Electromagnetic Waves, Antennas, and RF: Radiation from Time-Varying Currents and the Hertzian Dipole

Electromagnetic Waves, Antennas, and RF: Radiation from Time-Varying Currents and the Hertzian Dipole

EM22 described electromagnetic energy guided along a transmission line. EM23 asks what happens when a time-varying current distribution is allowed to launch fields into open space. The simplest source that contains the essential physics is the Hertzian dipole: an ideal current element whose length is much smaller than a wavelength. Although no physical antenna is literally infinitesimal, this source is the basic building block for thin-wire antennas, radiation integrals, arrays, and aperture methods [1, 2, 3, 4].

The central chain is

(1)

The most important physical distinction is that not every field surrounding an antenna is a radiated field. Close to the source, electric and magnetic energy can be stored and returned to the source each cycle. Far away, the surviving 1∕r terms carry net power outward.

1 Time convention and propagation convention

Throughout this article, sinusoidal quantities use the phasor convention

(2)

An outward traveling spherical wave therefore contains

(3)

where

(4)

and, in a lossless medium,

(5)

In vacuum,

(6)

Writing the convention explicitly is important because reversing the sign convention for time also reverses several i signs in the phasor formulas, while leaving all measurable powers unchanged.

2 Why radiation must be retarded

Maxwell’s equations do not allow a change in source current to influence distant points instantaneously. In a homogeneous medium with propagation speed

(7)

the field observed at position r depends on the source at the earlier retarded time

(8)

where

(9)

The time-domain magnetic vector potential is

(10)

For sinusoidal steady state, retardation becomes a phase factor. The phasor vector potential is

(11)

The factor 1∕R represents spherical amplitude spreading, while e^−ikR records propagation phase. This is the first mathematical appearance of the radiating 1∕r behavior that will dominate the far field.

3 The Hertzian current element

Place a very short current element of length ℓ at the origin and orient it along the z axis. Let its current phasor be I₀. The defining assumptions are

(12)

and

(13)

so the current element may be treated as spatially infinitesimal at the observation point.

Its current moment is

(14)

with units of ampere-meters. For an ideal Hertzian element the current is taken uniform over ℓ.

The associated end charges are not optional. Charge conservation requires

(15)

For harmonic time dependence,

(16)

Thus a time-varying short current element necessarily has a corresponding oscillating charge distribution. The electric near field will contain the effect of those charges.

For the ideal element, the vector potential is

(17)

Define

(18)

Since

(19)

the spherical components are

(20)

4 Deriving the magnetic field

The magnetic flux density is

(21)

For this axisymmetric source, only the ϕ component survives. The spherical-coordinate curl gives

(22)

First,

(23)

so

(24)

Also,

(25)

Therefore

(26)

Since B = μH,

(27)

Two radial dependences already appear:

(28)

The 1∕r² term is associated with the induction field; the 1∕r term survives into the radiation zone.

5 Deriving the electric field

Outside the source region, J = 0, so the phasor Ampere-Maxwell law is

(29)

Hence

(30)

Because H_ϕ is the only nonzero magnetic-field component,

(31)

and

(32)

Carrying out the derivatives and using

(33)

gives

(34)

(35)

and

(36)

Together with

(37)

these are the complete external fields of the ideal Hertzian current element under the stated phasor convention.

6 Three radial field scalings

The electric field contains three characteristic powers of distance:

(38)

They have different physical roles.

6.1 The 1∕r³ electrostatic-like term

The most rapidly decaying term appears in the electric field and is tied to the oscillating charge separation. It resembles the static electric dipole field and dominates sufficiently close to the source, while still outside the idealized source itself.

6.2 The 1∕r² induction term

The 1∕r² terms occur in both electric and magnetic fields. They are associated with induction and with energy exchanged between the source and nearby fields. They decay faster than radiation and therefore become progressively less important with distance.

6.3 The 1∕r radiation term

The 1∕r terms are special because their power density falls as

(39)

Multiplying by the area of a sphere,

(40)

produces a distance-independent total radiated power. This is exactly the geometric-spreading behavior developed earlier in EM17.

7 Near, intermediate, and far-field interpretation

For the infinitesimal dipole, the dimensionless parameter controlling the hierarchy is

(41)

When

(42)

the 1∕r³ and 1∕r² terms are important. This is the reactive near-field regime of the idealized source.

When

(43)

no single radial power need dominate. This is the transition or induction region.

When

(44)

the 1∕r terms dominate. This is the far-field or radiation regime for the point-source model.

For a real antenna of finite maximum dimension D, an additional far-field condition involving D is required. A commonly used engineering criterion is

(45)

provided the observation point is also many wavelengths and many antenna dimensions away when those constraints are relevant [1, 2]. The Hertzian element is idealized as D → 0, so the kr hierarchy is the important one here.

8 Far-field simplification

In the far field, retain only the 1∕r terms. Then

(46)

(47)

and

(48)

Therefore

(49)

The far field has become locally plane-wave-like:

(50)

Also,

(51)

The source launches a spherical wave globally, but over a small patch of a sufficiently large sphere the wavefront is approximately planar.

9 Why the near field is called reactive

The radial powers alone do not tell the whole story. Phase matters too. Near-field electric and magnetic terms are not generally in the same phase. Energy is alternately stored in the electric and magnetic fields and can flow back toward the source during part of a cycle.

In contrast, the far-field terms share the same propagation phase and satisfy

(52)

Their time-averaged Poynting vector is therefore outward and nonzero:

(53)

This is the physical distinction between merely having electromagnetic fields around a source and actually radiating net energy to infinity.

10 Angular radiation pattern

The far-field electric amplitude contains the angular factor

(54)

Thus the field amplitude is zero on the dipole axis,

(55)

and maximum broadside to the dipole,

(56)

Because power is proportional to field amplitude squared, the normalized power pattern is

(57)

The pattern is independent of azimuth ϕ. Rotating the two-dimensional figure-eight power cut about the z axis produces the familiar doughnut-shaped three-dimensional radiation pattern.

11 Far-field power density

Using the far-field relation

(58)

the radial time-average Poynting vector is

(59)

Since

(60)

we obtain

(61)

The 1∕r² dependence of power density is now explicit.

12 Radiation intensity

A useful distance-independent quantity is the radiation intensity

(62)

For the Hertzian dipole,

(63)

Its units are watts per steradian. Radiation intensity removes the trivial spherical-spreading factor and leaves only the angular distribution of radiated power.

13 Total radiated power

The total power passing through a large sphere is

(64)

Using

(65)

we have

Prad

= ∫ 02πdϕ∫ 0π sin 3𝜃 d𝜃.

(66)

The angular integrals are

(67)

and

(68)

Therefore

(69)

In free space, use

(70)

and

(71)

to obtain

(72)

Here I₀ is the peak phasor current amplitude; the factor 1∕2 associated with time averaging has already been included.

14 Radiation resistance

It is convenient to represent the radiated power by an equivalent resistance defined through

(73)

Comparing with the free-space radiated-power expression gives

(74)

This is not an ohmic resistor. It is a circuit representation of real power carried away by electromagnetic radiation.

A terminology warning is important. The formula above is for the ideal Hertzian or infinitesimal dipole with uniform current I₀ over length ℓ. A physically short center-fed dipole has an approximately triangular current distribution and a different radiation-resistance coefficient. The two models should not be mixed.

15 Directivity of the Hertzian dipole

Although antenna gain will be developed systematically in EM24, the radiation pattern already lets us calculate directivity.

The maximum radiation intensity occurs at

(75)

so

(76)

The average radiation intensity over all directions is

(77)

Therefore the maximum directivity is

(78)

Substitution gives

(79)

In decibels relative to isotropic,

(80)

This is directivity, not yet gain. Gain will require radiation efficiency as well as angular concentration.

16 Worked numerical example

Consider an ideal Hertzian element in free space at

(81)

The wavelength is

(82)

Let

(83)

and let the peak current amplitude be

(84)

The radiation resistance is

Rrad	= 80π22	(85)
	≈ 0.3158 Ω.	(86)

Thus

(87)

Now observe the antenna broadside at

(88)

The far-field electric amplitude is

(89)

The magnetic amplitude is

(90)

The average local power density is

(91)

The result is small locally because the radiated energy has spread over a large spherical area, but integration over all directions recovers the same 0.1579 W total radiated power.

17 General far-field current-distribution form

The Hertzian element is a building block for arbitrary antennas. For a localized current distribution observed far away, write

(92)

When

(93)

we use the far-zone approximations

(94)

and

(95)

Therefore

(96)

The vector potential becomes

(97)

This expression contains one of the central ideas of antenna theory: radiation in a given direction is a coherent sum of contributions from the entire current distribution, each weighted by its relative propagation phase. Antenna patterns and phased arrays ultimately grow from this phase-summation principle.

18 What changes between near field and far field

Several properties emerge together in the far zone:

• the 1∕r field terms dominate;
• the radial electric component becomes negligible;
• E, H, and r become mutually perpendicular;
• the field ratio approaches the medium impedance η;
• electric and magnetic far fields are in phase for a lossless medium;
• the average Poynting vector is directed radially outward;
• angular pattern becomes independent of distance;
• power density falls as 1∕r².

These are precisely the properties needed to connect Maxwell’s equations to practical antenna quantities such as radiation intensity, directivity, gain, effective aperture, EIRP, and eventually the Friis transmission equation.

19 Common mistakes

• Calling every 1∕r² quantity a near field. Far-field power density falls as 1∕r² even though the far-field field amplitudes fall as 1∕r.
• Assuming the near field carries no instantaneous power. Energy can flow locally in either direction. The distinction is that reactive energy is returned rather than producing a distance-independent net radiated power.
• Dropping E_r everywhere. The radial electric field is negligible in the far zone, not in the general Hertzian-dipole solution.
• Using a far-field formula at kr ≪ 1. The discarded 1∕r² and 1∕r³ terms can then dominate.
• Confusing field pattern with power pattern. The Hertzian field amplitude is proportional to sin 𝜃; the power pattern is proportional to sin ²𝜃.
• Mixing peak and RMS current conventions. Here I₀ is a peak phasor amplitude, so P = (1∕2)R|I₀|².
• Confusing Hertzian and physically short dipoles. Their current distributions, and therefore their radiation-resistance coefficients, differ.

20 What EM23 adds to the series

EM16–EM21 established electromagnetic waves, energy flow, propagation, polarization, and boundary behavior. EM22 showed how those waves can be guided on transmission lines. EM23 now makes the transition from guided waves to radiation from sources.

The key field results for the ideal z-directed Hertzian element are

(98)

(99)

and

(100)

In the radiation zone,

(101)

which leads to

(102)

(103)

and in free space

(104)

The next article can build directly on these results to define radiation intensity, beam solid angle, directivity, radiation efficiency, antenna gain, EIRP, and effective aperture systematically.

References