Dipolar antenna

The dipolar antenna , worked out by Heinrich Rudolph Hertz about 1886, is an antenna made up of two bits metal fed in its medium and intended to transmit or receive electromagnetic energy. This type of antenna is simplest to study from an analytical point of view.

Elementary dipole

An elementary dipole is a small length $\ scriptstyle {\ conducting delta \ ell}$ of (small in front of the Wavelength $\ scriptstyle {\ lambda}$ ) in which circulates a Alternative course:

\ scriptstyle {I=I_ \ circ e^ {J \ Omega T}}

In which $\ scriptstyle {\ omega=2 \ pi F}$ is the pulsation (and $\ scriptstyle {F}$ the frequency). $\ scriptstyle {J}$ is, as usual $\ scriptstyle {\ sqrt {- 1}}$ . This notation, employing complex numbers is the same one used when one works with the formalism of the impedances.

It should be noticed that this type of dipole cannot be practically manufactured. It is necessary well that the current comes from some share and that it left some share. Actually, this small piece of driver and the current which circulates there, will be simply one of the small pieces in which one will divide a macroscopic antenna, to be able to calculate it. The interest is that one can easily calculate the remote electric field of the electromagnetic Onde emitted by this short period of driver. We give the result directly:

$E_ \ theta= {\ delta \ ell \ over \ lambda} e^ {J \ left (\ Omega T-Kr \ right)}$

Here: ,

$\ scriptstyle {\ varepsilon_ \ circ}$ is the Permittivité vacuum.
$\ scriptstyle {C}$ is speed of light in the vacuum.
$\ scriptstyle {R}$ is the distance between the dipole and the point where the field $\ scriptstyle {E_ \ theta}$ is evaluated.
$\ scriptstyle {K}$ is the Nombre of wave $\ scriptstyle {k= {2 \ pi \ over \ lambda}}$

The exhibitor of $e \,$ gives an account of the variation of Phase of the Electric field of the wave with time and the distance to the dipole.

The remote electric field $\ scriptstyle {E_ \ theta}$ of the electromagnetic wave is coplanar with the driver and perpendicular to the line which connects the point where it is evaluated with the driver. If we imagine the dipole in the center of a sphere and parallel with the North-South axis, the electric field of the radiated electromagnetic wave will be parallel to the meridian and the Magnetic field of the wave will have the same parallel direction as the geographical.

Dipole runs

A short dipole is a realizable dipole practically formed by two drivers overall length $\ very small scriptstyle {L}$ compared with the wavelength $\ scriptstyle {\ lambda}$ . The two drivers are fed in the center of the dipole (see drawing). One takes as assumption that the current is maximum there in the middle of the dipole (or it is fed) and that it decrease linearly up to zero at the ends of the dipole. Notice that the current circulates in the same direction in the two arms of the dipole: towards the right-hand side on both or the left on both. The far field $\ scriptstyle {E_ \ theta}$ of the electromagnetic wave radiated by this dipole is:

E_ \ theta= {- jI_ \ circ \ sin \ theta \ over 4 \ varepsilon_ \ circ C R} {L \ over \ lambda} e^ {J \ left (\ Omega T-Kr \ right)}

The emission is maximum in the plan perpendicular to the dipole and zero in the direction of the drivers, which is the same one as the direction of the current. The send-out chart to the form of a circular Torus of section (image of left) and of ray interns no one. In the image on the right the dipole is vertical and it is in the center of the torus.

Starting from this electric field one can calculate the total power emitted by this dipole and from that, to calculate the resistive part of the impedance series of this dipole:

R_ {series} =80 \ pi^2 \ left ({L \ over \ lambda} \ right) ^2

ohms for

\ scriptstyle {L \ L \ lambda}

but on the other hand:

R_ {series} =20 \ pi^2 \ left ({L \ over \ lambda} \ right) ^2

ohms for

\ scriptstyle {L < \ lambda}

Profit of an antenna

The profit of an antenna is defined like the report/ratio of the powers per unit of area of the antenna given and an isotropic hypothetical antenna:

G= {\ left ({P \ over S} \ right) _ {ant} \ over {\ left ({P \ over S} \ right) _ {Iso}}}

The power per unit of area transported by an electromagnetic wave is:

\ textstyle {\ left ({P \ over S} \ right) _ {ant}} = \ textstyle {1 \ over2} C \ varepsilon_ \ circ E_ \ theta^2 \ simeq \ textstyleE_ \ theta^2

The power per unit of area emitted by an isotropic radiator supplied with the same power is:

\ textstyle {\ left ({P \ over S} \ right) _ {Iso}} = \ textstyle

While replacing the values in the case of a dipole runs, the end result is:

G= \ textstyle

= 1,5 = 1,76 dBi

The dBi are Décibel S with a I added to recall that it is about a profit compared to an antenna I sotrope.

Dipole half-wave

A dipole $\ scriptstyle {\ lambda \ over 2}$ or dipole half-wave is an antenna formed by two drivers overall length equal to a half wavelength. This length should be announced does not have anything remarkable from the electric point of view. The impedance of the antenna corresponds neither to a maximum nor with a minimum. The impedance is not real although it becomes it for a length of the close dipole (towards $\ scriptstyle {0,46 \ lambda}$ ). It should be recognized that the only characteristic this length is that the mathematical formulas are simplified as by miracle.

In the case of the dipole $\ scriptstyle {\ lambda \ over 2}$ one takes as assumption that the amplitude of the current along the dipole with a sinusoidal form

\ textstyle {I=I_ \ circ e^ {J \ Omega T} \ cos {K \ ell}}

It is easy to note that for

\ scriptstyle {\ ell=0}

the current is worth

\ scriptstyle {I_ \ circ}

and for

\ scriptstyle {\ ell= {\ lambda \ over4}}

the current is worth zero.

In spite of simplifications of this particular case, the formula of the distant field is unpleasant:

\ textstyle {E_ \ theta= {- jI_ \ circ \ over 2 \ pi \ varepsilon_ \ circ C R}} {\ cos \ left (\ scriptstyle {\ pi \ over 2} \ cos \ theta \ right) \ over \ sin \ theta} e^ {J \ left (\ Omega T-Kr \ right)}

Nevertheless the fraction

\ textstyle

is not very different from

\ scriptstyle {\ sin \ theta}

. The result is a a little flattened digraph of emission.

The image of left shows the section of the send-out chart. For comparison, we drew in dotted lines the section of the send-out chart of a short dipole. It is noted that they are not very different. The image of right-hand side shows the same send-out chart in prospect.

This time we cannot calculate the total power analytically emitted by the antenna. A simple numerical calculation leads us to una value of resistance series of:

\ textstyle {R_ {series} =73}

ohms

But it is not sufficient to characterize the impedance of the dipole which comprises also an imaginary part. Simplest is to measure it. In the image of right-hand side one finds the parts real and imaginary of the impedance for lengths of dipole which go from

$\ scriptstyle {0,4 \, \ lambda}$ has $\ scriptstyle {0,6 \, \ lambda}$

The profit of this antenna is:

\ textstyle {G= {120 \ over R_ {series}} = {120 \ over 73}}

= 1,64 = 2,14 dBi

Here a table with the profits of antennas dipole other lengths (notice that the profits are not given in dBi):

Effective height of the antenna

When the dipole bathes in an electric field generated by the transmitting antenna, a tension is induced in the circuit connected on the outlet side of the antenna. This one is expressed by:

V=h_e E_ {EFF}

where $E_ {EFF}$ is the effective value of the electric field in V/m to which is subjected the antenna and $h_e$ its height effective in m, the latter being different its physical length. Thus for a dipole half-wave:

h_e= \ frac {\ lambda} {\ pi}

References

Elementary, short dipole and

\ scriptstyle

Frederick E. Terman, Electronic Radio operator and Engineering , MacGraw-Hill
Richard Philips Feynman, Robert Leighton and Matthew Sands, Readings one physics , Addison-Wesley
Wolfgang Panofsky and Melba Philips, Classical Electricity and Magnetism , Addison-Wesley

See too

electrostatic Dipole

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