Electrical length

In radio frequency applications, when a delay is introduced due to a conductor, it is often the phase shift ${\displaystyle \phi }$, the difference in phase of the sinusoidal wave between the two ends of the conductor, that is of importance.[5] The length of a sinusoidal wave is commonly expressed as an angle, in units of degrees (with 360° in a wavelength) or radians (with 2π radians in a wavelength). So alternately the electrical length can be expressed as an angle which is the phase shift of the wave between the ends of the conductor[1][3][5]

${\displaystyle \phi =360^{\circ }{l \over \lambda }\,{\text{degrees}}}$

${\displaystyle \quad =2\pi {l \over \lambda }\,{\text{radians}}}$

The electrical length of a conductor determines when wave effects (phase shift along the conductor) are important.[4]^{: p.12–14} If the electrical length ${\displaystyle G}$ is much less than one, that is the physical length of a conductor is much shorter than the wavelength, say less than one tenth of the wavelength (${\displaystyle l<\lambda /10}$) it is called electrically short. In this case the voltage and current are approximately constant along the conductor, so it acts as a simple connector which transfers alternating current with negligible phase shift. In circuit theory the connecting wires between components are usually assumed to be electrically short, so the lumped element circuit model is only valid for alternating current when the circuit is electrically small, much smaller than a wavelength.[4]^{: p.12–14}[5] When the electrical length approaches or is greater than one, a conductor will have significant reactance, inductance or capacitance, depending on its length. So simple circuit theory is inadequate and transmission line techniques (the distributed-element model) must be used.

In a vacuum an electromagnetic wave (radio wave) travels at the speed of light ${\displaystyle v_{p}=c=}$ 2.9979×10⁸ meters per second, and very close to this speed in air, so the free space wavelength of the wave is ${\displaystyle \lambda _{\text{0}}=c/f}$.[5] (in this article free space variables are distinguished by a subscript 0) Thus a physical length ${\displaystyle l}$ of a radio wave in space or air has an electrical length of

${\displaystyle G_{\text{0}}={l \over \lambda _{\text{0}}}={lf \over c}}$ wavelengths.

In the SI system of units, empty space has a permittivity of ${\displaystyle \epsilon _{\text{0}}=}$ 8.854×10⁻¹² F/m (farads per metre) and a magnetic permeability of ${\displaystyle \mu _{\text{0}}=}$ 1.257×10⁻⁶ H/m (henries per meter). These universal constants determine the speed of light[5][7]

${\displaystyle c={1 \over {\sqrt {\epsilon _{\text{0}}\mu _{\text{0}}}}}}$

Equivalent circuit of a lossless transmission line. ${\displaystyle L}$ and ${\displaystyle C}$ represent the inductance and capacitance per unit length of a small section of line

In an electrical cable, for a cycle of the alternating current to move a given distance along the line, it takes time to charge the capacitance between the conductors, and the rate of change of the current is slowed by the series inductance of the wires. This determines the phase velocity ${\displaystyle v_{p}}$ at which the wave moves along the line. In cables and transmission lines an electrical signal travels at a rate determined by the effective shunt capacitance ${\displaystyle C}$ and series inductance ${\displaystyle L}$ per unit length of the transmission line

${\displaystyle v_{p}={1 \over {\sqrt {LC}}}}$

Some transmission lines consist only of bare metal conductors, if they are far away from other high permittivity materials their signals propagate at very close to the speed of light, ${\displaystyle c}$. In most transmission lines the material construction of the line slows the velocity of the signal so it travels at a reduced phase velocity[5]

${\displaystyle v_{p}=\kappa c}$

where ${\displaystyle \kappa }$ (kappa) is a dimensionless number between 0 and 1 called the velocity factor (VF), characteristic of the type of line, equal to the ratio of signal velocity in the line to the speed of light.[8][9][6]

Most transmission lines contain a dielectric material (insulator) filling some or all of the space in between the conductors. The permittivity ${\displaystyle \epsilon }$ or dielectric constant of that material increases the distributed capacitance ${\displaystyle C}$ in the cable, which reduces the velocity factor below unity. If there is a material with high magnetic permeability (${\displaystyle \mu }$) in the line such as steel or ferrite which increases the distributed inductance ${\displaystyle L}$, it can also reduce ${\displaystyle \kappa }$, but this is almost never the case. If all the space around the transmission line conductors containing the near fields was filled with a material of permittivity ${\displaystyle \epsilon }$ and permeability ${\displaystyle \mu }$, the phase velocity on the line would be[5]

The effective permittivity ${\displaystyle \epsilon }$ and permeability ${\displaystyle \mu }$ per unit length of the line are frequently given as dimensionless constants; relative permittivity: ${\displaystyle \epsilon _{\text{r}}}$ and relative permeability: ${\displaystyle \mu _{\text{r}}}$ equal to the ratio of these parameters compared to the universal constants ${\displaystyle \epsilon _{\text{0}}}$ and ${\displaystyle \mu _{\text{0}}}$

${\displaystyle \epsilon _{\text{r}}={\epsilon \over \epsilon _{\text{0}}}\qquad \mu _{\text{r}}={\mu \over \mu _{\text{0}}}}$

so the phase velocity is

${\displaystyle v_{\text{p}}={1 \over {\sqrt {\epsilon \mu }}}={1 \over {\sqrt {\epsilon _{\text{0}}\epsilon _{\text{r}}\mu _{\text{0}}\mu _{\text{r}}}}}=c{1 \over {\sqrt {\epsilon _{\text{r}}\mu _{\text{r}}}}}}$

So the velocity factor of the line is

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

In many lines only a fraction of the space surrounding the line is occupied by a solid dielectric. With only part of the electromagnetic field effected by the dielectric, there is less reduction of the wave velocity. In this case an effective permittivity ${\displaystyle \epsilon _{\text{eff}}}$ can be calculated which if it filled all the space around the line would give the same phase velocity. This is computed as a weighted average of the relative permittivity of free space, unity, 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 around the line occupied by dielectric.

In most transmission lines there are no materials with high magnetic permeability, so ${\displaystyle \mu =\mu _{\text{0}}}$ and ${\displaystyle \mu _{\text{r}}=1}$ and so

Since the electromagnetic waves travel slower in the line than in free space, the wavelength of the wave in the transmission line ${\displaystyle \lambda }$ is shorter than the free space wavelength by the factor kappa: ${\displaystyle \lambda =v_{\text{p}}/f=\kappa c/f=\kappa \lambda _{\text{0}}}$. Therefore, more wavelengths fit in a transmission line of a given length ${\displaystyle l}$ than in the same length of wave in free space, so the electrical length of a transmission line is longer than the electrical length of a wave of the same frequency in free space[5]

Reduction factor of physical length of a resonant dipole from a half-wavelength electrical length as a function of element thickness

A thin-element antenna can be thought of as a transmission line with the conductors separated,[15] so the near-field electric and magnetic fields extend further into space than in a transmission line, in which the fields are mainly confined to the vicinity of the conductors. Near the ends of the antenna elements the electric field is not perpendicular to the conductor axis as in a transmission line but spreads out in a fan shape (fringing field).[16] As a result, the end sections of the antenna have increased capacitance, storing more charge, so the current waveform departs from a sine wave there, decreasing faster toward the ends.[17] When approximated as a sine wave, the current does not quite go to zero at the ends; the nodes of the current standing wave, instead of being at the ends of the element, occur somewhat beyond the ends.[18] Thus the electrical length of the antenna is longer than its physical length.

The electrical length of an antenna element also depends on the length-to-diameter ratio of the conductor.[19][15][20][21] As the ratio of the diameter to wavelength increases, the capacitance increases, so the node occurs farther beyond the end, and the electrical length of the element increases.[19][20] When the elements get too thick, the current waveform becomes significantly different from a sine wave, so the entire concept of electrical length is no longer applicable, and the behavior of the antenna must be calculated by electromagnetic simulation computer programs like NEC.

As with a transmission line, an antenna's electrical length is increased by anything that adds shunt capacitance or series inductance to it, such as the presence of high permittivity dielectric material around it. In microstrip antennas which are fabricated as metal strips on printed circuit boards, the dielectric constant of the substrate board increases the electrical length of the antenna. Proximity to the Earth or a ground plane, a dielectric coating on the conductor, nearby grounded towers, metal structural members, guy lines and the capacitance of insulators supporting the antenna also increase the electrical length.[20]

These factors, called "end effects", cause the electrical length of an antenna element to be somewhat longer than the length of the same wave in free space. In other words, the physical length of the antenna at resonance will be somewhat shorter than the resonant length in free space (one-half wavelength for a dipole, one-quarter wavelength for a monopole).[19][20] As a rough generalization, for a typical dipole antenna, the physical resonant length is about 5% shorter than the free space resonant length.[19][20]

In many circumstances for practical reasons it is inconvenient or impossible to use an antenna of resonant length. An antenna of nonresonant length at the operating frequency can be made resonant by adding a reactance, a capacitance or inductance, either in the antenna itself or in a matching network between the antenna and its feedline.[20] A nonresonant antenna appears at its feedpoint electrically equivalent to a resistance in series with a reactance. Adding an equal but opposite type of reactance in series with the feedline will cancel the antenna's reactance; the combination of the antenna and reactance will act as a series resonant circuit, so at its operating frequency its input impedance will be purely resistive, allowing it to be fed power efficiently at a low SWR without reflections.

In a common application, a monopole antenna with an electrical length shorter than a quarter-wavelength (${\displaystyle \lambda /4}$), or a dipole antenna shorter than a half-wavelength (${\displaystyle \lambda /2}$) will have capacitive reactance. Adding an inductor (coil of wire), called a loading coil, at the feedpoint in series with the antenna, with inductive reactance equal to the antenna's capacitive reactance at the operating frequency, will cancel the capacitance of the antenna, so the combination of the antenna and coil will be resonant at the operating frequency. An antenna shorter than resonant length is called electrically short, and since adding inductance is equivalent to increasing the electrical length, this technique is called electrically lengthening the antenna. This is the usual technique for matching an electrically short transmitting antenna to its feedline, so it can be fed power efficiently. However, an electrically short antenna that has been lengthened in this way still has the same radiation pattern; it does not radiate as much power, and therefore has lower gain than a full-sized antenna.

Conversely, an antenna longer than resonant length at its operating frequency, such as a monopole longer than a quarter wavelength but shorter than a half wavelength, will have inductive reactance. This can be cancelled by adding a capacitor of equal but opposite reactance at the feed point to make the antenna resonant. This is called electrically shortening the antenna.

Two antennas that are similar (scaled copies of each other), fed with different frequencies, will have the same radiation resistance and radiation pattern and fed with equal power will radiate the same power density in any direction if they have the same electrical length at the operating frequency; that is, if their lengths are in the same proportion as the wavelengths.[22][4]^{: p.12–14}

${\displaystyle {l_{\text{1}} \over l_{\text{2}}}={\lambda _{\text{1}} \over \lambda _{\text{2}}}={f_{\text{2}} \over f_{\text{1}}}}$

This means the length of antenna required for a given antenna gain scales with the wavelength (inversely with the frequency), or equivalently the aperture scales with the square of the wavelength.

An electrically short conductor, much shorter than one wavelength, makes an inefficient radiator of electromagnetic waves. As the length of an antenna is made shorter than its fundamental resonant length (a half-wavelength for a dipole antenna and a quarter-wavelength for a monopole), the radiation resistance the antenna presents to the feedline decreases with the square of the electrical length, that is the ratio of physical length to wavelength, ${\displaystyle (l/\lambda )^{2}}$. As a result, other resistances in the antenna, the ohmic resistance of metal antenna elements, the ground system if present, and the loading coil, dissipate an increasing fraction of transmitter power as heat. A monopole antenna with an electrical length below .05${\displaystyle \lambda }$ or 18° has a radiation resistance of less than one ohm, making it very hard to drive.

A second disadvantage is that since the capacitive reactance of the antenna and inductive reactance of the required loading coil do not decrease, the Q factor of the antenna increases; it acts electrically like a high Q tuned circuit. As a result, the bandwidth of the antenna decreases with the square of electrical length, reducing the data rate that can be transmitted. At VLF frequencies even the huge toploaded wire antennas that must be used have bandwidths of only ~10 hertz, limiting the data rate that can be transmitted.