Transmission Line Theory
Transmission line theory is a fundamental framework in electrical engineering and applied physics that describes the propagation of electrical signals along conductors, accounting for distributed parameters such as inductance, capacitance, and resistance [2]. This theory extends beyond simple circuit analysis by modeling conductors as structures with finite dimensions, where voltage and current can vary as functions of both position and time [4]. It provides the mathematical and conceptual basis for understanding signal integrity, impedance matching, and power transfer in systems where the physical length of the interconnecting line is significant compared to the wavelength of the signals involved [3]. The theory is essential for the design and analysis of systems operating at radio frequencies (RF), microwave frequencies, and in high-speed digital circuits, forming a critical bridge between lumped-element circuit theory and electromagnetic wave propagation. A core principle of transmission line theory is the representation of a line using a distributed-element model, where properties are not concentrated at discrete points but are spread continuously along its length [2]. This model is characterized by key parameters: the characteristic impedance, which determines how voltage and current waves relate; the propagation constant, which describes the attenuation and phase shift per unit length; and the phase velocity, the speed at which a wave propagates down the line [4]. Transmission lines come in various physical forms, including coaxial cables, microstrips, striplines, and waveguides, each suited to different frequency ranges and applications [5]. A significant related concept is the artificial transmission line, a discrete electrical structure composed of periodically arranged lumped inductors and capacitors that emulates the propagation characteristics of a continuous transmission line, such as characteristic impedance and phase velocity, while allowing for compact designs in high-frequency applications [8]. These can be constructed from discrete components for educational or experimental purposes, such as on a solderless breadboard [7], and find everyday examples in delay lines and compact filters [1]. The applications of transmission line theory are vast and underpin modern technology. It is indispensable in the design of telecommunications infrastructure, including antenna feed networks, cable television systems, and radar [3]. In computing, it ensures signal integrity in the high-speed interconnects within microprocessors and between computer components [5]. The theory guides the layout of printed circuit boards (PCBs) for RF and microwave circuits, where controlled impedance traces are mandatory [5]. Furthermore, advanced concepts derived from the theory enable innovative devices, such as self-matching rectifiers based on artificial transmission lines for enhanced dynamic range in energy harvesting and RF power conversion [6]. The ongoing development of faster digital systems and higher-frequency wireless communications continues to drive research and refinement in transmission line theory, making its understanding a cornerstone of electrical engineering education and practice [3][4].
Overview
Transmission line theory provides the fundamental framework for analyzing and designing electrical systems where signal propagation time becomes comparable to or exceeds the signal period, necessitating treatment as distributed-parameter networks rather than simple lumped-element circuits. This theory bridges the gap between conventional circuit analysis, valid at lower frequencies where electromagnetic wave propagation effects are negligible, and full-wave electromagnetic analysis required at optical frequencies. At its core, transmission line theory models conductors as having distributed series resistance and inductance along their length, with distributed shunt conductance and capacitance between them, creating a structure that supports guided electromagnetic wave propagation [13]. This distributed model becomes essential when physical dimensions approach a significant fraction (typically one-tenth or more) of the signal wavelength, where voltage and current magnitudes vary along the line's length and wave reflection phenomena dominate system behavior.
Fundamental Principles and Telegrapher's Equations
The mathematical foundation of transmission line theory is encapsulated in the telegrapher's equations, a pair of coupled partial differential equations derived from applying Kirchhoff's laws to an infinitesimal section of the transmission line. For a lossless line—an idealization where series resistance (R) and shunt conductance (G) are zero—these equations simplify to the wave equation, demonstrating that signals propagate as waves along the line. The characteristic impedance (Z₀), a fundamental property of any transmission line, represents the ratio of voltage to current for a traveling wave and is determined solely by the distributed inductance (L) and capacitance (C) per unit length: Z₀ = √(L/C) [13]. For a typical coaxial cable like RG-58, this value is approximately 50 ohms, while television coaxial cables often use 75 ohms. The phase velocity (vₚ), at which signals propagate along the line, is given by vₚ = 1/√(LC), which for transmission lines with dielectric materials is slower than the speed of light in vacuum by a factor of 1/√(εᵣ), where εᵣ is the relative permittivity of the dielectric [13]. These structures implement the distributed L and C parameters of theoretical transmission line models using physical, discrete components arranged in ladder networks (either T-sections or Π-sections). Each repeating unit cell contributes a specific amount of series inductance and shunt capacitance, with the overall behavior approximating a continuous line as the number of sections increases and the electrical length of each section decreases. The characteristic impedance of such an artificial line is given by Z₀ = √(Lₛ/Cₛ), where Lₛ and Cₛ are the inductance and capacitance values in each unit cell [14]. This approach enables the creation of transmission line effects on circuit boards where physical wavelength constraints would normally require impractically long traces, facilitating miniaturization in radio frequency (RF) and microwave systems.
Everyday Example and Practical Modeling
A common, everyday example of an artificial transmission line structure is the humble power strip or extension cord with multiple outlets. While not designed as a transmission line, its physical construction—conductors running in parallel with periodic tap points—creates a structure that exhibits transmission line effects at sufficiently high frequencies. More deliberately engineered examples include delay lines in analog oscilloscopes and pulse-forming networks in radar systems, where controlled propagation delay is essential. For practical circuit analysis, a simple model using resistors, inductors, and capacitors provides an effective representation of real transmission lines [13]. The series resistance accounts for conductor losses (skin effect and proximity effect), the series inductance represents magnetic energy storage around the conductors, the shunt conductance models dielectric losses in the insulation, and the shunt capacitance represents electric energy storage between conductors. This lumped-element model, when applied to sufficiently small segments of the line, accurately predicts frequency-dependent attenuation, dispersion, and impedance characteristics observed in practical cables like twisted-pair, coaxial, and microstrip lines [13].
Applications and Design Considerations
Transmission line theory finds application across numerous engineering domains. In telecommunications, it governs the design of impedance-matched networks to maximize power transfer and minimize reflections in systems operating from megahertz to gigahertz frequencies. In digital electronics, with edge rates in modern processors below 100 picoseconds, printed circuit board traces must be treated as transmission lines to prevent signal integrity issues like ringing, overshoot, and intersymbol interference. The theory also underpins the operation of:
- Impedance transformers using quarter-wavelength sections (Z₀ = √(Z_in * Z_out))
- Resonators and filters in RF/microwave engineering
- Antenna feed networks and matching circuits
- High-speed serial data links (PCI Express, USB, HDMI)
Critical design parameters derived from transmission line theory include the propagation constant (γ = α + jβ, where α is attenuation and β is phase constant), reflection coefficient (Γ = (Z_L - Z₀)/(Z_L + Z₀)), voltage standing wave ratio (VSWR = (1+|Γ|)/(1-|Γ|)), and electrical length (θ = βl, often measured in degrees at a specific frequency). Proper application of these concepts enables engineers to design systems that maintain signal fidelity over required distances and bandwidths while managing electromagnetic interference and emissions [13][14].
History
The theoretical foundations of transmission line theory emerged in the late 19th century as electrical telegraphy and telephony expanded over long distances, necessitating a deeper understanding of signal propagation on wires. Early telegraph systems operated under the assumption of instantaneous signal travel, but engineers observed significant signal distortion, attenuation, and time delays over hundreds of miles, prompting the development of distributed circuit models.
Early Foundations and the Distributed Parameter Model
The pivotal conceptual breakthrough was the recognition that a pair of parallel conductors could not be accurately modeled as a simple series circuit with lumped resistance. Instead, it was more accurately represented as a distributed structure with electrical properties per unit length. This led to the development of the distributed parameter model, where a transmission line is characterized by four primary parameters per unit length:
- Series resistance (R) in ohms per meter (Ω/m), representing conductor losses. - Series inductance (L) in henries per meter (H/m), arising from the magnetic field around the conductors. - Shunt conductance (G) in siemens per meter (S/m), representing dielectric losses in the insulation. - Shunt capacitance (C) in farads per meter (F/m), arising from the electric field between the conductors. A fundamental everyday example of this distributed structure is a simple two-wire power cord connecting a lamp to a wall outlet; while short enough that propagation effects are negligible at power frequencies, it inherently possesses these distributed R, L, G, and C parameters [16]. The argument that a circuit composed of these discrete resistors, inductors, and capacitors serves as an excellent model for a transmission line is central to the theory. By cascading an infinite series of infinitesimally small segments, each containing these lumped elements, one can derive the telegrapher's equations that describe voltage and current wave propagation [16].
Pupin and the Advent of Loading
A major practical challenge in early long-distance telephony was excessive attenuation and distortion of voice-frequency signals (below 4 kHz) on lines exceeding 50 km. The primary issue was the unbalanced relationship between the distributed inductance (L) and capacitance (C); typical open-wire lines had insufficient inductance, leading to high signal loss. In 1899, Serbian-American engineer Mihajlo Idvorski Pupin developed a revolutionary solution and its corresponding theory [16]. He proposed periodically loading the transmission line with discrete inductors (loading coils) spaced at regular intervals. This technique, which became known as "Pupinization," artificially increased the per-unit-length inductance of the line. By raising the inductance, Pupin's method reduced the attenuation constant and minimized distortion for the critical voice band, dramatically extending the usable range of telephone circuits without increasing wire gauge. At the time, the process of installing these coils was widely referred to as "Pupinizing" the line [16].
Development of Modern Theory and Applications
The work of Pupin and his contemporaries formalized the analysis of periodically loaded structures, which are a form of artificial transmission line. An artificial transmission line is a discrete electrical network composed of periodically arranged lumped inductors and capacitors that emulates the propagation characteristics—such as characteristic impedance and phase velocity—of a continuous transmission line, while allowing for a more compact or tunable design. This concept became fundamental to later developments in electrical engineering. The solution to the telegrapher's equations reveals that signals propagate as waves along the line. The characteristic impedance (Z₀), a key property already discussed in prior sections, is defined by the ratio of the distributed parameters: Z₀ = √((R + jωL)/(G + jωC)). For a lossless line (where R=0 and G=0), this simplifies to Z₀ = √(L/C). The propagation constant (γ) determines how the signal is attenuated and phase-shifted: γ = α + jβ = √((R + jωL)(G + jωC)), where α is the attenuation constant in nepers per meter and β is the phase constant in radians per meter. The phase velocity (v_p), the speed at which a single frequency travels, is given by v_p = ω/β. A critical insight from this theory is the behavior under different terminal conditions. When a line of finite length is terminated by a load impedance (Z_L) equal to its characteristic impedance (Z₀), all incident power is absorbed by the load, and no signal is reflected—this is a matched condition. If the termination is not matched, reflections occur, leading to standing waves. The reflection coefficient (Γ) quantifies this: Γ = (Z_L - Z₀)/(Z_L + Z₀). The theory also covers special cases like short-circuit (Z_L=0, Γ=-1) and open-circuit (Z_L=∞, Γ=+1) terminations, which result in total reflection with specific phase inversions. The application of transmission line theory expanded far beyond telephony. In power engineering, while alternating current (AC) dominates distribution networks, high-voltage direct current (HVDC) transmission lines are used for bulk power transfer over very long distances or between asynchronous AC grids, as DC transmission avoids issues of capacitive and inductive reactance that plague long AC lines [15]. The most profound impact, however, began in the early 20th century with the advent of radio frequencies (RF). At these higher frequencies, the wavelengths become comparable to the physical length of the conductors, making transmission line effects paramount. The theory became essential for designing:
- Antenna feed lines
- Impedance matching networks
- Filters and delay lines
- Resonators
The development of the Smith chart in 1939 by Phillip H. Smith provided a powerful graphical tool for solving complex impedance transformation and matching problems, cementing transmission line theory as a cornerstone of RF and microwave engineering. Today, the principles first developed for telegraph wires are integral to the design of every high-speed digital circuit, satellite communication system, and radar installation, governing signal integrity in everything from computer motherboards to fiber-optic data links.
Description
Transmission line theory provides a comprehensive framework for analyzing the propagation of electrical signals along conductors whose physical dimensions are significant relative to the signal wavelength. This theory models these structures using distributed circuit elements—resistance (R), inductance (L), conductance (G), and capacitance (C) per unit length—to accurately predict signal behavior, including propagation velocity, attenuation, and impedance characteristics [2][4]. The classical approach to analyzing wave propagation on transmission lines forms the foundation for understanding how discrete lumped-element circuits can emulate continuous transmission line behavior [13].
The Distributed-Parameter Model and Telegrapher's Equations
The core of transmission line theory rests on the distributed-parameter model, which treats the line as an infinite series of infinitesimally small segments. Each segment contains series resistance (R) and inductance (L), and shunt conductance (G) and capacitance (C), representing the losses and energy storage mechanisms along the line's length [2][4]. This model leads to the derivation of the telegrapher's equations, a pair of coupled partial differential equations that describe the spatial and temporal variation of voltage (V) and current (I) along the line:
∂V/∂z = - (R + jωL)I ∂I/∂z = - (G + jωC)V
where z is the position along the line and ω is the angular frequency [2]. Solving these equations yields solutions in the form of traveling waves, characterized by a propagation constant (γ) and a characteristic impedance (Z₀). The propagation constant γ = α + jβ = √((R + jωL)(G + jωC)) defines the attenuation constant (α) in nepers per meter and the phase constant (β) in radians per meter [2][4]. The characteristic impedance, given by Z₀ = √((R + jωL)/(G + jωC)), represents the ratio of voltage to current for a traveling wave and is a fundamental property of the line's physical construction [4]. This concept has a significant historical precedent. In 1899, Serbian-American engineer Mihajlo Idvorski Pupin developed the theory of periodically loading transmission lines with discrete inductors to increase per-unit-length inductance, reducing attenuation and distortion in voice-frequency signals below 4 kHz, a process termed "Pupinization" [14]. At the time, his technique became known as "Pupinizing" [1]. This technique directly addressed the primary issue of unbalanced distributed parameters in early telephone lines, as noted earlier, by artificially increasing inductance at regular intervals. A common everyday example of an artificial transmission line is the delay line found in some older television sets or electronic test equipment, where a cascade of inductors and capacitors is used to introduce a precise time delay to a signal. The theory demonstrates that a circuit with discrete resistors, inductors, and capacitors is a valid and effective model for a transmission line, especially when the lumped elements are arranged in a periodic ladder network [13]. For a lossless artificial line (where R=0 and G=0), the characteristic impedance simplifies to Z₀ = √(L/C), and the phase velocity is v_p = 1/√(LC), where L and C are the inductance and capacitance per section [13].
Physical Structures and Parameter Derivation
Transmission lines exist in various physical forms, each with distinct electromagnetic field configurations and derived RLGC parameters. A microstrip transmission line, a common planar structure, comprises a conductor placed above a ground plane, separated by a dielectric layer [5]. Its characteristics depend critically on the conductor width, the dielectric thickness, and the relative permittivity (εᵣ) of the substrate material. Another common planar structure is the stripline, where the conductor is embedded within a dielectric substrate between two ground planes. If the RLGC parameters are known for the transmission line based on their physical structure, then several key properties can be calculated, including characteristic impedance, propagation delay, and signal attenuation [4]. For instance, the characteristic impedance of a microstrip line can be approximated by formulas involving the width-to-height ratio and the effective dielectric constant.
Applications and Broader Relevance
The principles of transmission line theory extend far beyond simple signal interconnection. Due to the scalability of electromagnetic theories, transmission line modelling can also be adapted to explaining the working principle of metamaterial absorbers with proper modifications [3]. In these applications, the theory helps design structures that exhibit tailored absorption of electromagnetic waves by modeling them as circuits with specific impedance profiles. Furthermore, the theory is essential for:
- Designing impedance-matching networks to maximize power transfer and minimize reflections. - Analyzing signal integrity in high-speed digital circuits, where traces on printed circuit boards behave as transmission lines. - Understanding the behavior of antennas, which can be viewed as terminated transmission lines. - Modeling and designing filters and other microwave components using distributed elements. The analysis of reflections, building on the concept of impedance mismatch mentioned previously, is a direct application of the theory. When a line is terminated in an impedance other than Z₀, reflected waves are generated, leading to standing wave patterns quantified by the voltage standing wave ratio (VSWR) [2]. The theory provides the tools to calculate reflection coefficients and manage these effects in practical system design.
Conclusion
Transmission line theory provides the essential mathematical and conceptual tools for understanding and engineering systems where signal propagation time is comparable to or exceeds the signal period. From its roots in solving the practical problems of long-distance telephony, exemplified by Pupin's work, to its modern applications in high-frequency electronics, microwave engineering, and metamaterials, the theory remains a cornerstone of electrical engineering. By modeling conductors as distributed-parameter networks and solving the resulting wave equations, engineers can predict and control signal behavior, enabling the design of everything from compact radio-frequency circuits to sophisticated electromagnetic materials.
Significance
Transmission line theory provides the fundamental framework for analyzing and designing the interconnections that carry electrical signals across distances ranging from microscopic integrated circuits to continental power grids. Its significance extends beyond the basic analysis of wave propagation on uniform conductors to encompass the design of specialized components, the modeling of complex systems, and the conceptual foundation for modern metamaterials and computational techniques. The theory's application to both continuous and artificial structures has enabled technological advancements in radio frequency engineering, telecommunications, and signal processing.
Foundational Role in Filter and Network Design
A core application of transmission line theory is in the design of frequency-selective networks, particularly bandpass filters (BPFs), which are critical for controlling the required frequency band in numerous RF and microwave applications [18]. The theory provides the mathematical basis for understanding how distributed inductance (L) and capacitance (C) interact to create passbands and stopbands. This is exemplified in the analysis of phase-shifting networks, where the series impedance is represented as being equally divided, with each half having a value of Z/2, a conceptual model derived directly from transmission line segmentation [19]. The principles of impedance matching and reflection, as noted earlier, are paramount in ensuring that filters deliver their intended frequency response without deleterious signal degradation. Furthermore, the design of derived filters, such as the m-derived low pass filter, relies on manipulating the line's characteristic equations; for instance, the calculation of a key frequency parameter f∞ involves substituting specific component values into the governing equation derived from transmission line theory [23].
Enabling Artificial and Metamaterial Transmission Lines
The theory's abstraction of a line into distributed parameters enabled the development of artificial transmission lines. These are discrete electrical structures composed of periodically arranged lumped inductors and capacitors that emulate the propagation characteristics—namely characteristic impedance and phase velocity—of a continuous transmission line while allowing for compact, tunable designs [14]. An everyday example of such an artificial line is the delay line found in vintage analog television sets or audio equipment, used to create precise timing intervals for signal synchronization or echo effects [24]. These lines function as low-pass filters with a defined cutoff frequency, known as the Bragg frequency, beyond which signal distortion occurs, limiting their operational bandwidth to approximately one-third of this value for optimal performance [14]. The concept validates the argument that a circuit with resistors, inductors, and capacitors serves as an effective model for a transmission line [18]. This modeling approach has profound implications, as it bridges classical network theory with modern metamaterial science. Research as early as 2002 proposed using non-resonant, low-loss transmission line structures to implement and practice the concept of metamaterials, which exhibit properties not found in nature, such as negative refractive index [22]. The analysis of ladder-network transmission lines with capacitive and inductive lumped elements is a direct application of this theory [22].
Historical Impact on Long-Distance Communication
The practical significance of transmission line theory was decisively demonstrated in solving the attenuation problem in early long-distance telephone lines. As noted earlier, the primary issue was unbalanced distributed parameters. The theoretical insight that signal loss could be reduced by increasing the inductance per unit section led to the invention of load coils. This technique, which involved inserting inductors at regular intervals, directly engineered the line's distributed L and C parameters based on theoretical predictions to minimize distortion and loss for voice frequencies [20]. This application was a triumph of applied theory over "perpetually laborious computing" methods used in electric power transmission analysis before the advent of electronic computers [21]. It transformed telephony from a local service into a continent-spanning network.
Basis for Advanced Simulation and Modern Implementations
Transmission line theory continues to evolve and find new applications in computational methods and system design. In 2020, an efficient electromagnetic transient simulation method based on the discrete similarity principle was proposed, which further expanded the application scope of root-matching techniques derived from transmission line modeling [18]. These advanced simulations are crucial for designing reliable power grids and high-speed digital circuits. Furthermore, the theory underpins modern implementations of signal delay. While artificial lumped-element delay lines were once common in hardware, the preferred implementation of measured delays today is often in software after sampling with an analog-to-digital converter (ADC) [24]. This digital signal processing approach still relies on the theoretical models of phase shift and group delay originally developed for physical transmission lines. The theory thus provides a continuous thread from the physical design of coaxial cables and loaded telephone lines to the algorithmic manipulation of digital signals, cementing its role as an indispensable tool in electrical engineering.
Applications and Uses
Transmission line theory provides the fundamental framework for designing and analyzing a vast array of components and systems in radio frequency (RF), microwave, and high-speed digital engineering. Its principles are applied to create functional circuits that manipulate signal propagation, filter specific frequency bands, and model complex electromagnetic behavior, moving beyond the simple interconnection of components.
Filter Design and Implementation
A primary application of transmission line theory is in the design of frequency-selective filters. By constructing networks of transmission line segments with specific characteristic impedances and electrical lengths, engineers can create low-pass, high-pass, band-pass, and band-stop filters. These structures are essential for controlling the required frequency band in numerous RF and microwave applications, allowing desired signals to pass while rejecting others [6]. For instance, bandpass filters (BPFs) are critical components in transmitters and receivers to select specific communication channels. One classical design approach involves the m-derived filter. This technique improves the attenuation characteristics beyond basic constant-k filter sections by introducing a tuned shunt or series element that creates an infinite attenuation pole at a specific frequency outside the passband. The m-derived low-pass filter can be implemented in both T and π section configurations, offering designers flexibility in matching to source and load impedances [23]. More advanced and compact filter designs leverage non-uniform transmission line structures. An example is the interdigital bandpass filter, where coupled line segments act as resonators. Research has demonstrated compact nonuniform transmission line interdigital BPFs designed for fifth-generation (5G) low-frequency bands, such as 3.4–3.6 GHz and 4.8–5.0 GHz, showcasing their relevance in modern wireless systems [7].
Artificial Transmission Lines and Lumped-Element Modeling
Building on the concept of loading lines to correct parameter imbalance, the creation of artificial transmission lines is a direct application of the theory. These are circuits composed of discrete inductors and capacitors arranged in a ladder network to mimic the propagation characteristics of a distributed transmission line over a limited frequency range [14]. They are invaluable for realizing large phase shifts or time delays in a compact physical space, a technique discussed in publications on artificial delay lines as early as the 1950s [24]. Artificial lines also enable novel circuit functions, such as in the design of self-matching rectifiers for enhanced dynamic range in power harvesting applications [6]. The analysis of such structures relies on lumped-element modeling. A transmission line is modeled as a cascade of identical unit cells, each containing a series impedance (Z) and a shunt admittance (Y). This ladder-network approach allows for the modeling and analysis of lines that incorporate both distributed properties and discrete capacitive or inductive lumped elements [22]. This model is distinct from full-wave electromagnetic simulations and provides a circuit-oriented understanding of wave propagation, cutoff frequencies, and characteristic impedance. For lossless lines, this lumped-circuits model serves as a foundational tool for digital simulation, offering an alternative to classical finite-difference time-domain algorithms by focusing on equivalent circuit behavior [8].
Phase-Shifting and Delay Networks
Transmission line theory is essential for designing networks that provide a specific phase shift or time delay without attenuating the signal. These phase-shifting networks are crucial in applications like beamforming for antenna arrays, where the relative phase between antenna elements must be precisely controlled to steer a radiation pattern. The design requirements for such networks often include achieving a constant characteristic impedance across the operating band to prevent reflections, as noted earlier, and a specific phase response [19]. By using sections of transmission line of calculated electrical length (θ = βl, where β is the phase constant and l is physical length), a predictable phase shift (θ) or time delay (τ = l/v_p, where v_p is phase velocity) can be introduced. Artificial transmission lines, implemented with lumped LC elements, are particularly useful for creating large delays in a small volume [24][14].
Simulation and Numerical Analysis Techniques
Advanced applications of transmission line theory extend into the realm of computer-aided simulation and analysis. Efficient simulation of systems containing transmission lines, especially in power systems or mixed-signal circuits, requires specialized numerical models. The lumped-circuits model translates the distributed parameters into a discrete network suitable for nodal analysis in simulation software [8]. Furthermore, root-matching techniques, used for creating stable and accurate digital models of transmission lines in electromagnetic transient (EMT) simulation programs, have seen expanded application scope. Research in 2020 proposed an efficient EMT simulation method based on the discrete similarity principle, enhancing the modeling of line dynamics and transients [Source: 2020 method based on discrete similarity principle].
Specialized Components and Antenna Integration
The principles are further applied to create specialized components that integrate seamlessly with modern systems. For example, the design of a compact 5G nonuniform transmission line interdigital bandpass filter demonstrates direct integration with a reconfigurable antenna for 5G and ultra-wideband (UWB) applications. This approach uses the filter's transmission line structure as part of the antenna feeding network, showcasing a system-level application [7]. Additionally, artificial transmission line concepts are employed to design components like baluns, impedance transformers, and leaky-wave antennas, where the controlled propagation constant of the artificial line dictates the radiation properties [14].