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Antenna factor

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In electromagnetics, the antenna factor (AF, units: m⁻¹, reciprocal meter) is defined as the ratio of the electric field E (units: V/m or μV/m) to the voltage V (units: V or μV) induced across the terminals of an antenna:

AF={\frac {E}{V}}

If all quantities are expressed logarithmically in decibels instead of SI units, the above equation becomes

AF_{\mathrm {dB/m} }=E_{\mathrm {\mathrm {dBV/m} } }-V_{\mathrm {dBV} }

The voltage measured at the output terminals of an antenna is not the actual field intensity due to actual antenna gain, aperture characteristics, and loading effects.^[1]^{[clarification needed]}

For a magnetic field, with units of A/m, the corresponding antenna factor is in units of A/(V⋅m). For the relationship between the electric and magnetic fields, see the impedance of free space.

For a 50 Ω load, knowing that P_D A_e = P_r = V²/R and E²= ${\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}$ P_D ~ 377P_D (E and V noted here are the RMS values averaged over time), the antenna factor is developed as:

AF={\frac {\sqrt {377P_{D}}}{\sqrt {50P_{D}A_{e}}}}={\frac {2.75}{\sqrt {A_{e}}}}={\frac {9.73}{\lambda {\sqrt {G}}}}

Where

A_e = (λ²G)/4π : the antenna effective aperture
P_D is the power density in watts per unit area
P_r is the power delivered into the load resistance presented by the receiver (normally 50 ohms)
G: the antenna gain
$\mu _{0}$ is the magnetic constant
$\varepsilon _{0}$ is the electric constant

For antennas which are not defined by a physical area, such as monopoles and dipoles consisting of thin rod conductors, the effective length (units: meter) is used to measure the ratio between voltage and electric field.

Notes

[edit]

^ Electronic Warfare and Radar Systems - Engineering Handbook (4th ed.). US Naval Air Warfare Center Weapons Division. 2013. p. 192.

References

[edit]

Interpreting Antenna Performance Parameters for EMC Applications
Interpreting Antenna Performance Parameters for EMC Applications An excellent page with summaries of all the relevant equations
Antenna Theory
Antenna Factor Calculator

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The antenna factor (AF) is a fundamental parameter in electromagnetics that describes the relationship between the electric field strength incident on a receiving antenna and the voltage induced across its load terminals. It is mathematically defined as the ratio of the field strength

E

(in V/m) to the output voltage

V

(in V), expressed as

\text{AF} = E / V

, with units of reciprocal meters (m⁻¹) or, more commonly in practice, decibels per meter (dB/m) using the formula

\text{AF (dB/m)} = 20 \log_{10} (E / V)

.^[1] This factor accounts for the antenna's effective length and losses, enabling precise field strength determinations without direct field measurement.^[1] In electromagnetic compatibility (EMC) testing, the antenna factor plays a pivotal role in evaluating radiated emissions and immunity of electronic equipment, as required by standards such as those from the International Electrotechnical Commission (IEC) and the Federal Communications Commission (FCC).^[2] By applying the AF to the measured voltage from a spectrum analyzer or receiver, testers can compute the incident electric field strength, which is essential for compliance assessments across frequencies from 9 kHz to 40 GHz.^[2] For instance, in open-area test sites (OATS) or semi-anechoic chambers, the AF corrects for antenna-specific responses, ensuring measurements reflect true environmental electromagnetic interference (EMI) levels rather than instrument artifacts.^[1] Calibration of the antenna factor is standardized to achieve uncertainties typically below ±1 dB, using methods outlined in ANSI C63.5, which includes the reference antenna technique and the three-antenna method for broadband validation.^[2] The reference method involves substituting a calibrated standard antenna in a known field generated by a transmitting antenna, while the three-antenna approach leverages mutual coupling between pairs of antennas to derive AF values independently of site imperfections.^[3] These procedures, often performed at facilities like the National Institute of Standards and Technology (NIST), emphasize free-space conditions to minimize multipath effects, with AF values provided in calibration certificates for specific frequency bands and polarizations.^[1]

Fundamentals

Definition

Antenna factor (AF) is defined as the ratio of the incident electric field strength

E

(in V/m) to the voltage

V

(in V) measured across the antenna terminals when connected to a standard load impedance (typically 50 Ω), expressed mathematically as

AF = \frac{E}{V}

, with units of inverse meters (m⁻¹).^[4]^[5]^[6]^[1] This definition captures the antenna's response to an electromagnetic wave, where the field strength represents the intensity of the incoming signal and the induced voltage is the electrical output generated at the antenna's receiving port. The antenna factor serves as a key conversion parameter in electromagnetic measurements, enabling the transformation of the voltage signal measured at the antenna's output into the corresponding incident electric field strength.^[7]^[8] This capability is essential for non-contact evaluation of electromagnetic fields, allowing precise determination of field levels without direct probing, which is particularly valuable in scenarios requiring accurate field strength assessments.^[9] Primarily defined and applied in the receiving mode of antennas, the antenna factor quantifies the device's field-to-voltage sensitivity, distinguishing it from transmitting parameters that focus on power radiation patterns.^[5]^[10] In this context, it provides a standardized measure of how effectively an antenna converts an impinging field into a usable voltage signal, independent of the specific transmitting characteristics. The concept of antenna factor emerged from early 20th-century radio engineering practices for field strength measurements and was formalized in the mid-20th century amid the development of electromagnetic compatibility (EMC) standards, as radio frequency interference concerns grew with advancing technology.^[7]^[11] This evolution integrated the parameter into standardized testing protocols to ensure reliable electromagnetic assessments.^[12]

Units and notation

The antenna factor (AF) is conventionally expressed in linear units of reciprocal meters (m⁻¹), representing the ratio of the incident electric field strength to the received voltage across the antenna terminals.^[13] In logarithmic form, it is given in decibels per meter (dB/m), calculated as AF(dB/m) = 20 log₁₀(AF(m⁻¹)), which facilitates measurements and comparisons in electromagnetic compatibility testing.^[14] Standard notation uses AF for the linear antenna factor, though variants such as K or f are occasionally employed in technical literature.^[15] The received voltage V is typically specified as root-mean-square (RMS), while the electric field strength E is also taken as RMS, ensuring consistency in field strength calculations.^[15] The logarithmic conversion formula accounts for the base-10 logarithm applied to the linear ratio, with the factor of 20 arising from the voltage-squared relationship in power terms. For many antenna types, such as dipoles or log-periodics, the AF exhibits frequency dependence, generally increasing with frequency due to the inverse wavelength scaling in effective aperture.^[16] In practical measurements, the antenna factor assumes a standard load impedance of 50 Ω at the receiver terminals, which influences the interpretation of the output voltage; deviations from this impedance require mismatch corrections to maintain accuracy.^[17]

Theoretical Relations

Relation to effective length

The antenna factor (AF) of a receiving antenna is directly related to its effective length

l_{\text{eff}}

through the reciprocal relation

\text{AF} = \frac{1}{l_{\text{eff}}}

, where

l_{\text{eff}}

has units of meters and AF has units of reciprocal meters (m

^{-1}

). This connection stems from the fundamental definition of effective length, which characterizes the antenna's response to an incident electromagnetic wave. Specifically, the open-circuit voltage

V_{\text{oc}}

induced at the antenna terminals by an incident electric field

\mathbf{E}

is given by

V_{\text{oc}} = \mathbf{l}_{\text{eff}} \cdot \mathbf{E}

, where

\mathbf{l}_{\text{eff}}

is the vector effective length. For the case of aligned polarization between the antenna and the field, this simplifies to the scalar form

V_{\text{oc}} = l_{\text{eff}} E

, so

l_{\text{eff}} = \frac{V_{\text{oc}}}{E}

. Since the antenna factor is defined as the ratio

\text{AF} = \frac{E}{V_{\text{oc}}}

under open-circuit conditions, the reciprocal relationship follows immediately. This derivation holds under the reciprocity principle, linking the receiving properties of the antenna to its transmitting characteristics.^[18]^[19] Physically, the effective length

l_{\text{eff}}

represents the equivalent length of a uniform-current infinitesimal dipole that would induce the same terminal voltage in the given incident field, thereby quantifying the fraction of the electric field "captured" by the antenna and converted to voltage. It accounts for the spatial integration of the field over the antenna structure and the current distribution along its elements. For a short dipole antenna, where the physical length

L

is much smaller than the wavelength

\lambda

(

L \ll \lambda

), the current distribution is approximately triangular, yielding

l_{\text{eff}} = \frac{L}{2}

. In this regime, the induced voltage is thus

V_{\text{oc}} = \frac{L}{2} E

, and the corresponding antenna factor is

\text{AF} = \frac{2}{L}

. This interpretation emphasizes

l_{\text{eff}}

as a measure of the antenna's field-to-voltage conversion efficiency, independent of power considerations.^[16]/10%3A_Antennas_and_Radiation/10.03%3A_Antenna_gain_effective_area_and_circuit_properties) The effective length

l_{\text{eff}}

, and thus the antenna factor, exhibits frequency dependence due to the fixed physical dimensions of the antenna relative to the varying wavelength. At low frequencies, where the antenna operates in the electrically short regime (

L \ll \lambda

l_{\text{eff}}

remains approximately constant at

\frac{L}{2}

, resulting in a frequency-independent AF. As frequency increases and the antenna approaches resonance,

l_{\text{eff}}

grows with

\lambda

; for example, a thin half-wave dipole (

L = \frac{\lambda}{2}

) has

l_{\text{eff}} \approx \frac{\lambda}{\pi}

in the direction of maximum response, yielding

\text{AF} \approx \frac{\pi}{\lambda}

. This variation illustrates how AF decreases (becomes more sensitive) at higher frequencies near resonance for fixed-size dipoles, reflecting the antenna's improved coupling to the field as its dimensions align better with the wave.^[20]^[16] This theoretical link assumes far-field conditions with plane-wave incidence, perfect alignment of the incident field polarization with the antenna's principal polarization (effectively isotropic in the sense of uniform plane-wave exposure for the relevant component), and a lossless transmission line connecting the antenna terminals for open-circuit voltage measurement. Deviations from these, such as near-field effects or mismatch, would require corrections to the basic relation.^[18]^[16]

Relation to antenna gain

The reciprocity principle in antenna theory states that, for a linear, passive antenna in free space, the transmitting and receiving properties are identical, allowing the same gain parameter to characterize performance in both modes. This equivalence enables the derivation of the antenna factor (AF) from the antenna gain (G) under lossless and matched conditions, linking the incident electric field strength to the output voltage through power transfer mechanisms.^[21]^[7] The physical relationship stems from adaptations of the Friis transmission equation, which relates received power to effective aperture

A_e = \frac{\lambda^2 G}{4\pi}

, where

\lambda

is the wavelength and

G

is the power gain (dimensionless). The power density

S = \frac{E^2}{\eta_0}

(with

\eta_0 = 120\pi \, \Omega

the free-space impedance) delivers power

P_r = A_e S

to a matched load, yielding output voltage

V = \sqrt{P_r R_L}

V=PrRL

for load resistance $R_L = 50 \, \Omega$ . Thus, AF = $\frac{E}{V} = \frac{9.73}{\lambda \sqrt{G}}$

9.73 (in m $^{-1}$ ), incorporating radiation efficiency implicitly in $G$ for lossless cases.^[7]^[22] In logarithmic terms, for practical computations with frequency $f$ in MHz, the relation simplifies to AF (dB/m) $\approx 20 \log_{10} f - G$ (dBi) - 29.79, providing a direct link for broadband antenna analysis. This formula assumes a 50-Ω system and isotropic reference for gain, with the constant arising from $c = 3 \times 10^8$ m/s and unit conversions.^[22]^[23] While AF quantifies receiving sensitivity (field-to-voltage conversion), gain emphasizes transmitting directivity (power density enhancement), yet reciprocity ensures their equivalence for identical antennas without ohmic losses or impedance mismatch. Deviations occur in lossy or mismatched scenarios, requiring efficiency corrections (e.g., radiation efficiency $\eta_r < 1$ reduces effective G) or site-specific adjustments, such as over ground planes where transmit and receive AF differ by up to 3 dB.^[21]^[23]

Measurement and Calibration

Measurement techniques

The three-antenna method is a primary technique for determining the antenna factor (AF) of an antenna under test (AUT) by measuring the mutual couplings between three antennas, including the antenna under test (AUT), in a controlled environment such as an anechoic chamber or an open-area test site (OATS).^[24] This method involves transmitting signals from each antenna pair while recording received voltages, then solving a system of equations derived from the measured transmission coefficients to extract the AF, accounting for site-specific effects like reflections.^[25] It is particularly effective for broadband antennas, providing absolute calibration without relying on external field strength references.^[26] Alternative approaches include the substitution method, where the AUT replaces a calibrated reference antenna or source in the measurement setup, and the received voltage is adjusted to match the reference level to compute the AF.^[27] For low-frequency applications, direct field injection techniques utilize transverse electromagnetic (TEM) cells to generate a uniform electric field, allowing the AF to be calculated from the known field strength and the voltage induced in the AUT placed within the cell.^[28] These methods are selected based on frequency and antenna type, with TEM cells suitable below 30 MHz due to their ability to produce well-defined fields in compact setups.^[29] Measurements typically cover broadband frequency ranges from 30 MHz to 18 GHz, though extensions to higher frequencies up to 40 GHz are possible in fully lined anechoic chambers.^[30] On OATS, site attenuation corrections are applied to compensate for ground plane effects and environmental reflections, ensuring measurement accuracy within ±1 dB for most setups.^[31] Common equipment includes a spectrum analyzer to measure received voltages, a signal generator for transmission, and field probes for independent verification of the electric field strength.^[32] Post-processing involves averaging data over horizontal and vertical polarizations to mitigate orientation errors and improve reliability.^[33] These techniques align with established calibration standards for traceability.^[27]

Calibration standards

Calibration of antenna factors relies on established international standards to ensure consistency, accuracy, and reproducibility in electromagnetic compatibility (EMC) and antenna performance assessments. The American National Standards Institute (ANSI) C63.5 standard outlines procedures for calibrating antennas used in radiated emission measurements, covering frequencies from 9 kHz to 40 GHz and including methods such as the three-antenna method, reference antenna method, and equivalent capacitance substitution for low frequencies.^[2] Similarly, the International Electrotechnical Commission (IEC) CISPR 16-1-6 standard specifies procedures for determining antenna factors in EMC contexts, emphasizing validation for disturbance and immunity measurements across broadband antennas like biconicals and log-periodics.^[34] For broader antenna measurements, IEEE Std 149 provides recommended practices for evaluating transmitting and receiving properties, including gain and pattern, which support antenna factor derivation in controlled environments.^[35] Traceability to national metrology institutes, such as the National Institute of Standards and Technology (NIST) in the United States, is essential for validating antenna factor measurements. Calibrations are typically achieved using reference antennas with known factors, establishing an unbroken chain from primary standards like calculable dipoles or standard gain horns.^[36] Uncertainty budgets for these calibrations generally range from ±1.5 dB to ±2.5 dB at a coverage factor k=2, accounting for contributions from instrumentation, site imperfections, and environmental factors, ensuring measurements meet ISO/IEC 17025 accreditation requirements.^[36] Frequency-specific protocols address challenges in different bands to maintain measurement reliability. For low frequencies (DC to 30 MHz), gigahertz transverse electromagnetic (GTEM) cells provide a shielded, controlled environment for calibrating loop and monopole antennas, simulating free-space conditions while minimizing external interference.^[37] At higher frequencies (above 1 GHz), compact antenna test ranges (CATRs) utilize parabolic reflectors to generate plane waves, enabling precise factor determinations for directive antennas like horns, with quiet zone dimensions supporting tests up to 110 GHz.^[38] Periodic recalibration is recommended every 1 to 2 years, depending on usage and antenna type, to account for potential degradation in materials or alignment.^[3] Common error sources in antenna factor calibration include multipath reflections from the test site, insertion losses in feed cables, and imbalances introduced by baluns, which can distort voltage-to-field conversions. These are mitigated through corrections in the uncertainty budget and site validation using voltage standing wave ratio (VSWR) measurements to confirm reflection coefficients below -20 dB across the frequency range.^[3]

Applications

EMC testing

In electromagnetic compatibility (EMC) testing, the antenna factor (AF) plays a crucial role in quantifying radiated emissions from equipment under test (EUT) to ensure compliance with regulatory limits. During emissions measurements, the received voltage

V

at the antenna terminals is converted to electric field strength

E

using the relation

E = AF \times V

, where

E

is in V/m and AF is in 1/m.^[39] In logarithmic form, commonly used for decibel-based specifications, this becomes

E(\mathrm{dB}\mu\mathrm{V/m}) = V(\mathrm{dB}\mu\mathrm{V}) + AF(\mathrm{dB/m})

, often adjusted for cable losses and preamplifier gains.^[39] This conversion enables direct comparison of measured field strengths against limits, such as those in FCC Part 15 Subpart B, which specify maximum radiated emissions for unintentional radiators like digital devices, with field strength limits varying by frequency and class (e.g., Class B limits at 3 m distance). For immunity or susceptibility testing, the AF is employed to establish the incident field levels required for the test, deriving

E

from the power injected into the transmitting antenna. The transmit antenna factor (TAF), a related parameter, specifically relates the input voltage to the generated electric field at a specified distance, allowing precise setup of uniform fields to assess the EUT's resilience to electromagnetic disturbances.^[40] This is essential in standards like IEC 61000-4-3, where forward power is adjusted to achieve the desired field strength using TAF.^[40] Typical EMC test setups utilize broadband antennas such as biconical types for frequencies from 30 MHz to 300 MHz and log-periodic antennas for 300 MHz to 1 GHz, ensuring coverage across common emission bands.^[41] Measurements often employ quasi-peak detectors in EMI receivers to mimic human perception of interference, as mandated by CISPR standards and FCC Part 15. Test distances are standardized at 3 m for most commercial equipment or 10 m for higher-power devices, with extrapolation formulas applied for alternative distances to maintain far-field conditions.^[42] In regulatory contexts, AF supports compliance with CISPR 11 for industrial, scientific, and medical (ISM) equipment emissions, specifying radiated limits from 9 kHz to 400 GHz, and CISPR 22 (now harmonized with CISPR 32) for information technology equipment (ITE).^[43] The 2024 edition of CISPR 11 extended radiated emission limits above 1 GHz for Group 1 equipment, addressing higher-frequency operations in modern ISM devices.^[43] As of 2025, ongoing EMC standard evolutions incorporate broader frequency ranges to accommodate emerging technologies, including extensions toward millimeter-wave bands relevant to 6G systems.^[43]

Antenna performance evaluation

Antenna factor (AF) plays a crucial role in the design phase of antennas by enabling engineers to evaluate and optimize key performance characteristics such as bandwidth and overall efficiency. By plotting AF against frequency, designers can identify the operational bandwidth, defined as the range where AF variations remain minimal and suitable for the target application, allowing for adjustments in geometry or materials to extend this range without compromising signal integrity. This visualization also facilitates comparison with theoretical effective length (l_eff), where discrepancies between measured or simulated AF and predicted l_eff (since AF ≈ 1/l_eff for many configurations) validate antenna models and highlight modeling inaccuracies, such as those arising from mutual coupling or substrate effects. In performance metrics assessment, AF serves as a foundational parameter for deriving realized gain and radiation efficiency, providing insights into how effectively the antenna converts input power to radiated fields. The antenna gain G in dBi can be estimated from the antenna factor as

G \, (\mathrm{dBi}) = 20 \log_{10} (f) - \mathrm{AF} \, (\mathrm{dB/m}) - 29.8

, where

f

is the frequency in MHz, assuming a matched and lossless antenna.^[5] Similarly, variations in AF versus frequency can indicate resonances, where the antenna achieves maximum coupling to the incident field, guiding optimizations for applications requiring narrowband high-efficiency operation, such as radar systems. Integration of AF into simulation workflows enhances design iteration speed and accuracy. In electromagnetic simulation software like Ansys HFSS or Dassault Systèmes CST Studio Suite, AF is derived from S-parameters obtained via finite element or time-domain solvers, allowing prediction of far-field behavior from near-field port data without physical prototypes. Experimental verification then compares these simulated AF curves with measured values in anechoic chambers, quantifying discrepancies due to fabrication tolerances or unmodeled losses, typically achieving agreement within 1-2 dB for well-designed structures. For advanced antenna systems, AF evaluation extends to array configurations and feed structures. In phased arrays, AF is assessed per element to ensure uniform response across the aperture, mitigating grating lobes and optimizing beamforming efficiency by adjusting element spacing or excitation phases based on individual AF profiles. The impact of baluns or feeds on measured AF is particularly significant, as mismatches can introduce common-mode currents that alter the effective aperture, necessitating their inclusion in both simulations and tests to achieve balanced radiation patterns.

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Antenna factor

Fundamentals

Definition

Units and notation

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Relation to effective length

Relation to antenna gain

Measurement and Calibration

Measurement techniques

Calibration standards

Applications

EMC testing

Antenna performance evaluation

References

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Relation to antenna gain

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