Twin-lead

Part of a series on
Antennas
Common types Dipole Fractal Loop Monopole Satellite dish Television Whip
Components Balun Block upconverter Coaxial cable Counterpoise (ground system) Feed Feed line Low-noise block downconverter Passive radiator Receiver Rotator Stub Transmitter Tuner Twin-lead
Systems Antenna farm Amateur radio Cellular network Hotspot Municipal wireless network Radio Radio masts and towers Wi-Fi Wireless
Safety and regulation Wireless device radiation and health Wireless electronic devices and health International Telecommunication Union (Radio Regulations) World Radiocommunication Conference
Radiation sources /regions Boresight Focal cloud Ground plane Main lobe Near and far field Side lobe Vertical plane
Characteristics Array gain Directivity Efficiency Electrical length Equivalent radius Factor Friis transmission equation Gain Height Radiation pattern Radiation resistance Radio propagation Radio spectrum Signal-to-noise ratio Spurious emission
Techniques Beam steering Beam tilt Beamforming Small cell Bell Laboratories Layered Space-Time (BLAST) Massive Multiple-input multiple-output (MIMO) Reconfiguration Spread spectrum Wideband Space Division Multiple Access (WSDMA)
v t e

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 or solid copper or copper-clad steel wires, held a precise distance apart by a plastic (usually polyethylene) 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. It is available with several different values of characteristic impedance, the most common type is 300 ohm.

Contents

• Characteristics and uses
• How it works
• Other similar types
• Window line
• Ladder line
• Impedance matching
• Characteristic impedance
• Antennas
• References

Twin lead is mainly used as an antenna feedline at shortwave and VHF frequencies, to connect radio receivers and transmitters to their antennas. It can have significantly lower signal loss than miniature flexible coaxial cable, the main alternative type of feedline at these frequencies; for example, type RG-58 coaxial cable loses 6.6 dB per 100 metres (330 ft) at 30 MHz, while 300 ohm twin-lead loses only 0.55 dB. ^[1] 300 ohm twin lead is widely used to connect FM radios to their antennas, and was previously used to connect television antennas to televisions until it was replaced by coaxial cable. However, it is more vulnerable to interference; proximity to metal objects will inject signals into twin-lead that would be blocked out by coaxial cable. It therefore requires spacing around rain gutters, and standoff insulators along metal support masts.

Characteristics and uses

Twin lead and other types of parallel-conductor transmission line are mainly used to connect radio transmitters and receivers to their antennas. Parallel transmission line has the advantage that its losses per unit length are an order of magnitude smaller than that of coaxial cable, the main alternative form of transmission line. Its disadvantages are that it is more vulnerable to interference, and must be kept away from metal objects which can cause power losses. For this reason, when installed along the outside of buildings and on antenna masts, standoff insulators must be used. It is also common practice to twist the twin lead on long free standing lengths to further reject any induced imbalances to the line.

Twin-lead is supplied in several different sizes, with values of 600, 450, 300, and 75  ohm characteristic impedance. The most common, 300 ohm twin-lead, was once widely used to connect television sets and FM radios to their receiving antennas. 300 ohm twin-lead for television installations has been largely replaced with 75 ohm coaxial cable feedlines. Twin-lead is also used in amateur radio stations as a transmission line for balanced transmission of radio frequency signals.

The characteristic impedance of twin-lead is a function of the wire diameter and its spacing; in 300 ohm twin-lead, the most common type, the wire is usually 20 or 22  gauge (0.52 or 0.33 mm²), about 7.5 mm (0.30 inches) apart. ^[2] This is well matched with the natural impedance of a folded dipole antenna, which is normally around 275 ohms. Twin-lead generally has higher impedance than the other common transmission wiring, coaxial cable (coax). The widely used RG-6 coax has a characteristic impedance of 75 ohms, which requires the use of a balun to match impedance when used with common antenna types.

How it works

Twin lead is a form of parallel-wire balanced transmission line. The separation between the two wires in twin-lead is small compared to the wavelength of the radio frequency (RF) signal carried on the wire. ^[3] The RF current in one wire is equal in magnitude and opposite in direction to the RF current in the other wire. Therefore, in the far field region far from the transmission line, the radio waves radiated by one wire are equal in magnitude but opposite in phase (180° out of phase) to the waves radiated by the other wire, so they superpose and cancel each other. ^[3] The result is that almost no net radio energy is radiated by the line.

Similarly, any interfering external radio waves will induce equal, in phase RF currents, traveling in the same direction, in the two wires. Since the load at the destination end is connected across the wires, only differential, oppositely-directed currents in the wires create a current in the load. Thus the interfering currents are canceled out, so twin lead does not tend to pick up radio noise.

However, if a piece of metal is located sufficiently close to a twin-lead line, within a distance comparable to the wire spacing, it will be significantly closer to one wire than the other. As a result, the RF current induced in the metal object by one wire will be greater than the opposing current induced by the other wire, so the currents will no longer cancel. Thus nearby metal objects can cause power losses in twin lead lines, through energy dissipated as heat by induced currents. Similarly, radio noise originating in cables or metal objects located near the twin-lead line can induce unbalanced currents in the wires, coupling noise into the line. Therefore, the line must be kept at a distance from metal objects such as gutters and masts.

In order to prevent power from being reflected from the load end of the line, causing high SWR and inefficiency, the load must have an impedance which matches the characteristic impedance of the line. This causes the load to appear electrically identical to a continuation of the line, preventing reflections. Similarly, to transfer power efficiently into the line, the source must also match the characteristic impedance. To connect a balanced transmission line to an unbalanced line like coaxial cable, a device called a balun must be used.

Other similar types

In addition to Twin-lead, two-conductor parallel wire line comes in two additional forms

Window line

Window line is a variation on Twin-lead; it is similarly built, except that the polyethylene ribbon between the wires holding them apart has regularly spaced rectangular openings ("windows") cut in it. ^{[2] [4]}

Among the advantages to cutting "windows" in the ribbon is that the manufacturer can adjust the size of the cuts to make fine adjustments to the feedline's electrical properties. The windows lighten the line, and reduce the amount of surface on which dirt and moisture can accumulate, making window line somewhat less vulnerable to weather-induced changes in its characteristic impedance. ^[2] The most common type is nominal 450  Ω window line, which has a conductor spacing of about an inch (25 mm); its actual impedance may be closer to 400 Ω. ^[2] It is also made in nominal 350 Ω impedance.

Ladder line

Ladder line is an older, simpler form of parallel-wire line, often called open wire line. The configuration looks like a rope ladder, hence the name. It can either be purchased already made by a cable company, or home-made; the construction is easy, though tedious, and originally all radio amateurs made their own open wire line. It consists of two wires, usually either clad in a DC insulator or coated with a durable lacquer, held a constant distance apart by insulated spacers.

These "ladder rungs" can be made out of any convenient insulating material – at present, usually short pieces cut from plastic plumbing pipe – formerly either long, thin ceramic insulators or dowel stubs of water-proofed wood were used. The "rungs" hold the wires at constant spacing, somewhere between a constant two to five inches (13 cm) apart, depending on the wire and the desired impedance. The rungs are spaced about every five to twelve inches (13–30 cm). Although it is important to ensure the wire separation is uniform, it doesn't matter if distances between the rungs is erratic, as long as the separation remains nearly constant in gusts of wind.

The ratio of chosen wire spacing to wire diameter determines the line's characteristic impedance – usually 500~600  Ω, but also depends on the wire insulation's relative permittivity and conductive loss in the rung insulators, if significant. ^[5] For example, for a 500 Ω line, two bare or lacquered wires should be spaced 32× the wires' individual diameters apart – about four inches (10 cm) in the case of common  1 / 8 ″ diameter wire (0.125 inches (3 mm)). If the wires are insulated, the separation may need to be a little different, depending on the insulating plastic. To get 600 Ω parallel line, the same wires would be spaced nine inches (23 cm); 600 Ω is roughly the practical limit for transmission lines made with wire rather than thicker copper-clad steel cable, or with rigid aluminum or copper pipe.

Impedance matching

As a transmission line, transmission efficiency will be maximum when the impedance of the antenna, the characteristic impedance of the twin-lead line and the impedance of the equipment are the same. For this reason, when attaching a twin-lead line to a coaxial cable connection, such as the 300 ohm twin-lead from a domestic television antenna to the television's 75 ohm coax antenna input, a balun with a 4:1 ratio is commonly used. Its purpose is double: first, it transforms twin-lead's 300 ohm impedance to match the 75 ohm coaxial cable impedance; and second, it transforms the balanced, symmetric transmission line to the unbalanced coax input. In general, when used as a feedline, twin-lead (especially ladder line versions) has a higher efficiency than coaxial cable when there is an impedance mismatch between the feedline and the source (or sink). For receive-only use this merely implies that the system can communicate under slightly less optimal conditions; for transmit use, this can often result in significantly less energy lost as heat in the transmission line.

Twin-lead also can serve as a convenient material with which to build a simple folded dipole antenna. Such antennas may be fed either by using a 300 ohm twin-lead feeder or by using a 300-to-75-ohm balun and using coaxial feedline and will usually handle moderate power loads without overheating.

Characteristic impedance

The characteristic impedance of a parallel-wire transmission line like twin lead or ladder line depends on its dimensions; the diameter of the wires  d  and their separation  D . This is derived below.

The characteristic impedance  Z_o  of any transmission line operating at frequency f or radian frequency ${\displaystyle \;\omega =2\ \pi \ f\;,}$ is given by

${\displaystyle Z_{\mathsf {o}}={\sqrt {{\frac {R+j\ \omega \ L}{\;G+j\ \omega \ C\;}}\ }}}$

where for twin-lead line the primary line constants are

${\displaystyle R={\frac {\ 2\ R_{\mathsf {s}}\ }{\pi \ d}}}$

${\displaystyle L={\frac {\mu }{\ \pi \ }}\ \operatorname {arcosh} \left({\frac {\ D\ }{d}}\right)}$

${\displaystyle G={\frac {\pi \ \sigma }{\ \operatorname {arcosh} \left({\frac {\ D\ }{d}}\right)\ }}}$

${\displaystyle C={\frac {\ \pi \ \varepsilon \ }{\ \operatorname {arcosh} \left({\frac {\ D\ }{d}}\right)\ }}}$

where  d  is the wire diameter and  D  is the separation of the wires measured between their centre-lines;  ε  is the absolute permittivity between the wires;  μ  is the absolute permeability;  σ  is the surface conductivity between the wires; and where the surface resistance of the wires is given by

${\displaystyle R_{\mathsf {s}}={\sqrt {{\frac {\ \pi \ f\ \mu _{\mathsf {c}}\ }{\sigma _{\mathsf {c}}}}\;}}~.}$

Neglecting the wire resistance  R  and the leakage conductance  G , this gives

${\displaystyle Z_{\mathsf {o}}={\frac {\zeta _{\mathsf {o}}}{\ \pi {\sqrt {\varepsilon _{\mathsf {R}}\ }}\ }}\ \operatorname {arcosh} \left({\frac {\ D\ }{d}}\right)={\frac {\zeta _{\mathsf {o}}}{\ \pi {\sqrt {\varepsilon _{\mathsf {R}}\ }}\ }}\ \ln {\Biggl [}\ {\frac {\ D\ }{d}}+{\sqrt {\left({\tfrac {\ D\ }{d}}\right)^{2}-1~}}\ {\Biggr ]}\ }$ ^[6]

${\displaystyle \quad ={\frac {\zeta _{\mathsf {o}}}{\ \pi {\sqrt {\varepsilon _{\mathsf {R}}\ }}\ }}\ {\Biggl [}\ \ln \left({\frac {\ 2\ D\ }{d}}\right)+\ln \left(\ {\tfrac {\ 1\ }{2}}\left[\ 1+{\sqrt {1-\left({\tfrac {d}{\ D\ }}\right)^{2}~}}\;\right]\right){\Biggr ]}\ }$

${\displaystyle \quad ={\frac {\zeta _{\mathsf {o}}}{\ \pi {\sqrt {\varepsilon _{\mathsf {R}}\ }}\ }}\left[\ln \left({\frac {\ 2\ D\ }{d}}\right)-\left({\frac {d}{\ 2\ D\ }}\right)^{2}-\operatorname {\mathcal {O}} \left\{\left({\tfrac {d}{\ 2\ D\ }}\right)^{4}\right\}\right]}$

${\displaystyle \quad \approx {\frac {\ 119.92\ {\mathsf {\Omega }}\ }{\sqrt {\varepsilon _{\mathsf {R}}\ }}}\left[\ln \left({\frac {\ 2\ D\ }{d}}\right)-\left({\frac {d}{\,2\,D\,}}\right)^{2}\right]\approx {\frac {\ 119.92\ {\mathsf {\Omega }}\ }{\sqrt {\varepsilon _{\mathsf {R}}\ }}}\ \ln \left({\frac {\ 2\ D\ }{d}}\right)~.}$

where  ζ_o  is the impedance of free space (approximately 376.74 Ω),  ε_R  is the relative permittivity (which for air is 1.00054).

When the separation  D  is many times greater than the wire diameter  d  then the  arcosh  function can be approximately replaced by a natural logarithm (with its argument doubled):

${\displaystyle Z_{\mathsf {o}}={\frac {119.92\ {\mathsf {\Omega }}}{\ {\sqrt {\varepsilon _{\mathsf {R}}\ }}\ }}\ \operatorname {arcosh} \left({\frac {\ D\ }{d}}\right)\approx {\frac {\ 119.92\ {\mathsf {\Omega }}\ }{\ {\sqrt {\varepsilon _{\mathsf {R}}\ }}\ }}\ \ln \left({\frac {\ 2\ D\ }{d}}\right)\approx {\frac {\ 276\ {\mathsf {\Omega }}\ }{\sqrt {\varepsilon _{\mathsf {R}}\ }}}\ \log _{10}\left({\frac {\ 2\ D\ }{d}}\right)~.}$ ^[7]

The exact and approximate formulas for the separation needed to achieve some given characteristic impedance ${\displaystyle \ Z_{\mathsf {o}}\ }$ through a pair of wires are therefore

${\displaystyle D=d\ \cosh \left({\frac {\ \pi \ Z_{\mathsf {o}}{\sqrt {\varepsilon _{\mathsf {R}}\ }}\ }{\zeta _{\mathsf {o}}}}\right)\approx {\tfrac {\ 1\ }{2}}\ d\ \exp \left({\frac {\ \pi \ Z_{\mathsf {o}}{\sqrt {\varepsilon _{\mathsf {R}}\ }}\ }{\zeta _{\mathsf {o}}}}\right)~.}$

The dielectric material between the two conductors with either twin-lead or ladder line is not all air. The effect of a "mixed" dielectric, part air and part polyethylene or other plastic, is that the actual impedance will fall somewhere between the value calculated assuming all air or all polyethylene. Published, carefully measured values for Z_o will typically be more accurate than estimates from formulas.

Antennas

Twin-lead can be connected directly to a suitably designed antenna:

Windom antenna

A multiply-resonant antenna whose resonant impedances cluster around 300 Ω.

Folded dipole

Double-wire dipole whose characteristic impedance in free space is around 300 Ω.

Dipole

Although the center impedance at resonance is approximately 73 Ω in free space, in actual use it varies between 30 and 100 Ω, depending on height above ground, so with high-impedance feedline a T-match or Y-match feed will probably be necessary.

Yagi–Uda antenna

Yagi and the simpler Moxon antenna, and other directional antennas; some special impedance matching arrangement at the feedpoint is necessary for any cabling, since interference between the typically close-spacing of the parallel near-resonant antenna segments causes low feedpoint resistance as well as making the antenna more directional.

Wikimedia Commons has media related to Twin-lead cables .

Related Research Articles

In telecommunications and professional audio, a balanced line or balanced signal pair is an electrical circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths, to ground, and to other circuits. The primary advantage of the balanced line format is good rejection of common-mode noise and interference when fed to a differential device such as a transformer or differential amplifier.

In electrical engineering, electrical length is a dimensionless parameter equal to the physical length of an electrical conductor such as a cable or wire, divided by the wavelength of alternating current at a given frequency traveling through the conductor. In other words, it is the length of the conductor measured in wavelengths. It can alternately be expressed as an angle, in radians or degrees, equal to the phase shift the alternating current experiences traveling through the conductor.

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 dimensionless change in magnitude or phase per unit length. 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.

In radio engineering and telecommunications, standing wave ratio (SWR) is a measure of impedance matching of loads to the characteristic impedance of a transmission line or waveguide. Impedance mismatches result in standing waves along the transmission line, and SWR is defined as the ratio of the partial standing wave's amplitude at an antinode (maximum) to the amplitude at a node (minimum) along the line.

In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmission must be taken into account. This applies especially to radio-frequency engineering because the short wavelengths mean that wave phenomena arise over very short distances. However, the theory of transmission lines was historically developed to explain phenomena on very long telegraph lines, especially submarine telegraph cables.

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 consisting of an inner conductor surrounded by a concentric conducting shield, with the two separated by a dielectric ; many coaxial cables also have a protective outer sheath or jacket. The term coaxial refers to the inner conductor and the outer shield sharing a geometric axis.

In electromagnetics, an antenna's gain is a key performance parameter which combines the antenna's directivity and radiation efficiency. The term power gain has been deprecated by IEEE. In a transmitting antenna, the gain describes how well the antenna converts input power into radio waves headed in a specified direction. In a receiving antenna, the gain describes how well the antenna converts radio waves arriving from a specified direction into electrical power. When no direction is specified, gain is understood to refer to the peak value of the gain, the gain in the direction of the antenna's main lobe. A plot of the gain as a function of direction is called the antenna pattern or radiation pattern. It is not to be confused with directivity, which does not take an antenna's radiation efficiency into account.

In physics, the Poynting vector represents the directional energy flux or power flow of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m²); kg/s³ in base SI units. It is named after its discoverer John Henry Poynting who first derived it in 1884. Nikolay Umov is also credited with formulating the concept. Oliver Heaviside also discovered it independently in the more general form that recognises the freedom of adding the curl of an arbitrary vector field to the definition. The Poynting vector is used throughout electromagnetics in conjunction with Poynting's theorem, the continuity equation expressing conservation of electromagnetic energy, to calculate the power flow in electromagnetic fields.

In electromagnetism, skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor. It is caused by opposing eddy currents induced by the changing magnetic field resulting from the alternating current. The electric current flows mainly at the skin of the conductor, between the outer surface and a level called the skin depth. Skin depth depends on the frequency of the alternating current; as frequency increases, current flow becomes more concentrated near the surface, resulting in less skin depth. Skin effect reduces the effective cross-section of the conductor and thus increases its effective resistance. At 60 Hz in copper, skin depth is about 8.5 mm. At high frequencies, skin depth becomes much smaller.

Radiation resistance is that part of an antenna's feedpoint electrical resistance caused by the emission of radio waves from the antenna. A radio transmitter excites with a radio frequency alternating current an antenna, which radiates the exciting energy as radio waves. Because the antenna is absorbing the energy it is radiating from the transmitter, the antenna's input terminals present a resistance to the current from the transmitter.

In radio and telecommunications a dipole antenna or doublet is one of the two simplest and most widely-used types of antenna; the other is the monopole. 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 far 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. All dipoles are electrically equivalent to two monopoles mounted end-to-end and fed with opposite phases, with the ground plane between them made "virtual" by the opposing monopole.

An antenna tuner is a passive electronic device inserted between a radio transmitter and its antenna. Its purpose is to optimize power transfer by matching the impedance of the radio to the signal impedance on the feedline to the antenna.

Microstrip is a type of electrical transmission line which can be fabricated with any technology where a conductor is separated from a ground plane by a dielectric layer known as "substrate". Microstrip lines are used to convey microwave-frequency signals.

A ‘T’-antenna, ‘T’-aerial, or flat-top antenna is a monopole radio antenna consisting of one or more horizontal wires suspended between two supporting radio masts or buildings and insulated from them at the ends. A vertical wire is connected to the center of the horizontal wires and hangs down close to the ground, connected to the transmitter or receiver. The shape of the antenna resembles the letter "T", hence the name. The transmitter power is applied, or the receiver is connected, between the bottom of the vertical wire and a ground connection.

Antenna measurement techniques refers to the testing of antennas to ensure that the antenna meets specifications or simply to characterize it. Typical parameters of antennas are gain, bandwidth, radiation pattern, beamwidth, polarization, and impedance.

A radio transmitter or receiver is connected to an antenna which emits or receives the radio waves. The antenna feed system or antenna feed is the cable or conductor, and other associated equipment, which connects the transmitter or receiver with the antenna and makes the two devices compatible. In a radio transmitter, the transmitter generates an alternating current of radio frequency, and the feed system feeds the current to the antenna, which converts the power in the current to radio waves. In a radio receiver, the incoming radio waves excite tiny alternating currents in the antenna, and the feed system delivers this current to the receiver, which processes the signal.

The telegrapher's equations are a set of two coupled, linear equations that predict the voltage and current distributions on a linear electrical transmission line. The equations are important because they allow transmission lines to be analyzed using circuit theory. The equations and their solutions are applicable from 0 Hz to frequencies at which the transmission line structure can support higher order non-TEM modes. The equations can be expressed in both the time domain and the frequency domain. In the time domain the independent variables are distance and time. The resulting time domain equations are partial differential equations of both time and distance. In the frequency domain the independent variables are distance and either frequency, or complex frequency, The frequency domain variables can be taken as the Laplace transform or Fourier transform of the time domain variables or they can be taken to be phasors. The resulting frequency domain equations are ordinary differential equations of distance. An advantage of the frequency domain approach is that differential operators in the time domain become algebraic operations in frequency domain.

Space cloth is a hypothetical infinite plane of conductive material having a resistance of η ohms per square, where η is the impedance of free space. η ≈ 376.7 ohms. If a transmission line composed of straight parallel perfect conductors in free space is terminated by space cloth that is normal to the transmission line then that transmission line is terminated by its characteristic impedance. The calculation of the characteristic impedance of a transmission line composed of straight, parallel good conductors may be replaced by the calculation of the D.C. resistance between electrodes placed on a two-dimensional resistive surface. This equivalence can be used in reverse to calculate the resistance between two conductors on a resistive sheet if the arrangement of the conductors is the same as the cross section of a transmission line of known impedance. For example, a pad surrounded by a guard ring on a printed circuit board (PCB) is similar to the cross section of a coaxial cable transmission line.

Magnetic current is, nominally, a current composed of moving magnetic monopoles. It has the unit volt. The usual symbol for magnetic current is , which is analogous to for electric current. Magnetic currents produce an electric field analogously to the production of a magnetic field by electric currents. Magnetic current density, which has the unit V/m², is usually represented by the symbols and . The superscripts indicate total and impressed magnetic current density. The impressed currents are the energy sources. In many useful cases, a distribution of electric charge can be mathematically replaced by an equivalent distribution of magnetic current. This artifice can be used to simplify some electromagnetic field problems. It is possible to use both electric current densities and magnetic current densities in the same analysis.

References