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Theory
Introduction to Electromagnetics Maxwell's Equations Near-Field and Far-Field Transmission Line Models Scale modelling References Symbols and units Constants Mutual Impedance Between Wire Elements
5 Models
Following are models derived to, in theory, determine the important factors affecting the coupling between an aerial telecommunication line and an antenna. We will also try to find the main contributions to the radiated electromagnetic field in different situations. Since these models are going to be used in the study of xDSL signals we will focus on frequencies up to 30MHz. The models are divided into eight different coupling paths. They are initially divided into four near-field models and four far-field models. The near-field models, based on the mutual impedance, will have more significant contributions to the coupling when the distance between the emitting line and the receiving antenna r is much less than and the far-field models, based on the radiated field, will dominate when the distance is much greater than (see Equation 43 to Equation 45). At distances between the near- field and the far-field the contribution will be a combination between the far-field models and the near- field models.
Figure 15 The major contribution to the coupling at frequencies up to 30MHz (FAR-FIELD = radiated, NEAR-FIELD = mutual impedance) Both the near-field models and the far-field models are divided into looking at the magnetic coupling or fields and the electric coupling or fields separately. This is because of the different characteristics of the two. All cases are finally divided into one model for the common-mode signals and one model for the differential-mode signals. For the common-mode signals, the signals on a two-wire line are approximated with one signal on a single conductor. In almost all cases the differential-mode models show much less tendency to radiate than the common-mode model, but since the differential-mode signals usually are of a much greater magnitude than the common-mode signals in a balanced transmission system, the contributions from the differential-mode signals can not always be neglected. Notice that the far-field is said to start at the distance if the size of the source, in this case the length of the line D, is greater than a wavelength. This has nothing to do with the different types of coupling, i.e. if the induction terms or the radiation terms will dominate the coupling (see Equation 43 to Equation 45). The distance is, however, often used as the boundary between the near-field and the far-field because at a shorter distance maxima and minima would appear due to interference caused by different distances to different parts of the source. That means that it is preferred to do discrete measurements beyond this distance to get repeatable results. The distance, r, is calculated for a couple of line-lengths in Figure 16 below. Figure 16 Distances to a far-field without fluctuations using a long line compared to the wavelength The coupling between the telecommunication line radiating electromagnetic energy and a receiving antenna can be modeled with an equivalent two-port network as in the following figure:
Equation 121 We can write the produced voltage and current in the antenna due to the current in the telecommunication line as: Equation 122 Equation 123 These relations will be used in the following near-field models, where we look at the magnetic coupling considering the coupling impedance to be purely inductive and looking at the electric coupling considering the coupling to be purely capacitive. We will also look at common-mode signals and differential-mode signals separately. The mutual impedance, Z 12 or Z 21 , between two arbitrary positioned conductors above ground can be calculated according to the equations in Appendix C. These equations are however quite difficult to solve. There for we have decided to look at different situations to get simplified but restricted solutions.
5.1 Near-Field Models
In the near-field, the main coupling between a telecommunication line and an antenna is the mutual impedance, mainly consisting of mutual inductance from magnetic field coupling and mutual capacitance from the electric field coupling. The models for the near-field are divided in magnetic or electric coupling in which both common-mode coupling and differential-mode coupling are studied. The models are limited to looking at aerial telecommunication lines since they radiate more than buried cables, and represent a worst case situation. The results from each model could then be superimposed to determine the overall coupling. This can however produce errors when the length of the cable is much less than the wavelength. In our case this is almost always true. The purpose with the models will then be to find the main cause of radiation and the characteristic of the radiation in a particular case. This can be used to derive the important factors for this particular kind of radiation. Knowing these factors is very important when building realistic set-ups to measure the radiation and also when looking at different reduction techniques. Notice that the results and reduction techniques can also be applied to cable-to-cable cross talk problems or antenna to cable disturbances. 5.1.1 Magnetic Coupling 5.1.1.1 Common-mode Coupling In this chapter we will study the magnetic coupling of the common-mode current to an antenna due to mutual inductance, M , shown in the following figure:
Figure 18 Common-mode magnetic (mutual inductance) coupling between a telecommunication line above and an antenna below. The common-mode current on the two wire line is approximated with one current on a single conductor. The distance between the conductors is considered very small compared to the distance to the antenna. Therefore the two conductors carrying the common-mode current are considered as a single conductor. The common-mode current in an aerial telecommunication line will follow a closed loop, denoted C 1. The currents will follow the copper leads and then be coupled to ground via the common-mode impedance, which can be purely capacitive, and return in the ground-plane back to the source producing a closed loop bounding the surface S 1. The common-mode current I 1 flowing around the circumference C 1 of S 1 , will produce a magnetic field B 1. A part of the magnetic flux caused by B 1 will link with the antenna, which bounds the surface S 2. The mutual flux is then: Equation 124 If we take the surface integral of the fundamental postulate for electromagnetic induction (Equation 17) we get: Equation 125 where the left side is the induced electromotive force in the antenna, U 2. Notice that C 2 does not have to be a physical closed loop. The right side can be rewritten using Equation 124 and the result is the Faraday’s law of electromagnetic induction, which states that the electromotive force induced in a stationary closed loop circuit is equal to the negative rate of inverse of the magnetic flux linking the circuit [ 8 ]:
where the vector magnetic potential for a thin wire, using Equation 26, is: Equation 132 where R is the distance to the point of observation, in this case the distance between the telecommunication line and the antenna. Combining Equation 131 and Equation 132 we get the mutual inductance as: Equation 133 where we can see that in this situation the mutual inductance would vary inversely with the distance R between the antenna and the telecommunication line. We can also see that for a linear medium, it is proportional to the permeability and independent of the currents in the circuits. The contour integrals over C 1 and C 2 is however hard to calculate since the contour of a dipole antenna is non-obvious. Interchanging the subscripts would not change the value of the double integral which means that the reciprocity relations hold as discussed earlier, Z 12 = Z 21. Equation 133 is called the Neumann formula for mutual inductance. The mutual inductance M as calculated in Equation 128 or Equation 133 would represent the value of a coupling inductor Z 12 in Figure 17. The self-inductances, as calculated in Equation 129, would be part of the impedances Z 11 and Z 22 in the same figure. If Z 12 is purely inductive then - Z 12 in the upper impedances in Figure 17 would represent pure capacitances and the schematic in Figure 17 could be illustrated as in the following figure: Figure 19 Magnetic coupling with the mutual inductance M From Figure 19 we can clearly see the origin of the term mutual inductance, since both the current I 1 and I 2 are going through the same inductance M. From Equation 122 we get the induced current in the antenna as: Equation 134
The mutual inductance as illustrated in Figure 19 can also be illustrated as a transformer coupling using a lumped parameter circuit as in the following figure: Figure 20 Equivalent circuit of the magnetic coupling between a telecommunication line above and an antenna below, showing the mutual inductance M and the self inductances L 1 and L 2. U 2 is induced emf in the antenna due to the common-mode current I 1. The mutual inductance can be written as: Equation 135 where L 1 is the self inductance of the telecommunication line, L 2 is the self inductance of the antenna and the constant k is called the coefficient of coupling. This coefficient will depend on how much of the magnetic flux from the common-mode current that is coupled to the antenna. If there is no leakage flux this coefficient will be one, but in our case it will be much less than one. Using the circuit in Figure 20 the induced emf could be written as: Equation 136 which is directly proportional to the common-mode current. It also depends on the geometry by k and the self inductances of the telecommunication line and of the antenna. We also know from Equation 133 that the mutual inductance vary inversely with the distance between the line and the antenna, R. We could extract the distance from the coupling coefficient, k , and define a new constant which will depend on the geometry of the loops only as K. Then we can rewrite Equation 136 as: Equation 137 5.1.1.2 Reduction of Common-mode Magnetic Coupling
Equation 139 or with the distance extracted as in Equation 137: Equation 140 where we can see that if the mutual inductances are equal, there would not be an induced emf in the antenna. Due to slight differences in the distances between the antenna and the two lines and differences in the self inductances in the lines there will however be some coupling. 5.1.1.4 Reduction of Differential-mode Magnetic Coupling The reduction techniques for reducing differential-mode magnetic coupling is based on getting the mutual inductance from the two wires to be as equal as possible. This is done by twisting the wire, which is equal to reducing the loop area between the wires. It is important to keep the wires as close as possible. Also here we can see that the induced voltage in the antenna is directly proportional to the differential- mode current and varies inversely with the distance between the antenna and the line, R. 5.1.2 Electric Coupling 5.1.2.1 Common-mode Coupling In this chapter we will study the electric coupling of the common-mode current to an antenna due to mutual capacitance, C , shown in the following figure:
Figure 22 Common-mode electric (mutual capacitance) coupling between a telecommunication line above and an antenna below. The two wire line carrying the common-mode signal is approximated with a single conductor. As in the case of magnetic coupling, the distance between the conductors are considered very small compared to the distance to the antenna. Therefore the two conductors carrying the common-mode current are considered as a single conductor. The capacitance between the telecommunication line and the antenna is a physical property. It depends on the geometry of the line and the antenna and of the permittivity of the medium between them. The capacitance of an isolated conducting body is the electric charge that must be added to the body per unit of increase in its electrical potential. The capacitance was defined from the observation that the ratio between the charge, Q , and the voltage, V , is a proportionality constant which remains constant. Equation 141 where the unit is coulomb per volt or farad, F. Recall that for a parallel plate capacitor of area S the capacitance C is expressed as: Equation 142 where d and are the distance between the plates and the permittivity of the dielectric that space. Consider an infinitely long line charge with a charge density [C/m]. It will cause a cylindrical electric field with intensity E at the perpendicular distance r from the line charge. Equation 143 This relationship can be used to approximate the electric field intensity of both the dipole antenna and the transmission line. At a distance r from the line charge an electric potential can be calculated by integrating the electric field intensity E over the distance from the line charge to the point where the potential is to be calculated. Figure 23 Cross section of a line charge, in P, and its image in a parallel conductor If the equivalent diameter of the transmission line is approximately equal to the diameter of the dipole antenna wire, both diameters can be written as the same variable a.
Equation 147 and then the same with the circuit to the right: Equation 148 The equivalent circuit will then be the same as in Figure 19 with the capacitance C as the value of M. The resistance 1/ R will be neglectible compared to Z 11 in Figure 19 since R is very large. Then we get the following relation of the current in the antenna due to the common-mode current in the telecommunication line (as in Equation 134): Equation 149 which is directly proportional to the common-mode voltage in the telecommunication line and to the capacitance between them. Notice that this current has the same sign as that from mutual inductance in Equation 134 and thus these two contributions will be additive and not cancel each other. 5.1.2.2 Reduction of Common-mode Electric Coupling Since we can not control the antenna’s ground impedance, the only factors we can change are either the common-mode voltage or the capacitance between the line and the antenna. The capacitance is hard to reduce since we can not separate the line and the antenna or reduce the radius of the wires. A possibility is of course to operate at lower frequencies to increase the reactance caused by C. Reducing the common-mode voltage is done by increasing the balance of the cable (see Equation 91), increasing the common-mode impedance and the use of different common-mode rejection techniques (these methods also increases LCL as seen in Equation 101). The common-mode voltage is also reduced by reducing the differential-mode voltage since that is the source of the common-mode voltages. 5.1.2.3 Differential-mode Coupling With differential-mode voltages on the line, the two wires will cause almost equal and opposite currents in the antenna which will tend to cancel each other. We denote the mutual capacitance from one wire to the antenna C 13 and the mutual capacitance from the other wire to the antenna C 23 , as in the following figure
Figure 26 The electric (capacitive) coupling between a two wire telecommunication line carrying differential-mode signals above and an antenna below. The current in the antenna due to the differential-mode voltages will then be: Equation 150 where I 1 is the current in one conductor in the line and I 2 is the current in the other conductor. But since these are differential currents only, I 2 will have the same magnitude as I 1 but opposite direction: Equation 151 Then we can write Equation 150 as: Equation 152 which is directly proportional to the differential-mode currents on the line and to the differences in mutual capacitance to the two conductors in the line. We can see that if the mutual capacitances are equal, the induced currents in the antenna will cancel each other. Due to slight differences in the distances between the antenna and the two wires and differences in the geometry between the two wires, there will however be some nonzero capacitive coupling even here. 5.1.2.4 Reduction of Differential-mode Magnetic Coupling The most common reduction technique is to twist the wires to keep them as close as possible to each other. Reducing the differential-mode currents on the line will have a direct effect as seen in Equation
- Equal diameters of the wires are important. The frequencies should be kept as low as possible. Separating the antenna and the line is very favorable, but hard to accomplish.
Figure 28 The common-mode current loop causing magnetic fields, with aerial telecommunication lines. The common-mode signals on the two wire line are approximated with one signal on a single conductor. As before the distance between the conductors are considered very small compared to the distance to the point of observation. Therefore the two conductors carrying the common-mode current are considered as a single conductor carrying a current. Consider a small loop of radius b carrying a uniform time-harmonic current i(t)=Icos w t around its circumference as in the following figure: Figure 29 A magnetic dipole or current loop. This is called an elemental magnetic dipole, which has a vector magnetic moment, m , as: [Am^2 ] Equation 153 To determine the electromagnetic field from this current loop, we need the vector magnetic potential, A , since the magnetic flux density, B , is the curl of the vector magnetic potential. Assuming that we have a thin wire and the current is flowing entirely along the wire we have: Equation 154 This integral is however hard to calculate exactly, because R 1 will change with the location of dl’ on the loop. If we assume that we have a small loop we can solve the vector magnetic potential as: Equation 155 Then the electric and magnetic field intensities, E and H , can be solved by deriving the magnetic flux from the vector magnetic potential, A , for the magnetic field and then the electric field can be calculated from the curl of the magnetic field intensity as: Equation 156 Equation 157
From this we get the electric and magnetic field intensities as: Equation 158 Equation 159 Equation 160 Notice the similarity with the equations for the electric dipole, derived in Equation 43, Equation 44 and Equation 45, and that the nature of the near and far-field discussed earlier also applies to these equations. For the far-field ( R >> l /2p ) these equations will simplify to: [V/m] Equation 161 [A/m] Equation 162 where w =2p c /l. We can see that the far-field intensities vary inversely as R and their ratio E f / H q equals the intrinsic impedance of free space, h 0. We can also see that the maximum fields are produced in the same plane as the current loop, where q is p /2. The vector magnetic moment, m , is the current, I , times the area of the loop which we denote S. That means that the electric (and the magnetic) field intensity vary linearly with the current in the loop and the area of the loop. If we look at the electric field intensity in a point in the same plane as the loop, in the x-y plane, where we have the maximum field intensity, we could write the electric field intensity as a constant depending only on the frequency and the distance R times the current and the area of the loop (in free space): Equation 163 where the area, S , is in m^2 and the current, I , is the peak amplitude current in amperes at appropriate frequency, then the constant K is derived as [ 5 ]: Equation 164
distribution along the telecommunication line is generally in the form of standing waves with current maxima and minima along the line resulting in directional radiation field patterns. The amplitude of the standing waves is determined by the match between the characteristic impedance of the line and the load impedance. Notice that in the common-mode situation, the load impedance is the common-mode impedance, which could be reactive (capacitive coupling to ground). If the load impedance is purely reactive, ZL=jXL , then the input impedance can be written as: Equation 168 and we will have a purely reactive input impedance with poles and zeros at different frequencies depending on the relation between the line length, load impedance and the characteristic impedance of the line. If we look at the common-mode current loop as illustrated in Figure 28 we know that the electric and magnetic field intensity in the far-field is proportional to the loop area which is the medium height over ground times the length of the line and to the common-mode current flowing in the loop. Notice that in the aerial telecommunication line case, the loop will often become a large loop. 5.2.1.2 Reduction of Common-mode Magnetic Radiation The most obvious way would be to reduce the loop area by reducing the height over ground or reducing the length of the cable, but since this would be to expensive, the reduction techniques are focusing on reducing the common-mode current amplitude. This is mainly done by reducing the ground loop coupling. The ground loop coupling can be reduced by on or more of the following techniques: The use of ferrite beads on signal lines (common-mode choke) Float (ungrounded) box shields inside equipment enclosures Install RF chokes in the case-to-ground path Use an isolation transformer Use an optical isolator Using a balanced system, with a balanced transmitter and receiver Float (unground) either or both the transmitter and receiver Shielding the loop area by grounding it is, at present, quite expensive. We can see that a buried cable would not have as large loop area as an aerial telecommunication cable. So if burying the cable is too expensive, the only way to reduce the radiation is to have as good balance as possible to reduce the amount of common-mode current and then to make it as hard as possible for the common-mode current to flow around the loop. 5.2.1.3 Differential-mode Radiation The theory presented above for common-mode signals will hold in this case too except for the fact that the loop area will be much less than in the case of common-mode current in aerial cables. The loop area for the differential mode signal will be the area between the conductors of the cable, as illustrated in the following figure:
Figure 31 The differential-mode current loop causing magnetic fields. Note that this area will be even less in a real cable since the cables are twisted, intentionally or unintentionally. The different parts of a twisted cable will not radiate in the same direction and will in some cases tend to cancel each other. Also what was said about the input impedance, see Figure 30, depending on the length of the cable can be applied to this situation, see Equation 166 and about the maxima and minima in Equation 168. 5.2.1.4 Reduction of Differential-mode Magnetic Coupling Also here the radiation will be proportional to the current in the loop, see Equation 163, which means that a reduction of the differential-mode current would have a direct effect on the amount of radiated fields, by the same amount. It is also direct proportional to the loop area, Equation 163, and inversely proportional to the distance, R , to the point of observation (Equation 161 and Equation 162). The most common reduction technique is the use of twisted pairs. The reduction of the differential mode coupling that is offered by twisted pair can be written as [ 8 ]: [dB] Equation 169 where n is the number of twists per meter, l is the length of the wire in meters and l is the wavelength in meters. The reduction of differential-mode coupling offered by twisted pair is plotted in the following figure:
