What Is a Butler Matrix?

A Butler matrix is a passive microwave beamforming feed network that accepts a signal at one of N input ports and distributes it with equal amplitude and a linear phase gradient across N output ports connected to the elements of a phased array antenna. The linear phase gradient steers the array's radiation pattern to a specific angular direction determined by which input port is driven. With N inputs, the network produces N orthogonal beam positions, each corresponding to one active input. The circuit contains no active components and no variable elements: beam selection is achieved by switching which input port is excited, rather than by adjusting individual phase shifters.

The device was introduced by Jesse Butler and Ralph Lowe in 1961 at Sanders Associates, described in a paper in IEEE Transactions on Antennas and Propagation. The original motivation was to provide a simple, passive alternative to the mechanically steered antennas and complex active phase-control systems of the period. Because the matrix operates reciprocally, the same physical network serves equally for transmit and receive applications, a property that makes it attractive wherever antenna system hardware is shared between those functions.

Circuit Design and Operating Principle

The internal topology of a Butler matrix is built from three types of passive elements: 90-degree hybrid couplers, fixed phase shifters, and signal crossovers. A 90-degree hybrid coupler splits power equally between two output ports while introducing a 90-degree phase difference between them. For a 4x4 matrix, the standard topology uses two hybrid couplers at the input stage, two at the output stage, two fixed 45-degree phase shifters, and two crossovers. Each combination of hybrid outputs and phase shifter delays creates a unique phase progression across the four antenna ports, corresponding to one of four beam directions. Larger matrices of order 8x8 or 16x16 repeat this pattern recursively, adding stages of couplers and shifters, and the number of crossovers grows accordingly, making planar layout increasingly complex at higher orders. Design details and circuit configurations are documented in Microwaves101's engineering reference on Butler matrices, including worked examples for 4x4 and 8x8 implementations.

Bandwidth Constraints and Design Variants

A standard Butler matrix achieves its ideal phase relationships only at the design frequency. At other frequencies, the quadrature hybrids deviate from 90 degrees and the fixed phase line lengths introduce excess delay, shifting beam positions and increasing sidelobes. The usable bandwidth of a basic implementation using branchline couplers is typically less than 10 percent of the center frequency. Designers extend bandwidth by substituting wideband couplers, such as Lange couplers or multi-section coupled-line structures, and by replacing simple transmission-line phase shifters with Schiffman sections that maintain a more constant differential phase over frequency. Substrate-integrated waveguide (SIW) technology provides an alternative fabrication approach for millimeter-wave frequencies, reducing radiation losses compared to microstrip. A study published in ScienceDirect on Butler matrix realization for phased array systems demonstrates how SIW-based designs achieve more than twice the fractional bandwidth of equivalent microstrip implementations at X-band.

Beam Coverage and Scan Angle

For a half-wavelength spaced array fed by a 4x4 Butler matrix, the four beam positions fall at approximately ±14.5 and ±48.6 degrees from broadside. No input produces a broadside beam, which is a structural property of the orthogonal beam set, not a design flaw. When broadside coverage is required, designers add a power divider with a phase offset to synthesize an additional beam or combine the Butler matrix with a separate fixed-beam element. The Microwave Journal's coverage of wideband Butler matrix applications discusses hybrid architectures that combine Butler matrices with switched-beam overlays to fill coverage gaps.

Applications

A Butler matrix has applications in a wide range of disciplines, including:

• 5G millimeter-wave base stations, where passive beamforming reduces active component count and power consumption
• Radar systems, where multiple simultaneous beams enable electronic scan without digitally controlled phase shifters
• Satellite ground terminals, where switchable beam positions allow tracking of multiple satellites with a single array aperture
• Indoor wireless access points, where compact planar Butler matrix designs support spatial diversity