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In microwave and radio-frequency engineering, a **stub** or **resonant stub** is a length of transmission line or waveguide that is connected at one end only. The free end of the stub is either left open-circuit or (always in the case of waveguides) short-circuited. Neglecting transmission line losses, the input impedance of the stub is purely reactive; either capacitive or inductive, depending on the electrical length of the stub, and on whether it is open or short circuit. Stubs may thus function as capacitors, inductors and resonant circuits at radio frequencies.

**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.

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.

In electromagnetics and communications engineering, the term **waveguide** may refer to any linear structure that conveys electromagnetic waves between its endpoints. However, the original and most common meaning is a hollow metal pipe used to carry radio waves. This type of waveguide is used as a transmission line mostly at microwave frequencies, for such purposes as connecting microwave transmitters and receivers to their antennas, in equipment such as microwave ovens, radar sets, satellite communications, and microwave radio links.

- Short circuited stub
- Open circuited stub
- Resonant stub
- Stub matching
- Radial stub
- References
- See also

Stubs work by means of standing waves of radio waves along their length. Their reactive properties are determined by their physical length in relation to the wavelength of the radio waves. Therefore, stubs are most commonly used in UHF or microwave circuits in which the wavelengths are short enough that the stub is conveniently small.^{ [1] } They are often used to replace discrete capacitors and inductors, because at UHF and microwave frequencies lumped components perform poorly due to parasitic reactance.^{ [1] } Stubs are commonly used in antenna impedance matching circuits, frequency selective filters, and resonant circuits for UHF electronic oscillators and RF amplifiers.

In physics, a **standing wave**, also known as a **stationary wave**, is a wave which oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with time, and the oscillations at different points throughout the wave are in phase. The locations at which the amplitude is minimum are called nodes, and the locations where the amplitude is maximum are called antinodes.

In physics, the **wavelength** is the **spatial period** of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter *lambda* (λ). The term *wavelength* is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

In radio engineering, an **antenna** is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies an electric current to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves. In reception, an antenna intercepts some of the power of a radio wave in order to produce an electric current at its terminals, that is applied to a receiver to be amplified. Antennas are essential components of all radio equipment.

Stubs can be constructed with any type of transmission line: parallel conductor line (where they are called Lecher lines), coaxial cable, stripline, waveguide, and dielectric waveguide. Stub circuits can be designed using a Smith chart, a graphical tool which can determine what length line to use to obtain a desired reactance.

**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 physicist, engineer, and mathematician Oliver Heaviside, who patented the design in 1880.

**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.

The **Smith chart**, invented by Phillip H. Smith (1905–1987), is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.

The input impedance of a lossless short circuited line is,

where j is the imaginary unit, is the characteristic impedance of the line, is the phase constant of the line, and is the physical length of the line.

The **imaginary unit** or **unit imaginary number** is a solution to the quadratic equation *x*^{2} + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3*i*.

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.

Thus, depending on whether is positive or negative, the stub will be inductive or capacitive, respectively.

The length of a stub to act as a capacitor *C* at an angular frequency of is then given by:

In physics, **angular frequency***ω* is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

The length of a stub to act as an inductor *L* at the same frequency is given by:

The input impedance of a lossless open circuit stub is given by

It follows that depending on whether is positive or negative, the stub will be capacitive or inductive, respectively.

The length of an open circuit stub to act as an inductor *L* at an angular frequency of is:

The length of an open circuit stub to act as a capacitor *C* at the same frequency is:

Stubs are often used as resonant circuits in oscillators and distributed element filters. An open circuit stub of length will have a capacitive impedance at low frequency when . Above this frequency the impedance is inductive. At precisely the stub presents a short circuit. This is qualitatively the same behaviour as a series resonant circuit. For a lossless line the phase change constant is proportional to frequency,

where *v* is the velocity of propagation and is constant with frequency for a lossless line. For such a case the resonant frequency is given by,

While stubs function as resonant circuits, they differ from lumped element resonant circuits in that they have multiple resonant frequencies; in addition to the fundamental resonant frequency , they resonate at multiples of this frequency: . The impedance will not continue to rise monotonically with frequency after resonance as in a lumped tuned circuit. It will rise until the point where at which point it will be open circuit. After this point (which is actually an anti-resonance point) the impedance will again become capacitive and start to fall. It will continue to fall until at it again presents a short circuit. At this point the filtering action of the stub has totally failed. This response of the stub continues to repeat with increasing frequency alternating between resonance and anti-resonance. It is not only a characteristic of stubs, but of all distributed element filters, that there is some frequency beyond which the filter fails and multiple unwanted passbands are produced.^{ [2] }

Similarly, a short circuit stub is an anti-resonator at , that is, it behaves as a parallel resonant circuit, but again fails as is approached.^{ [2] }

Stubs can be used to match a load impedance to the transmission line characteristic impedance. The stub is positioned a distance from the load. This distance is chosen so that at that point the resistive part of the load impedance is made equal to the resistive part of the characteristic impedance by impedance transformer action of the length of the main line. The length of the stub is chosen so that it exactly cancels the reactive part of the presented impedance. That is, the stub is made capacitive or inductive according to whether the main line is presenting an inductive or capacitive impedance respectively. This is not the same as the actual impedance of the load since the reactive part of the load impedance will be subject to impedance transformer action as well as the resistive part. Matching stubs can be made adjustable so that matching can be corrected on test.^{ [3] }

A single stub will only achieve a perfect match at one specific frequency. For wideband matching several stubs may be used spaced along the main transmission line. The resulting structure is filter-like and filter design techniques are applied. For instance, the matching network may be designed as a Chebyshev filter but is optimised for impedance matching instead of passband transmission. The resulting transmission function of the network has a passband ripple like the Chebyshev filter, but the ripples never reach 0 dB insertion loss at any point in the passband, as they would do for the standard filter.^{ [4] }

Radial stubs are a planar component that consists of a sector of a circle rather than a constant-width line. They are used with planar transmission lines when a low impedance stub is required. Low characteristic impedance lines require a wide line. With a wide line the junction of the stub with the main line is not at a well defined point. Radial stubs overcome this difficulty by narrowing to a point at the junction. Filter circuits using stubs often use them in pairs, one connected to each side of the main line. A pair of radial stubs so connected is called a butterfly stub or a bowtie stub.^{ [5] }

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In physics and electrical engineering, a **cutoff frequency**, **corner frequency**, or **break frequency** is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

The **propagation constant** of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, **propagation constant **measures the change undergone by the source quantity as it propagates from one port to the next.

**Electrical impedance** is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term *complex impedance* may be used interchangeably.

In electric and electronic systems, **reactance** is the opposition of a circuit element to a *change* in current or voltage, due to that element's inductance or capacitance. The notion of reactance is similar to electric resistance, but it differs in several respects.

An **LC circuit**, also called a **resonant circuit**, **tank circuit**, or **tuned circuit**, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.

The **Heaviside condition**, named for Oliver Heaviside (1850–1925), is the condition an electrical transmission line must meet in order for there to be no distortion of a transmitted signal. Also known as the **distortionless condition**, it can be used to improve the performance of a transmission line by adding loading to the cable.

**Electrical resonance** occurs in an electric circuit at a particular *resonant frequency* when the impedances or admittances of circuit elements cancel each other. In some circuits, this happens when the impedance between the input and output of the circuit is almost zero and the transfer function is close to one.

**Foster's reactance theorem** is an important theorem in the fields of electrical network analysis and synthesis. The theorem states that the reactance of a passive, lossless two-terminal (one-port) network always strictly monotonically increases with frequency. It is easily seen that the reactances of inductors and capacitors individually increase with frequency and from that basis a proof for passive lossless networks generally can be constructed. The proof of the theorem was presented by Ronald Martin Foster in 1924, although the principle had been published earlier by Foster's colleagues at American Telephone & Telegraph.

**Zobel networks** are a type of filter section based on the image-impedance design principle. They are named after Otto Zobel of Bell Labs, who published a much-referenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the input impedance is fixed in the design independently of the transfer function. This characteristic is achieved at the expense of a much higher component count compared to other types of filter sections. The impedance would normally be specified to be constant and purely resistive. For this reason, Zobel networks are also known as constant resistance networks. However, any impedance achievable with discrete components is possible.

**Constant k filters**, also **k-type filters**, are a type of electronic filter designed using the image method. They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive components. Historically, they are the first filters that could approach the ideal filter frequency response to within any prescribed limit with the addition of a sufficient number of sections. However, they are rarely considered for a modern design, the principles behind them having been superseded by other methodologies which are more accurate in their prediction of filter response.

**m-derived filters** or **m-type filters** are a type of electronic filter designed using the image method. They were invented by Otto Zobel in the early 1920s. This filter type was originally intended for use with telephone multiplexing and was an improvement on the existing constant k type filter. The main problem being addressed was the need to achieve a better match of the filter into the terminating impedances. In general, all filters designed by the image method fail to give an exact match, but the m-type filter is a big improvement with suitable choice of the parameter m. The m-type filter section has a further advantage in that there is a rapid transition from the cut-off frequency of the pass band to a pole of attenuation just inside the stop band. Despite these advantages, there is a drawback with m-type filters; at frequencies past the pole of attenuation, the response starts to rise again, and m-types have poor stop band rejection. For this reason, filters designed using m-type sections are often designed as composite filters with a mixture of k-type and m-type sections and different values of m at different points to get the optimum performance from both types.

A **quarter-wave impedance transformer**, often written as **λ/4 impedance transformer**, is a transmission line or waveguide used in electrical engineering of length one-quarter wavelength (λ), terminated with some known impedance. It presents at its input the dual of the impedance with which it is terminated.

The **gyrator–capacitor model** is a lumped-element model for magnetic fields, similar to magnetic circuits, but based on using elements analogous to capacitors rather than elements analogous to resistors to represent the magnetic flux path. Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.

The **primary line constants** are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

**Metal-mesh optical filters** are optical filters made from stacks of metal meshes and dielectric. They are used as part of an optical path to filter the incoming light to allow frequencies of interest to pass while reflecting other frequencies of light.

**Commensurate line circuits** are electrical circuits composed of transmission lines that are all the same length; commonly one-eighth of a wavelength. Lumped element circuits can be directly converted to distributed element circuits of this form by the use of **Richards' transformation**. This transformation has a particularly simple result; inductors are replaced with transmission lines terminated in short-circuits and capacitors are replaced with lines terminated in open-circuits. Commensurate line theory is particularly useful for designing distributed element filters for use at microwave frequencies.

An **RLC circuit** is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

A **frequency-selective surface** (**FSS**) is any thin, repetitive surface designed to reflect, transmit or absorb electromagnetic fields based on the frequency of the field. In this sense, an FSS is a type of optical filter or metal-mesh optical filters in which the filtering is accomplished by virtue of the regular, periodic pattern on the surface of the FSS. Though not explicitly mentioned in the name, FSS's also have properties which vary with incidence angle and polarization as well - these are unavoidable consequences of the way in which FSS's are constructed. Frequency-selective surfaces have been most commonly used in the radio frequency region of the electromagnetic spectrum and find use in applications as diverse as the aforementioned microwave oven, antenna radomes and modern metamaterials. Sometimes frequency selective surfaces are referred to simply as periodic surfaces and are a 2-dimensional analog of the new periodic volumes known as photonic crystals.

- 1 2 Shuart, George W. (October 1934). "New high impedance lines replace coils" (PDF).
*Short Wave Craft*. New York: Popular Book Corp.**5**(6): 332–333. Retrieved March 24, 2015. - 1 2 Ganesh Prasad Srivastava, Vijay Laxmi Gupta,
*Microwave Devices and Circuit Design*, pp.29-31, PHI Learning, 2006 ISBN 81-203-2195-2. - ↑ F.R. Connor,
*Wave Transmission*, pp.32-34, Edward Arnold Ltd., 1972 ISBN 0-7131-3278-7. - ↑ Matthaei, G.; Young, L.; Jones, E. M. T.,
*Microwave Filters, Impedance-Matching Networks, and Coupling Structures*, pp.681-713, McGraw-Hill 1964. - ↑ Jia-Shen G. Hong, M. J. Lancaster,
*Microstrip Filters for RF/Microwave Applications*, pp. 188-190, Wiley, 2004 ISBN 0471464201.

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