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Microstrip

A microstrip is a planar consisting of a flat conducting strip placed on one side of a , with a on the opposite side, used to transmit and radio-frequency signals. This open structure allows electromagnetic fields to extend into both the and the surrounding air, resulting in a quasi-transverse electromagnetic (quasi-TEM) mode of propagation that supports frequencies from up to several gigahertz, though with frequency-dependent effective constants. Developed in the early as a compact alternative to and lines, microstrip enables the of passive and active components in microwave integrated circuits (MICs) and monolithic microwave integrated circuits (MMICs). Its typically ranges from 20 to 120 ohms, determined by the strip width, thickness, and constant (usually 2 to 10 times that of free space), making it suitable for applications despite losses from radiation, conductors, and dielectrics. Key advantages include ease of fabrication using techniques, small size, low weight, and compatibility with devices for hybrid and monolithic , though challenges like unwanted higher-order modes limit upper-frequency operation to below the point where thickness exceeds one-quarter . Microstrips are widely employed in antennas, filters, couplers, and RF front-ends for communications, systems, and .

Fundamentals

Definition and Configuration

A microstrip is a planar transmission line consisting of a conducting strip placed on one side of a dielectric substrate, with a ground plane on the opposite side of the substrate. This configuration enables the propagation of electromagnetic waves in a quasi-TEM mode and is widely used in microwave integrated circuits due to its simplicity and compatibility with planar fabrication processes. The primary components of a microstrip line include the strip conductor, defined by its width w and thickness t; the dielectric , characterized by its height h, \varepsilon_r, and \mu_r \approx 1; and the , which serves as the reference conductor. These elements form an open structure where the electromagnetic fields are partially confined within the substrate and partially in the air above it. The strip conductor is typically thin compared to the substrate height (t \ll h), minimizing its impact on overall performance. Common materials for the substrate include for cost-effective applications at lower frequencies, RT/Duroid (such as Rogers 5880 with \varepsilon_r \approx 2.2) and alumina (with \varepsilon_r \approx 9.8) for high-frequency microwave use due to their stable properties and low loss tangents. Conductors are usually for its conductivity and ease of deposition, or for corrosion resistance in demanding environments. The choice of materials influences mechanical stability, thermal performance, and in the final device. Geometric parameters such as the w/h are critical in , as they allow precise over the line's electrical characteristics, including to standard values like 50 Ω. Typical dimensions for frequencies feature substrate heights h ranging from 0.1 to 1 mm, with strip widths w scaled accordingly to achieve desired performance. Microstrip lines are commonly fabricated using for high-precision patterning in integrated circuits or PCB processes for circuits, enabling reproducible structures at these scales.

Historical Development

The origins of microstrip technology trace back to the early 1950s, when researchers at ITT Laboratories sought alternatives to bulky coaxial cables for high-frequency transmission. In 1952, D. D. Grieg and H. F. Engelmann introduced microstrip as a planar transmission line consisting of a strip conductor on a dielectric substrate backed by a ground plane, demonstrating its potential for kilomegacycle (microwave) frequencies in a seminal paper published in the Proceedings of the IRE. This innovation, patented and developed at ITT, addressed the need for compact, low-loss waveguides suitable for emerging radar and communication systems, marking the first practical realization of what would become a cornerstone of microwave engineering. During the and , microstrip evolved rapidly as a key element in integrated circuits (s), enabling the of passive and active components on a single substrate. Harold A. Wheeler advanced the theoretical foundation in 1965 with his analysis of transmission-line properties for parallel strips separated by a dielectric sheet, providing formulas for that facilitated precise design and widespread adoption. Concurrently, M. V. Schneider's 1969 work detailed microstrip lines specifically for MIC applications, emphasizing fabrication techniques using thin- and thick-film deposition on substrates like alumina, which supported circuits operating up to millimeter waves and spurred commercial MIC production for and . These contributions, building on early MIC concepts from the mid-, transformed microstrip from a novel line into the backbone of planar circuitry by the late . The 1980s saw microstrip's transition to monolithic microwave integrated circuits (MMICs), particularly with (GaAs) substrates that offered superior high-frequency performance over silicon. Early GaAs MMIC efforts, starting around 1972 but accelerating in the early , integrated microstrip interconnects with MESFET transistors, enabling compact amplifiers, mixers, and phase shifters for and systems. This shift reduced assembly costs and improved reliability, with microstrip's planar geometry proving ideal for GaAs processing, leading to the first commercial MMIC products by mid-decade. In the , microstrip expanded beyond pure analog applications into high-speed circuits, where its low and ease of with printed circuit boards (PCBs) addressed challenges in emerging gigabit data rates. Researchers developed coupled microstrip models for mixed-signal environments, supporting applications in GaAs-based logic and interconnects for computers and networks. Wheeler's impedance formulations remained influential, adapted for transmission lines. Recent milestones from the 2000s to 2025 have integrated microstrip with advanced structures like substrate-integrated waveguides (SIW), metamaterials, and / systems. SIW, pioneered in the early by D. Deslandes and K. Wu, embeds microstrip-like vias in substrates to mimic rectangular waveguides, enhancing low-loss performance for millimeter-wave circuits. By the 2010s, microstrip antennas incorporating metamaterials improved gain and bandwidth for base stations, with ongoing developments in reconfigurable designs for terahertz bands. As of 2025, hybrid microstrip-metamaterial patches enable compact, high-efficiency arrays for and beyond-5G networks, reflecting microstrip's enduring adaptability.

Electromagnetic Characteristics

Inhomogeneity and Quasi-TEM Mode

Microstrip transmission lines exhibit inherent inhomogeneity arising from the , where the region above the conducting strip is filled with air (permittivity ε = 1) and the region below consists of a with ε_r > 1. This configuration leads to hybrid electromagnetic fields that are not confined solely to either medium, with lines partially embedded in the and partially fringing into the air. Consequently, microstrip cannot support a pure transverse electromagnetic (TEM) mode, as the differing wave velocities in air and the prevent uniform transverse field ; instead, the dominant mode is quasi-TEM, featuring small but non-zero longitudinal components of both electric (E_z) and magnetic (H_z) fields. The quasi-TEM approximation holds well for narrow strips, where the width-to-substrate-height ratio (w/h) is low, and at low frequencies, typically f < 10–20 GHz, provided the operating wavelength significantly exceeds the cross-sectional dimensions. In the quasi-TEM mode, the field distribution is predominantly transverse, with concentrated electric fields between the strip and ground plane in the dielectric, but substantial fringing fields extend laterally and upward into the air, blending characteristics of both TEM and higher-order TE/TM modes without fully aligning with either. This fringing effect arises from the sharp discontinuity at the dielectric-air boundary, distorting the field lines and contributing to the hybrid nature of the mode. Beyond the quasi-TEM regime, higher-order modes such as the lowest-order TM mode emerge when the frequency surpasses their cutoff values, which are determined by the strip geometry and substrate parameters, potentially leading to multimode propagation and signal distortion. These cutoffs, often in the range of several GHz for typical substrates, define the practical upper frequency limit for reliable quasi-TEM operation in microstrip designs. The structural inhomogeneity further manifests in dispersion, where the phase velocity of the quasi-TEM mode decreases with frequency, transitioning from a higher value at low frequencies (with fields more extended into air) toward the slower velocity in the dielectric at higher frequencies, thereby altering the effective propagation characteristics.

Effective Dielectric Constant

In microstrip transmission lines, the effective dielectric constant, denoted as \epsilon_\text{eff}, represents the weighted average permittivity experienced by the propagating quasi-TEM wave due to the inhomogeneous dielectric environment, where part of the electromagnetic field resides in the substrate with relative permittivity \epsilon_r and the rest in air with \epsilon_r = 1. This parameter bridges the propagation characteristics between homogeneous and inhomogeneous media, enabling accurate prediction of wave behavior without full field solutions. For low frequencies where the quasi-static approximation holds, \epsilon_\text{eff} can be estimated using a closed-form expression derived from electrostatic analysis: \epsilon_\text{eff} = \frac{\epsilon_r + 1}{2} + \frac{\epsilon_r - 1}{2} \cdot \frac{1}{\sqrt{1 + 12 \frac{h}{w}}} where h is the substrate thickness and w is the strip width; this formula assumes zero strip thickness and provides accuracy within 1% for typical geometries. For finite strip thickness t, a correction term is added to account for increased fringing fields: \Delta \epsilon_\text{eff} = \frac{t}{2\pi h} \cdot \frac{\epsilon_r (\epsilon_r - 1)}{2} \left( \ln \left( \frac{2\pi e (w + 1.25 t/ \epsilon_r)}{t} \right) + 0.04 (1 - w/t)^2 \right) yielding \epsilon_\text{eff}^\prime = \epsilon_\text{eff} + \Delta \epsilon_\text{eff}, which improves precision for practical conductor dimensions where t/h > 0.01. At higher frequencies, \epsilon_\text{eff} becomes frequency-dependent due to dispersion effects from the hybrid nature of the modes, where transverse electric and magnetic field components vary with frequency. Getsinger's model treats the microstrip as a parallel-plate waveguide filled with an effective dielectric that increases with frequency, providing a dispersion relation: \epsilon_\text{eff}(f) = \epsilon_r - \frac{\epsilon_r - \epsilon_\text{eff}(0)}{1 + \left( \frac{f}{f_{50}} \right)^p} with f_{50} a knee frequency scaling as h^{-1} and p \approx 2, capturing the transition to asymptotic \epsilon_r behavior; this model agrees with measurements to within 0.5% up to millimeter waves. Precise computation of \epsilon_\text{eff} for zero-thickness strips relies on numerical methods such as conformal mapping, which transforms the inhomogeneous geometry into a parallel-plate equivalent to derive exact capacitance ratios, yielding \epsilon_\text{eff} = 1 + q (\epsilon_r - 1) where q is the filling factor obtained via elliptic integrals. Hammerstad and Jensen refined this approach into highly accurate closed-form expressions valid for $1 < \epsilon_r < 128 and wide w/h ratios, with errors below 0.2%. The effective dielectric constant directly influences the phase velocity v_p of the propagating wave, given by v_p = c / \sqrt{\epsilon_\text{eff}}, where c is the speed of light in vacuum; thus, \epsilon_\text{eff} determines the effective electrical length and timing in circuits, with typical values ranging from 1.5 to 8 for common substrates like alumina (\epsilon_r \approx 10).

Characteristic Impedance

Calculation Methods

The calculation of the characteristic impedance Z_0 of a microstrip line can be approached through analysis, where Z_0 is computed from given physical dimensions such as strip width w, substrate height h, and strip thickness t, or through synthesis, where dimensions like the width-to-height ratio w/h are determined to achieve a target Z_0. These methods rely on the effective dielectric constant \epsilon_{eff}, which accounts for the inhomogeneous field distribution in the microstrip structure. One of the earliest and widely used empirical formulas for Z_0 was developed by Wheeler for zero-thickness strips. For narrow strips where w/h < 1, the formula is Z_0 = \frac{60}{\sqrt{\epsilon_{eff}}} \ln \left( \frac{8h}{w} + \frac{w}{4h} \right), while for wide strips where w/h > 1, it approximates to Z_0 = \frac{120\pi}{\sqrt{\epsilon_{eff}} \left[ \frac{w}{h} + 1.393 + 0.667 \ln \left( \frac{w}{h} + 1.444 \right) \right]}. These expressions provide good accuracy for many practical designs, with errors typically below 1% compared to measured values. For improved precision, especially when accounting for finite strip thickness t, the Hammerstad-Jensen model offers a more accurate closed-form . This approach modifies the per unit length to include thickness effects through correction factors, yielding Z_0 values with errors under 0.2% over a wide range of w/h ratios and substrate permittivities. The model is particularly useful for where conductor thickness impacts fringing fields significantly. In cases involving complex geometries or higher frequencies where analytical approximations falter, numerical methods such as the Method of Moments (MoM) or (FEM) are employed to solve directly. MoM discretizes the strip surface into basis functions to compute charge distribution and , enabling precise Z_0 extraction. FEM, on the other hand, meshes the cross-section and iteratively solves for field solutions, suitable for inhomogeneous dielectrics. These techniques, introduced in seminal works on planar structures, achieve accuracies better than 0.1% but require computational resources. Commercial software tools facilitate these calculations in practice. Keysight's Advanced Design System (ADS) integrates empirical models like Hammerstad-Jensen with MoM solvers for rapid analysis and synthesis. Similarly, Ansys HFSS employs FEM for full-wave simulations, allowing verification of Z_0 including dispersion effects. These tools streamline design workflows while maintaining traceability to underlying analytical foundations.

Influence of Enclosure

The presence of a metallic enclosure around a microstrip line significantly alters its electromagnetic behavior by confining the fringing fields that normally extend into the air region above the substrate. This confinement increases the proportion of the electric field lines within the dielectric substrate, thereby raising the effective dielectric constant ε_eff compared to an open structure. Consequently, the characteristic impedance Z_0 decreases for the same strip width and substrate thickness, as Z_0 is inversely related to the square root of ε_eff. In partial enclosures, such as those with only a top metallic (covered microstrip), the primary impact is on the upper fringing fields, leading to a moderate increase in ε_eff and reduction in Z_0, particularly when the cover height is comparable to the substrate thickness. Full enclosures, or boxed microstrip, incorporate side walls in addition to the top , further restricting lateral fringing fields and providing more uniform field confinement. These shielding types suppress higher-order modes by defining boundary conditions that prevent spurious waveguide-like propagation, which is beneficial for maintaining quasi-TEM operation up to higher frequencies. However, if the enclosure dimensions are not sufficiently large, resonances can arise, introducing frequency-dependent variations in Z_0 and potential unwanted between circuit elements. The trade-offs of enclosing a microstrip line include reduced radiation losses and improved (EMI) shielding, which enhance in dense integrated circuits, but at the cost of potentially higher ohmic and losses at elevated frequencies due to increased current crowding near the metal walls. Experimental measurements on shielded structures, using techniques like ring resonators for dispersion characterization, have validated theoretical models, showing deviations in predicted Z_0 of less than 5% when hybrid-mode analysis is employed. Design guidelines for enclosed microstrip emphasize dimensioning the to minimize perturbations while achieving shielding benefits: the height from the to the top cover should exceed 5 to 8 times the substrate thickness h, and the lateral distance from the strip edges to side walls should be at least 4 to 5 times the strip width w to render enclosure effects negligible on Z_0. For tighter enclosures where corrections are needed, empirical adjustments to open-line Z_0 formulas are applied, often incorporating geometric ratios like cover height to substrate thickness; full-wave electromagnetic simulations are recommended for precise optimization in high-frequency applications.

Variations

Suspended Microstrip

Suspended microstrip is a variant of the microstrip in which the conducting strip is positioned above a with an intervening air gap, often supported by low-ε_r , bridges, or support posts. This configuration places the substrate directly on the ground plane, but the air gap between the strip and substrate reduces the portion of the confined within the . The primary benefits of suspended microstrip over standard microstrip include a lower effective constant, which enables higher impedances, reduced and losses, wider , and minimized effects. These improvements arise from the increased field presence in the low-loss air region, leading to higher Q-factors and better performance in high-frequency circuits. For instance, suspended structures exhibit significantly lower transmission loss at millimeter-wave frequencies, such as 60 GHz. Characteristic impedance calculations for suspended microstrip adapt standard microstrip formulas by incorporating a modified effective constant that accounts for the air gap height d. Specifically, \varepsilon_\text{eff} is approximated as \varepsilon_\text{eff} = 1 + \frac{h}{h + d} (\varepsilon_r - 1), where h is the substrate thickness and \varepsilon_r is the substrate constant, reflecting the reduced dielectric filling factor. The impedance Z_0 then follows the conventional microstrip expression using this \varepsilon_\text{eff}. Fabrication of suspended microstrip presents challenges, including the precise creation and maintenance of the air gap using support posts or bridges to avoid mechanical instability or additional losses from support materials. These structures require careful alignment and may involve multi-layer processing for in complex circuits. Suspended microstrip finds applications in high-power components, such as filters, directional couplers, and transitions, particularly up to mm-wave frequencies, where its low-loss characteristics and high power-handling capability—enhanced by better heat dissipation through the air gap—are critical.

Inverted Microstrip

The inverted microstrip consists of a conducting strip positioned directly beneath a , with the located on the top surface of the substrate, thereby placing the material primarily between the strip and the , unlike the standard microstrip where the strip resides atop the substrate. This configuration results in stronger field confinement within the , leading to an effective dielectric constant \epsilon_\text{eff} that approaches the substrate's \epsilon_r, particularly for wider strips where fringing fields into the air below the strip are minimized. Key advantages of the inverted microstrip include enhanced shielding against radiation losses due to the ground plane's position above the , which reduces spurious emissions compared to the open structure of conventional microstrip lines. It also exhibits lower because of the more homogeneous environment for the propagating fields, making it suitable for applications, and supports integration in multilayer printed circuit boards (PCBs) where the structure can be between layers for improved . The characteristic impedance Z_0 of an inverted microstrip can be approximated using the parallel-plate transmission line model, given by Z_0 \approx \frac{377}{\sqrt{\epsilon_r}} \cdot \frac{h}{w} for wide strips (w \gg h), where h is the substrate thickness and w is the strip width; this reflects the near-full immersion of fields in the dielectric. More precise calculations often require numerical methods, as closed-form expressions account for fringing effects but vary with geometry. Despite these benefits, the inverted microstrip presents drawbacks such as increased fabrication complexity, particularly in multilayer environments where precise alignment of the embedded is needed, and potential for higher conductor losses due to challenges in achieving uniform contact between the strip and . Additionally, it may support higher-order modes more readily in embedded forms, complicating single-mode operation in integrated circuits. In integrated circuits, the inverted microstrip is employed for its superior isolation properties, enabling compact routing in silicon-based RFICs or modules where minimizing between adjacent lines is critical.

Discontinuities

Bends

Bends in microstrip lines are essential for routing signals in compact circuits, but they introduce discontinuities that disrupt the and generate unwanted radiation. These effects arise primarily from abrupt changes in the current path, leading to excess electric and at the bend region. Sharp right-angle bends, whether unmitered or mitered, and smooth curved arcs represent the main types, each with distinct modeling approaches and performance characteristics. Right-angle bends exhibit discontinuity effects modeled by an consisting of a shunt due to charge accumulation at the inner corner and a series from current crowding at the outer edge. The shunt is given by C_b \approx 0.02 \left( \frac{w}{h} \right) pF for substrates with relative \varepsilon_r \leq 9.6, where w is the strip width and h is the thickness. The associated series is approximately L_b \approx 0.23 \left( \frac{w}{h} \right)^{-0.5} nH, with accuracy within ±3% for $0.1 \leq w/h \leq 2.0. These lumped elements cause impedance mismatches, resulting in reflections that increase with frequency; for unmitered bends, can exceed -10 above several GHz depending on the line width. Radiation losses in bends stem from the excitation of transverse electromagnetic fields, with power loss increasing as the bend angle sharpens due to enhanced fringing. For 90-degree right-angle bends on typical substrates, radiated power can account for up to 11% of the incident energy, compared to 6-7% for straight sections, as measured using TEM cell techniques and numerical simulations. Curved arc bends mitigate some by distributing the turn over a , but small radii still induce losses proportional to the ; equivalent circuit models for radial bends incorporate distributed and along the arc to predict these effects. Compensation techniques focus on minimizing reflections and radiation while preserving compact layouts. Mitered bends, featuring a 45° cut at the corner with a miter length of approximately $0.5 \sqrt{2} w, reduce excess by nearly half and improve by over 16 dB across 2-20 GHz compared to unmitered designs. For optimal performance, a 50% miter ratio is recommended for wider lines (w/h > 1), as it aligns the path more smoothly and limits errors to under 2.5° at 0.2\lambda_g, where \lambda_g is the guided . In curved bends, increasing the bend radius beyond three times the strip width decreases discontinuity and , though it trades off circuit size; S-parameter analysis reveals insertion losses below 0.5 dB and return losses exceeding -20 dB for radii optimized via full-wave simulations. These methods ensure bend performance aligns closely with the nominal of the straight microstrip sections.

Junctions

Junctions in microstrip transmission lines represent abrupt changes in structure, such as variations in strip width or branching configurations, which introduce discontinuities that disrupt the quasi-TEM mode propagation. These elements are essential in circuits for power division, filtering, and matching, but they generate parasitic reactances leading to reflections and insertion losses if not properly modeled. Common types include step junctions, where the strip width changes suddenly; T-junctions, featuring a main line with a ; and cross-junctions, involving two intersecting lines. Step junctions occur at transitions between microstrip sections of differing widths, primarily inducing a shunt due to fringing fields at the wider strip's edge. This arises from excess charge accumulation and can be approximated using fringing field models, such as C_p = \frac{\varepsilon_0 (\varepsilon_{eff} + 1)}{2} (w_2 - w_1) \ln\left(1 + \frac{2h}{w_1}\right) for small steps (where w_2 > w_1), or more accurate empirical expressions accounting for width ratio and substrate permittivity. A series also arises from the constriction at the narrower strip, modeled empirically as L_s \approx \mu_0 h \left[ 0.2 \ln\left(\frac{w_2}{w_1}\right) + 0.2235 \right] \left( \frac{Z_{01}}{Z_{02}} - 1 \right), where Z_{01} and Z_{02} are the impedances of the narrower and wider lines, respectively. models, typically pi- or T-networks, represent these as lumped elements: a shunt across the junction flanked by series inductors for the step. T-junctions and cross-junctions exhibit more complex behavior due to their three- or four-port nature, exciting higher-order modes and orthogonal fields that amplify parasitic . In T-junctions, the branch arm introduces a shunt and series inductances in a T-network , with values depending on arm widths and parameters, leading to frequency-dependent . Cross-junctions similarly model as pi-networks with central shunt and arm inductances, where parasitic elements are determined by junction geometry and effective constant using planar analysis or methods, causing mode conversion and losses. These effects are analyzed using full-wave methods like techniques for accurate multi-mode characterization up to millimeter-wave frequencies. To mitigate reflections from these junctions, tapered transitions are employed, such as linear ramps or curves that gradually vary the strip width over a length of several wavelengths, achieving with below -20 dB across octave bands. For complex or high-frequency designs, full-wave electromagnetic simulations using tools like finite element or method-of-moments solvers are recommended to capture dispersive effects and influences not accounted for in quasi-static models.

Coupled Microstrips

Even and Odd Mode Analysis

Coupled microstrip lines consist of two parallel conducting strips separated by a small s on a above a , forming a structure that supports two primary quasi-TEM propagation modes: the even mode, characterized by symmetric field distributions, and the odd mode, characterized by antisymmetric field distributions. In the even mode, the voltages along the two strips are in and equal in magnitude, resulting in a magnetic wall at the plane of between the strips, where the tangential is maximum and no transverse current flows across this plane. For the odd mode, the voltages are equal in magnitude but 180 degrees out of , establishing an electric wall at the symmetry plane, where the potential is zero and the tangential is maximum. These conditions simplify the by allowing the coupled system to be modeled as two independent single microstrip lines with modified geometries. The inhomogeneous dielectric environment of microstrip lines leads to distinct effective permittivities for each , causing differences in velocities. The even has a higher effective dielectric constant (\epsilon_{\text{eff,even}}) because more lines penetrate the to the , whereas the odd exhibits a lower \epsilon_{\text{eff,odd}} as fields concentrate primarily in the air region between the strips, reducing influence. Consequently, the odd-mode (v_{\text{odd}}) exceeds the even-mode (v_{\text{even}}), with v_{\text{even}} < v_{\text{odd}}. This velocity disparity is quantified by the velocity coupling coefficient k_v = \frac{v_{\text{odd}} - v_{\text{even}}}{v_{\text{odd}} + v_{\text{even}}}, which measures the modal mismatch and impacts device performance in applications like directional couplers. The even- and odd-mode analysis framework for coupled transmission lines emerged from the foundational developments in coupled mode theory during the 1950s, pioneered for microwave applications by researchers such as and .

Characteristic Impedances

The characteristic impedances of symmetric coupled microstrip lines are determined separately for the even and odd modes, reflecting the distinct field distributions in each case. The even-mode characteristic impedance Z_{0e} is given by Z_{0e} = \sqrt{L_e / C_e}, where L_e and C_e are the per-unit-length inductance and capacitance under even-mode excitation, respectively. Similarly, the odd-mode characteristic impedance Z_{0o} is Z_{0o} = \sqrt{L_o / C_o}, with L_o and C_o defined analogously for odd-mode excitation. In symmetric coupled microstrips, Z_{0e} > Z_{0o} because the odd mode experiences higher effective capacitance due to enhanced fringing fields across the gap between the strips. Analytical approximations for these impedances leverage the single microstrip Z_{0,\text{single}}. For weak , the even-mode impedance can be estimated as Z_{0e} \approx Z_{0,\text{single}} (1 + k), where k is the voltage , defined as k = (Z_{0e} - Z_{0o}) / (Z_{0e} + Z_{0o}). This arises from perturbation analysis of the mutual and contributions. For the odd mode, the image plane method exploits the : the midway between the strips acts as an electric , effectively modeling the structure as two isolated microstrips with mirrored charges, allowing Z_{0o} to be computed using single-line formulas adjusted for the effective strip separation s/2. These methods provide closed-form estimates suitable for initial design, with accuracy improving for low levels (k < 0.2). More precise calculations rely on determining the capacitance matrix [C] via numerical solution of the two-dimensional electrostatic problem. The elements C_{11} (self-capacitance) and C_{12} (mutual capacitance) per unit length yield the mode capacitances as C_e = C_{11} + C_{12} and C_o = C_{11} - C_{12}, from which the impedances follow after accounting for the effective dielectric permittivity in each mode: Z_{0e,o} = \frac{1}{c_0 \sqrt{\varepsilon_{\text{eff},e,o}} C_{e,o}}, where c_0 is the speed of light in vacuum. A Fourier series solution for the capacitance matrix employs the Green's function approach in the spectral domain, expanding the charge distribution on the strips as a series of basis functions (e.g., Chebyshev polynomials or rooftops) and the dyadic Green's function as a Fourier integral over the substrate layers. This method solves the integral equation for the charge density, enabling extraction of [C] with convergence controlled by the number of terms (typically 20–50 for 1% accuracy). The technique, foundational to quasi-TEM analysis, handles the inhomogeneity of microstrip effectively. For high-accuracy modeling, the conformal mapping solution detailed by Chen provides explicit expressions for both modes, incorporating the substrate permittivity \varepsilon_r and normalized dimensions w/h and s/h. These derive from mapping the microstrip geometry to a parallel-plate equivalent, yielding Z_{0e} and Z_{0o} as functions of elliptic integrals adjusted for fringing. The approach achieves errors below 3% compared to rigorous numerical methods for typical geometries. As an illustrative example, consider symmetric coupled microstrips with strip width w = 1 mm, gap s = 0.5 mm, substrate height h = 0.635 mm, and \varepsilon_r = 4.5. The single-line Z_{0,\text{single}} \approx 48.5 Ω; using Chen's formulas, Z_{0e} \approx 62.3 Ω and Z_{0o} \approx 36.8 Ω, while full-wave simulation (e.g., via method of moments) yields 63.1 Ω and 36.2 Ω, respectively, demonstrating formula accuracy within 1.3%.

Asymmetric and Multiple Coupling

In asymmetric coupled microstrip lines, where the strip widths differ (w_1 \neq w_2), the coupling cannot be simplified to pure even and odd modes as in symmetric pairs; instead, a full 2×2 impedance matrix [Z] is required, comprising self-impedances Z_{11} and Z_{22} (which differ due to asymmetry) and mutual impedance Z_{12} = Z_{21}. These elements are derived from self-capacitances C_{11}, C_{22}, mutual capacitance C_{12}, and corresponding inductances, with empirical models providing closed-form expressions for design, such as C_{ii} = \varepsilon_0 \varepsilon_r \left[ A + B \left( \frac{w_i}{h} \right) + C \left( \frac{s}{h} \right) \right] where A, B, and C are coefficients dependent on substrate thickness h, separation s, and relative permittivity \varepsilon_r. The resulting modes, often termed c-mode and π-mode, exhibit mixed field distributions that account for the unequal line geometries. Analysis of these structures typically employs numerical methods, such as the spectral domain approach, which transforms the problem into the Fourier domain to solve for mixed-mode propagation constants and impedances via Galerkin's method or integral equations. This method efficiently handles the hybrid nature of microstrip modes (quasi-TEM with higher-order effects) and provides accurate results for dispersion and coupling coefficients, validated against measurements with errors below 2% in effective permittivity for lines with w_1/h = 0.5, w_2/h = 1.5, and s/h = 0.2. For multiple coupled microstrip lines (N > 2), the analysis extends to an N \times N impedance matrix, often using parameters to model voltage and current propagation along the coupled system, or supermode decomposition where the structure supports N orthogonal modes derived from eigenvalue solutions of the per-unit-length and matrices. These supermodes represent collective excitations, enabling efficient simulation of and in dense interconnects. Such configurations are commonly applied in power dividers, where multiple coupled sections achieve compact designs with isolation resistors, as in π-type structures. In directional couplers, asymmetric designs with unequal line widths enable controlled unequal power splitting (e.g., 10:1 ratios) by tailoring the coupling coefficient k = Z_{12}/\sqrt{Z_{11}Z_{22}}, offering flexibility for applications like signal where one output requires significantly lower . Accuracy comparisons for multi-line cases show that Fourier-based spectral domain methods yield capacitance values within 1-3% of benchmarks for N=3 lines on \varepsilon_r = 10.8 substrates, comparable to method-of-moments results but with lower computational cost due to closed-form Green's functions.

Losses and Dispersion

Conductor and Dielectric Losses

In microstrip transmission lines, conductor losses stem from the ohmic resistance of the metallic strip and ground plane, dominated by the skin effect at microwave frequencies. The skin effect confines currents to a thin layer near the conductor surface, yielding a surface resistance R_s = \sqrt{\pi f \mu / \sigma}, where f is the operating frequency, \mu is the magnetic permeability, and \sigma is the conductor conductivity (typically for copper, \sigma \approx 5.8 \times 10^7 S/m). This resistance contributes to the attenuation constant \alpha_c \approx R_s / (Z_0 w) for the dominant strip contribution, where Z_0 is the characteristic impedance and w is the strip width; contributions from the ground plane are smaller but can be included via an effective width adjustment. Dielectric losses arise from energy dissipation in the material, characterized by its loss tangent \tan \delta. The due to these losses is approximated as \alpha_d = \frac{k_0}{2} \sqrt{\epsilon_\mathrm{eff}} \tan \delta \frac{\epsilon_\mathrm{eff} - 1}{\epsilon_r - 1}, where k_0 = 2\pi f / c is the free-space , \epsilon_\mathrm{eff} is the effective , \epsilon_r is the of the , and c is the . This expression accounts for the quasi-TEM mode where fields partially occupy air, reducing the effective loss compared to fully embedded lines via the filling factor \frac{\epsilon_\mathrm{eff} - 1}{\epsilon_r - 1}, which represents the fraction of electric energy in the . The total attenuation \alpha = \alpha_c + \alpha_d exhibits frequency dependence: \alpha_c scales with \sqrt{f} due to the skin depth decreasing as $1/\sqrt{f}, while \alpha_d increases linearly with f as dissipation grows proportionally to the electric field energy stored in the dielectric. To separate these components experimentally, the Wheeler incremental inductance method measures the change in quality factor or resonant frequency of a microstrip resonator upon small dimensional perturbations, isolating conductor effects from dielectric ones. In coupled microstrips, the odd mode suffers slightly higher attenuation than the even mode owing to enhanced electric fields in the coupling region, amplifying both conductor and dielectric contributions.

Radiation and Surface Wave Losses

In microstrip transmission lines, radiation losses manifest as power leakage into free space, primarily triggered by discontinuities such as abrupt open ends, bends, and junctions, where asymmetric fringing fields disrupt guided wave propagation. These effects are exacerbated in regions of high curvature, like sharp corners, leading to unintended that reduces and efficiency. For instance, an abruptly open-circuited microstrip line can radiate significantly due to the sudden termination, with the radiated fields behaving similarly to those of a . Radiation losses increase with operating , as higher frequencies enhance field fringing and scaling, and with strip width, which broadens the radiating aperture. Mitigation strategies include the use of guard rings around discontinuities to confine fields and metallic shielding enclosures to suppress outward , thereby reducing spurious emissions. Surface wave losses occur when guided energy couples to substrate-bound waves, particularly the dominant TM_0 , which lacks a and propagates at all frequencies within the . Excitation of this is prominent in structures with thick (large height h) or wide strips (w), as these dimensions increase the overlap between the microstrip fields and the substrate interface, promoting power transfer to non-radiating that the signal. models for surface waves typically involve solving for the 's and power distribution, showing that losses rise with frequency due to enhanced coupling and with strip width via greater extent into the . These radiation and mechanisms contribute to in microstrip lines, where the effective \varepsilon_{\text{eff}} varies with frequency, resulting in group velocity fluctuations that distort . primarily arises from the frequency dependence of the effective constant due to the nature of modes and to higher-order modes, modeled using analytical approaches such as the Schneider model. The inherent inhomogeneity of the microstrip—spanning air and regions—facilitates this to both radiative space waves and substrate-bound modes. Liquid crystal polymers (LCP) offer low tangent (tan \delta < 0.002) and dimensional stability up to millimeter waves, enabling microstrip designs with insertion losses as low as 0.11 dB/mm at 40 GHz (as measured around 2007), with ongoing applications in 5G circuits as of 2020. Hybrid LCP formulations further reduce transmission losses compared to traditional polyimides, supporting high-frequency applications while minimizing surface wave excitation.

Applications

Microwave Circuits and PCBs

Microstrip transmission lines play a pivotal role in monolithic microwave integrated circuits (MMICs) and microwave integrated circuits (MICs), enabling the design of compact components such as filters, low-noise amplifiers, and oscillators that operate across frequencies from 1 to 100 GHz. In MMICs, microstrip structures facilitate high-integration levels for applications like X-band amplifiers, where they support low insertion loss and precise impedance matching essential for signal amplification in radar and communication systems. Similarly, microstrip-based oscillators and filters leverage the planar geometry to achieve tunable performance in microwave subsystems, minimizing parasitic effects at these frequencies. In printed circuit boards (PCBs), microstrip lines are employed as impedance-controlled traces to ensure signal integrity in high-speed digital interfaces, including and , supporting data rates up to 32 GT/s per lane. For implementations, microstrip traces maintain a differential impedance of 90 Ω ±5 Ω, reducing reflections and crosstalk in environments with bit rates exceeding 5 Gbps. Characteristic impedance control in these traces is critical for preserving waveform quality over extended lengths. In designs, microstrip routing on outer layers helps mitigate losses while adhering to 100 Ω differential standards, enabling reliable operation in server and networking hardware. Key design considerations for microstrip in PCBs include via transitions and multilayer stacking, where microstrip is typically placed on outer layers to interface with embedded striplines. Via transitions, such as microstrip-to-stripline conversions, require careful impedance matching to minimize parasitic inductance, often using blind or buried vias with diameters of 0.2–0.4 mm and spacing at least λ/20 for RF isolation. In multilayer configurations, stitching vias provide continuous ground paths across layers, enhancing shielding and reducing radiation in high-frequency signals. Recent advancements in 5G mm-wave PCBs incorporate microstrip lines with low-loss dielectrics, such as those with dielectric constants (Dk) below 3.5 and loss tangents (Df) under 0.003, to support frequencies from 24 to 53 GHz while meeting 2025 3GPP standards for sub-6 GHz and mm-wave bands. These materials, like Rogers RO3000 series, enable hybrid integration of mm-wave modules with minimal insertion loss in smartphone and base station PCBs. A notable case study involves microstrip directional couplers in Doppler motion detection sensors, such as the MURENA system operating at 10–11 GHz, where the coupler achieves a coupling factor of -6.1 dB, directivity over 28 dB, and 84% efficiency for coherent signal mixing in radar-like security applications. This integration demonstrates microstrip's utility in compact, low-cost radar subsystems by cascading coupled lines on standard PCBs with ε_r = 3.5.

Antennas and Filters

Microstrip antennas, commonly implemented as patch antennas, consist of a metallic —typically rectangular or circular—etched on one side of a substrate with a ground plane on the opposite side. Rectangular patches resonate in the fundamental TM10 mode when the patch length equals approximately half the guided (λg/2), determined by the substrate's effective constant and thickness. Circular patches operate similarly, with occurring when the patch is approximately 0.293 λ_g (or 0.586 λ_g / 2) for the dominant TM11 mode, offering potential. These configurations provide broadside radiation patterns with gains around 6-7 dBi, though inherent bandwidths are narrow, typically 1-5% for below -10 dB, limited by the substrate's low Q-factor and thin profile. In filter applications, microstrip structures enable compact frequency-selective devices, such as edge-coupled bandpass formed by parallel coupled-line sections that act as λ/2 resonators. These leverage coupled-line for even- and odd-mode propagation to achieve bandpass responses, with design parameters like coupling gaps and line widths controlling the center and . Stub-loaded use open- or short-circuited stubs—typically λ/4 long at the center —connected in shunt or series to create bandpass or bandstop characteristics, offering simplicity for narrowband selectivity. Interdigital employ interleaved finger-like resonators in a comb structure, providing higher-order responses and sharper roll-offs in a compact layout, suitable for integration in RF front-ends. The primary advantages of microstrip antennas and filters stem from their planar, low-profile nature, facilitating easy fabrication on printed circuit boards and seamless integration with feed networks like microstrip lines or corporate feeds for array configurations. This conformability supports applications in and mobile systems where space and weight constraints are critical. Post-2010 advances have focused on bandwidth enhancement through loading, where structures like split-ring resonators or complementary split-ring resonators are incorporated as superstrates or defected grounds to introduce negative or permeability, effectively broadening the impedance match. For instance, -loaded patches can increase by 2-3 times compared to conventional designs by exciting additional resonances and improving radiation efficiency. A representative example is a 2.4 GHz rectangular microstrip for applications, designed on an substrate (εr = 4.4, h = 1.6 mm) with dimensions of length 28.61 mm and width 37.93 mm, fed via an inset microstrip line of width 2.52 mm and depth 10.69 mm. This configuration achieves resonance at 2.393 GHz with a of 58 MHz (∼2.4%), of -38.86 dB, and VSWR of 1.02, demonstrating efficient performance for S-band communications.