1 Introduction to Electromagnetics

Electromagnetic fields are caused by electric charges at rest and in motion. Positive and negative electric charges are sources of the electric fields and moving electric charges yielding a current is the source of magnetic fields. Time-varying electric and magnetic fields are coupled in an electromagnetic field radiating from the source.

Figure 1 Positive and negative electric charges are sources of an electric field

Electromagnetic fields are divided into four different quantities:

the magnetic flux density B with the unit T (Tesla or volt-second per square meter)

the magnetic field intensity H with the unit A/m (Ampere per meter)

the electric field intensity E with the unit V/m (Volt per meter)

the electric flux density D with the unit C/m2 (Coulomb per square meter)

A time-varying E and D will give rise to B and H, and vice versa where the relation depends on the properties of the medium. Far enough from the source the magnetic field, H, will be perpendicular to the electric field, E, and both are normal to the direction of propagation, as shown in the following figure:

Figure 2 A time-varying electric field, E, will give rise to a perpendicular magnetic field, H, and vice versa. Far enough from the source it will become a uniform plane wave and the ratio between E and H will be the intrinsic impedance of the medium.

Far enough from the source, the wave-front, which will become almost spherical, can be seen as an almost plane front if the sphere is large enough. Then we have a uniform plane wave where the ratio between the electric field and the magnetic field, called the wave impedance:

Equation 1

is a constant named the intrinsic impedance of the medium, . The electromagnetic theory is also based on three universal constants:

the velocity of an electromagnetic wave in free space c (the speed of light 3.108 m/s)

the permittivity of free space 0 ( 8.854.10-12 F/m)

the permeability of free space 0 (= 4 .10-7 H/m)

These constants are related by [8]:

Equation 2

The permittivity is a proportionality constant between the electric flux density D and the electric field intensity E, in free space as:

Equation 3

and the permeability is the proportionality constant between the magnetic flux density B and the magnetic field intensity H, in free space as:

Equation 4

From these constants, the intrinsic impedance of free space can be calculated as [8]:

[ ]

Equation 5

2 Maxwells Equations

James Clerk Maxwell (1837-1879) gathered all prior knowledge in electromagnetics and summoned the whole theory of electromagnetics in four equations, called the Maxwells equations.

To evolve the Maxwells equations we start with the fundamental postulates of electrostatics and magnetostatics. These fundamental relations are considered laws of nature from which we can build the whole electromagnetic theory.

According to Helmholtzs theorem, a vector field is determined to within an additive constant if both its divergence and its curl are specified everywhere [8]. From this an electrostatic model and a magnetostatic model are derived only by defining two fundamental vectors, the electric field intensity E and the magnetic flux density B, and then specifying their divergence and their curls as postulates. Written in their differential form we have for the electrostatic model the following two relations' [8]:

Equation 6

Equation 7

where is the volume charge density:

[C/m3]

Equation 8

These are based on the electric field intensity vector, E, as the only fundamental field quantity in free space. Then to account for the effect of polarisation in a medium the electric flux density, D, is defined by the constitutive relation:

Equation 9

where the permittivity is a scalar (if the medium is linear and isotropic). Similarly for the magnetostatic model we have the following two relations, based on the magnetic flux density vector, B, as the fundamental field quantity:

Equation 10

Equation 11

where J is the current density. To account for the material here as well, we define another fundamental field quantity, the magnetic field intensity, H, and we get the following constitutive relation:

Equation 12

where is the permeability of the medium. Using the constitutive relations we can rewrite the postulates and the relations derived is gathered in the following table:

Table 1 Fundamental Relations for Electrostatic and Magnetostatic Models (The Governing Equations)

The Governing Equations

Electrostatic Model

Equation 13

Equation 14

Magnetostatic Model

Equation 15

Equation 16

These equations must, however, be revised for calculation of time varying fields. The electrostatic model must be modified due to the observed fact that a time varying magnetic field gives rise to an electric field and vice versa and the magnetostatic model must be modified in order to be consistent with the equation of continuity.

The complete model for electromagnetic fields (Maxwells equations) is gathered in the following table (Table 2), where the integral forms of the equations are added [8]:

Table 2 Maxwell's Equations, both in differential and integral form

Maxwells Equations

Faradays law

Equation 17

Equation 18

Ampres circuital law

Equation 19

Equation 20

Gausss law

Equation 21

Equation 22

No isolated magnetic charge

Equation 23

Equation 24

We can see in Equation 17 that the electric field intensity vector (Equation 13) is replaced with according to Faradays law of electromagnetic induction [8]. In Equation 19, the term is called displacement current density and its introduction was one of the major contributions by Maxwell. The displacement current density is necessary in order to make the equations consistent with the principle of conservation of charge in the time varying case.

There are many ways of solving and using these equations. One technique to make the solution of Maxwells equations easier, which we will use later, is to use potential functions. It is known that if a vector field is divergenceless, then it can be expressed as the curl of another vector field. For instance since the divergence of B is zero, , then B can be expressed as the curl of the vector field A:

Equation 25

where A is called the vector magnetic potential and it can be determined from the current distribution J:

Equation 26

where:

is the wave number

r is the distance between the radiating source and the point of observation

Using the constitutive relations, Equation 9 and Equation 12, the magnetic field intensity H can then be calculated as:

Equation 27

The electric field intensity E can also be calculated using potential functions by introduction of the scalar electric potential V:

Equation 28

When both the vector magnetic potential A and the scalar electric potential V are known, the electric field intensity E is derived by:

Equation 29

It is however not necessary to calculate both the magnetic field intensity H and the electric field intensity E since they are related by the equation:

Equation 30

3 Near-Field and Far-Field

In this chapter we will solve the Maxwells equations for a radiating wire and by analysing the solution we will define the near-field and the far-field. The electromagnetic field radiating from a wire can be calculated by solving Maxwells equations for a short current element and then placing the current elements end-to-end.

Figure 3 A small current element or an electric dipole.

To be able to derive the field of a conducting element acting as an electrical dipole, the current in the element is defined to flow along an axis z. This gives that the magnetic vector potential can be expressed as (see Equation 26):

Equation 31

where:

Equation 32

It is wise to use spherical coordinates when deriving the field of a small current element that can be approximated with a point source

Equation 33

If A is then expressed in the form

Equation 34

it can apparently be divided into the spherical components:

Equation 35

Equation 36

Equation 37

The intensity ofq he magnetic field is then (using Equation 27):

Equation 38

According to Equation 30 above E can be determined when H is known:

Equation 39

Which results in:

Equation 40

Equation 41

Equation 42

By rewriting those equations we can identify the wavelength and get the following equations in Table 3. A summation of the theory above states that:

If the length of one element is much less than a wavelength (dl the last term proportional to r-1 in Equation 43 and Equation 45 will dominate and the wave impedance will approach the free space impedance Z0 = 377 . This is called the far-field or radiation field. The E and the H fields will then be in phase and orthogonal to each other producing plane waves. This is illustrated in Figure 5 below:

Figure 5 Wave impedance at different distances from either an electric source or a magnetic source

When such current elements are placed end-to-end to produce a model of a radiating wire, the charge at the ends of the elements will cancel and the term due to the electrostatic field (the one proportional to r-3) will disappear. This is however only true with a constant current distribution on the line. With a varying current distribution the electrostatic fields will not cancel entirely. However, if the wire is divided into a sufficient number of segments per wavelength then this error will be small.

The length of telecommunication lines and the size of telecommunication systems are often greater than one wavelength, D . If the dimension of the field source D is greater than a wavelength the near-field/far-field boundary is said to be at the distance [5]:

Equation 48

At a shorter distance maxima and minima would appear due to interference caused by different distances to different parts of the source.

4 Transmission Line

4.1 Lossy Transmission Line Model

A transmission line can be approximated by a distributed-parameter network with the circuit parameters distributed throughout the line. One line segment with the length z can be approximated with an electric circuit as in the following figure:

Figure 6 Equivalent circuit of an element of a transmission line with a length of z

Note that this model is actually for an unbalanced line but the model for a balanced line is the same except for R and L which is divided symmetrically on both conductors with R/2 and L/2 both on the upper conductor and the lower conductor in Figure 6. This lossy transmission line model is described by four lumped parameters, which will be derived later in this section:

R is the resistance in both conductors per unit length in /m

L is the inductance in both conductors per unit length in H/m

G is the conductance of the dielectric media per unit length in S/m

C is the capacitance between the conductors per unit length in F/m

where R and G is zero under lossless conditions. We will also derive the propagation constant and the characteristic impedance in this section.

To derive the parameters we begin by using Kirchhoffs voltage law on the circuit in Figure 6:

Equation 49

which can be written as:

Equation 50

and then letting z 0 we get:

Equation 51

Then we have one equation containing R and L. To get another equation relating G and C we apply Kirchhoffs current law on the circuit and get:

Equation 52

and letting z 0 in this equation also we get:

Equation 53

The first order partial differential equations, Equation 51 and Equation 53, are called the general transmission line equations. These equations can be simplified if the voltage v(z,t) and the current i(z,t) are time-harmonic cosine functions:

Equation 54

Equation 55

Using Equation 54 and Equation 55, the general transmission line equations in Equation 51 and Equation 53 can be written as:

Equation 56

Equation 57

These equations are called the time-harmonic transmission line equations. These equations (Equation 56 and Equation 57) can be used to derive the propagation constant and the characteristic impedance of the line. By combining Equation 56 and Equation 57 we get:

Equation 58

Equation 59

where is the propagation constant:

Equation 60

The real part, , of the propagation constant is called the attenuation constant in Np/m and the imaginary part, , is called the phase constant of the line in rad/m. In a transmission line the series resistance of the conductors are so low that it can be neglected in the computation of the propagation constant. Equation 60 can then be simplified to:

Equation 61

The solution of Equation 58 and Equation 59 is:

Equation 62

Equation 63

where the two terms in each equation denotes travelling waves in positive and negative direction along z respectively. The characteristic impedance of a transmission line is the ratio between the voltage and the current for an infinitely long line. For an infinitely long line with the source at the left end there are no reflecting waves so only the waves travelling in the positive z direction exist (only V+ and I+ in Equation 62 and Equation 63). Using Equation 56 and Equation 57 the characteristic impedance can be written as:

[ ]

Equation 64

Note that it is independent of z.

If the distance D between the two wires is much greater than the radius, a, of a conductor, the capacitance per unit length, C, can be written as:

[F/m]

Equation 65

And then by comparing Equation 61 with the propagation constant for a transverse electromagnetic wave in a medium with the constitutive parameters , and written as:

[m-1]

Equation 66

and assuming that the following relationship is known:

Equation 67

we get:

Equation 68

Using Equation 67 and Equation 68 we can get L and G for the transmission line from Equation 65:

[H/m]

Equation 69

and

[S/m]

Equation 70

To derive the series resistance R we look at the ohmic power dissipated per unit length of both conductors. Assuming the surface current Js to flow in a very thin surface layer and to be uniform over the circumference of both conductors the current in each conductor is then:

Equation 71

Since copper, and other materials used in transmission lines, are good conductors (i.e. c >> c) and the radius a is much larger than the skin depth the above assumptions will hold for transmission lines and then the surface impedance can then be written as the intrinsic impedance, c, of a good conductor:

[ ]

Equation 72

The dissipated power per unit length, p , assuming the presence of a non-vanishing axial electric field at the surface, will then be:

[W/m2]

Equation 73

Using Equation 71 the total ohmic power dissipated in a unit length of the line (in both conductors) will then be:

[W/m]

Equation 74

The series resistance per unit length is then:

[ /m]

Equation 75

where the parameters c and c is that of the material in the conductors. The real part of the surface impedance, Rs, is calculated as:

[ ]

Equation 76

Having the distance, D, between the conductors and the radius, a, of the conductors, the characteristic impedance from Equation 64 can be simplified using Equation 65, Equation 69, Equation 70 and Equation 75 into:

[ ]

Equation 77

It can also be shown that for a balanced pair of wires near ground the characteristic impedance will also depend on the height over ground, h, as:

[ ]

Equation 78

The derived equations is collected in the following table:

Table 4 Basic transmission line equations

Description

Parameter

Equation

Unit

Resistance

R

/m

Inductance

L

H/m

Conductance

G

S/m

Capacitance

C

F/m

Characteristic Impedance

Z0

Characteristic Impedance (simplified)

Z0

Characteristic Impedance

(at height h above ground)

Z0

Propagation Constant

m-1

4.2 Termination and Reflections

In an infinitely long line there are only forward travelling waves and no reflected waves. The second term in Equation 62 and Equation 63 will be zero. This is however also true for a line terminated with its characteristic impedance. A line is called a matched line when the load impedance is equal to the characteristic impedance. If we consider a line with the characteristic impedance Z0, a propagation constant and with the length l terminated with a load impedance ZL connected to a sinusoidal voltage source, and then the voltage and current distribution on the line can be calculated as:

[V]

Equation 79

[A]

Equation 80

where z = l-z is the distance measured backward from the load. These equations are derived from Equation 62 and Equation 63 by looking at z=0 and z=l. These equations are solved using the fact that:

[ ]

Equation 81

and then using the hyperbolic functions to simplify. Having the voltage and current distribution the input impedance, Zi, can be calculated, which is the impedance that the source sees at z = 0 or z= l:

[ ]

Equation 82

Notice that when ZL equals Z0 the input impedance will be equal to the characteristic impedance. So the voltage source only sees an impedance Zi and the input current, Ii. The voltage, Vi, can then easily be calculated from the source voltage, Vg, and the internal impedance of the source Zg as:

Equation 83

Equation 84

Note that if the line is matched, Equation 79 and Equation 80 reduces to:

Equation 85

Equation 86

This means that the voltage and current distribution on a matched line are exactly the same as though the line has been extended to infinity.

The average input power delivered by the source can then be calculated as:

Equation 87

where * denotes the complex conjugate. The average power delivered to the load is calculated as:

Equation 88

In the area of telecommunication the absolute power is often measured in dBm. That is the power in decibel, with the reference power one milliwatt.

Equation 89

Where P is the power in milliwatt, V is the circuit voltage in volts and R is the circuit impedance in across which V is measured. When describing a voltage in decibel, measured in dBu, the reference is said to be the power of 1 mW over 600 which represents a voltage of 0.775 V. Note that dBm and dBu will be the same if measured over 600 . If the impedance, Z, is not 600 , however, the absolute power in dBm can be computed from the measured voltage, U, as:

Equation 90

4.3 Balanced and Unbalanced Transmission

A good balance of the telecommunication line is important for reducing common-mode signals. Common-mode signals are the main cause of the electromagnetic field around an aerial wire since the distance between the conductors in the wire is much less than the distance between the wire and ground. For differential-mode signals the phase difference is 180 and the amplitudes are equal, therefore fields generated by the two conductors will tend to cancel each other.

INCLUDEPICTURE "http://www.sm.luth.se/~urban/master/Theory/image238.gif" \* MERGEFORMATINET

Figure 7 Radiation from common-mode signals and radiation from differential-mode signals.

Figure 7 illustrates simulated radiation from a wire, carrying a common-mode current to the left and a differential-mode current to the right. The magnitudes of the currents were the same and the differences in field strength between the two lines are also the same in both figures. The existing radiation from the line to the right is due to the distance between the conductors.

If the wire is not perfectly balanced a common-mode signal will appear due to introduced differences in amplitude and phase. The effect of this phenomenon can be calculated with the use of the longitudinal conversion loss (LCL). In recommendation G.117 from ITU-T a standardised method is described for determination of LCL [4]. First EL is applied and VT is measured according to the following figure:

Figure 8 Method for measurement of LCL [4]

Then LCL can be calculated as [4]:

[dB]

Equation 91

To see the relation between the longitudinal conversion loss (LCL), as measured in Figure 8, the unbalance of the line and the conversion from differential signals to common-mode signals, we will study a model of a telecommunication line as in the following figure:

Figure 9 Equivalent circuit of equipment and line

where ZC is the source impedance to ground and Z1 + Z2 is the differential mode impedance (normally the termination which is customarily the same as the characteristic impedance of the line). The common-mode impedance is the impedance for the common mode signals to ground, (Z1 in parallel with Z2) + ZL. If the measurement set-up for the LCL-measurements shown in Figure 8 is applied to the line in Figure 9, we get the following equivalent circuit:

Figure 10 Equivalent circuit of the LCL measurement set-up (Figure 8) applied to the telecommunication line in Figure 9.

Then the common-mode current, I, in Figure 10, is given by:

Equation 92

Furthermore the currents, I1 and I2, in each conductor of the line are given by:

Equation 93

Equation 94

The differential mode voltage, VT, can then be expressed as

Equation 95

Combining Equation 92, Equation 93, Equation 94 and Equation 95 and substituting Equation 91 leads to:

Equation 96

Because the differential mode impedance is fixed and the source is perfectly balanced the following assumptions can be made:

Equation 97

Equation 98

We can express the cable unbalance as the difference between the differential impedances Z1 and Z2:

Equation 99

Substituting Equation 97, Equation 98 and Equation 99 in Equation 96 gives the following relation for the longitudinal conversion loss:

Equation 100

Finally by assuming a reasonable balance on the line, , give the approximation:

Equation 101

In the next step of the study this expression is more practical to use than Equation 91. From Equation 101 it can easily be observed that LCL is inversely dependent of the unbalance Z. In addition larger load impedance to ground, ZL, will increase LCL and thus less current will be converted from differential-mode to common-mode.

To be able to investigate the relationship between differential mode signals and common mode signals a general model in Figure 9 is refined below.

Figure 11 Equivalent circuit of Figure 9 showing differential mode and common mode currents

Note that the signal source is primarily producing a differential signal in this case. By applying Kirchhoffs voltage law on the equivalent circuit in Figure 11, the following two equations are obtained:

Equation 102

Equation 103

Combining these equations will result in the following common-mode current:

Equation 104

Substituting Equation 97, Equation 98 and Equation 99 in Equation 104 leads to:

Equation 105

Assuming that we have a reasonable balanced line, , results in:

Equation 106

or if we instead write this in decibel:

Equation 107

where the factor Q is:

Equation 108

But we want an expression containing the LCL so we rewrite Equation 108 and then use Equation 101 to insert the LCL. Q can then be rewritten as:

Equation 109

Substituting the CM-impedance, which can be written as , gives:

Equation 110

For the CM-current in Equation 107 this leads to:

Equation 111

However, if it is the common mode voltage (VC) that is the wanted variable, is simply subtracted from ICM in Equation 111 and we get VC in dB V.

In most cases this relationship can under some assumptions be simplified, for instance assuming that the line is approximately balanced, i.e. ES in Equation 111 is equal to the logarithmic differential-mode voltage (VD).

Having the longitudinal conversion loss, the common-mode voltage (VC) can then be calculated from the differential-mode voltage (VD) [1]:

[dB V]

Equation 112

Where ZCM is the common-mode impedance and Z0 is the differential-mode impedance. This relation between the differential-mode voltage (VD) and the common-mode voltage (VC) is the most commonly used for calculations on balanced transmission lines.

When the common-mode voltage is known it is possible to estimate the interference field strength (E) by the use of a conversion factor (FAV) from common-mode voltage to an electric field determined at a specified distance from the cable [1].

[dB V/m]

Equation 113

or if we use the relationship in Equation 112 and replace the common-mode voltage in Equation 113 we get:

[dB V/m]

Equation 114

The relation between LCL and the radiated electric field strength have also been measured [1] (in Figure 12 below) and it have been shown that the radiated field strength is approximately inversely proportional to LCL.

Figure 12 Relationship between LCL of the telecommunication line and the radiated field strength

The conversion from an unbalanced signal from the source to a balanced signal on the line and back to an unbalanced signal again is done with balanced to unbalanced transformers (baluns). There are mainly three kinds of broadband baluns, shown in Figure 13 below.

Figure 13 Three different kinds of baluns with their equivalent schematic to the right

These can be divided into two distinct categories. The first two baluns are voltage baluns causing equal and opposite voltages to appear at the two output terminals. The third balun is a current balun, or choke balun, forcing equal and opposite currents to flow on the line. If the line and the terminating load is perfectly symmetrical, the voltage baluns will force the voltages to be equal and opposite and thus the currents flowing from the balun output terminals will be equal and opposite and there will be no CM-current on the line. If, however, the line is not perfectly symmetrical unequal currents, resulting in CM-currents, will appear leading to line radiation.

A solution, that is good in the aspect of reducing line radiation, would be to use a current balun, which makes it possible for opposite currents to flow and rejects CM-currents. Another solution is to use a voltage balun in combination with a current balun. Notice that even if the voltage is perfectly balanced at the ends of the line, some CM-currents will still appear due to non-symmetrical line and load. When using a current balun on a balanced line it will function as a CM-suppressor. The wire is wrapped around a ferrite core as in Figure 13. Then the differential currents of opposite phase will produce magnetic fluxes of opposite phase in the core that cancel each other and there will be no remaining reactance in the core for differential-mode signals. The common-mode current will however produce a magnetic flux that appears as a reactance in series with the line.

Figure 14 Impedance in different ferrite beads

Ferrite cores are well suited as core material but the characteristic will vary a lot with the frequency and with the ferrite material as shown in Figure 14 above.

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 modelled with an equivalent two-port network as in the following figure:

Figure 17 Equivalent two-port network of the coupling between the telecommunication line and an antenna, with the antenna to the right and the coupling impedance Z12 (which is assumed to be equal to Z21)

The voltage on the line V1 principally represents the common-mode voltage of the line but could also be made up by differences in distance to the differential-mode voltages. I1 is the current in the telecommunication line and I2 is the current in the antenna. V2 is the voltage induced across the load impedance of the antenna. These voltages and currents are linearly related as:

Equation 115

Equation 116

where Z11, Z12, Z21 and Z22 are open circuit impedance coefficients. When the medium between the line and the antenna is bilateral, governing reciprocity relations, the coupling impedances Z12 and Z21 are equal. This equivalent network is used in the following theory only by the near-field models but it will hold for the far-field also since at large distances, r, between the line and the antenna, the coupling impedances will become very small:

Equation 117

Then the impedances Z11 and Z22 will be nearly equal to the input impedances of the line and the antenna ZA and ZB. That means that with weak coupling Equation 115 could be written as:

Equation 118

If we apply Thvenins theorem to the left of the load impedance of the antenna, ZL, we can determine the open-circuit voltage VOC and internal impedance Zg as:

Equation 119

Equation 120

Also here we can see that if we are in the far-field and have a weak coupling, the internal impedance, Zg, for the antenna will be approximately equal to its input impedance as:

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, Z12 or Z21, 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. Therefor 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 crosstalk 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 C1. 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 S1. The common-mode current I1 flowing around the circumference C1 of S1, will produce a magnetic field B1. A part of the magnetic flux caused by B1 will link with the antenna, which bounds the surface S2. 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, U2. Notice that C2 does not have to be a physical closed loop. The right side can be rewritten using Equation 124 and the result is the Faradays 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]:

Equation 126

Since the emf U2 is induced by a time-varying magnetic field it is called a transformer emf. The negative sign states that the induced emf will cause a current, which in turn will produce an opposite magnetic flux. This is known as Lenzs law.

Then by using Biot-Savarts law, which is obtained by taking the curl of the vector magnetic potential A (as in Equation 25 and Equation 26):

Equation 127

where aR is the unit vector from the source point to the field point and R is the distance from the wire element dl to the field point. We can see that B1 is directly proportional to the common-mode current I1 and from Equation 124 we can see that the mutual flux 12 is also proportional to I1. If we define that proportionality factor as the mutual inductance, M, between the telecommunication line and the antenna we have:

Equation 128

which states that the mutual inductance is the magnetic flux linkage with one circuit per unit current in the other. In a similar way the self inductance is defined as the magnetic flux linkage per unit current in the loop itself as:

Equation 129

For a linear medium, the self inductance does not depend on the current in the loop and exists regardless of whether the loop is open or closed. The inductances depend on the geometrical shape of the elements constituting the current and on the permeability of the medium.

If we consider the wire as one long conductor and a parallel dipole antenna as another long conductor where both conductors have the same radius, a, for simplicity reasons. These are separated by a distance d, which is much larger than a. Both conductors are in free space. To calculate the mutual inductance we first combine Equation 124 and Equation 128 into:

Equation 130

where we use the subscript 1 to denote the telecommunication line and the subscript 2 for the antenna. Since B1 can be calculated from the curl of the vector magnetic potential A1 we can write (Equation 25):

Equation 131

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 C1 and C2 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, Z12 = Z21. 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 Z12 in Figure 17. The self-inductances, as calculated in Equation 129, would be part of the impedances Z11 and Z22 in the same figure. If Z12 is purely inductive then -Z12 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 I1 and I2 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 L1 and L2.

U2 is induced emf in the antenna due to the common-mode current I1. The mutual inductance can be written as:

Equation 135

where L1 is the self inductance of the telecommunication line, L2 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

The most obvious reduction techniques like reducing the loop area (i.e. reducing the height and the length of the telecommunication line) or increasing the distance, R, between the interfering and the interfered object is however hard to do in practice.

Reducing the common-mode current in the line will of cause have a direct effect on induced voltage in the antenna, as seen in Equation 137. Increasing the balance of the cable would reduce the common-mode current (see Equation 91). Increasing the common-mode impedance and the use of different common-mode rejection techniques could also be utilised to reduce the common-mode current.

The common-mode current is also reduced by reducing the differential-mode current since that is the source of the common-mode current. Operating at lower frequencies, if possible, would decrease the reactance of M.

5.1.1.3 Differential-mode Coupling

With differential-mode currents on the line, the two wires will cause almost equal and opposite magnetic fluxes that will tend to cancel each other. We denote the mutual inductance from one wire to the antenna M13 and the mutual inductance from the other wire to the antenna M23, as in the following figure:

Figure 21 The magnetic (mutual inductance) coupling between a two wire telecommunication line carrying differential-mode currents above and an antenna below.

The differential-mode currents are equal and have opposite direction. We can then write the induced current by using Equation 134 and the law of superposition:

Equation 138

or with the emf-method as in Equation 136 (superposition):

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 ekvivalent 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.

Figure 24 Cross section of a telecommunication cable in parallel with the conductor of a dipole antenna and equivalent line charges

The potential in the dipole antenna at a distance d from a line charge in the transmission line is:

Equation 144

and the corresponding potential caused by a line charge in the dipole antenna is:

Equation 145

If l is the length for which the transmission line and the dipole antenna are parallel, the mutual capacitance can simply be expressed as (using Equation 141):

Equation 146

To use the mutual capacitance in the coupling impedance concept as in Figure 17, we have to rewrite the mutual capacitance coupling as a mutual inductance by assuming a very large resistance to ground as in the following figure:

Figure 25 Equivalent circuit of mutual capacitance, using mutual inductance instead

That the left side is the same as the right side in Figure 25 is realised when calculating the voltage out on the right, V2, from applying a voltage, V1, to the left (voltage dividing), first on the left circuit (in Figure 25):

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 Z11 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 antennas 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 ofcourse 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 C13 and the mutual capacitance from the other wire to the antenna C23, 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 I1 is the current in one conductor in the line and I2 is the current in the other conductor. But since these are differential currents only, I2 will have the same magnitude as I1 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 152. Equal diameters of the wires are important. The frequencies should be kept as low as possible. Separating the antenna and the line is very favourable, but hard to accomplish.

5.2 Far-Field Models

In the far-field the main cause of coupling between a telecommunication line and an antenna is the radiation term as derived in chapter 3. We chose here to look at the electromagnetic fields generated instead of the inductive and capacitive coupling as in the near-field models. The models are divided into magnetic fields from current loops and electric fields from time-varying voltages. These models are divided into common-mode radiation and differential-mode radiation. The results could, with care, be superimposed, but the purpose is to locate the main cause of radiation in a particular case as with the near-field models. Also here it is very important to know the main factors in each type of radiation to be able to build a realistic case for measuring, or to know which factor to focus on to reduce a specific kind of radiation.

In the models for both the magnetic and the electric radiation we will focus on a straight aerial telecommunication line as shown in the following figure:

Figure 27 The main radiation direction and polarisations for a rectangular loop

5.2.1 Magnetic Fields

5.2.1.1 Common-mode Radiation

In this chapter we will study the magnetic radiation due to common-mode current flowing as shown in the following figure:

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 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:

[Am2]

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 R1 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 >> /2 ) these equations will simplify to:

[V/m]

Equation 161

[A/m]

Equation 162

where =2 c/ . We can see that the far-field intensities vary inversely as R and their ratio E /H equals the intrinsic impedance of free space, 0. We can also see that the maximum fields are produced in the same plane as the current loop, where is /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 m2 and the current, I, is the peak amplitude current in amperes at appropriate frequency, then the constant K is derived as [5]:

Equation 164

The current loop can be either a small loop or a large loop which will affect the input impedance of the loop and therefor the current flowing in the loop. We have earlier derived the input impedance for a transmission line (Equation 82) repeated here:

Equation 165

If we consider an ideal transmission line and assume negligible line losses it can be shown that the input impedance can be written as:

Equation 166

where Z0 is the characteristic impedance of the line, ZL is the load impedance, l is the length of the line and is the wavelength as in the following figure:

Figure 30 The input impedance for a line of length l

From Equation 166 we can see that for small loops with l > /2 ) the contribution to the far field radiation from each element can be calculated according to the following simplified equations:

[A/m]

Equation 170

[V/m]

Equation 171

By examining the relationships in these equations we see that the magnitude of the fields in the far field is proportional to the current through the element I, the length of the element dl and inversely dependent of the distance R and that the ratio between E and H equals the intrinsic impedance of free space, 0. Another important observation is that the magnitude of the field is maximised perpendicular to the conducting element where = /2, and minimised along the axis of the conducting element where =0, see Figure 3.

Consider a typical aerial telecommunication line with horizontally and vertically oriented segments according to Figure 33. According to the theory above, the common mode current will produce an electromagnetic field. The current distribution will also give rise to a charge distribution that also will produce an electromagnetic field. The value of the common-mode impedance will decide whether the common-mode current or the common-mode voltage will be the main cause of radiation.

Figure 33 The model for the electric fields generated by common-mode signals on the line. The common-mode signals on the two wire line are approximated with one signal on a single conductor.

The distribution of currents along a telecommunication line above a perfectly conducting ground has a corresponding image of an opposite current distribution equidistant to, but at the opposite side of the ground plane. This is called method of images. The vertical segments (downleads) in Figure 33 represent the common mode current path to ground. This path may consist of a capacitive coupling as well as a conductor and a common-mode impedance.

If the common-mode impedance is relatively small then the resulting radiation is mainly due to currents. Radiation caused by common-mode currents in the horizontal oriented segment of the line and its image will tend to cancel each other at some point in the far-field where the distance to the line and its image is equal or differs by a multiple of a wavelength. Since we are only interested in the fields at almost ground level the distance to the line and its image will be almost equal and the fields will cancel, see Figure 34 below.

Figure 34 Simulated electric fields from a telecommunication line above ground using the method of images

If the horizontal telecommunication line is several wavelength long and properly terminated it behaves as a travelling wave antenna with the current distribution:

Equation 172

At points where the distance to the line and its image differs by an uneven fractions of a wavelength, radiation will occur.

Figure 35 Radiation pattern of a travelling wave antenna ( is the elevation angle above earth)

Again, this is not the case at ground level in the far field.

If the common-mode impedance instead is relatively large then the resulting radiation is mainly due to the charge distribution. In this case the radiation caused by the potential of the horizontal oriented segment of the line and its image will form a electrical dipole radiating a vertical polarised electric field.

Figure 36 A telecommunication line over ground containing a charge distribution, and its corresponding image.

Instead of expressing the electric field in terms of current distribution, as in Equation 171, a time varying field can be expressed in terms of charge distribution as:

Equation 173

where

Equation 174

Equation 175

To be able to derive the electric field intensity in the far field we need to integrate Equation 173 to get:

Equation 176

The capacitance per unit length between the telecommunication line and earth can be calculated as:

Equation 177

where D is the average ekvivalent cable diameter of the telecommunication line and h is the heigth of the line above earth. This results in that an aerial telecommunication line of length L will have a total capacitance to earth of:

Equation 178

The fundamental expression for a charge

Equation 179

can by substitution, using Equation 178, be rewritten as:

Equation 180

By examining Equation 176 we can identify a dependence of the derivative of the charge. The derivative of the charge can be written as:

Equation 181

This is illustrated in the following figure:

Figure 37 A telecommunication line over ground acting as a vertical dipole

It can can also be expressed in terms of the potential V and the capacitance per unit length of the telecommunication line.

Equation 182

This can be used to express the radiation as:

Equation 183

A quick analyse of Equation 183 confirms that the electric field intensity in the far field will increase when:

the potential of the telecommunication line increases

the length of the telecommunication line increases

the capacitance per unit length between the line and earth increases

the height of the telecommunication line above earth increases

the distance between the line and the observation point decreases

the angle between a normal to the ground plane and a line from the telecommunication line to the observation point increases towards 90

The resulting magnitude of the radiation will at ground level also depend on the azimuth angle to the communication line. Perpendicular to the line the magnitude will be maximised and in parallel with the line the magnitude will depend on the distance between the downleads in relation to the wavelength as in the situation of a two element antenna array.

The dependence of the distance between the communication line and the observation point can also be calculated as the free space propagation loss. By assuming that far-field propagation and a spherical wavefront applies and neglecting effects on the propagation of polarisation, antenna heights, curvature of earth, atmospheric conditions, reflections and interfering objects along the path, the free space propagation loss can be approximated with the following formula, also shown in Figure 38.

[dB]

Equation 184

where FSPL is the Free Space Propagation Loss, R is the distance to the point of observation and f is the frequency in MHz.

Figure 38 The Free Space Propagation Loss

Notice that this is intended only to be used as a rule of thumb when evaluating the propagation loss at a distance from the source.

5.2.2.2 Reduction of Common-mode Electric Radiation

The signal source will ideally only produce differential-mode signals, common-mode signals will then arise from differential-mode signals due to longitudinal conversion loss (Equation 91). Hence, a way to reduce the common-mode electric radiation is to increase LCL (increase balance) to prevent conversion from differential-mode signals to common-mode signals. There will however always be some remaining common-mode signal that will cause radiation. To minimise the electric radiation we then have to reduce the common mode impedance, an undesired consequence of this is that the magnetic radiation will increase.

5.2.2.3 Differential-mode Radiation

The electric field radiation due to differential-mode signals is caused by opposite charges in the two wires of the telecommunication line.

Figure 39 The model for the electric fields generated by differential-mode signals on the line.

The theory for common-mode radiation above applies to this case too, if we consider having the opposite charges located in the two wires instead of in the line and its image. This will cause the distance between the opposite charges to be much smaller in this case for an aerial communication line. The capacitance between the two wires (Equation 65) is larger than the capacitance between the line and earth. As in the case of differential-mode magnetic radiation in the far field the magnitude of the radiation is also dependent on how the telecommunication line is twisted. As in illustrated in Figure 7 the electrical fields generated from the two conductors will tend to cansel each other.

5.2.2.4 Reduction of Differential-mode Electric Radiation

Because of the similarities with the case of differential-mode magnetic field radiation in the far field, the same reduction techniques are applicable. Twisting the cable is the best reduction technique (Equation 169).

Appendix A - Symbols and Units

Table 6 Fundamental SI Units

Quantity

Symbol

Unit

Abbreviation

Length

l

meter

m

Mass

m

kilogram

kg

Time

t

second

s

Current

I, i

ampere

A

Table 7 Derived Quantities

Quantity

Symbol

Unit

Abbreviation

Admittance

Y

siemens

S

Angular frequency

radian/second

rad/s

Attenuation constant

neper/meter

Np/m

Capacitance

C

farad

F

Charge

Q, q

coulomb

C

Charge density (linear)

l

coulomb/meter

C/m

Charge density (surface)

s

coulomb/meter2

C/m2

Charge density (volume)

coulomb/meter3

Conductance

G

siemens

S

Conductivity

siemens/meter

S/m

Current density (surface)

Js

siemens/meter

A/m

Current density (volume)

J

siemens/meter2

A/m2

Dielectric constant

r

Electric flux density

D

coulomb/meter2

C/m2

Electric field intensity

E

volt/meter

V/m

Electric potential

V

volt

V

Electromotive force

V

volt

V

Energy

W

joule

J

Energy density

w

joule/meter3

J/m3

Frequency

f

hertz

Hz

Impedance

Z,

ohm

Inductance

L

henry

H

Magnetic dipole moment

m

ampere-meter2

A.m2

Magnetic field intensity

H

ampere/meter

A/m

Magnetic flux

weber

Wb

Magnetic flux density

B

tesla

T

Magnetic potential

A

weber/meter

Wb/m

Permeability

0

henry/meter

H/m

Permittivity

0

farad/meter

F/m

Phase

radian

rad

Phase constant

radian/meter

rad/m

Power

P

watt

W

Propagation constant

meter-1

m-1

Reactance

X

ohm

Relative permeability

r

Relative permittivity

r

Resistance

R

ohm

Voltage

V

volt

V

Wavelength

meter

m

Wavenumber

k

radian/meter

rad/m

Work

W

joule

J

Appendix B - Constants

Table 8 Constants of Free Space

Constant

Symbol

Value

Velocity of light

c

3.108 [m/s]

Permittivity

0

[F/m]

Permeability

0

[H/m]

Intrinsic impedance

0

120 or 377 [ ]

Table 9 Physical Constants

Constant

Symbol

Value

Rest mass of electron

me

9.107.10-31 [kg]

Charge of electron

-e

-1.602.10-19 [C]

Charge-to-mass ratio of electron

-e/me

-1.759.1011 [C/kg]

Radius of electron

Re

2.81.10-15 [m]

Rest mass of proton

mp

1.673.10-27 [kg]

Table 10 Relative Permittivities (average low-frequency values at room temperature)

Material

Value

Material

Value

Air

1.0

Plexiglas

3.4

Backelite

5.0

Polyethylene

2.3

Glass

4-10

Polystyrene

2.6

Mica

6.0

Porcelain

5.7

Oil

2.3

Rubber

2.3-4.0

Paper

2-4

Soil

3-4

Paraffin wax

2.2

Teflon

2.1

Table 11 Relative Permeabilities (average low-frequency values at room temperature)

Material

Value

Material

Value

Ferromagnetic:

Diamagnetic:

Nickel

250

Bismuth

0.99983

Cobalt

600

Gold

0.99996

Iron

4000

Silver

0.99998

Mumetal

100000

Copper

0.99999

Paramagnetic:

Aluminium

1.000021

Magnesium

1.000012

Palladium

1.00082

Titanium

1.00018

Table 12 Conductivities (average low-frequency values at room temperature)

Material

Value [S/m]

Material

Value [S/m]

Silver

6.17.107

Distilled water

2.10-4

Copper

5.80.107

Dry soil

10-5

Gold

4.10.107

Transformer oil

10-11

Aluminium

3.54.107

Glass

10-12

Brass

1.57.107

Porcelain

2.10-13

Bronze

107

Rubber

10-15

Iron

107

Fused quartz

10-17

Table 13 Ground Conductivity and Dielectric Constants

Surface Type

Dielectric Constant

Conductivity [S/m]

Comment

Salt water

81

5.0

Excellent Ground

Fresh water

80

0.001

Pastoral, low hills, rich soil type

20

0.03

Very good

Flat country, marchy, densely wooded

13

0.007

Pastoral, medium hills, few trees

13

0.006

Pastoral, heavy clay soil

13

0.005

Average

Rocky soil, steep hills, mountainous

12

0.002

Poor

Sandy, dry, flat

10

0.002

Cities

5

0.001

Very poor

Cities, high buildings, industrial areas

3

0.001