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Initialise

The Hertzian Dipole antenna

The Hertzian Dipole is a conceptual antenna that carries a constant current along its length.

By laying a number of these small current elements end to end, it is possible to model a physical antenna (such as a half-wave dipole for example).  But since we are only interested in obtaining an expression for the effective area of an Isotropic Antenna (in order to derive The Friis Transmission Equation) the Hertzian Dipole will be sufficient for our needs.

Maxwell's Equations

Since the purpose of a radio antenna is to either launch or to receive radio waves, we know that both the antenna, and the space surrounding the antenna, must satisfy Maxwell's Equations. We define Maxwell's Equations in terms of vector functions using spherical coordinates:

Maxwell–Faraday equation:

Ampère's circuital law (with Maxwell's addition):

Gauss' Law:

Gauss' Law for Magnetism:

Where:

        E is the electric field strength [Volts/m]

        H is the magnetic field strength [Amperes/m]

        J is the current density (current per unit area) [Amperes/m²]

        ρ is the charge density (charge per unit volume) [Coulombs/m³]

        ε is Electric Permittivity

        μ is Magnetic Permeability

Helmholtz decomposition

The Helmholtz Decomposition Theorem states that providing a vector field, (F) satisfies appropriate smoothness and decay conditions, it can be decomposed as the sum of components derived from a scalar field, (Φ) called the "scalar potential", and a vector field (A) called the "vector potential".

F = -VΦ + V×A

And that the scalar (Φ) and vector (A) potentials can be calculated from the field (F) as follows (image from https://en.wikipedia.org/wiki/Helmholtz_decomposition):

Where:

        r is the vector from the origin to the observation point (P) at which we wish to know the scalar or vector potential.

        r' is the vector from the origin to the source of the scalar or vector potential (i.e. a point on the Hertzian Dipole antenna).

        V'·F(r')  is the Divergence of the vector field (F) at source position r'.

        V'×F(r')  is the Curl of the vector field (F) at source position r'.

Calculating the Poynting Vector

We know that the magnitude of the Poynting Vector (S) can be calculated as the cross product of the electric field vector (E) and the magnetic field vector (H) :

        S = -E x H which is analogous to a resistive circuit where power is the product of voltage and current: P=V*I.

We also know that the impedance of free space (Z) can be calculated as the ratio of the electric field (E) and magnetic field (H) vectors: Z = E /H =

This is analogous to a resistive circuit where resistance is the ratio of voltage and current: R=V/I.

This provides two more methods for calculating the Poynting Vector (S):

        S = -E·E/Z which is analogous to a resistive circuit where power, P=V²/R, and:

        S = -H·H*Z which is analogous to a resistive circuit where power, P=I²R.

Since we have obtained an expression for the magnetic field vector (H), we can derive an expression for the Poynting Vector (S):

We can separate out the time variable part to yield:

Where:

And we can visualise this radiation pattern using Maple's plotting tools:

So the Hertzian Dipole produces a electromagnetic radiation pattern with a pleasing doughnut shape :-)

Calculating Antenna Gain

We can calculate the total power radiated by the Hertzian Dipole by integrating the power flux density over all solid angles dΩ=sin(θ) dθ dφ.  Since we have expressed power flux density in terms of watts per square meter, we multiply the solid angle by r² to convert the solid angle expressed in steradians into an area expressed in m².

We can now use this power to calculate the power flux density that would be produced by an isotropic antenna by dividing the total transmitted power by the area of a sphere with radius r:

The Gain of the Hertzian Dipole is defined as the ratio between the maximum power flux density produced by the Hertzian Dipole and the maximum power flux density produced by the isotropic antenna:

Calculating Radiation Resistance

The input impedance of the Hertzian Dipole will have both a real and a reactive part.  The reactive part will be associated with energy storage in the near field and will not contribute to the Poynting Vector in the far-field.  For an ideal antenna (with no resistive power loss) the real part will be responsible for the radiated power:

Calculating the power received by a Hertzian Dipole

If an electromagnetic field (E) is incident on the Hertzian Dipole antenna, it will generate an Electro-Motive Force (EMF) at the antenna terminals.  The EMF will be at a maximum when the transmitter is on the x-y-plane (that is when θ=π/2) and will be zero when the transmitter is on the z-axis.

For and incident E-field:

The z-axis component will be:

The z-axis component of the E-field will create an EMF at the antenna terminals that will draw charge out of the receiver to each tip of the antenna. We can calculate the work done per unit charge by integrating the z-axis component of (E) over the length of the antenna (L):

In order to extract the maximum possible power from the antenna, we will form a conjugate match between the impedance of the antenna and the load.  This means that the load resistance must be the same as the radiation resistance of the antenna.  The voltage developed across the load resistance will therefore be half of the open circuit EMF:

And so the power delivered to the load will be:

Calculating the Effective area of an Isotropic Antenna

We can also calculate the power received by the Hertzian Dipole by multiplying the power flux density arriving at the antenna with the effective area of an isotropic antenna and the gain of the Hertzian Dipole relative to an isotropic antenna:

We can express the incident power flux density in terms of electric field strength and wave impedance:

Rearranging, we obtain an expression for the effective area of the isotropic antenna:

The Friis Transmission Equation

We can calculate the power flux density that would be produced by an isotropic antenna at a distance r from the antenna by dividing the total transmitted power P_tx by the area of a sphere with radius r:

And so the power flux density that would be produced by an antenna with gain G_tx is:

We can calculate the power received by an isotropic antenna by multiplying the power flux density incident onto the antenna with the effective area of an isotropic antenna:

And so the power that would be received by an antenna with gain G_rx is:

The free space path loss is defined as the ratio between the received power and the transmitted power:

And so:

restart;

interface(imaginaryunit=II):

with(LinearAlgebra):

The Euler-Bernoulli beam equation
.

We wish to determine the natural modes of vibration of

a possibly multi-span Euler-Bernoulli beam.

Separate the variables by setting .   We get
-

whence
.

Let .  Then

 and

The idea behind the Krylov-Duncan technique is to express  

in terms an alternative (and equivalent) set of basis
functions  through ,, as
,

where the functions  through  are defined in the next section.

In some literature the symbols , are used for these

functions but I find it more sensible to use the indexed function

notation.

The Krylov-Duncan approach is particularly effective in formulating
and finding a multi-span beam's natural modes of vibration.

The Krylov-Duncan functions

The K[i](x) defined by this proc evaluates to the th

Krylov-Duncan function.

Normally the index  will be in the set, however the proc is

set up to accept any integer index (positive or negative).  The proc evaluates

the index modulo 4 to bring the index into the set .   For

instance, K[5](x) and K[-3](x)i are equivalent to K[1](x) .

Here are the Krylov-Duncan basis functions:

and here is what they look like.  All grow exponentially for large
but are significantly different near the origin.

The cyclic property of the derivatives: 
We have .  Let's verify that:

The fourth derivative of each   function equals itself. This is a consequence of the cyclic property:

The essential property of the Krylov-Duncan basis function is that their

zeroth through third derivatives at  form a basis for :

As noted earlier, in the case of a single-span beam, the modal  shapes

are expressed as
.

Then, due to the cyclic property of the derivatives of the Krylov-Duncan

functions, we see that:
.
.
.
Let us note, in particular, that
,
,
,
.

A general approach for solving multi-span beams

In a multi-span beam, we write  for the deflection of the th span, where

 and where  is the span's length.  The  coordinate indicates the

location within the span, with  corresponding to the span's left endpoint.

Thus, each span has its own  coordinate system.

 
We assume that the interface of the two adjoining spans is supported on springs

which (a) resist transverse displacement proportional to the displacement (constant of

proportionality of   (d for displacement), and (b) resist rotation proportional to the
slope (constant of proportionality of   (t for torsion or twist). The spans are numbered

from left to right. The interface conditions between spans  and +1 are

The special case of a pinned support corresponds to  and .
In that case, condition 3 above implies that ,
and condition 4 implies that

Let us write the displacements  and  in terms of the Krylov-Duncan

functions as:

Then applying the cyclic properties of the Krylov-Duncan functions described

earlier, the four interface conditions translate to the following system of four
equations involving the eight coefficients