What is Ludwig-3 co-polarization definition?

Ludwig's third definition (1973) defines co-polarization as the polarization of a Huygens source (combined electric and magnetic dipole) located at the observation point and oriented to maximize the received signal from the antenna under test. For a linearly polarized antenna with the E-field in the y-direction at boresight, the Ludwig-3 co-pol component in spherical coordinates is E_co = E_theta*cos(phi) - E_phi*sin(phi) and the cross-pol is E_cross = E_theta*sin(phi) + E_phi*cos(phi). In the principal planes (phi = 0 and phi = 90 degrees), co-pol equals E_theta in the E-plane and E_phi in the H-plane. In the diagonal planes (phi = 45 degrees), both E_theta and E_phi contribute to both co-pol and cross-pol, which is where cross-pol is typically highest. Ludwig-3 is the standard used by antenna measurement facilities because it corresponds to the physical measurement made by a rotating probe antenna.

Why does cross-pol matter in the co-pol pattern?

Cross-polarization represents energy radiated in the unintended polarization, which causes several problems. In dual-polarized communication systems (CCDP microwave links, satellite dual-pol operation), cross-pol from the antenna couples energy between the H and V channels, limiting the cross-polar discrimination (XPD) and requiring XPIC to cancel the interference. The co-pol to cross-pol ratio at boresight must exceed 25 to 30 dB for dual-pol operation with 256-QAM. In radar, cross-pol degrades polarimetric measurements used for weather classification and target identification. In satellite systems, cross-pol from the earth station antenna couples into the orthogonal polarization transponder of the same satellite or adjacent satellites, creating interference. Antenna specifications typically require cross-pol below -25 dB relative to co-pol peak across the main beam (within the 3 dB beamwidth) and below -15 dB across the first sidelobes.

Measurement Techniques

Co-Pol Pattern

Q: How is the co-pol pattern measured?

Co-pol patterns are measured in anechoic chambers (compact range or far-field) or outdoor antenna ranges. The antenna under test (AUT) is mounted on a positioner that rotates it in azimuth and elevation. A source antenna transmits a known polarization (e.g., vertical), and the AUT receives while rotating. The received power versus angle gives the co-pol pattern in one plane cut. The source is then rotated 90 degrees (horizontal) and the measurement repeated to obtain the cross-pol pattern. For full 3D characterization, the AUT is scanned over a full sphere of angles using phi-over-theta or theta-over-phi coordinate systems. Near-field scanning (planar, cylindrical, or spherical) measures amplitude and phase across a scan surface and applies near-field-to-far-field transformation to compute the pattern. This is preferred for electrically large antennas where far-field distances (2*D^2/lambda) exceed practical range lengths. Measurement accuracy depends on range reflections (below -40 dB), probe calibration, positioner accuracy (0.01 to 0.1 degrees), and cable flexure compensation.

/koh-pohl pat-urn/

The co-pol pattern shows antenna gain in the intended polarization direction, defined by Ludwig's third definition (1973). A Huygens source reference at each observation point determines co-pol and cross-pol components. Well-designed antennas achieve cross-pol 20 to 40 dB below co-pol peak in the main beam. Specifications require cross-pol < -25 dB within the 3 dB beamwidth for dual-polarized systems using 256-QAM or higher modulation.

Category: Measurement Techniques

Cross-pol ratio: 20 to 40 dB

Standard: Ludwig-3 definition

Understanding Co-Pol Patterns

Every antenna radiates electromagnetic fields with both intended and unintended polarization components. A vertically polarized antenna, for example, radiates a dominant vertical E-field (the co-pol component) along with a smaller horizontal E-field (the cross-pol component) that arises from design imperfections, feed asymmetry, and edge diffraction. The co-pol pattern plots the intended component versus observation angle, revealing the main beam shape, sidelobes, and nulls that determine the antenna's performance in its designated polarization. The cross-pol pattern, measured simultaneously, reveals polarization purity as a function of angle.

Ludwig's third definition provides the standard mathematical framework for separating co-pol and cross-pol in measured data. Unlike simpler definitions that use fixed coordinate axes (Ludwig-1) or spherical unit vectors (Ludwig-2), Ludwig-3 defines co-pol and cross-pol relative to a reference antenna (Huygens source) at each observation point. This matches the physical measurement process where a calibrated probe antenna is rotated to measure each polarization component. The resulting co-pol pattern is smooth and well-defined across the full hemisphere, while cross-pol shows characteristic nulls along the principal planes (E-plane and H-plane) and peaks in the diagonal (45°) planes where the field decomposition is most sensitive to asymmetry.

Co-Pol Pattern Equations

Ludwig-3 Co-Polarization:
E_co = E_θ cosφ - E_φ sinφ

Ludwig-3 Cross-Polarization:
E_cross = E_θ sinφ + E_φ cosφ

Cross-Polar Discrimination:
XPD = 20 log(|E_co| / |E_cross|) (dB)

Where E_θ, E_φ = spherical field components, φ = azimuthal angle. In E-plane (φ=0): E_co=E_θ, E_cross=E_φ. In H-plane (φ=90°): E_co=-E_φ, E_cross=E_θ. In 45° plane: both components mix.

Co-Pol Specifications by Application

Application	Co-Pol Sidelobes	Cross-Pol (main beam)	Measurement Method	Standard
SATCOM earth station	< -14 dB (29-25logθ)	< -25 dB	Far-field range	ITU-R S.580
Dual-pol microwave	< -20 dB	< -30 dB	Compact range	ETSI EN 302 217
Weather radar	< -25 dB	< -35 dB (boresight)	Near-field scan	WMO No. 8
5G base station	< -15 dB	< -20 dB	OTA chamber	3GPP 38.141
Automotive radar	< -15 dB	Not specified	Compact range	ETSI EN 301 091

Common Questions

Frequently Asked Questions

What is Ludwig-3 co-polarization?

Ludwig-3 (1973) defines co-pol as the component received by a Huygens source at each observation point maximized for the AUT's polarization. E_co = E_θcosφ - E_φsinφ. In principal planes, co-pol equals E_θ (E-plane) or -E_φ (H-plane). Cross-pol peaks in 45° planes where both components mix.

Why does cross-pol matter?

In CCDP links, antenna cross-pol limits XPD and requires XPIC for >256-QAM. Co-pol to cross-pol must exceed 25 to 30 dB at boresight. In polarimetric radar, cross-pol degrades weather classification. In satellite systems, it couples into orthogonal transponders. Specs require cross-pol < -25 dB within the 3 dB beamwidth.

How is the co-pol pattern measured?

AUT rotates on positioner in anechoic chamber; source transmits known polarization. Power vs angle gives co-pol cut. Source rotated 90° for cross-pol. Near-field scanning (planar/cylindrical/spherical) preferred for large antennas where far-field distance (2D²/λ) is impractical. Accuracy needs: reflections < -40 dB, positioner < 0.1°.

Related Terms

← Co Located Mimo Co Simulation →