denoted by a number “X″” in  FIG. 8D , and thus will not be re-described herein. The slow-wave transmission line  76 ″ includes T-shaped first ground bars  140 ( 1 ),  140 ( 2 ) connected to the first ground structure  88 ″ and disposed within the loop structures  80 ″( 1 ),  80 ″( 2 ), respectively. The slow-wave transmission line  76 ″ also includes a T-shaped second ground bar  142  connected to the second ground structure  90 ″ and disposed between the loop structures  80 ″( 1 ),  80 ″( 2 ). 
     Additionally,  FIG. 8E  illustrates an exemplary P-shaped slow-wave transmission line  92 ″ that employs certain common components with the slow-wave transmission line  92 ′ in  FIG. 8C . Such common components that have an associated number “X′” in  FIG. 8C  are denoted by a number “X″” in  FIG. 8E , and thus will not be re-described herein. The slow-wave transmission line  92 ″ includes L-shaped first ground bars  144 ( 1 ),  144 ( 2 ) connected to the first ground structure  104 ″ and disposed within the loop structures  96 ″( 1 ),  96 ″( 2 ), respectively. The slow-wave transmission line  92 ″ also includes an L-shaped second ground bar  146  connected to the second ground structure  106 ″ and disposed between the loop structures  96 ″( 1 ),  96 ″( 2 ). 
     To provide further illustration,  FIG. 9A  illustrates a cross-sectional diagram of the slow-wave transmission line  76 ″ in  FIG. 8D . The slow-wave transmission line  76 ″ includes similar components as those described in reference to the slow-wave transmission line  76  in  FIG. 7A , and thus will not be re-described herein. Notably, as previously described, the slow-wave transmission line  76 ″ includes the T-shaped first ground bars  140 ( 1 ),  140 ( 2 ) and the T-shaped second ground bar  142 . Further,  FIG. 9B  illustrates a cross-sectional diagram of the slow-wave transmission line  92 ″ in  FIG. 8E . The slow-wave transmission line  92 ″ includes similar components as those described in reference to the slow-wave transmission line  92  in  FIG. 7B , and thus will not be re-described herein. Notably, as previously described, the slow-wave transmission line  92 ″ includes the L-shaped first ground bars  144 ( 1 ),  144 ( 2 ) and the L-shaped second ground bar  146 . 
     In addition to the U-shaped, T-shaped, and P-shaped slow-wave transmission lines  40 ,  76 , and  92  previously described, other embodiments may employ slow-wave transmission lines in alternative shapes.  FIG. 10A  illustrates an exemplary slow-wave transmission line  148  disposed in a double-L-shaped pattern (also referred to as a “double-P-shaped pattern”). The slow-wave transmission line  148  includes an undulating signal path  150 , and a first and second ground structure  152 ,  154  disposed on opposite sides of the undulating signal path  150 . Notably, although only one loop structure  156  is illustrated in  FIG. 10A , the slow-wave transmission line  148  may employ multiple loop structures  156 ( 1 )- 156 (N). 
     The loop structure  156  employs via structures  158 ( 1 )- 158 ( 5 ) connected by intra-loop traces  160 ( 1 )- 160 ( 4 ). In this manner, the via structures  158 ( 1 ),  158 ( 2 ) are connected by the intra-loop trace  160 ( 1 ), and the via structures  158 ( 2 ),  158 ( 3 ) are connected by the intra-loop trace  160 ( 2 ). Further, the via structures  158 ( 3 ),  158 ( 4 ) are connected by the intra-loop trace  160 ( 3 ), and the via structures  158 ( 4 ),  158 ( 5 ) are connected by the intra-loop trace  160 ( 4 ). Additionally, an inter-loop trace  162  is employed to connect the loop structure  156  to an adjacent loop structure (not shown). 
       FIG. 10B  illustrates an exemplary slow-wave transmission line  164  disposed in a double-T-shaped pattern. The slow-wave transmission line  164  includes an undulating signal path  166 , and a first and second ground structure  168 ,  170  disposed on opposite sides of the undulating signal path  166 . Notably, although only one loop structure  172  is illustrated in  FIG. 10B , the slow-wave transmission line  164  may employ multiple loop structures  172 ( 1 )- 172 (N). 
     Further, the loop structure  172  employs via structures  174 ( 1 )- 174 ( 7 ) connected by intra-loop traces  176 ( 1 )- 176 ( 6 ). In this manner, the via structures  174 ( 1 ),  174 ( 2 ) are connected by the intra-loop trace  176 ( 1 ), and the via structures  174 ( 2 ),  174 ( 3 ) are connected by the intra-loop trace  176 ( 2 ). Further, the via structures  174 ( 3 ),  174 ( 4 ) are connected by the intra-loop trace  176 ( 3 ), and the via structures  174 ( 4 ),  174 ( 5 ) are connected by the intra-loop trace  176 ( 4 ). The via structures  174 ( 5 ),  174 ( 6 ) are connected by the intra-loop trace  176 ( 5 ), and the via structures  174 ( 6 ),  174 ( 7 ) are connected by the intra-loop trace  176 ( 6 ). Additionally, an inter-loop trace  178  is employed to connect the loop structure  172  to adjacent loop structures (not shown). 
       FIG. 10C  illustrates an exemplary slow-wave transmission line  180  disposed in a polygonal-shaped pattern. The slow-wave transmission line  180  includes an undulating signal path  182 , and a first and second ground structure  184 ,  186  disposed on opposite sides of the undulating signal path  182 . Notably, although only two loop structures  188 ( 1 ),  188 ( 2 ) are illustrated in  FIG. 10C , the slow-wave transmission line  180  may employ multiple loop structures  188 ( 1 )- 188 (N). 
     Further, the loop structure  188 ( 1 ) employs via structures  190 ( 1 )- 190 ( 14 ) connected by intra-loop traces  192 ( 1 )- 192 ( 13 ). In this manner, the via structures  190 ( 1 ),  190 ( 2 ) are connected by the intra-loop trace  192 ( 1 ), and the via structures  190 ( 2 ),  190 ( 3 ) are connected by the intra-loop trace  192 ( 2 ). Further, the via structures  190 ( 3 ),  190 ( 4 ) are connected by the intra-loop trace  192 ( 3 ), and the via structures  190 ( 4 ),  190 ( 5 ) are connected by the intra-loop trace  192 ( 4 ). The via structures  190 ( 5 ),  190 ( 6 ) are connected by the intra-loop trace  192 ( 5 ), and the via structures  190 ( 6 ),  190 ( 7 ) are connected by the intra-loop trace  192 ( 6 ). The via structures  190 ( 7 ),  190 ( 8 ) are connected by the intra-loop trace  192 ( 7 ), and the via structures  190 ( 8 ),  190 ( 9 ) are connected by the intra-loop trace  192 ( 8 ). The via structures  190 ( 9 ),  190 ( 10 ) are connected by the intra-loop trace  192 ( 9 ), and the via structures  190 ( 10 ),  190 ( 11 ) are connected by the intra-loop trace  192 ( 10 ). The via structures  190 ( 11 ),  190 ( 12 ) are connected by the intra-loop trace  192 ( 11 ), and the via structures  190 ( 12 ),  190 ( 13 ) are connected by the intra-loop trace  192 ( 12 ). The via structures  190 ( 13 ),  190 ( 14 ) are connected by the intra-loop trace  192 ( 13 ). Additionally, an inter-loop trace  194  is employed to connect the loop structures  188 ( 1 ),  188 ( 2 ). 
     Additionally, the loop structure  188 ( 2 ) employs via structures  190 ( 15 )- 190 ( 28 ) connected by intra-loop traces  192 ( 14 )- 192 ( 26 ). In this manner the via structures  190 ( 15 ),  190 ( 16 ) are connected by the intra-loop trace  192 ( 14 ), and the via structures  190 ( 16 ),  190 ( 17 ) are connected by the intra-loop trace  192 ( 15 ). Further, the via structures  190 ( 17 ),  190 ( 18 ) are connected by the intra-loop trace  192 ( 16 ), and the via structures  190 ( 18 ),  190 ( 19 ) are connected by the intra-loop trace  192 ( 17 ). The via structures  190 ( 19 ),  190 ( 20 ) are connected by the intra-loop trace  192 ( 18 ), and the via structures  190 ( 20 ),  190 ( 21 ) are connected by the intra-loop trace  192 ( 19 ). The via structures  190 ( 21 ),  190 ( 22 ) are connected by the intra-loop trace  192 ( 20 ), and the via structures  190 ( 22 ),  190 ( 23 ) are connected by the intra-loop trace  192 ( 21 ). The via structures  190 ( 23 ),  190 ( 24 ) are connected by the intra-loop trace  192 ( 22 ), and the via structures  190 ( 24 ),  190 ( 25 ) are connected by the intra-loop trace  192 ( 23 ). The via structures  190 ( 25 ),  190 ( 26 ) are connected by the intra-loop trace  192 ( 24 ), and the via structures  190 ( 26 ),  190 ( 27 ) are connected by the intra-loop trace  192 ( 25 ). The via structures  190 ( 27 ),  190 ( 28 ) are connected by the intra-loop trace  192 ( 26 ). 
       FIG. 10D  illustrates an exemplary slow-wave transmission line  196  disposed in a rounded pattern. The slow-wave transmission line  196  includes an undulating signal path  198 , and a first and second ground structure  200 ,  202  disposed on opposite sides of the undulating signal path  198 . Loop structures  204 ( 1 ),  204 ( 2 ) are disposed adjacent to one another and connected by an inter-loop trace  206 , thus forming a rounded pattern between the loop structures  204 ( 1 ),  204 ( 2 ). The loop structure  204 ( 1 ) includes via structures  208 ( 1 ),  208 ( 2 ) connected by an intra-loop trace  210 ( 1 ). Similarly, the loop structure  204 ( 2 ) includes via structures  208 ( 3 ),  208 ( 4 ) connected by an intra-loop trace  210 ( 2 ). Notably, to help regulate the impedance within the slow-wave transmission line  196  as previously described, first ground bars  212 ( 1 ),  212 ( 2 ) connected to the first ground structure  200  are disposed within the loop structures  204 ( 1 ),  204 ( 2 ), respectively. A second ground bar  214  connected to the second ground structure  202  is disposed between the loop structures  204 ( 1 ),  204 ( 2 ). 
     Therefore, the slow-wave transmission lines  148 ,  164 ,  180 , and  196  in  FIGS. 10A-10D , respectively, are designed to transmit slow-wave signals at speeds compatible with tunable filters by forcing the slow-wave signal to travel a further distance, as well as by slowing down the slow-wave signal using corresponding LC networks (not shown). 
     In addition to forming an LC network within a slow-wave transmission line as previously described, shielding may be disposed around a slow-wave transmission line so as to form an LC network along an entire undulating signal path. In this manner,  FIG. 11  illustrates an exemplary slow-wave transmission line  216  employing a shield structure  218  along an undulating signal path  220 . The slow-wave transmission line  216  includes loop structures  222 ( 1 )- 222 ( 3 ). The loop structure  222 ( 1 ) includes two via structures  224 ( 1 ),  224 ( 2 ) connected by an intra-loop trace  226 ( 1 ). Similarly, the loop structure  222 ( 2 ) includes via structures  224 ( 3 ),  224 ( 4 ) connected by an intra-loop trace  226 ( 2 ), while the loop structure  222 ( 3 ) includes via structures  224 ( 5 ),  224 ( 6 ) connected by an intra-loop trace  226 ( 3 ). The slow-wave transmission line  216  also employs inter-loop traces  228 ( 1 ),  228 ( 2 ) that connect loop structures  222 ( 1 ),  222 ( 2 ) and  222 ( 2 ),  222 ( 3 ), respectively. 
     Further, the shield structure  218  is formed so that each shield section  230 ( 1 )- 230 ( 6 ) provides shielding around each corresponding via structure  224 ( 1 )- 224 ( 6 ). Thus, the shield section  230 ( 1 ) provides shielding for the via structure  224 ( 1 ), the shield section  230 ( 2 ) provides shielding for the via structure  224 ( 2 ), and the shield section  230 ( 3 ) provides shielding for the via structure  224 ( 3 ). Additionally, the shield section  230 ( 4 ) provides shielding for the via structure  224 ( 4 ), the shield section  230 ( 5 ) provides shielding for the via structure  224 ( 5 ), and the shield section  230 ( 6 ) provides shielding for the via structure  224 ( 6 ). By providing the shielding in this manner, the shield structure  218  forms an LC network along the undulating signal path  220  that reduces the speed of a transmitted slow-wave signal in the slow-wave transmission line  216 . 
     In addition to via structures and traces as described above, slow-wave transmission lines disclosed herein may also be formed using metal bands.  FIG. 12  illustrates an exemplary double-folded slow-wave transmission line  232  having an undulating signal path  234 . The double-folded slow-wave transmission line  232  includes a metal band  236  that is folded in a U-shaped pattern with alternating turns along an X-axis (e.g., X-folding). The metal band  236  is constructed of a conductive material that is adapted to propagate a transmitted slow-wave signal. The double-folded slow-wave transmission line  232  also includes a ground band  238  that is folded in a U-shaped pattern with alternating turns along a Z-axis (e.g., Z-folding). The metal band  236  is disposed 90 degrees counter-clockwise relative to the ground band  238 . Further, the metal band  236  is interlaced with the ground band  238  so that the folds of the metal band  236  alternate with the folds of the ground band  238 . Interlacing of the metal band  236  and the ground band  238  causes an LC network to form within the double-folded slow-wave transmission line  232 . Thus, the double-folded slow-wave transmission line  232  is designed to transmit slow-wave signals at speeds compatible with tunable filters by forcing the slow-wave signal to travel a further distance, as well as by slowing down the slow-wave signal using the LC network. 
     Notably, slow-wave transmission lines with undulating signal paths as disclosed herein may be fabricated as discrete devices and mounted onto other devices.  FIG. 13  illustrates an exemplary slow-wave transmission line  240  with an undulating signal path  242 , wherein the slow-wave transmission line  240  is employed as a discrete surface-mounted device. In this manner, the slow-wave transmission line  240  is mounted to a PCB  244 . Further, in some embodiments, an integrated circuit (IC) die may be stacked on top of a slow-wave transmission line, which may increase the overall height of a device. Alternatively, an IC die may be embedded within the layers of a slow-wave transmission line to retain a lower profile with reduced height. Connections between a slow-wave transmission line and such IC die may be realized vertically or horizontally. 
     Notably, slow-wave transmission lines as disclosed herein may also be employed using a wire in a solenoid-type fashion.  FIG. 14A  illustrates an exemplary solenoid-type slow-wave transmission line  246  with an undulating signal path  248 . The solenoid-type slow-wave transmission line  246  includes a conductive wire  250  disposed around a ground structure  252 . Further,  FIG. 14B  illustrates an exemplary solenoid-type slow-wave transmission line  254  with an undulating signal path  256 . The solenoid-type slow-wave transmission line  254  includes a conductive wire  258  disposed between a first ground structure  260  and a second ground structure  262 . The solenoid-type slow-wave transmission lines  246 ,  254  are designed to transmit slow-wave signals at speeds compatible with tunable filters by forcing the slow-wave signal to travel a further distance, as well as by slowing down the slow-wave signal using an LC network. 
     In addition to the embodiments described above, slow-wave transmission lines may also be formed using both semiconductor and multi-layer laminate processes so as to include a high permittivity material and/or a high permeability material to further reduce speeds of transmitted waves. 
     Notably, a high permittivity material is defined herein as a material that has a relative permittivity ∈(r) greater than or equal to 10 at 2.5 GHz, room temperature, and 50% humidity, wherein ∈(r)=∈/∈(0), ∈(0) is the permittivity of free space (∈(0)=8.85×10E−12 F/m), and c is the absolute permittivity of the material. Relative permittivity ∈(r) is also referred to as the dielectric constant. In select embodiments, relative permittivity ∈(r) of the high permittivity material may have an upper bound of 100, 1,000, and 10,000, respectively. 
     Further, a high permeability material is defined herein as a material that has a relative permeability μ(r) greater than or equal to 2 at 2.5 GHz, room temperature, and 50% humidity, wherein μ(r)=μ/μ(0), μ(0) is the permeability of free space (μ(0)=4π×10E−7 F/m), and μ is the absolute permeability of the material. In select embodiments, relative permeability μ(r) of the high permeability material may have an upper bound of 1,000, 10,000, and 100,000, respectively. 
     For example,  FIG. 15A  illustrates an exemplary slow-wave transmission line  264  similar to the slow-wave transmission line  76  described in  FIG. 6A . However, the slow-wave transmission line  264  includes a first insulator layer  266  and a second insulator layer  268  made from a higher permittivity material. Notably, the first insulator layer  266  is formed between a first ground structure  270  and an undulating signal path  272  that includes loop structures  274 ( 1 ),  274 ( 2 ). The second insulator layer  268  is formed between a second ground structure  276  and the undulating signal path  272 . Forming the first and second insulator layers  266 ,  268  in this manner increases the capacitive component of the slow-wave transmission line  264 . Further, because the speed at which a wave signal is transmitted (the velocity factor (Vf) (not shown)) by the slow-wave transmission line  264  is inversely proportional to the square root of the relative permittivity (Vf=1/√∈(r)), the high permittivity material of the first and second insulator layers  266 ,  268  further reduces the speed of transmitted waves. 
     Additionally,  FIG. 15B  illustrates an exemplary slow-wave transmission line  278  similar to the slow-wave transmission line  76  described in  FIG. 6A . However, the slow-wave transmission line  278  includes a first insulator layer  280 , a second insulator layer  282 , and a third insulator layer  284  made from a high permeability material. Notably, the first insulator layer  280  is formed within an interior cavity of loop structure  286 ( 1 ), while the second insulator layer  282  is formed within an interior cavity of loop structure  286 ( 2 ). Further, the third insulator layer  284  is formed between the loop structures  286 ( 1 ),  286 ( 2 ). Forming the first, second, and third insulator layers  280 ,  282 ,  284  in this manner increases the inductive component of the slow-wave transmission line  278 . Further, because the speed at which a wave signal is transmitted (the velocity factor (Vf) (not shown)) by the slow-wave transmission line  278  is inversely proportional to the square root of the relative permeability (μ(r)) (Vf=1/√μ(r)), the permeability (μ) of the first, second, and third insulator layers  280 ,  282 , and  284  further reduces the speed of transmitted waves. 
     Further, the slow-wave transmission lines as disclosed herein may be implemented using processes other than laminate technology. As non-limiting examples, the slow-wave transmission lines may be implemented using three-dimensional (3-D) printing, spraying, or metal bending. 
     In this manner, slow-wave transmissions lines as described herein may be implemented using a combination of IC and multi-layer laminate processes. Notably, such an IC process includes multiple metal layers, wherein one or more metal layers may have a relatively high thickness. For example,  FIG. 16A  illustrates a portion of a slow-wave transmission line  287  employed using IC and laminate processes. The slow-wave transmission line  287  includes a loop structure  288  formed between first and second ground structures  290 ,  292 . The first ground structure  290  and inter-loop traces  294 ( 1 ),  294 ( 2 ) are formed using laminate. Further, via structures  296 ( 1 ),  296 ( 2 ) are formed using copper pillars, while an intra-loop trace  298  and the second ground structure  292  are formed from thick metal layers using the IC process. Using the copper pillars and the thick metal layers in this manner helps to realize an inductance  300  of the slow-wave transmission line  287 . 
     Further,  FIG. 16B  illustrates a portion of another slow-wave transmission line  302  employed using IC and laminate processes. The slow-wave transmission line  302  includes a loop structure  304  formed between first and second ground structures  306 ,  308 . The first ground structure  306  and inter-loop traces  310 ( 1 ),  310 ( 2 ) are formed using laminate. Further, via structures  312 ( 1 ),  312 ( 2 ) are formed using copper pillars. Intra-loop traces  314 ( 1 ),  314 ( 2 ),  314 ( 3 ) and the second ground structure  308  are formed from thick metal layers using the IC process. Using the copper pillars and the thick metal layers in this manner help to realize a capacitance  316  of the slow-wave transmission line  302 . 
     Notably, metal capture pads between consecutive vias may have a certain overhang or may be coincident with the via footprint (i.e., a zero capture pad). Alternatively, the via and the metal capture pads may have a certain offset, or the metal capture pads may not be present. 
     In addition to the embodiments described above, the slow-wave transmission lines described herein have certain magnetic properties that may be taken advantage of so as to magnetically couple multiple slow-wave transmission lines to form filters. Before discussing details of such filters, the magnetic properties of the slow-wave transmission lines will first be discussed. 
     In this manner,  FIG. 17  illustrates a slow-wave transmission line  318  similar to the slow-wave transmission line  76  in  FIG. 6A . Notably, the slow-wave transmission line  318  includes loop structures  80 ( 1 )- 80 ( 3 ) formed between the first and second ground structures  88 ,  90 . The loop structure  80 ( 1 ) includes four via structures  84 ( 1 )- 84 ( 4 ) and three intra-loop traces  86 ( 1 )- 86 ( 3 ). The via structure  84 ( 1 ) is connected to the via structures  84 ( 2 ),  84 ( 3 ) by the intra-loop traces  86 ( 1 ),  86 ( 2 ), respectively, and the via structure  84 ( 2 ) is connected to the via structure  84 ( 4 ) by the intra-loop trace  86 ( 3 ). Similarly, the loop structure  80 ( 2 ) includes four via structures  84 ( 5 )- 84 ( 8 ) and three intra-loop traces  86 ( 4 )- 86 ( 6 ). The via structure  84 ( 5 ) is connected to the via structures  84 ( 6 ),  84 ( 7 ) by the intra-loop traces  86 ( 4 ),  86 ( 5 ), respectively, and the via structure  84 ( 6 ) is connected to the via structure  84 ( 8 ) by the intra-loop trace  86 ( 6 ). Further, the loop structure  80 ( 3 ) includes four via structures  84 ( 9 )- 84 ( 12 ) and three intra-loop traces  86 ( 7 )- 86 ( 9 ). The via structure  84 ( 9 ) is connected to the via structures  84 ( 10 ),  84 ( 11 ) by the intra-loop traces  86 ( 7 ),  86 ( 8 ), respectively, and the via structure  84 ( 10 ) is connected to the via structure  84 ( 12 ) by the intra-loop trace  86 ( 9 ). Additionally, the inter-loop trace  82 ( 1 ) connects the loop structures  80 ( 1 ),  80 ( 2 ), while the inter-loop trace  82 ( 2 ) connects the loop structures  80 ( 2 ),  80 ( 3 ). 
     When a signal is transmitted in the slow-wave transmission line  318 , a current (I) flows through the undulating signal path  78 . Notably, the current (I) induces magnetic fields  320 ( 1 )- 320 ( 5 ) within the slow-wave transmission line  318 . The direction of each magnetic field  320 ( 1 )- 320 ( 5 ) is based on the direction in which the current (I) flows at a corresponding point in the undulating signal path  78 , wherein the direction of current (I) flow is illustrated using arrows in  FIG. 17 . In this manner, the magnetic fields  320 ( 1 )- 320 ( 3 ) induced within the corresponding loop structures  80 ( 1 )- 80 ( 3 ) each have a first direction based on the direction of the current (I) flow. However, the magnetic fields  320 ( 4 ),  320 ( 5 ) generated between the loop structures  80 ( 1 ),  80 ( 2 ) and  80 ( 2 ),  80 ( 3 ), respectively, have a second direction different from the first direction due to the direction of the current (I) at those points in the undulating signal path  78 . As described in more detail below, the varying directions of the magnetic fields  320 ( 1 )- 320 ( 5 ) may be used to couple multiple instances of the slow-wave transmission line  318  to form filters. As a non-limiting example, such coupling may have a coupling factor between about 0.1% and 99.9%, and includes strong coupling, moderate coupling, and weak coupling, wherein the coupling factor is partly dependent on the distance between and alignment or non-alignment of two transmission lines. As used herein, weakly coupled slow-wave transmission lines have a coupling factor of less than about 40%. 
     In this manner,  FIG. 18A  illustrates a top-view of a transmission line structure  322  that includes two instances of the slow-wave transmission line  318  in  FIG. 17 . Thus, a first slow-wave transmission line  318 A includes a first undulating signal path  78 A, while a second slow-wave transmission line  318 B includes a second undulating signal path  78 B. The first undulating signal path  78 A includes loop structures  80 A( 1 )- 80 A( 3 ) (also referred to herein as first loop structures  80 A( 1 )- 80 A( 3 )), while the second undulating signal path  78 B includes loop structures  80 B( 1 )- 80 B( 3 ) (also referred to herein as second loop structures  80 B( 1 )- 80 B( 3 )). Although not shown in  FIG. 18A , the transmission line structure  322  includes a first ground structure  88  disposed below and a second ground structure  90  disposed above the first and second undulating signal paths  78 A,  78 B. However, other embodiments may employ separate first and second ground structures  88 ,  90  for each first and second undulating signal path  78 A,  78 B. 
     Similar to the slow-wave transmission line  318  in  FIG. 17 , a current (I) flowing through the first slow-wave transmission line  318 A induces magnetic fields  320 A( 1 )- 320 A( 5 ). Additionally, a current (I) flowing through the second slow-wave transmission line  318 B induces magnetic fields  320 B( 1 )- 320 B( 5 ). The direction of each magnetic field  320 A( 1 )- 320 A( 5 ) and  320 B( 1 )- 320 B( 5 ) is based on the direction in which the current (I) flows at a corresponding point in the first and second undulating signal paths  78 A,  78 B. Based on factors such as, but not limited to, the distance between the first and second undulating signal paths  78 A,  78 B and the alignment of the first and second loop structures  80 A( 1 )- 80 A( 3 ) and  80 B( 1 )- 80 B( 3 ), the magnetic fields  320 A( 1 )- 320 A( 3 ) on one side and the magnetic fields  320 A( 4 ),  320 A( 5 ) on another side may constructively and destructively couple at the second undulating signal path  78 B. Similarly, the magnetic fields  320 B( 1 )- 320 B( 3 ) on one side and the magnetic fields  320 B( 4 ),  320 B( 5 ) on another side may constructively and destructively couple at the first undulating signal path  78 A. 
     In this manner, in the transmission line structure  322  in  FIG. 18A , the first undulating signal path  78 A is disposed a distance DS 1  from the second undulating signal path  78 B such that the first undulating signal path  78 A is immediately adjacent to and electrically isolated from the second undulating signal path  78 B. Notably, the first and second slow-wave transmission lines  318 A,  318 B are positioned such that the second undulating signal path  78 B is disposed alongside of the first undulating signal path  78 A. Further, the first and second undulating signal paths  78 A,  78 B are disposed such that the first loop structures  80 A( 1 )- 80 A( 3 ) are aligned with the second loop structures  80 B( 1 )- 80 B( 3 ). A current (I) is driven in a first direction on the first undulating signal path  78 A, which generates the magnetic fields  320 A( 1 )- 320 A( 5 ). Further, the magnetic fields  320 A( 1 )- 320 A( 5 ) induce a current (I) in the first direction in the second undulating signal path  78 B, which in turn induces the magnetic fields  320 B( 1 )- 320 B( 5 ). Because the first and second undulating signal paths  78 A,  78 B are aligned, are only separated by the distance DS 1 , and each include a current (I) flowing in the first direction, the magnetic fields  320 A( 1 )- 320 A( 5 ) experience constructive coupling at the second undulating signal path  78 B, as illustrated by corresponding arrows  324 ( 1 )- 324 ( 5 ). For example, the constructive coupling of the magnetic field  320 A( 1 ) at the loop structure  80 B( 1 ) of the second undulating signal path  78 B is illustrated by the arrow  324 ( 1 ), constructive coupling of the magnetic field  320 A( 2 ) at the loop structure  80 B( 2 ) of the second undulating signal path  78 B is illustrated by the arrow  324 ( 2 ), and constructive coupling of the magnetic field  320 A( 3 ) at the loop structure  80 B( 3 ) of the second undulating signal path  78 B is illustrated by the arrow  324 ( 3 ). Further, constructive coupling of the magnetic field  320 A( 4 ) at the second undulating signal path  78 B between the loop structures  80 B( 1 ),  80 B( 2 ) is illustrated by the arrow  324 ( 4 ), and constructive coupling of the magnetic field  320 A( 5 ) at the second undulating signal path  78 B between the loop structures  80 B( 2 ),  80 B( 3 ) is illustrated by the arrow  324 ( 5 ). 
     Additionally,  FIG. 18B  illustrates a transmission line structure  326  similar to the transmission line structure  322  in  FIG. 18A . However, rather than being separated by the distance DS 1 , the first undulating signal path  78 A is disposed a distance DS 2  from to the second undulating signal path  78 B, wherein the distance DS 2  is less than or equal to two (2) times a width W 1  of the first undulating signal path  78 A. Further, the distance DS 2  is greater than the distance DS 1 . Notably, in other embodiments, the distance DS 2  is less than or equal to one (1) times the width W 1  of the first undulating signal path  78 A. Further, the first and second undulating signal paths  78 A,  78 B are disposed such that the first loop structures  80 A( 1 )- 80 A( 3 ) are aligned with the second loop structures  80 B( 1 )- 80 B( 3 ). 
     In this manner, due to the first and second undulating signal paths  78 A,  78 B being separated by the distance DS 2 , which is larger than the distance DS 1  in  FIG. 18A , the magnetic fields  320 A( 1 )- 320 A( 5 ) experience both constructive and partial destructive coupling at the second undulating signal path  78 B. For example, constructive coupling of the magnetic field  320 A( 1 ) at the second undulating signal path  78 B is illustrated by arrow  328 ( 1 ), constructive coupling of the magnetic field  320 A( 2 ) at the second undulating signal path  78 B is illustrated by arrow  328 ( 2 ), and constructive coupling of the magnetic field  320 A( 3 ) at the second undulating signal path  78 B is illustrated by arrow  328 ( 3 ). However, partial destructive coupling of the magnetic field  320 A( 4 ) at the second undulating signal path  78 B is illustrated by arrow  328 ( 4 ), partial destructive coupling of the magnetic field  320 A( 1 ) at the second undulating signal path  78 B is illustrated by arrow  328 ( 5 ), partial destructive coupling of the magnetic field  320 A( 5 ) at the second undulating signal path  78 B is illustrated by arrow  328 ( 6 ), and partial destructive coupling of the magnetic field  320 A( 2 ) at the second undulating signal path  78 B is illustrated by arrow  328 ( 7 ). 
       FIG. 19A  illustrates a transmission line structure  330  similar to the transmission line structure  322  in  FIG. 18A . However, rather than aligning, the first and second undulating signal paths  78 A,  78 B are disposed such that the first loop structures  80 A( 1 )- 80 A( 3 ) are not aligned with the second loop structures  80 B( 1 )- 80 B( 3 ). Notably, the first and second undulating signal paths  78 A,  78 B are separated by the distance DS 1 . A current (I) is driven in the first direction on the first undulating signal path  78 A, which generates the magnetic fields  320 A( 1 )- 320 A( 5 ). Further, the magnetic fields  320 A( 1 )- 320 A( 5 ) induce a current (I) in a second direction that is different from the first direction in the second undulating signal path  78 B, which in turn induces the magnetic fields  320 B( 1 )- 320 B( 5 ). In this manner, due to the first and second undulating signal paths  78 A,  78 B not being aligned, and thus, the current (I) induced on the second undulating signal path  78 B flowing in the second direction, the magnetic fields  320 A( 1 )- 320 A( 5 ) experience destructive coupling at the second undulating signal path  78 B. For example, destructive coupling of the magnetic field  320 A( 4 ) at the second undulating signal path  78 B is illustrated by arrow  332 ( 1 ), while destructive coupling of the magnetic field  320 A( 2 ) at the second undulating signal path  78 B is illustrated by arrow  332 ( 2 ). Further, destructive coupling of the magnetic field  320 A( 5 ) at the second undulating signal path  78 B is illustrated by arrow  332 ( 3 ), and destructive coupling of the magnetic field  320 A( 3 ) at the second undulating signal path  78 B is illustrated by arrow  332 ( 4 ). Notably, the level of non-alignment of the first and second undulating signal paths  78 A,  78 B partly determines the level of both constructive and destructive coupling. Thus, disposing the first and second undulating signal paths  78 A,  78 B in a non-aligned manner provides a level of control over the coupling factor. 
     Additionally,  FIG. 19B  illustrates a transmission line structure  334  similar to the transmission line structure  330  in  FIG. 19A . However, rather than being separated by the distance DS 1 , the first undulating signal path  78 A is disposed a distance DS 2  from the second undulating signal path  78 B, wherein the distance DS 2  is greater than the distance DS 1 . Further, the first and second undulating signal paths  78 A,  78 B are disposed such that the first loop structures  80 A( 1 )- 80 A( 3 ) are not aligned with the second loop structures  80 B( 1 )- 80 B( 3 ). In this manner, due to the first and second undulating signal paths  78 A,  78 B not being aligned, the magnetic fields  320 A( 1 )- 320 A( 5 ) experience destructive coupling and partial constructive coupling at the second undulating signal path  78 B, although such coupling is weaker than the coupling illustrated in  FIG. 19A  due to the distance DS 2  being greater than the distance DS 1 . For example, destructive coupling of the magnetic field  320 A( 1 ) at the second undulating signal path  78 B is illustrated by arrow  336 ( 1 ), while destructive coupling of the magnetic field  320 A( 4 ) at the second undulating signal path  78 B is illustrated by arrow  336 ( 2 ). Further, destructive coupling of the magnetic field  320 A( 2 ) at the second undulating signal path  78 B is illustrated by arrow  336 ( 3 ), and destructive coupling of the magnetic field  320 A( 4 ) at the second undulating signal path  78 B is illustrated by arrow  336 ( 4 ). However, partial constructive coupling of the magnetic field  320 A( 1 ) at the second undulating signal path  78 B is illustrated by arrow  336 ( 5 ), partial constructive coupling of the magnetic field  320 A( 4 ) at the second undulating signal path  78 B is illustrated by arrows  336 ( 6 ),  336 ( 7 ), partial constructive coupling of the magnetic field  320 A( 2 ) at the second undulating signal path  78 B is illustrated by arrows  336 ( 8 ),  336 ( 9 ), and partial constructive coupling of the magnetic field  320 A( 5 ) at the second undulating signal path  78 B is illustrated by arrow  336 ( 10 ). 
     In addition to the embodiments described above, additional elements may be introduced into a transmission line structure to control or alter the coupling factor. In this regard,  FIG. 20A  illustrates a top-view of a transmission line structure  338  similar to the transmission line structure  322  in  FIG. 18A . Notably,  FIG. 20B  illustrates a cross-section of the transmission line structure  338 . Although the first undulating signal path  78 A is shown in  FIG. 20B , the second undulating signal path  78 B is understood to have similar elements and features. The transmission line structure  338  includes a wall structure  340  disposed between the first and second undulating signal paths  78 A,  78 B. Further, the wall structure  340  is perpendicular to the first and second ground structures  88 ,  90  (not shown). As illustrated in  FIG. 20B , the wall structure  340  includes window openings  342 ( 1 )- 342 ( 3 ) aligned with first loop portions  344 A( 1 )- 344 A( 3 ) of the first loop structures  80 A( 1 )- 80 A( 3 ). The window openings  342 ( 1 )- 342 ( 3 ) also align with first loop portions (not shown) of the second loop structures  80 B( 1 )- 80 B( 3 ). In this manner, the window openings  342 ( 1 )- 342 ( 3 ) allow portions of the magnetic fields  320 A( 1 )- 320 A( 3 ) and  320 B( 1 )- 320 B( 3 ) to flow between the first and second undulating signal paths  78 A,  78 B. However, sections of the wall structure  340  not corresponding to the window openings  342 ( 1 )- 342 ( 3 ) are solid, and thus, prevent flow of the magnetic fields  320 A( 1 )- 320 A( 5 ) and  320 B( 1 )- 320 B( 5 ) in those sections. Therefore, the wall structure  340  may be employed to control or alter the coupling factor of the transmission line structure  338 . 
     Further,  FIG. 21A  illustrates a top-view of a transmission line structure  348  similar to the transmission line structure  338  in  FIG. 20A . Notably,  FIG. 21B  illustrates a cross-section of the transmission line structure  348 . Although only the first undulating signal path  78 A is shown in  FIG. 21B , the second undulating signal path  78 B is understood to have similar elements and features. The transmission line structure  348  includes a wall structure  350  that includes the window openings  342 ( 1 )- 342 ( 3 ) aligned with the first loop portions  344 A( 1 )- 344 A( 3 ) of the first loop structures  80 A( 1 )- 80 A( 3 ) and the first loop portions (not shown) of the second loop structures  80 B( 1 )- 80 B( 3 ). However, the wall structure  350  also includes window openings  342 ( 4 ),  342 ( 5 ) aligned with intermediate portions  352 A( 1 ),  352 A( 2 ) of the first undulating signal path  78 A. Notably, the window openings  342 ( 4 ),  342 ( 5 ) also align with intermediate portions of the second undulating signal path (not shown). The intermediate portion  352 A( 1 ) is between the first loop structures  80 A( 1 ),  80 A( 2 ), and the intermediate portion  352 A( 2 ) is between the first loop structures  80 A( 2 ),  80 A( 3 ). The intermediate portions of the second undulating signal path  78 B are similarly positioned. Thus, the window openings  342 ( 1 )- 342 ( 5 ) allow portions of the magnetic fields  320 A( 1 )- 320 A( 5 ) and  320 B( 1 )- 320 B( 5 ) to flow between the first and second undulating signal paths  78 A,  78 B. Therefore, the wall structure  350  may be employed to control or alter the coupling factor of the transmission line structure  348 . 
     As another example of an element that may be introduced to control or alter the coupling factor,  FIG. 22A  illustrates a transmission line structure  354  similar to the transmission line structure  322  in  FIG. 18A . However, the transmission line structure  354  includes a floating loop structure  356  that alters the magnetic coupling factor. In this manner, a first portion  358  of the floating loop structure  356  resides within a space  360 A of the first loop structure  80 A( 1 ). Further, a second portion  362  of the floating loop structure  356  resides within a space  360 B of the second loop structure  80 B( 1 ). Notably, the first portion  358  and the second portion  362  are aligned with one another and connected so that the floating loop structure  356  forms a closed loop. Additionally, the first portion  358  is electrically isolated from the first undulating signal path  78 A, while the second portion  362  is electrically isolated from the second undulating signal path  78 B. 
     Similar to the description provided in relation to  FIG. 17 , a current (I) flowing in the first undulating signal path  78 A induces the corresponding magnetic field  320 A( 1 ) (not shown) in the first undulating signal path  78 A. Further, the magnetic field  320 A( 1 ) induces a current (I) to flow in the first portion  358  of the floating loop structure  356  such that the induced current (I) induces a magnetic field  364 A( 1 ) corresponding to the first portion  358 . Additionally, a current (I) flowing in the second undulating signal path  78 B induces the corresponding magnetic field  320 B( 1 ) (not shown) in the second undulating signal path  78 B. Further, the magnetic field  320 B( 1 ) induces a current (I) to flow in the second portion  362  of the floating loop structure  356  such that the induced current (I) induces a magnetic field  364 B( 1 ) corresponding to the second portion  362 . As a result, the induced magnetic fields  364 A( 1 ),  364 B( 1 ) affect the coupling factor of the transmission line structure  354 . The extent to which the coupling factor is affected is based on the strength and directionality of the induced magnetic fields  364 A( 1 ),  364 B( 1 ). 
     In this manner,  FIG. 22B  illustrates a transmission line structure  366  similar to the transmission line structure  354  in  FIG. 22A . The transmission line structure  366  includes a floating loop structure  368  similar to the floating loop structure  356  in  FIG. 22A . However, rather than including the second portion  362  in the space  360 B of the second loop structure  80 B( 1 ), the floating loop structure  368  includes a second portion  370 . Notably, the second portion  370  is disposed in the space  360 B of the second loop structure  80 B( 1 ) such that a current (I) induced in the second portion  370  flows in an opposite direction from the current (I) induced in the second portion  362  in  FIG. 22A . Thus, a magnetic field  372 B( 1 ) has an opposite direction as compared to the magnetic field  364 B( 1 ) in  FIG. 22A . Therefore, because the directionality of the magnetic fields  364 B( 1 ),  372 B( 1 ) differ in this manner, the floating loop structure  368  in  FIG. 22B  affects the coupling factor of the transmission line structure  366  differently as compared to how the floating loop structure  356  affects the coupling factor of the transmission line structure  354  in  FIG. 22A . 
     Additionally,  FIG. 23A  illustrates a transmission line structure  374  similar to the transmission line structure  354  in  FIG. 22A . Further, the transmission line structure  374  includes a floating loop structure  376  similar to the floating loop structure  356  in  FIG. 22A . However, the floating loop structure  376  includes a switch  378  configured to control current (I) flow through the floating loop structure  376 . In other words, activation of the switch  378  allows the floating loop structure  376  to form a closed loop and affects the magnetic coupling of the transmission line structure  374  in a similar manner as described in relation to the floating loop structure  356 . In contrast, deactivation of the switch  378  prevents the floating loop structure  376  from forming a closed loop, thus preventing current (I) flow within the floating loop structure  376  and inducement of corresponding magnetic fields (not shown). Thus, activation of the switch  378  enables the floating loop structure  376  to alter the magnetic coupling factor, while deactivation of the switch  378  disables the floating loop structure  376  from altering the magnetic coupling factor of the transmission line structure  374 . 
       FIG. 23B  illustrates a transmission line structure  380  similar to the transmission line structure  374  in  FIG. 23A . Further, the transmission line structure  380  includes a floating loop structure  382  similar to the floating loop structure  368  in  FIG. 22B . However, the floating loop structure  382  includes a switch  384  configured to control current (I) flow through the floating loop structure  382 . In other words, activation of the switch  384  allows the floating loop structure  382  to form a closed loop and affects the magnetic coupling of the transmission line structure  380  in a similar manner as described in relation to the floating loop structure  368 . In contrast, deactivation of the switch  384  prevents the floating loop structure  382  from forming a closed loop, thus preventing current (I) flow within the floating loop structure  382  and inducement of corresponding magnetic fields (not shown). Thus, activation of the switch  384  enables the floating loop structure  382  to alter the magnetic coupling factor, while deactivation of the switch  384  disables the floating loop structure  382  from altering the magnetic coupling factor. 
     As another example of an element that may be introduced to control or alter the coupling factor,  FIG. 24  illustrates a transmission line structure  386  similar to the transmission line structure  322  in  FIG. 18A . However, the transmission line structure  386  includes a floating ring structure  388  that alters the magnetic coupling factor. In this manner, a first portion  390  of the floating ring structure  388  resides within a space  392 A of the first loop structure  80 A( 1 ) and is electrically isolated from the first loop structure  80 A( 1 ). Further, a second portion  394  of the floating ring structure  388  resides within a space  392 B of the second loop structure  80 B( 1 ) and is electrically isolated from the second loop structure  80 B( 1 ). Notably, the first portion  390  and the second portion  394  are aligned with one another. Further, a switch  396  is included in the floating ring structure  388  that is configured to control current (I) flow through the floating ring structure  388 . 
     In this manner, similar to the description provided in relation to  FIG. 17 , a current (I) flowing in the first undulating signal path  78 A induces the corresponding magnetic field  320 A( 1 ) (not shown) in the first undulating signal path  78 A. Thus, when the switch  396  is activated so as to allow the floating ring structure  388  to form a closed loop, the magnetic field  320 A( 1 ) induces a current (I) to flow in the first portion  390  of the floating ring structure  388  such that the induced current (I) generates a magnetic field (not shown) corresponding to the first portion  390 . Additionally, a current (I) flowing in the second undulating signal path  78 B induces the corresponding magnetic field  320 B( 1 ) (not shown) in the second undulating signal path  78 B. Further, the magnetic field  320 B( 1 ) induces a current (I) to flow in the second portion  394  of the floating ring structure  388  such that the induced current (I) generates a magnetic field (not shown) corresponding to the second portion  394 . As a result, the induced magnetic fields affect the coupling factor of the transmission line structure  386 . The extent to which the coupling factor is affected is based on the strength and directionality of the induced magnetic fields corresponding to the first and second portions  390 ,  394 . Deactivation of the switch  396  disables the floating ring structure  388  from altering the magnetic coupling factor of the transmission line structure  386 . Notably, although this embodiment includes the floating ring structure  388  disposed in the first and second loop structures  80 A( 1 ),  80 B( 1 ), other embodiments that do not include the second ground structure  90  (not shown) may include the floating ring structure  388  disposed above the first and second undulating signal paths  78 A,  78 B. 
     Further, the transmission line structure  386  includes the first undulating signal path  78 A aligned with the second undulating signal path  78 B such that the floating ring structure  388  is disposed in the aligned first and second loop structures  80 A( 1 ),  80 B( 1 ). However, other embodiments may include the first undulating signal path  78 A not aligned with the second undulating signal path  78 B, wherein the floating ring structure  388  is employed with an angle so as to be disposed in the first and second loop structures  80 A( 1 ),  80 B( 1 ), which are not aligned. 
     Notably, the floating loop structures  356 ,  368 ,  376 , and  384 , and the floating ring structure  388  are described herein as disposed in the first loop structure  80 A( 1 ) and the second loop structure  80 B( 1 ). However, other embodiments may include the floating loop structures  356 ,  368 ,  376 , and  384 , and the floating ring structure  388  in alternative or multiple first and second loop structures  80 A( 1 )- 80 A( 3 ),  80 B( 1 )- 80 B( 3 ). 
     As another example of an element that may be introduced to control or alter the coupling factor,  FIG. 25  illustrates a transmission line structure  398  similar to the transmission line structure  322  in  FIG. 18A . However, the transmission line structure  398  includes a first plate structure  400  and a second plate structure  402 . In this embodiment, the second plate structure  402  is narrower than the first plate structure  400 . Further, a first portion  404  of the first plate structure  400  resides within a space  406 A of the first loop structure  80 A( 1 ). A second portion  408  of the first plate structure  400  resides within a space  406 B of the second loop structure  80 B( 1 ), wherein the first portion  404  and the second portion  408  are aligned with one another. In this manner, the first plate structure  400  forms a capacitance  410  between the first portion  404  and the first loop structure  80 A( 1 ), and a capacitance  412  between the second portion  408  and the second loop structure  80 B( 1 ). Similarly, a first portion  414  of the second plate structure  402  resides within a space  416 A of the first loop structure  80 A( 3 ). A second portion  418  of the second plate structure  402  resides within a space  416 B of the second loop structure  80 B( 3 ), wherein the first portion  414  and the second portion  418  are aligned with one another. In this manner, the second plate structure  402  forms a capacitance  420  between the first portion  414  and the first loop structure  80 A( 3 ), and a capacitance  422  between the second portion  418  and the second loop structure  80 B( 3 ). Thus, the first and second plate structures  400 ,  402  capacitively couple the first and second undulating signal paths  78 A,  78 B, wherein the wider width of the first plate structure  400  allows for more coupling than the narrower width of the second plate structure  402 . Notably, various methods known in the art may be used to make the capacitances  410 ,  412 ,  420 , and  422  either constant or variable. 
     Notably, the embodiments described in  FIGS. 18A-25  include the first loop structures  80 A( 1 )- 80 A( 3 ) and the second loop structures  80 B( 1 )- 80 B( 3 ) disposed in a T-shaped pattern. Further, a T-shaped pattern is formed between each of the first loop structures  80 A( 1 )- 80 A( 3 ) and between each of the second loop structures  80 B( 1 )- 80 B( 3 ). However, other embodiments may employ alternative patterns. 
     Further, the transmission line structure  398  includes the first undulating signal path  78 A aligned with the second undulating signal path  78 B such that the first plate structure  400  is disposed in the aligned first and second loop structures  80 A( 1 ),  80 B( 1 ) and the second plate structure  402  is disposed in the aligned first and second loop structures  80 A( 3 ),  80 B( 3 ). However, other embodiments may include the first undulating signal path  78 A not aligned with the second undulating signal path  78 B, wherein the first plate structure  400  and the second plate structure  402  are each employed with an angle so as to be disposed in the first and second loop structures  80 A( 1 ),  80 B( 1 ) and  80 A( 3 ),  80 B( 3 ), respectively, wherein the first and second loop structures are not aligned. 
     Further, although not illustrated in  FIGS. 18A-25 , the transmission line structures  322 ,  326 ,  330 ,  334 ,  338 ,  348 ,  354 ,  366 ,  374 ,  380 ,  386 , and  398  each include the first ground structure  88  disposed below and the second ground structure  90  disposed above the first and second undulating signal paths  78 A,  78 B as described in  FIG. 17 . However, other embodiments may employ separate first and second ground structures  88 ,  90  for each first and second undulating signal path  78 A,  78 B, or employ only one of the first and second ground structures  88 ,  90 . 
     Those skilled in the art will recognize improvements and modifications to the embodiments of the present disclosure. All such improvements and modifications are considered within the scope of the concepts disclosed herein and the claims that follow.