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# Quadrature (90 degree) property of symmetric coupled lines

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The quadrature property of couplers is when the coupled section and through section have transmission phase angles that are 90 degrees apart. This is a useful property, and ideally it extends over infinite bandwidth, well outside of the useful amplitude band of a real coupler.

Symmetric coupled line structures include single-section couplers and any coupling factor: couplers used for power amplifier combine are tweaked for "3 dB" power coupling so that amplifiers are equally combined and one of the amps does not do all the work. These are called 3 dB hybrid couplers. The coupled-line section is one-quarter wavelength at center frequency. Wider-band couplers can be created by adding quarterwave sections, this is an art that has been documented in literature well before the age of EDA software. "Symmetric" couplers means that there is X and Y symmetry: a typical cascaded coupler might have a -10 dB section followed by a -2 dB section followed by a -10 dB section. Once symmetry is lost, the 90 degree property is usually lost, too.

There are other forces at work besides symmetry that ruin the desired quadrature property. Any coupler that is not built in a purely TEM media will go out of quadrature. A Lange coupler is built on microstrip, and therefore only has limited bandwidth. The 90 degree property of a Lange falls away particularly above the center frequency. The degree that it falls off is controlled by the disparity in dielectric constant of the substrate and air, for example a Lange on Quartz (DK=3.5) will outperform a Lange on GaAs (DK=12.9).

For the broadest-band quadrature couplers, stripline or coax must be used, as they offer TEM media. Microstrip is cheap and easy, but in the case of couplers it is not your friend!

When couplers fall out of quadrature, directivity is degraded, and the coupler becomes lossy as energy is routed to the normally isolated port.

Why do symmetric coupled lines produce a natural 90 degree phase shift between "through" and "coupled" paths? This is a great question and it takes a little explanation to do it justice.

The quadrature property of the coupled lines is a subset of the amazing properties of lossless, symmetric four-port microwave circuits. For double symmetry circuits (like two coupled lines) the 16 S-parameters of the scattering matrix of the four-port reduce to four independent S-parameters, S11, S12, S13 and S14.

Because the circuit is lossless (no heat or radiation losses), conservation of power is applicable to the four ports. This means that the multiplication of the scattering matrix by its complex conjugate transposed matrix will be equal to the unity matrix, [S],sup>T x [S]=I. This results in 16 equations-one for each element of the unity matrix. Twelve of these equations are not independent and are just variations of the first four. Click here to see the bloody details.

For more information on this approach to the properties of a lossless 4-port, check out the textbook by Mongia, Bahl & Bhartia, * RF and Microwave Coupled-Line Circuits, *Artech House, 1999, pp 40-46.

**Author : **The Other Editor

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