Electrical length
Electrical length determines when wave effects (phase shift along conductors) become important in a circuit.
In radio frequency applications, when a delay is introduced due to a conductor, it is often the phase shift
So simple circuit theory is inadequate and transmission line techniques (the distributed-element model) must be used.
This means the inductance and capacitance per unit length of the line determine the phase velocity.
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.
In cables and transmission lines an electrical signal travels at a rate determined by the effective shunt capacitance
This is computed as a weighted average of the relative permittivity of free space, unity, and that of the dielectric:
where the fill factor F expresses the effective proportion of space around the line occupied by dielectric.
A transmission line is a specialized cable designed for carrying electric current of radio frequency.
The distinguishing feature of a transmission line is that it is constructed to have a constant characteristic impedance along its length and through connectors and switches, to prevent reflections.
Electrical length is widely used with a graphical aid called the Smith chart to solve transmission line calculations.
The equation for the voltage as a function of time along a transmission line with a matched load, so there is no reflected power, is where In a matched transmission line, the current is in phase with the voltage, and their ratio is the characteristic impedance
[14] A thin antenna element is resonant at frequencies at which the standing current wave has a node (zero) at the ends (and in monopoles an antinode (maximum) at the ground plane).
A dipole antenna is resonant at frequencies at which its electrical length is a half wavelength (
A monopole antenna is resonant at frequencies at which its electrical length is a quarter wavelength (
If the resistance of the antenna is matched to the characteristic resistance of the feedline, it absorbs all the power supplied to it, while at other frequencies it has reactance and reflects some power back down the line toward the transmitter, causing standing waves (high SWR) on the feedline.
Since only a portion of the power is radiated this causes inefficiency, and can possibly overheat the line or transmitter.
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.
The electrical length of an antenna element also depends on the length-to-diameter ratio of the conductor.
[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.
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.
[19][20] In many circumstances for practical reasons it is inconvenient or impossible to use an antenna of resonant length.
[20] A nonresonant antenna appears at its feedpoint electrically equivalent to a resistance in series with a reactance.
This is the usual technique for matching an electrically short transmitting antenna to its feedline, so it can be fed power efficiently.
This can be cancelled by adding a capacitor of equal but opposite reactance at the feed point to make the antenna resonant.
An electrically short conductor, much shorter than one wavelength, makes an inefficient radiator of electromagnetic waves.
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.
These equations are mathematically difficult to solve in all generality, so approximate methods have been developed that apply to situations in which the electrical length of the apparatus is very short (
