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Electromagnetic Equations

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1. Introduction

 

 

2. Content

Electromagnetic waves or “radio waves” have an number of associated physical constants that are used to describe their properties. We will consider radio signal propagation in a vacuum (i.e. “free space”) for now. We will also assume “line of sight” propagation from an energy radiating radio source antenna (i.e. a transmit antenna) to an energy absorbing electromagnetic load (i.e. a receive antenna). This approach allows basic relationships to be established initially and then the real world imperfections to be included as modifications later on.

The first physical constant is generally called the “permeability of free space” identified as the Greek letter μ0 and related to magnetic properties. This physical constant is a defined coefficient and has a value defined as

      …(1)

This value is used as a reference for many other related constants that can be determined by experiment (i.e. μ0 can be thought of as a scaling factor) and has a numerical value of         

             …(2)

to 4 significant figures in units of Webers per Amps per meter. Also, the permittivity of free space labeled as the Greek letter ε0  has an experimentally determined numerical value of     

 …(3)

also to four significant figures in units of Coulombs-squared per Newton per meter. Permeability and permittivity are related to the speed of light c according to  

             …(4)

The value of c in a vacuum measured to 5 significant figures is

 …(5)

This (theoretical) relationship between permeability μ0, permittivity ε0, and the speed of light c implies that one coefficient needs to be defined and then the other two coefficients are “scaled” to suit this definition. This is no different than choosing a measure of distance as “meters” or “feet” or “cubits” etc and then measuring “length” in terms of such a reference unit. It is current practice to choose the “permeability of free space” μ0 as this “pivot” coefficient and therefore reference other measured coefficients against this defined value.  

It is also useful to introduce another term or concept called the “impedance of free space” that we will label as Rελ .This term is defined to have a value also related to permeability μ0 and permittivity ε0  defined as

             …(6)

The impedance of free space has a value to four significant figures of

             …(7)

Note: The value of Rελ can also be expressed in terms of μ0 and ε0 from equations (5) and (6) as Rελ  = μ0 ּ c

The impedance of free space Rελ can now be used to predict a relationship between the magnetic field component H and the transverse electrostatic component E in an electromagnetic wave propagating in free space. These radio signals components have RMS magnitudes related by

             …(8)

 This is analogous to Ohms Law for voltage, current and resistance.

             …(9)

We have similar energy predictions for these electromagnetic components. Let us use S to represent “energy density” in units of watts per square-meter i.e. W m-2 .The energy density S can be determined from either the magnetic component H or the electrostatic component E as

             …(10a)

or equally

             …(10b)

The total power Pr, total  flowing through an arbitrary surface is equal to the integrated power density S over this surface. Providing that no other losses (such as heat, light etc) are present, the total power Pr, total  flowing through a surface that totally encloses a radiating source will equal the power supplied to the radiating source. Although the size and shape of this surface does not affect the total integrated power (in free space without propagation losses as in a vacuum) it is mathematically convenient to define a surface shape that is familiar and convenient to work with. The most common surface shape is spherical and a given point z on its surface can be defined in polar co-ordinates in terms of a radius d,  a horizontal angular position θ and an vertical angular elevation Φ so that zzd, θ ,Ø.

This polar definition implies the following limits of 0 ≤ d  < ∞, 0 ≤ θ  < 2 ּ π and  -π \ 2  Φ <  π \ 2   .

This polar co-ordinate system now allows us to predict the total electromagnetic power flowing through the surface of this imaginary sphere from a radiating source as a double integral defined as

 …(11)

where  represents the electromagnetic energy density at a distance d from a radiating source placed at the origin of this sphere and at a horizontal angular position θ and with a vertical angular position Φ. We will drop the subscripts on S in further equations for convenience i.e.

 …(12)

If the energy density S is constant over the surface of this sphere then the integral simplifies to equal the energy density S multiplied by the surface area of this sphere i.e.

             …(13)

A hypothetical antenna that produces a constant energy density over the surface of a sphere originating from its position is called an “isotropic antenna”. Given that we generally accept that energy cannot be created nor destroyed, the total power applied to a transmitting isotropic antenna results in an equal total power flowing through the surface of such a sphere (or any other closed surface surrounding the transmitting antenna as previously mentioned). This is true for any antenna unless unwanted parasitic power loss mechanisms are present, for example power lost in conductor resistance or surrounding dielectric materials with dissipative loss.

It is useful to predict the propagation power loss between a transmitting antenna and a receiving antenna  separated by a distance d. Two isotropic antennas are typically used and form a basis for comparison with other antenna types. Since we immediately know that the power density Sd is equal to the transmitted power divided by the surface area of this sphere, the power received by an isotropic antenna will be proportional to its “effective receiving area” or aperture area.

The size of this aperture area depends of the wavelength λ of the electromagnetic radiation received and is usually considered to be given by

              …(14)

We also will note that the wavelength and frequency of electromagnetic wave in free space is

             …(15)

The free space propagation PowerRatio between received power and transmitted power is simply equal to a ratio of aperture area to the area of this imaginary sphere i.e.

             …(16)

When this loss is expressed in decibels (i.e. LossdB    -10 • Log10 { PowerRatio } ) as  a positive loss quantity we arrive at the well known “Fris Equation”

             …(17)

The isotropic antenna is used as a “reference” for comparison with other antenna that may exhibit a useful relative power gain GdBi (directional antennas) or a relative power loss (usually exhibited by compact or electrically small antennas) as a figure of merit.

A purely isotropic antenna is only a hypothetical concept and is theoretically impossible to construct in the “real world”. However these isotropic antennas can be used to form a reference path loss prediction. When substituted with two identical real antennas the measured path loss will be different.

If the loss is lower, the substitute antennas are said to have a power gain, shared equally between each. For example, if a measurement distance is chosen and the predicted path loss for isotropic antenna is 50 dB, for example, and the measured path loss for two real, identical antennas is only 45 dB, then the combined antenna gain GdBi is 5 dB. Since the gain is shared equally for transmit and receive antennas it follows that each antenna gain is GdBi = 2.5 dB in this example.

Antenna gain or loss performance can therefore be measured in reference to other known antennas. However this comparison gain is affected to some extent by the separation distance d between the transmitting and receiving antennas. If the separation distance is too small, the comparison gain estimate will exhibit an error. Conversely, if the distance is to great the propagation loss may result in received power levels that are far too low to measure accurately.

There is some requirement therefore to define some “transition” distance between “too close” and “too far”. The transition is usually described as “near field” radiation for small separations and “far field” radiation for large separations. The actual transition distance is “fuzzy” and depends on the actual antennas used, but a general guide is that

           

In practice, real antenna measurements are readily performed sufficiently in the far field without unwanted errors introduced and with ample available signal strength energy received.

Now let’s illustrate the previous concepts diagrammatically,

       

We consider an arbitrary transmitting antenna radiating electromagnetic energy that is dependent on direction. Let us define a polar co-ordinate system with the  transmitting antenna placed at the origin. The distance from this origin to a point on the sphere’s surface is equal to the radius d of this imaginary sphere and has a horizontal angular position  and a vertical  angular elevation . As previously mentioned, 0 ≤   < 2 ּ π and  -π \ 2   <  π \ 2.

The power density  at this point on the surface of this sphere is proportional to the power Ptx supplied to the transmitting antenna its radiation  pattern. For this analysis we will consider that dissipative power losses are not present so that the antenna efficiency = 100 %. (Any unwanted losses can be added later to represent less than ideal real antenna performance).

The total integrated power can be predicted by first predicting the power flowing through a circular horizontal cross section disk of this sphere i.e.

             …(18)

We can see that this circular disk having a vertical elevation defined by ø  has a radius r = d · cos{ ø } and infinitesimal thickness w = d · ∂ ø. Its surface area is therefore equal to the integral of r · w = d 2 · cos{ ø }  ∂ ø wrt the horizontal angular position . We can represent equation (18) a bit more elegantly as 

             …(19)

Now that we have determined the total power flowing through the surface area of this circular cross section, the total power flowing through the surface of  the sphere requires an additional integration wrt the angular elevation  leading to

            …(20)

How to Determine Radiation Resistance 1

The power supplied to the transmitting antenna can be stated as

             …(21a)

were Irms2 represents the RMS current flowing into the antenna’s input connections and Ra refers to its “radiation resistance”. This radiation resistance refers to a real load that converts electrical input power into electromagnetic radiation in the same way that a resistor converts electrical energy to heat. Although a real antenna will also have a frequency dependant series reactive component, this reactance can always be removed by adding an equal and opposite series reactive component (we always assume that an antenna is properly “matched” to its power source for predicting its characteristics).

The input power can also be expressed relative to a parallel equivalent radiation resistance and a RMS voltage applied to its input connection

             …(21a)         

In this case Vrms2 represents the RMS voltage applied to the antenna’s input terminals and R/a refers to the antenna’s parallel “radiation resistance”. Any reactive component is cancelled by connecting an equal and opposite parallel reactance across the antenna terminals.

We know that a loss-less antenna will radiate all its electrical input energy as electromagnetic energy without (losses wasted as heat) so that

 …(22)

We can therefore write

 …(23)

were 

         or .

If we replace Ptx as defined in equation (21a) and make the radiation resistance Ra the subject we arrive at

             …(24a)

The actual radiation resistance will depend on the radiation pattern described by its radiated energy Sd,θ,Φ for a given RMS input current Irms .Alternatively, the parallel form of antenna radiation resistance R/a could be used if applicable

             …(24b)

Equation (24a) (and potentially (24b)) provide an immediate solution for the radiation resistance Ra of a dipole antenna providing that the radiated energy density pattern Sd,θ,Φ = S{ Irms, d, θ, Φ } is known.

The usual derivation procedure for determining the energy density pattern  Sd,θ,Φ is to first predict the magnetic field strength  resulting from a RMS AC current Ix flowing through each incremental length of radiating conductor relative to some reference point x along its length and at a distance in polar coordinates defined in terms of d, θ and Φ. Although the shape of the conductor is not important, simple geometries result in simple mathematical expressions and are therefore preferred. The simplest geometry is a straight conductor e.g. a wire, with an distance dependant AC current Ix flowing along its length.

       

If we assume a straight vertical conductor then we can predict the radiated magnetic field  produced by a position dependant current Ix flowing in each incremental length element  at a distant point in space relative to this polar coordinate system as

 …(25)  

This magnetic field intensity  is constant with respect to the horizontal angular position . The total magnetic intensity is equal to the vector sum of all incremental magnetic field contributions at this distant point. We therefore need to integrate these incremental field contributions  over the length L of the conductor

             …(26)

We can now predict the energy density  at this distant point from the magnetic field intensity   magnitude using equation (10a)

           

i.e.

 …(27)

Substituting equation (27) into equation (24a) results in

 …(28)

Equation (28) provides a generic prediction tool for radiation resistance Ra for any center fed dipole antenna. The distance term d is constant for a sphere and might be predicted to cancel (an advantage for the polar coordinate system as opposed to a Cartesian coordinate system!) but beware – this assumption may be premature!

Although the distance from the center of this radiating conductor is constant, this is not true over the length of the radiating conductor – i.e. d depends to some extent on x. Although this “modified” polar distance d/ will have a decreasing effect on the magnitude of the magnetic field intensity as the distance d/ increases, the associated phase shift caused by propagation time delay will remain unchanged and will affect the integration outcome. This is why the distance term d will not be removed from inside the inner integral just yet!

We can however reasonably expect that the AC current term Irms will disappear if we define a relative AC current  as

 …(29)

The final solution for radiation resistance Ra is now

 …(30)

It should be pointed out that the relative current distribution  in the radiating conductor is a vector quantity with magnitude and phase components so that the total magnetic field at a given point in space is the result of the vector summation of all vector contributions from all incremental radiating elements of the conductor. The total power over a surface however is the result of a scalar integration over all points were the total magnetic field density has been evaluated. 

Radiated Energy Density For Dipole Antennas

1.1. Electrically short Dipole antenna

The magnetic field strength H caused by an AC RMS current Ix with a sinusoidal frequency f = c/λ flowing in an “short” conductor element with length δx at a distance d and vertical angular elevation  is predicted from

             …(25)

 

       

We will define the vertical position of a point x on this conductor to be x = 0 at its center electrical input and  x = ±L / 2 at its extremities. Now let’s define the total length as a proportion of a half wavelength i.e. L ≡ γ · (λ / 2).  

            …(26)

We will first assume a linear current distribution Ix in the radiating dipole elements that has a maximum RMS value of Ix = Irms at the antenna’s center input connection falling to Ix = 0 (zero current) at the ends. Since the dipole is considered to be electrically short compared to a half wavelength we also will assume that the current flowing in each length segment δx remains in phase  so that

           

Ix  =  Irms · (1  -  | 2 ·  x / L | )  were  -L / 2    x    L / 2  and  L ≡ γ · λ / 2

The integral identified in equation (26) becomes

Substituting this result into equation (6) predicts the magnetic H field for an electrically short dipole to be

             …(27)

From equation (10a) we had  …(10a) where the magnetic field H was presented as a magnitude. This allows equation (27) to be rewritten as

             …(28)

We can now substitute the predicted energy density Sd, into equation (24a)

…(29)

Equation (29) predicts the radiation resistance Ra for a dipole antenna that is electrically short compared to a half wave dipole (i.e. γ << 1). This radiation resistance falls to low values as the antenna length reduces and will have an additional capacitive series reactance –j · Xs that will have an increasingly  high value. This will need to be “resonated” with a corresponding positive inductive reactance at a given operating frequency f but the high resulting Q = Xs / Ra will result in a narrow band frequency characteristic.

MathCAD Example

 

Half wave dipole

The previous electrically short dipole becomes a half wave dipole when  γ = 1 but the current distribution will change from a linear relationship to a cosine one, i.e.

Ix  =  Irms · cos{ 2 · π · x / λ  )  were  - λ / 4  x    λ / 4 …(30)

       

Since the half wave dipole is now physically large compared to a wavelength we will need to consider a “propagation delay” corrective term.

       

This half wave dipole is (by definition) physically large compared to a wavelength resulting in a small path length difference Δx between magnetic field contributions created by current flow in each incremental conductor length. Although this small path length difference will not change the magnetic field appreciably (since far field prediction is assumed i.e. d is large), the propagation time delay caused by this path difference will result in a phase shift from field contributions arising from different positions x on the dipole.

 …(31a)

were

 …(31b)

This time delay results in a phase correction term that can be applied to the incremental magnetic fields as a multiplicative term prior to integration

 …(32)

Each incremental magnetic field contribution hλ,x,d,Φ  at a distant polar position (d, θ, Φ) has magnitude and phase components to account for, therefore

…(33)

The total magnetic field magnitude at (d, θ, Φ) becomes

…(34)

Recall equation (30) and this time we understand that the integration distance d = d{x}, restated below as

            …(30)

Substituting equation (34) into equation (30) results in the final prediction for radiation resistance Ra

     …(35)

I tried to compute this integral numerically on MathCAD and obtained the following prediction for radiation resistance using an arbitrary wavelength parameter λ

       

We can see that the λ term will cancel after the inner integration is performed. The  constant will also appear as a scaling term after this inner integration so will be taken outside the expression. This leads me to simply state a simplified, wavelength independent expression to now be

         …(36)

       

 Yes, MathCAD is a great program for checking the maths!

3. Summary

 

 

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