5.1 Timedomain modeling of wire antennas by method of momentsBasic theoryIn the chapters 4.1, 4.4, and 4.5
the wire dipole, microstrip
dipole and patch antenna are modeled by
method of moments in the frequency domain. The integral equation is transformed
to the system of algebraic equations and the current distribution on the
antenna is computed at the steady harmonic state at the given frequency. The
antenna parameters can be computed from the current distribution.
Numerical modeling of antennas in the frequency
domain is effective if the antenna parameters are desirable in the narrow band
of frequencies. If the wideband analysis is desirable, it is better to use
timedomain analysis. However, this one is suitable only for the structures
with low Qfactor, otherwise, the computed response is too long, and the
analysis is time consuming.
The basic principles of the timedomain modeling of wire antennas will be described
in this chapter. Again, as in the frequency domain, the formulations which come
out from Maxwell’s equations in the differential or integral form are solved.
In this chapter we focus our attention on the integral formulation, as in the
chapters 4.1, 4.4 and 4.5.
The integral formulation in the time domain
is more complicated than in the frequency domain due to the time derivative of
the vector potential, thus, the integrodifferential formulation is solved
instead of the pure integral formulation.
Due to the education purpose of this chapter, we focus our attention only on the solution of the symmetric wire
dipole, because the solution of more complicated structures is, from the point
of the derivation and implementation, more demanding.
TimeDomain Modeling of Wire Dipole by Method of Moments
For modeling a symmetric wire dipole in the
time domain we come out from a similar situation as in case the frequency
domain. Let’s suppose that the symmetric wire dipole
is made from the perfect electric conductive material, and placed in the
vacuum. The dipole axis is identical with z axis of the cylindrical
coordinate system, and the electromagnetic plane wave incidents on it (fig. 5.1A.1). The intensity of the electric field is parallel to z axis, and the
waveform is arbitrary. Further, the wire radius a is much smaller than
the wavelength.

Fig. 5.1A.1  Wire dipole placed along the z axis and excited by electromagnetic wave. 

The incident plane wave E^{I}
induces a current I_{z} (the dipole is placed along z axis)
on the dipole surface. Since at open ends of the antenna the electric currents
do not have a chance to flow, the electric charge σ accumulates. In the
next instants the accumulated charge flows off and in, depending on the phase
of each spectral component of the incident wave. The induced currents and
charges become the sources for the radiation of the secondary wave. The
influence of the current along the z axis can be described by the vector potential [37]

${A}_{z}\left(z,t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\mu}{4\pi \text{\hspace{0.17em}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\displaystyle \underset{\xi =h}{\overset{h}{\int}}\frac{{I}_{z}\left(\xi ,tR\left(z,\xi \right)/c\right)}{R\left(z,\xi \right)}d\xi}$.

( 5.1A.1 )

The influence of the charge density can be described by the scalar potential

$\phi \left(z,t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{4\pi \epsilon \text{\hspace{0.17em}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\displaystyle \underset{\xi =h}{\overset{h}{\int}}\frac{\sigma \left(\xi ,tR\left(z,\xi \right)/c\right)}{R\left(z,\xi \right)}d\xi}$.

( 5.1A.2 )

The value of the vector potential A_{z} at a arbitrary “target” point z, and instant t can be computed by
the addition of contributions of the current components from all “source” points ξ on the antenna axis. The source current
has to be retarded about time R/c which the wave needs for traveling the distance between the source ξ point and the target point z

$R\left(z,\xi \right)=\sqrt{{a}^{2}+{\left(z\xi \right)}^{2}}$.

( 5.1A.3 )

Further, c in (5.1A.1) is speed of light in the given environment. The contribution to the vector potential is not
proportional to the distance between the source and target point. The integration has to be carried out along the whole antenna structure. Finally,
the value of the integral is multiplied by the permeability of surrounding medium μ (the vacuum).
In case of the scalar potential, the
current distribution is replaced by the charge density σ. The result of the integration is, instead of the multiplication by
the permeability, divided by the permittivity of the surrounding medium ε. The rest in the equation (5.1A.2) stays the same as in (5.1A.1).
Since the current and the charge are mutually tied, the relations (5.1A.1) and (5.1A.2) has to be tied by the
continuity equation

$\frac{\partial \sigma \left(z,t\right)}{\partial t}=\frac{\partial {I}_{z}\left(z,t\right)}{\partial z}$.

( 5.1A.4 )

If the vector and scalar potentials
are known, the intensity of the electric field radiated by antenna E_{z}^{S} (the scattered field) can be computed

${E}_{z}^{S}\left(z,\text{\hspace{0.17em}}t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \text{}{A}_{z}\left(z,\text{\hspace{0.17em}}t\right)\text{\hspace{0.17em}}}{\partial \text{\hspace{0.17em}}t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \phi \left(z,\text{\hspace{0.17em}}t\right)}{\partial z}\text{\hspace{0.17em}}$.

( 5.1A.5 )

The derivative of the vector potential with respect to time describes “the dynamic contribution” of the electric charge on the antenna to the radiated wave (the electric currents,
charges in motion, are the source for the vector potential).
The partial derivative of the scalar potential with respect to the variable z express the “static
contribution” of the electric charge on the antenna to radiated field (the static charges, accumulated at ends of the antenna, are the source for the
scalar potential)
On the surface of the perfect electric conductor the intensity of the electric field has to satisfied the boundary condition

${E}_{z}^{S}\left(z,t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{E}_{z}^{I}\left(z,t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=0\text{\hspace{0.17em}}$.

( 5.1A.6 )

The system of equations (5.1A.1) to (5.1A.6) is solved
numerically in two steps. The first one consists in the space discretization of the problem and the second one consists in the application of time algorithms
on the result of the first step.
The first step is analogical to the solution in the frequency domain, where the
method of moments is used for the “transformation” of the integral equation to the
system to algebraic equations. Here, we use again the method of moments for the “transformation”
of the integrodifferential equation (5.1A.5) to a system of the retarded differential equations (the space discretization of the problem).
For the second step (the application of the
time algorithms), it is possible to use either the explicit, or the implicit
algorithm. At the explicit one, the length of the time step Δt has to satisfy the following condition: Δt ≤ R_{min}/c, where R_{min}_{ }is the smallest
distance of centers of the discretization elements and c is the speed of
the light in the surrounding medium. At the implicit algorithm, the length of
the time step can be chosen arbitrary. The larger length of the time step
increase the speed of computation, but the accuracy falls down. These
approaches belong to the marching on in time (MOT) methods.
The marching on in order method (MOO) is
the next method. It is based on the approximation of the transient response by
a set of weighted Laguerre polynomials. The advantage of MOO is its
unconditional stability, but the disadvantages are the higher demands on the
implementation, and less effectiveness in the comparison to MOT.
Due to above facts, we focus our attention
on MOT. Although we mentioned that the system of equations (5.1A.1) to (5.1A.6) is solved
in two steps (the space discretization and the application of time algorithms),
we will merge both these steps. It is appropriate to notify that by MOT the
system of retarded differential equations is transformed to the system of
difference equations (the derivative of the vector potential with respect to tome (5.1A.5) is replaced by appropriate
difference).
The advantage of the explicit algorithm is
its easier implementation and there is no need to compute an inverse matrix as
in case of the solution in the frequency domain, or as at the implicit
approach. However, the explicit approach suffers often from the late time
oscillation (the computed response is not steady, but grows exponentially.).
Therefore, the attention in this chapter is focused only on the implicit
approach.
Excitation pulse
Before, than we focus our attention on the
implicit approach, let’s choose an appropriate excitation pulse.
If we chose the Dirac pulse, the spectrum of the excitation signal would be equally spread
over the infinite frequency domain. Since in the technical practice the
solution is desirable over the limited bandwidth and we do not have unlimited computer
memory and the power of a computer, thus, the Dirac pulse is not an ideal choice.
Instead of the Dirac pulse it is better to use the Gaussian pulse modulated by a harmonic
signal. By change of its parameters, it is possible to analyze only the band of our interest.
The vector of the intensity of the electric field of the Gaussian excitation pulse modulated by harmonic signal is given

${\mathbf{E}}^{I}\left(\mathbf{r},t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathbf{E}}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{4}{cT\text{\hspace{0.17em}}\sqrt{\pi}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{exp}\left[{\left(\frac{4}{T}\left(t{t}_{0}\frac{\mathbf{r}\cdot {\mathbf{a}}_{k}}{c}\right)\right)}^{2}\right]\mathrm{cos}\left(2\pi {f}_{0}t\right)$,

( 5.1A.7 )

where E_{0} is the vector of the intensity of the electric field of the excitation wave at the origin of the
coordinate system, r is the positional vector of the point where the
intensity of the excitation wave is computed, a_{k } is the unit
vector in the direction of the propagation, c is the speed of the light
in the surrounding medium, T is the width of the Gaussian pulse, t_{0}
is the retardation of its peak, and f_{0} is the center frequency of the analyzed band. The example of the Gaussian pulse modulated by
the harmonic signal with its spectrum, defined by the following parameters: E_{0}=120π V/m, T=8
ns, t_{0}=10 ns and f_{0}= 500 MHz, is depicted in fig. 5.1A.2.

Fig. 5.1A.2  Gaussian pulse modulated by harmonic signal in time (leftside)
and frequency (rightside) domains. 

Implicit algorithm
The derivation of the implicit algorithm is
demanding (the system of the equations (5.1A.1) to (5.1A.6) is discretized in space
and time), and for the interested reader it is carried out in the layer B. In this layer the properties of the implicit
scheme are only discussed.
At the implicit algorithm, opposite to the
explicit one, the inverse matrix has to be solved and its implementation on the
computer is more demanding than in case of the explicit algorithm. However, its
advantage is that the length of the time step can be chosen arbitrarily (we
should keep in mind, the length of the time step influences the accuracy), and
this algorithm is more stable than the explicit one.
Using the implicit algorithm will be
demonstrated on the analysis of the wire dipole of the length 2 m, which is excited at its center by the Gaussian pulse modulated by a harmonic signal (5.1A.7)
defined by the following parameters: E_{0}=120π V/m, T=6
ns, t_{0}=8 ns, f_{0} = 0 Hz. Since the center
frequency of the analyzed band is 0 Hz, let’s further call this pulse as the
Gaussian pulse. Let’s chose the length of time analysis 400 ns. The body of the
dipole is divided to 40 segments (the space discretization of the task, layer B). The analysis is carried out two times for
different lengths of the time step. In case of the first analysis the length of
the time step is Δt = R_{min}/c = 0.166
ns, cΔt = 0,05 m, and in the
case of the second analysis the length of the time step is half of the first
case, Δt = R_{min}/(2c) = 0.083
ns, cΔt = 0,025 m (the time
discretization of the task).
The excitation Gaussian pulse is depicted
in fig. 5.1A.3. The computed current responses to this pulse at the center of the
dipole are depicted in fig. 5.1A.4. Comparing both figures it can be said that the
computed responses are significantly larger than the excitation pulse and the
difference between computed current is very small.
The wire dipole
is a narrow band antenna which is able to “transform” the energy to the free
space only within the limited range of the frequencies. Since the excitation
pulse carries the energy even at the frequencies, where the antenna has this
ability significantly smaller, the energy goes from the one arm of the dipole
to the other and back, and the energy is gradually radiated.

Fig. 5.1A.3  Excitation Gaussian pulse. 



Fig. 5.1A.4  Current responses on the Gaussian pulse (obr. 5.1A.3) in the
center of the wire antenna for different lengths of the time step 


Since the computed responses are stable,
the input impedance of the wire antenna can be computed (fig. 5.1A.5, denoted by
TD). In the same picture the input impedance computed by the method of moments
in the frequency domain is depicted (denoted by FD). The solution in the
frequency domain is taken as the reference solution because its accuracy is
known. Comparing this solution with the results for the implicit algorithm for
the length of the time step Δt = R_{min}/c = 0.166
ns, cΔt = 0,05 m (red line)
we can say that larger differences are apparent at higher frequencies. These
differences are mainly caused by the discretization of the system of the equations
(5.1A.1) to (5.1A.6) in time. If we decrease the length of the time step to the half
of the original value (green line), the differences are smaller (more accurate
approximation in the time). The length of the time step influences the accuracy
of the algorithm.


Fig. 5.1A.5  Real (left) a imaginary (right) part of the input impedance of
the analyzed wire dipole. 

The length of the time step influences the
stability of the algorithm too. This fact is demonstrated in fig. 5.1A.6, where
the current response of the same dipole antenna for the double length of the
time step (Δt = 2R_{min}/c = 0.333
ns, cΔt = 0,1 m.) is
depicted. The computed response is not stable, but it grows to infinity.

Fig. 5.1A.6  Unstable current response to the Gaussian pulse (fig. 5.1A.3) in
the center of the wire dipole (larger length of the time step). 

For stabilizing the time algorithm
different techniques can be used, however, these ones are behind the scope of
this chapter.
