Electromagnetic Waves, Antennas, and RF: Antenna Directivity, Radiation Efficiency, Gain, EIRP, and Effective Aperture

Electromagnetic Waves, Antennas, and RF: Antenna Directivity, Radiation Efficiency, Gain, EIRP, and Effective Aperture

EM23 derived the radiation field of a Hertzian dipole and showed that a localized time-varying current distribution launches electromagnetic power into different directions with different strengths. EM24 now turns that radiation pattern into the antenna quantities used throughout RF engineering: directivity, radiation efficiency, gain, equivalent isotropically radiated power, and effective aperture. The central objective is to connect every one of these quantities back to electromagnetic power flow rather than treat them as isolated link-budget definitions [1, 2, 3, 4].

The central chain is

(1)

The final result,

(2)

will be derived first for the Hertzian dipole and then generalized using antenna reciprocity. This relation is the missing bridge between a transmitting antenna’s directional gain and a receiving antenna’s ability to intercept power. EM25 will use it to derive the Friis transmission equation and free-space path loss.

1 Radiation intensity revisited

In the far field, the time-average Poynting vector is approximately radial,

(3)

Because a radiated field amplitude decreases as 1∕r, its power density decreases as 1∕r². It is therefore useful to remove this purely geometric spreading by defining the radiation intensity

(4)

Its units are watts per steradian,

(5)

The total radiated power is obtained by integrating over solid angle:

(6)

With spherical coordinates,

(7)

so

(8)

An isotropic radiator is the reference source that distributes the same power equally into every direction. Its radiation intensity is therefore

(9)

2 Directivity

The directional directivity is defined as

(10)

Directivity is dimensionless. It measures only how the radiated power is distributed in angle. It does not yet penalize Conductor loss, dielectric loss, or other dissipative mechanisms inside the antenna.

The maximum directivity is

(11)

Antenna values are often expressed logarithmically relative to an isotropic radiator:

(12)

The suffix dBi means decibels relative to an ideal isotropic radiator.

3 Beam solid angle

Let the normalized power pattern be

(13)

Then

(14)

Define the beam solid angle

(15)

Substituting into the maximum-directivity definition gives

(16)

so

(17)

This expression captures the physical idea of directivity: a pattern concentrated into a smaller solid angle has a larger peak directivity.

4 Example: Hertzian-dipole directivity

EM23 found the normalized power pattern of a Hertzian dipole,

(18)

Therefore

ΩA	= ∫ 02π ∫ 0π sin 2𝜃 sin 𝜃 d𝜃 dϕ	(19)
	= 2π ∫ 0π sin 3𝜃 d𝜃.	(20)

Using

(21)

we obtain

(22)

Hence

(23)

In logarithmic form,

(24)

This value describes angular concentration only. An ideal Hertzian dipole is being treated as lossless here.

5 Radiation efficiency

A real antenna may accept power at its terminals without radiating all of it. Let

(25)

be the time-average power accepted by the antenna terminals. This power is divided into radiated power and dissipative loss,

(26)

The radiation efficiency is

(27)

Therefore

(28)

For a simple equivalent circuit containing radiation resistance R_rad and loss resistance R_loss in series,

(29)

(30)

so

(31)

This makes the difficulty of electrically small antennas visible. If R_rad becomes very small while conductor and matching-network losses remain finite, the radiation efficiency can fall sharply.

6 Gain

Directivity normalizes radiation intensity by the power actually radiated. Gain instead normalizes by the power accepted at the antenna terminals:

(32)

Using

(33)

and

(34)

we obtain

(35)

Thus directivity answers

(36)

while gain answers

(37)

Maximum gain is usually quoted in dBi:

(38)

7 Mismatch and realized gain

The gain defined above uses power accepted by the antenna. If a source delivers incident power toward an antenna port with reflection coefficient Γ_A, the accepted fraction is

(39)

A useful quantity that includes this mismatch is the realized gain,

(40)

The distinction matters in practical link budgets:

(41)

Cable and feed losses before the antenna are normally accounted for separately unless they have explicitly been included in the quoted system gain.

8 Equivalent isotropically radiated power

Suppose an antenna accepts transmit power P_acc and has directional gain G(𝜃,ϕ). In its far field,

(42)

Therefore the power density at range r is

(43)

The product

(44)

is the equivalent isotropically radiated power: the power an isotropic radiator would need in order to produce the same far-field power density in that direction.

In logarithmic units,

(45)

If cable, filter, or connector loss L_feed,dB lies between the transmitter and antenna, a common system form is

(46)

For free-space propagation, the corresponding RMS electric-field magnitude is

(47)

Substituting

(48)

gives

(49)

Because

(50)

we obtain the useful free-space relation

(51)

Here EIRP is in watts, r in meters, and the result is in volts per meter.

9 Example: efficiency, gain, and EIRP

Consider an antenna with maximum directivity

(52)

and radiation efficiency

(53)

The maximum gain is

(54)

In dBi,

(55)

If the antenna accepts

(56)

then

(57)

This does not mean the antenna radiates 150 W of total power. The total radiated power is only

(58)

The larger EIRP expresses directional concentration relative to an isotropic radiator.

10 Receiving antennas and effective aperture

Now reverse the problem. Let a plane wave with time-average power density

(59)

arrive at a receiving antenna from some direction. The antenna produces electrical power at its terminals. For conjugate load matching and matched polarization, define the effective aperture by

(60)

where P_av is the maximum available receive power.

Therefore

(61)

Although A_e has units of square meters, it is not necessarily the literal geometric shadow of the antenna. A wire antenna may have negligible physical cross-sectional area yet possess a finite effective aperture.

11 The receive effective length

A receiving antenna can also be characterized by an effective length vector h_e. The open-circuit terminal voltage is

(62)

For an ideal Hertzian dipole oriented parallel to the incident Electric Field at broadside,

(63)

so

(64)

Here E is the peak phasor magnitude, consistent with the phasor convention used in EM23.

If the antenna input impedance is purely resistive at Resonance,

(65)

and a conjugate matched load is connected, the load voltage amplitude is V _oc∕2. The maximum time-average delivered power is therefore

(66)

For an ideal lossless Hertzian dipole,

(67)

12 Deriving the effective aperture of a Hertzian dipole

The time-average power density of a plane wave in a lossless medium is

(68)

For the broadside, polarization-matched Hertzian dipole,

(69)

so

(70)

Hence

Ae	=	(71)
	=	(72)
	= .	(73)

EM23 derived the radiation resistance of the ideal current element in a lossless medium,

(74)

Substituting gives

Ae	=	(75)
	= .	(76)

Using

(77)

we obtain

(78)

But EM23 also found

(79)

Therefore

(80)

Thus the transmit directivity and receive effective aperture satisfy

(81)

for the ideal lossless Hertzian dipole.

13 From the Hertzian dipole to the general aperture-gain theorem

For a linear reciprocal antenna, the Lorentz reciprocity theorem implies that the angular and polarization properties used to transmit a field are the same properties that govern reception from that direction [1, 2]. Therefore the directional receive aperture has the same normalized angular dependence as the transmit gain.

For a lossless antenna, the Hertzian result generalizes to

(82)

If the antenna has radiation efficiency less than unity, only the fraction η_rad of the receive-mode electromagnetic power associated with the antenna mode is available at the terminals. Since

(83)

the general matched-polarization, conjugate-match relation becomes

(84)

At the direction of maximum gain,

(85)

This is one of the most important antenna relations in RF engineering.

14 Physical meaning of the λ² factor

At fixed dimensionless gain,

(86)

Thus the same 0 dBi receiving antenna has a larger effective area at a lower frequency. This does not violate energy conservation. Gain is a dimensionless statement about angular coupling to a propagating spatial mode, while the transverse spatial scale of that mode grows with wavelength.

Conversely, for a fixed physical aperture whose effective area is nearly fixed, the gain rises approximately as

(87)

This is why a fixed-size dish antenna becomes more directive as frequency increases.

15 Example: effective aperture at GPS L1

For the GPS L1 carrier,

(88)

The free-space wavelength is approximately

(89)

For an antenna with

(90)

its maximum effective aperture is

Ae	=	(91)
	≈ 2.88 × 10−3 m2.	(92)

Therefore

(93)

The effective electromagnetic collection area is much more meaningful than the visible cross-sectional area of a small GNSS antenna.

16 Example: Hertzian-dipole effective aperture

At

(94)

the wavelength is approximately

(95)

For the ideal Hertzian dipole,

(96)

Hence

Ae,max	=	(97)
	≈ 0.119 m2.	(98)

So an electrically tiny ideal current element can possess an effective aperture of order a tenth of a square meter at a one-meter wavelength. Effective aperture is a coupling area, not a literal metal area.

17 Polarization mismatch

EM19 showed that a receiving antenna need not be polarization matched to the incident wave. If the normalized incident polarization vector is e_i and the receive polarization vector is e_r, the polarization loss factor is

(99)

The available receive power becomes

(100)

For two linearly polarized antennas misaligned by angle ψ,

(101)

Thus effective aperture should normally be interpreted as the aperture under matched-polarization conditions unless a polarization factor is stated separately.

18 Physical aperture and aperture efficiency

For aperture antennas such as horns, reflectors, or planar arrays, it is useful to compare effective aperture with physical aperture A_phys. Define the aperture efficiency

(102)

Then

(103)

Combining this with the aperture-gain theorem gives

(104)

For a circular aperture of diameter D_a,

(105)

so

(106)

This is the standard scaling behind parabolic-dish antenna gain. The aperture efficiency accounts for illumination taper, spillover, blockage, phase error, polarization effects, and other departures from ideal use of the physical opening.

19 Common conceptual mistakes

• Directivity is not gain. Directivity ignores dissipative antenna loss; gain includes radiation efficiency.
• Gain does not create power. A directional antenna redistributes radiated power so that some directions have larger radiation intensity than an isotropic reference.
• EIRP is not total radiated power. EIRP is a directional isotropic-equivalent quantity.
• Effective aperture is not necessarily geometric area. Wire antennas can have substantial A_e despite very small physical cross section.
• The A_e relation uses gain, not directivity, for a lossy antenna. Radiation efficiency must be included.
• Polarization and terminal mismatch still matter. The basic A_e = Gλ²∕(4π) expression assumes the stated gain direction, matched polarization, and the usual available-power definition.
• Realized gain includes mismatch; ordinary gain does not. Be explicit about which quantity a data sheet quotes.

20 The bridge to the Friis equation

EM17 established spherical power spreading,

(107)

while EM24 has now established the directional transmit form

(108)

and the receive relation

(109)

Using

(110)

immediately gives

(111)

EM25 will derive this carefully, identify the assumptions behind it, and separate the free-space spreading factor from antenna gain, polarization, mismatch, atmospheric loss, and other link-budget terms.

21 Summary

Radiation intensity removes the purely geometric 1∕r² spreading from far-field power density:

(112)

Directivity compares that intensity with an isotropic radiator having the same total radiated power:

(113)

Radiation efficiency measures how much accepted terminal power is actually radiated:

(114)

Gain combines the two:

(115)

Equivalent isotropically radiated power is

(116)

Effective aperture connects incident power density to maximum available receive power:

(117)

Finally, reciprocity connects transmit gain and receive aperture:

(118)

This identity completes the electromagnetic bridge from radiated fields to received power and prepares the series for Friis transmission, free-space path loss, and RF link budgets.

References

References