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**Microstrip** is a type of electrical transmission line which can be fabricated using printed circuit board technology, and is used to convey microwave-frequency signals. It consists of a conducting strip separated from a ground plane by a dielectric layer known as the substrate. Microwave components such as antennas, couplers, filters, power dividers etc. can be formed from microstrip, with the entire device existing as the pattern of metallization on the substrate. Microstrip is thus much less expensive than traditional waveguide technology, as well as being far lighter and more compact. Microstrip was developed by ITT laboratories as a competitor to stripline (first published by Grieg and Engelmann in the December 1952 IRE proceedings^{ [1] }).

In radio-frequency engineering, a **transmission line** is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

A **printed circuit board** (**PCB**) mechanically supports and electrically connects electronic components or electrical components using conductive tracks, pads and other features etched from one or more sheet layers of copper laminated onto and/or between sheet layers of a non-conductive substrate. Components are generally soldered onto the PCB to both electrically connect and mechanically fasten them to it.

**Microwaves** are a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter; with frequencies between 300 MHz (1 m) and 300 GHz (1 mm). Different sources define different frequency ranges as microwaves; the above broad definition includes both UHF and EHF bands. A more common definition in radio engineering is the range between 1 and 100 GHz. In all cases, microwaves include the entire SHF band at minimum. Frequencies in the microwave range are often referred to by their IEEE radar band designations: S, C, X, K_{u}, K, or K_{a} band, or by similar NATO or EU designations.

The disadvantages of microstrip compared with waveguide are the generally lower power handling capacity, and higher losses. Also, unlike waveguide, microstrip is not enclosed, and is therefore susceptible to cross-talk and unintentional radiation.

For lowest cost, microstrip devices may be built on an ordinary FR-4 (standard PCB) substrate. However it is often found that the dielectric losses in FR4 are too high at microwave frequencies, and that the dielectric constant is not sufficiently tightly controlled. For these reasons, an alumina substrate is commonly used.

**FR-4** is a NEMA grade designation for glass-reinforced epoxy laminate material. FR-4 is a composite material composed of woven fiberglass cloth with an epoxy resin binder that is flame resistant (*self-extinguishing*).

On a smaller scale, microstrip transmission lines are also built into monolithic microwave integrated circuits.

A **Monolithic Microwave Integrated Circuit**, or **MMIC**, is a type of integrated circuit (IC) device that operates at microwave frequencies. These devices typically perform functions such as microwave mixing, power amplification, low-noise amplification, and high-frequency switching. Inputs and outputs on MMIC devices are frequently matched to a characteristic impedance of 50 ohms. This makes them easier to use, as cascading of MMICs does not then require an external matching network. Additionally, most microwave test equipment is designed to operate in a 50-ohm environment.

Microstrip lines are also used in high-speed digital PCB designs, where signals need to be routed from one part of the assembly to another with minimal distortion, and avoiding high cross-talk and radiation.

Microstrip is one of many forms of planar transmission line, others include stripline and coplanar waveguide, and it is possible to integrate all of these on the same substrate.

**Planar transmission lines** are transmission lines with conductors, or in some cases dielectric (insulating) strips, that are flat, ribbon-shaped lines. They are used to interconnect components on printed circuits and integrated circuits working at microwave frequencies because the planar type fits in well with the manufacturing methods for these components. Transmission lines are more than simply interconnections. With simple interconnections the propagation of the electromagnetic wave along the wire is fast enough to be considered instantaneous, and the voltages at each end of the wire can be considered identical. If the wire is longer than a large fraction of a wavelength these assumptions are no longer true and transmission line theory must be used instead. With transmission lines, the geometry of the line is precisely controlled so that its electrical behaviour is highly predictable. At lower frequencies, these considerations are only necessary for the cables connecting different pieces of equipment, but at microwave frequencies the distance at which transmission line theory becomes necessary is measured in millimetres. Hence, transmission lines are needed *within* circuits.

**Stripline** is a transverse electromagnetic (TEM) transmission line medium invented by Robert M. Barrett of the Air Force Cambridge Research Centre in the 1950s. Stripline is the earliest form of planar transmission line.

**Coplanar waveguide** is a type of electrical planar transmission line which can be fabricated using printed circuit board technology, and is used to convey microwave-frequency signals. On a smaller scale, coplanar waveguide transmission lines are also built into monolithic microwave integrated circuits. Conventional coplanar waveguide (CPW) consists of a single conducting track printed onto a dielectric substrate, together with a pair of return conductors, one to either side of the track. All three conductors are on the same side of the substrate, and hence are *coplanar*. The return conductors are separated from the central track by a small gap, which has an unvarying width along the length of the line. Away from the cental conductor, the return conductors usually extend to an indefinite but large distance, so that each is notionally a semi-infinite plane.

A differential microstrip—a balanced signal pair of microstrip lines—is often used for high-speed signals such as DDR2 SDRAM clocks, USB Hi-Speed data lines, PCI Express data lines, LVDS data lines, etc., often all on the same PCB.^{ [2] }^{ [3] }^{ [4] } Most PCB design tools support such differential pairs.^{ [5] }^{ [6] }

**DDR2 SDRAM** is a double data rate synchronous dynamic random-access memory interface. It superseded the original DDR SDRAM specification, and is superseded by DDR3 SDRAM. DDR2 DIMMs are neither forward compatible with DDR3 nor backward compatible with DDR.

**PCI Express**, officially abbreviated as **PCIe** or **PCI-e**, is a high-speed serial computer expansion bus standard, designed to replace the older PCI, PCI-X and AGP bus standards. PCIe has numerous improvements over the older standards, including higher maximum system bus throughput, lower I/O pin count and smaller physical footprint, better performance scaling for bus devices, a more detailed error detection and reporting mechanism, and native hot-swap functionality. More recent revisions of the PCIe standard provide hardware support for I/O virtualization.

The electromagnetic wave carried by a microstrip line exists partly in the dielectric substrate, and partly in the air above it. In general, the dielectric constant of the substrate will be different (and greater) than that of the air, so that the wave is travelling in an inhomogeneous medium. In consequence, the propagation velocity is somewhere between the speed of radio waves in the substrate, and the speed of radio waves in air. This behaviour is commonly described by stating the effective dielectric constant (or effective relative permittivity) of the microstrip; this being the dielectric constant of an equivalent homogeneous medium (i.e., one resulting in the same propagation velocity).

A **dielectric** is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing **dielectric polarization**. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

Further consequences of an inhomogeneous medium include:

- The line will not support a true TEM wave; at non-zero frequencies, both the E and H fields will have longitudinal components (a hybrid mode).
^{ [7] }The longitudinal components are small however, and so the dominant mode is referred to as quasi-TEM.^{ [8] } - The line is dispersive. With increasing frequency, the effective dielectric constant gradually climbs towards that of the substrate, so that the phase velocity gradually decreases.
^{ [7] }^{ [9] }This is true even with a non-dispersive substrate material (the substrate dielectric constant will usually fall with increasing frequency). - The characteristic impedance of the line changes slightly with frequency (again, even with a non-dispersive substrate material). The characteristic impedance of non-TEM modes is not uniquely defined, and depending on the precise definition used, the impedance of microstrip either rises, falls, or falls then rises with increasing frequency.
^{ [10] }The low-frequency limit of the characteristic impedance is referred to as the quasi-static characteristic impedance, and is the same for all definitions of characteristic impedance. - The wave impedance varies over the cross-section of the line.
- Microstrip lines radiate and discontinuity elements such as stubs and posts, which would be pure reactances in stripline, have a small resistive component due to the radiation from them.
^{ [11] }

A closed-form approximate expression for the quasi-static characteristic impedance of a microstrip line was developed by Wheeler:^{ [12] }^{ [13] }^{ [14] }

where is the *effective width*, which is the actual width of the strip, plus a correction to account for the non-zero thickness of the metallization:

Here *Z*_{0} is the impedance of free space, *ε*_{r} is the relative permittivity of substrate, *w* is the width of the strip, *h* is the thickness ("height") of substrate, and *t* is the thickness of the strip metallization.

This formula is asymptotic to an exact solution in three different cases

- , any (parallel plate transmission line),
- , (wire above a ground-plane) and
- ,

It is claimed that for most other cases, the error in impedance is less than 1%, and is always less than 2%.^{ [14] } By covering all aspect-ratios in one formula, Wheeler 1977 improves on Wheeler 1965^{ [13] } which gives one formula for and another for (thus introducing a discontinuity in the result at ).

Curiously, Harold Wheeler disliked both the terms 'microstrip' and 'characteristic impedance', and avoided using them in his papers.

A number of other approximate formulae for the characteristic impedance have been advanced by other authors. However, most of these are applicable to only a limited range of aspect-ratios, or else cover the entire range piecewise.

In particular, the set of equations proposed by Hammerstad,^{ [15] } who modifies on Wheeler,^{ [12] }^{ [13] } are perhaps the most often cited:

where is the effective dielectric constant, approximated as:

In order to build a complete circuit in microstrip, it is often necessary for the path of a strip to turn through a large angle. An abrupt 90° bend in a microstrip will cause a significant portion of the signal on the strip to be reflected back towards its source, with only part of the signal transmitted on around the bend. One means of effecting a low-reflection bend, is to curve the path of the strip in an arc of radius at least 3 times the strip-width.^{ [16] } However, a far more common technique, and one which consumes a smaller area of substrate, is to use a mitred bend.

To a first approximation, an abrupt un-mitred bend behaves as a shunt capacitance placed between the ground plane and the bend in the strip. Mitring the bend reduces the area of metallization, and so removes the excess capacitance. The percentage mitre is the cut-away fraction of the diagonal between the inner and outer corners of the un-mitred bend.

The optimum mitre for a wide range of microstrip geometries has been determined experimentally by Douville and James.^{ [17] } They find that a good fit for the optimum percentage mitre is given by

subject to and with the substrate dielectric constant . This formula is entirely independent of . The actual range of parameters for which Douville and James present evidence is and . They report a VSWR of better than 1.1 (i.e., a return loss better than −26 dB) for any percentage mitre within 4% (of the original ) of that given by the formula. At the minimum of 0.25, the percentage mitre is 98.4%, so that the strip is very nearly cut through.

For both the curved and mitred bends, the electrical length is somewhat shorter than the physical path-length of the strip.

- Distributed element filter
- Spurline, a microstrip notch-filter

The **characteristic impedance** or **surge impedance** (usually written Z_{0}) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of reflections in the other direction. Alternatively and equivalently it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.

A **circulator** is a passive non-reciprocal three- or four-port device, in which a microwave or radio frequency signal entering any port is transmitted to the next port in rotation (only). A *port* in this context is a point where an external waveguide or transmission line, connects to the device. For a three-port circulator, a signal applied to port 1 only comes out of port 2; a signal applied to port 2 only comes out of port 3; a signal applied to port 3 only comes out of port 1, so to up to a phase-factor, the scattering matrix for an ideal three-port circulator is

In telecommunications and electrical engineering, **electrical length** refers to the length of an electrical conductor in terms of the phase shift introduced by transmission over that conductor at some frequency.

The **wave impedance** of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields. For a transverse-electric-magnetic (TEM) plane wave traveling through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the impedance of free space. The symbol *Z* is used to represent it and it is expressed in units of ohms. The symbol *η* (eta) may be used instead of *Z* for wave impedance to avoid confusion with electrical impedance.

**Coaxial cable**, or **coax**, is a type of electrical cable that has an inner conductor surrounded by a tubular insulating layer, surrounded by a tubular conducting shield. Many coaxial cables also have an insulating outer sheath or jacket. The term coaxial comes from the inner conductor and the outer shield sharing a geometric axis. Coaxial cable was invented by English engineer and mathematician Oliver Heaviside, who patented the design in 1880.

The **relative permittivity** of a material is its (absolute) permittivity expressed as a ratio relative to the permittivity of vacuum.

In electrostatics, **absolute permittivity**, often simply called **permittivity**, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium. Accordingly, a charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

The physical constant ** ε_{0}**, commonly called the

A **dielectric resonator** is a piece of dielectric (nonconductive) material, usually ceramic, that is designed to function as a resonator for radio waves, generally in the microwave and millimeter wave bands. The microwaves are confined inside the resonator material by the abrupt change in permittivity at the surface, and bounce back and forth between the sides. At certain frequencies, the resonant frequencies, the microwaves form standing waves in the resonator, oscillating with large amplitudes. Dielectric resonators generally consist of a "puck" of ceramic that has a large dielectric constant and a low dissipation factor. The resonant frequency is determined by the overall physical dimensions of the resonator and the dielectric constant of the material.

**Dielectric loss** quantifies a dielectric material's inherent dissipation of electromagnetic energy. It can be parameterized in terms of either the **loss angle***δ* or the corresponding **loss tangent** tan *δ*. Both refer to the phasor in the complex plane whose real and imaginary parts are the resistive (lossy) component of an electromagnetic field and its reactive (lossless) counterpart.

**Effective medium approximations** or **effective medium theory** (**EMT**) pertain to analytical or theoretical modeling that describes the macroscopic properties of composite materials. EMAs or EMTs are developed from averaging the multiple values of the constituents that directly make up the composite material. At the constituent level, the values of the materials vary and are inhomogeneous. Precise calculation of the many constituent values is nearly impossible. However, theories have been developed that can produce acceptable approximations which in turn describe useful parameters and properties of the composite material as a whole. In this sense, effective medium approximations are descriptions of a medium based on the properties and the relative fractions of its components and are derived from calculations.

**Uniform convergence in probability** is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the *empirical frequencies* of all events in a certain event-family converge to their *theoretical probabilities*. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory.

**Metamaterial antennas** are a class of antennas which use metamaterials to increase performance of miniaturized antenna systems. Their purpose, as with any electromagnetic antenna, is to launch energy into free space. However, this class of antenna incorporates metamaterials, which are materials engineered with novel, often microscopic, structures to produce unusual physical properties. Antenna designs incorporating metamaterials can step-up the antenna's radiated power.

The **non-radiative dielectric (NRD) waveguide ** has been introduced by Yoneyama in 1981. In Fig. 1 the cross section of NRD guide is shown: it consists of a dielectric rectangular slab of height a and width b, which is placed between two metallic parallel plates of suitable width. The structure is practically the same as the H waveguide, proposed by Tischer in 1953. Due to the dielectric slab, the electromagnetic field is confined in the vicinity of the dielectric region, whereas in the outside region, for suitable frequencies, the electromagnetic field decays exponentially. Therefore, if the metallic plates are sufficiently extended, the field is practically negligible at the end of the plates and therefore the situation does not greatly differ from the ideal case in which the plates are infinitely extended. The polarization of the electric field in the required mode is mainly parallel to the conductive walls. As it is known, if the electric field is parallel to the walls, the conduction losses decrease in the metallic walls at the increasing frequency, whereas, if the field is perpendicular to the walls, losses increase at the increasing frequency. Since the NRD waveguide has been devised for its implementation at millimeter waves, the selected polarization minimizes the ohmic losses in the metallic walls.

**Circuit quantum electrodynamics** provides a means of studying the fundamental interaction between light and matter. As in the field of cavity quantum electrodynamics, a single photon within a single mode cavity coherently couples to a quantum object (atom). In contrast to cavity QED, the photon is stored in a one-dimensional on-chip resonator and the quantum object is no natural atom but an artificial one. These artificial atoms usually are mesoscopic devices which exhibit an atom-like energy spectrum. The field of circuit QED is a prominent example for quantum information processing and a promising candidate for future quantum computation.

- ↑ Grieg, D. D.; Engelmann, H. F. (Dec 1952). "Microstrip-A New Transmission Technique for the Klilomegacycle Range".
*Proceedings of the IRE*.**40**(12): 1644–1650. doi:10.1109/JRPROC.1952.274144. ISSN 0096-8390. - ↑ Olney, Barry. "Differential Pair Routing" (PDF). p. 51.
- ↑ Texas Instruments (2015). "High-Speed Interface Layout Guidelines" (PDF). p. 10. SPRAAR7E.
When possible, route high-speed differential pair signals on the top or bottom layer of the PCB with an adjacent GND layer. TI does not recommend stripline routing of the high-speed differential signals.

- ↑ Intel (2000). "High Speed USB Platform Design Guidelines" (PDF). p. 7.
- ↑ Silicon Labs. "USB Hardware Design Guide" (PDF). p. 9. AN0046.
- ↑ Kröger, Jens (2014). "Data Transmission at High Rates via Kapton Flexprints for the Mu3e Experiment" (PDF). pp. 19–21.
- 1 2 Denlinger, E. J. (January 1971). "A frequency dependent solution for microstrip transmission lines".
*IEEE Transactions on Microwave Theory and Techniques*. MTT-19 (1): 30–39. Bibcode:1971ITMTT..19...30D. doi:10.1109/TMTT.1971.1127442. - ↑ Pozar, David M. (2017).
*Microwave Engineering*Addison–Wesley Publishing Company. ISBN 978-81-265-4190-4. - ↑ Cory, H. (January 1981). "Dispersion characteristics of microstrip lines".
*IEEE Transactions on Microwave Theory and Techniques*. MTT-29: 59–61. - ↑ Bianco, B.; Panini, L.; Parodi, M.; Ridetlaj, S. (March 1978). "Some considerations about the frequency dependence of the characteristic impedance of uniform microstrips".
*IEEE Transactions on Microwave Theory and Techniques*. MTT-26 (3): 182–185. Bibcode:1978ITMTT..26..182B. doi:10.1109/TMTT.1978.1129341. - ↑ Oliner, Arthur A. (2006). "The evolution of electromagnetic waveguides". In Sarkar, Tappan K.; Mailloux, Robert J.; Oliner, Arthur A.; Salazar-Palma, Magdalena; Sengupta, Dipak L.
*History of wireless*. Wiley Series in Microwave and Optical Engineering.**177**. John Wiley and Sons. p. 559. ISBN 978-0-471-71814-7. - 1 2 Wheeler, H. A. (May 1964). "Transmission-line properties of parallel wide strips by a conformal-mapping approximation".
*IEEE Transactions on Microwave Theory and Techniques*. MTT-12 (3): 280–289. Bibcode:1964ITMTT..12..280W. doi:10.1109/TMTT.1964.1125810. - 1 2 3 Wheeler, H. A. (March 1965). "Transmission-line properties of parallel strips separated by a dielectric sheet".
*IEEE Transactions on Microwave Theory and Techniques*. MTT-13 (2): 172–185. Bibcode:1965ITMTT..13..172W. doi:10.1109/TMTT.1965.1125962. - 1 2 Wheeler, H. A. (August 1977). "Transmission-line properties of a strip on a dielectric sheet on a plane".
*IEEE Transactions on Microwave Theory and Techniques*. MTT-25 (8): 631–647. Bibcode:1977ITMTT..25..631W. doi:10.1109/TMTT.1977.1129179. - ↑ E. O. Hammerstad (1975), "Equations for Microstrip Circuit Design",
*1975 5th European Microwave Conference*: 268–272, doi:10.1109/EUMA.1975.332206 - ↑ Lee, T. H. (2004).
*Planar Microwave Engineering*. Cambridge University Press. pp. 173–174. - ↑ Douville, R. J. P.; James, D. S. (March 1978). "Experimental study of symmetric microstrip bends and their compensation".
*IEEE Transactions on Microwave Theory and Techniques*. MTT-26 (3): 175–182. Bibcode:1978ITMTT..26..175D. doi:10.1109/TMTT.1978.1129340.

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