# Electrical length

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In telecommunications and electrical engineering, electrical length (or phase length) refers to the length of an electrical conductor in terms of the phase shift introduced by transmission over that conductor [1] at some frequency.

Electrical engineering is a professional engineering discipline that generally deals with the study and application of electricity, electronics, and electromagnetism. This field first became an identifiable occupation in the later half of the 19th century after commercialization of the electric telegraph, the telephone, and electric power distribution and use. Subsequently, broadcasting and recording media made electronics part of daily life. The invention of the transistor, and later the integrated circuit, brought down the cost of electronics to the point they can be used in almost any household object.

## Usage of the term

Depending on the specific usage, the term "electrical length" is used rather than simple physical length to incorporate one or more of the following three concepts:

• When one is concerned with the number of wavelengths, or phase, involved in a wave's transit over a segment of transmission line especially, one may simply specify that electrical length, while specification of a physical length, frequency, or velocity factor is omitted. The electrical length is then typically expressed as N wavelengths or as the phase φ expressed in degrees or radians. Thus in a microstrip design one might specify a shorted stub of 60° phase length, which will correspond to different physical lengths when applied to different frequencies. Or one might consider a 2-meter section of coax which has an electrical length of one quarter wavelength (90°) at 25 MHz and ask what its electrical length becomes when the circuit is operated at a different frequency.
• Due to the velocity factor of a particular transmission line, for instance, the transit time of a signal in a certain length of cable is equal to the transit time over a longer distance when travelling at the speed of light. So a pulse sent down a 2-meter section of coax (whose velocity factor is 2/3) would arrive at the end of the coax at the same time that the same pulse arrives at the end of a bare wire of length 3 meters (over which it propagates at the speed of light), and one might refer to the 2 meter section of coax as having an electrical length of 3 meters, or an electrical length of 1/2 wavelength at 50 MHz (since a 50 MHz radio wave has a wavelength of 6 meters).
• Since resonant antennas are usually specified in terms of the electrical length of their conductors (such as the half wave dipole), the attainment of such an electrical length is loosely equated with electrical resonance, that is, a purely resistive impedance at the antenna's input, as is usually desired. An antenna that has been made slightly too long, for instance, will present an inductive reactance, which can be corrected by physically shortening the antenna. Based on this understanding, a common jargon in the antenna trade refers to the achievement of resonance (cancellation of reactance) at the antenna terminals as electrically shortening that too-long antenna (or electrically lengthening a too-short antenna) when an electrical matching network (or antenna tuner) has performed that task without physically altering the antenna's length. Although a very inexact use of terminology, this usage is widespread, especially as applied to the use of a loading coil at the bottom of a short monopole (a vertical, or whip antenna) to "electrically lengthen" it and achieve electrical resonance as seen through the loading coil.

Phase is the position of a point in time on a waveform cycle. A complete cycle is defined as the interval required for the waveform to return to its arbitrary initial value. The graph to the right shows how one cycle constitutes 360° of phase. The graph also shows how phase is sometimes expressed in radians, where one radian of phase equals approximately 57.3°.

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.

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.

## Phase length

The first usage of the term "electrical length" assumes a sine wave of some frequency, or at least a narrowband waveform centered around some frequency f. The sine wave will repeat with a period of T = 1/f. The frequency f will correspond to a particular wavelength λ along a particular conductor. For conductors (such as bare wire or air-filled coax) which transmit signals at the speed of light c, the wavelength is given by λ=c/f. A distance L along that conductor corresponds to N wavelengths where N= L / λ.

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

In radio communications, narrowband describes a channel in which the bandwidth of the message does not significantly exceed the channel's coherence bandwidth.

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 the figure at the right, the wave shown is seen to be N=1.5 wavelengths long. A wave crest at the beginning of the graph, moving towards the right, will arrive at the end after a time 1.5T. The electrical length of that segment is said to be "1.5 wavelengths" or, expressed as a phase angle, "540°" (or 3π radians) where N wavelengths corresponds to φ = 360°N (or φ = 2πN radians). In radio frequency applications, when a delay is introduced due to a transmission line, it is often the phase shift φ that is of importance, so specifying a design in terms of the phase or electrical length allows one to adapt that design to an arbitrary frequency by employing the wavelength λ applying to that frequency.

Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around twenty thousand times per second to around three hundred billion times per second. This is roughly between the upper limit of audio frequencies and the lower limit of infrared frequencies; these are the frequencies at which energy from an oscillating current can radiate off a conductor into space as radio waves. Different sources specify different upper and lower bounds for the frequency range.

## Velocity factor

In a transmission line, a signal travels at a rate controlled by the effective capacitance and inductance per unit of length of the transmission line. Some transmission lines consist only of bare conductors, in which case their signals propagate at the speed of light, c. More often the signal travels at a reduced velocity κc, where κ is the velocity factor, a number less than 1, representing the ratio of that velocity to the speed of light. [2] [3]

Capacitance is the ratio of the change in an electric charge in a system to the corresponding change in its electric potential. There are two closely related notions of capacitance: self capacitance and mutual capacitance. Any object that can be electrically charged exhibits self capacitance. A material with a large self capacitance holds more electric charge at a given voltage than one with low capacitance. The notion of mutual capacitance is particularly important for understanding the operations of the capacitor, one of the three elementary linear electronic components.

In electromagnetism and electronics, inductance describes the tendency of an electrical conductor, such as coil, to oppose a change in the electric current through it. The change in current induces a reverse electromotive force (voltage). When an electric current flows through a conductor, it creates a magnetic field around that conductor. A changing current, in turn, creates a changing magnetic field, the surface integral of which is known as magnetic flux. From Faraday's law of induction, any change in magnetic flux through a circuit induces an electromotive force (voltage) across that circuit, a phenomenon known as electromagnetic induction. Inductance is specifically defined as the ratio between this induced voltage and the rate of change of the current in the circuit

Most transmission lines contain a dielectric material (insulator) filling some or all of the space in between the conductors. The relative permittivity or dielectric constant of that material increases the distributed capacitance in the cable, which reduces the velocity factor below unity. It is also possible for κ to be reduced due to a relative permeability (${\displaystyle \mu _{\text{r}}}$) of that material, which increases the distributed inductance, but this is almost never the case. Now, if one fills a space with a dielectric of relative permittivity ${\displaystyle \epsilon _{\text{r}}}$, then the velocity of an electromagnetic plane wave is reduced by the velocity factor:

In electromagnetism, 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.

In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself otherwise known as distributed inductance in Transmission Line Theory. Hence, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically represented by the (italicized) Greek letter µ. The term was coined in September 1885 by Oliver Heaviside. The reciprocal of magnetic permeability is magnetic reluctivity.

${\displaystyle \kappa ={\frac {v_{p}}{c}}={\frac {1}{\sqrt {\epsilon _{\text{r}}\mu _{\text{r}}}}}\approx {\frac {1}{\sqrt {\epsilon _{\text{r}}}}}}$.

This reduced velocity factor would also apply to propagation of signals along wires immersed in a large space filled with that dielectric. However, with only part of the space around the conductors filled with that dielectric, there is less reduction of the wave velocity. Part of the electromagnetic wave surrounding each conductor "feels" the effect of the dielectric, and part is in free space. Then it is possible to define an effective relative permittivity${\displaystyle \epsilon _{\text{eff}}}$ which then predicts the velocity factor according to

${\displaystyle \kappa ={\frac {1}{\sqrt {\epsilon _{\text{eff}}}}}}$

${\displaystyle \epsilon _{\text{eff}}}$ is computed as a weighted average of the relative permittivity of free space (1) and that of the dielectric:

${\displaystyle \epsilon _{\text{eff}}=(1-F)+F\epsilon _{\text{r}}}$

where the fill factor F expresses the effective proportion of space so affected by the dielectric.

In the case of coaxial cable, where all of the volume in between the inner conductor and the shield is filled with a dielectric, the fill factor is unity, since the electromagnetic wave is confined to that region. In other types of cable, such as twin lead, the fill factor can be much smaller. Regardless, any cable intended for radio frequencies will have its velocity factor (as well as its characteristic impedance) specified by the manufacturer. In the case of coaxial cable, where F=1, the velocity factor is solely determined by the sort of dielectric used as specified here.

For example, a typical velocity factor for coaxial cable is .66, corresponding to a dielectric constant of 2.25. Suppose we wish to send a 30 MHz signal down a short section of such a cable, and delay it by a quarter wave (90°). In free space, this frequency corresponds to a wavelength of λ0=10m, so a delay of .25λ would require an electrical length of 2.5 m. Applying the velocity factor of .66, this results in a physical length of cable 1.67 m long.

The velocity factor likewise applies to antennas in cases where the antenna conductors are (partly) surrounded by a dieletric. This particularly applies to microstrip antennas such as the patch antenna. Waves on microstrip are affected by the dielectric of the circuit board beneath them, but not the air above them. Their velocity factors thus depend not directly on the permittivity of the circuit board material but on the effective permittivity ${\displaystyle \epsilon _{\text{eff}}}$ which is often specified for a circuit board material (or can be calculated). Note that the fill factor and therefore ${\displaystyle \epsilon _{\text{eff}}}$ are somewhat dependent on the width of the trace compared to the thickness of the board.

## Antennas

While there are certain wideband antenna designs, many antennas are classified as resonant and perform according to design around a particular frequency. This applies especially to broadcasting stations and communication systems which are confined to one frequency or narrow frequency band. This includes the dipole and monopole antennas and all of the designs based on them (Yagi, dipole or monopole arrays, folded dipole, etc.). In addition to the directive gain in beam antennas suffering away from the design frequency, the antenna feedpoint impedance is very sensitive to frequency offsets. Especially for transmitting, the antenna is often intended to operate at the resonant frequency. At the resonant frequency, by definition, that impedance is a pure resistance which matches the characteristic impedance of the transmission line and the output (or input) impedance of the transmitter (or receiver). At frequencies away from the resonant frequency, the impedance includes some reactance (capacitance or inductance). It is possible for an antenna tuner to be used to cancel that reactance (and to change the resistance to match the transmission line), however that is often avoided as an extra complication (and needs to be controlled at the antenna side of the transmission line).

The condition for resonance in a monopole antenna is for the element to be an odd multiple of a quarter-wavelength, λ/4. In a dipole antenna both driven conductors must be that long, for a total dipole length of (2N+1)λ/2.

The electrical length of an antenna element is, in general, different from its physical length[ better source needed ] [4] [5] [6] For example, increasing the diameter of the conductor, or the presence of nearby metal objects, will decrease the velocity of the waves in the element, increasing the electrical length. [7] [8]

An antenna which is shorter than its resonant length is described as "electrically short" [9] , and exhibits capacitive reactance. Similarly, an antenna which is longer than its resonant length is described as "electrically long" and exhibits inductive reactance.

An antenna's effective electrical length can be changed without changing its physical length by adding reactance, (inductance or capacitance) in series with it. [10] This is called lumped-impedance matching or loading.

For example, a monopole antenna such as a metal rod fed at one end, will be resonant when its electrical length is equal to a quarter wavelength, λ/4, of the frequency used. If the antenna is shorter than a quarter wavelength, the feedpoint impedance will include capacitive reactance; this causes reflections on the feedline and a mismatch at the transmitter or receiver, even if the resistive component of the impedance is correct. To cancel the capacitive reactance, an inductance, called a loading coil, is inserted in between the feedline and the antenna terminal. Selecting an inductance with the same reactance as the (negative) capacitative reactance seen at the antenna terminal, cancels that capacitance, and the antenna system (antenna and coil) will again be resonant. The feedline sees a purely resistive impedance. Since an antenna which had been too short now appears as if it were resonant, the addition of the loading coil is sometimes referred to as "electrically lengthening" the antenna.

Similarly, the feedpoint impedance of a monopole antenna longer than λ/4 (or a dipole with arms longer than λ/4) will include inductive reactance. A capacitor in series with the antenna can cancel this reactance to make it resonant, which can be referred to as "electrically shortening" the antenna.

Inductive loading is widely used to reduce the length of whip antennas on portable radios such as walkie-talkies and short wave antennas on cars, to meet physical requirements.

The electrical lengthening allows the construction of shorter aerials. It is applied in particular for aerials for VLF, longwave and medium-wave transmitters. Because those radio waves are several hundred meters to many kilometers long, mast radiators of the necessary height cannot be realised economically. It is also used widely for whip antennas on portable devices such as walkie-talkies to allow antennas much shorter than the standard quarter-wavelength to be used. The most widely used example is the rubber ducky antenna.

The electrical lengthening reduces the bandwidth of the antenna if other phase control measures are not undertaken. An electrically extended aerial is less efficient than the equivalent, full-length antenna.

### Technical realization

There are two possibilities for the realisation of the electric lengthening.

1. switching in inductive coils in series with the aerial
2. switching in metal surfaces, known as roof capacitance, at the aerial ends which form capacitors to earth.

Often both measures are combined. The coils switched in series must sometimes be placed in the middle of the aerial construction. The cabin installed at a height of 150-metres on the Blosenbergturm in Beromünster is such a construction, in which a lengthening coil is installed for the supply of the upper tower part (the Blosenbergturm has in addition a ring-shaped roof capacitor on its top)

#### Application

Transmission aerials of transmitters working at frequencies below the longwave broadcasting band always apply electric lengthening. Broadcasting aerials of longwave broadcasting stations apply it often. However, for transmission aerials of NDBs electrical lengthening is extensively applied, because these use antennas which are considerably less tall than a quarter of the radiated wavelength.

## Related Research Articles

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.

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.

A balun is an electrical device that converts between a balanced signal and an unbalanced signal. A balun can take many forms and may include devices that also transform impedances but need not do so. Transformer baluns can also be used to connect lines of differing impedance. Sometimes, in the case of transformer baluns, they use magnetic coupling but need not do so. Common-mode chokes are also used as baluns and work by eliminating, rather than ignoring, common mode signals.

Twin-lead cable is a two-conductor flat cable used as a balanced transmission line to carry radio frequency (RF) signals. It is constructed of two stranded copper or copper-clad steel wires, held a precise distance apart by a plastic ribbon. The uniform spacing of the wires is the key to the cable's function as a transmission line; any abrupt changes in spacing would reflect some of the signal back toward the source. The plastic also covers and insulates the wires.

A helical antenna is an antenna consisting of one or more conducting wires wound in the form of a helix. In most cases, directional helical antennas are mounted over a ground plane, while omnidirectional designs may not be. The feed line is connected between the bottom of the helix and the ground plane. Helical antennas can operate in one of two principal modes — normal mode or axial mode.

In radio and telecommunications a dipole antenna or doublet is the simplest and most widely used class of antenna. The dipole is any one of a class of antennas producing a radiation pattern approximating that of an elementary electric dipole with a radiating structure supporting a line current so energized that the current has only one node at each end. A dipole antenna commonly consists of two identical conductive elements such as metal wires or rods. The driving current from the transmitter is applied, or for receiving antennas the output signal to the receiver is taken, between the two halves of the antenna. Each side of the feedline to the transmitter or receiver is connected to one of the conductors. This contrasts with a monopole antenna, which consists of a single rod or conductor with one side of the feedline connected to it, and the other side connected to some type of ground. A common example of a dipole is the "rabbit ears" television antenna found on broadcast television sets.

A single-wire transmission line is a method of transmitting electrical power or signals using only a single electrical conductor. This is in contrast to the usual use of a pair of wires providing a complete circuit, or an electrical cable likewise containing two conductors for that purpose.

A T-antenna, T-aerial, flat-top antenna, or top-hat antenna is a capacitively loaded monopole wire radio antenna used in the VLF, LF, MF and shortwave bands. T-antennas are widely used as transmitting antennas for amateur radio stations, long wave and medium wave broadcasting stations. They are also used as receiving antennas for shortwave listening.

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 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.

A monopole antenna is a class of radio antenna consisting of a straight rod-shaped conductor, often mounted perpendicularly over some type of conductive surface, called a ground plane. The driving signal from the transmitter is applied, or for receiving antennas the output signal to the receiver is taken, between the lower end of the monopole and the ground plane. One side of the antenna feedline is attached to the lower end of the monopole, and the other side is attached to the ground plane, which is often the Earth. This contrasts with a dipole antenna which consists of two identical rod conductors, with the signal from the transmitter applied between the two halves of the antenna.

In electronics, a Lecher line or Lecher wires is a pair of parallel wires or rods that were used to measure the wavelength of radio waves, mainly at UHF and microwave frequencies. They form a short length of balanced transmission line. When attached to a source of radio-frequency power such as a radio transmitter, the radio waves form standing waves along their length. By sliding a conductive bar that bridges the two wires along their length, the length of the waves can be physically measured. Austrian physicist Ernst Lecher, improving on techniques used by Oliver Lodge and Heinrich Hertz, developed this method of measuring wavelength around 1888. Lecher lines were used as frequency measuring devices until frequency counters became available after World War 2. They were also used as components, often called "resonant stubs", in UHF and microwave radio equipment such as transmitters, radar sets, and television sets, serving as tank circuits, filters, and impedance-matching devices. They are used at frequencies between HF/VHF, where lumped components are used, and UHF/SHF, where resonant cavities are more practical.

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.

Dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy. It can be parameterized in terms of either the loss angleδ 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.

A microwave cavity or radio frequency (RF) cavity is a special type of resonator, consisting of a closed metal structure that confines electromagnetic fields in the microwave region of the spectrum. The structure is either hollow or filled with dielectric material. The microwaves bounce back and forth between the walls of the cavity. At the cavity's resonant frequencies they reinforce to form standing waves in the cavity. Therefore, the cavity functions similarly to an organ pipe or sound box in a musical instrument, oscillating preferentially at a series of frequencies, its resonant frequencies. Thus it can act as a bandpass filter, allowing microwaves of a particular frequency to pass while blocking microwaves at nearby frequencies.

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.

Nominal impedance in electrical engineering and audio engineering refers to the approximate designed impedance of an electrical circuit or device. The term is applied in a number of different fields, most often being encountered in respect of:

In radio systems, many different antenna types are used with specialized properties for particular applications. Antennas can be classified in various ways. The list below groups together antennas under common operating principles, following the way antennas are classified in many engineering textbooks.

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