Derivation of Coupler Scattering Matrix Parameters
A scattering matrix (or S parameter matrix) can be used to describe coupler performance with any passive termination. The S parameters within the matrix are measured using terminations which are equal to the characteristic impedance ZØ of the coupler. This means that all the ports will be matched. The S parameters are the reflected and transmitted voltage wave components divided by the (ZØ)^1/2, making them equal to the square root of power and proportional to voltage. The coupler S parameters can be derived using the following port numbering assignments.

The input port is Port 1, the coupled port is Port 2, the isolated port is Port 3 and the through port is Port 4. From these port assignments the following S parameter matrix is given as:

The port input waves are A1, A2, A3, and A4. These port input wave components can also be termination reflection coefficients which are not matched to Z_Ø. A matched termination would be equal to 0. The port output waves are B1, B2, B3, and B4. The matched port condition for the S parameter measurement also means that S11= S22 = S33 = S44 = 0 signifying no reflections when all the port terminations are Z_Ø. The ports which are isolated from each other have no transmitted energy between them and this means that S13=S31= S42 =S24= 0.

At this point, it should be mentioned that all port terminations are seen as being connected through transmission lines with the characteristic impedance of Z_Ø. From a mathematical point of view, they can be considered infinitely short; however, the actual physical matched coupler realization contains a finite length of single mode line with an impedance of Z_Ø. As the coupled lines are bent gradually away from each other, a net impedance of Z_Ø must be maintained until the uncoupled single mode matched condition is reached.

What this means is that the instantaneous reflections at the isolated ports which add up to 0 volts will continue to be 0 volts regardless of the termination connected to the port which is isolated. This is because the instantaneous reflection encounters a line impedance of Z_Ø and this instantly causes all voltage components to sum to 0 volts. The wave transmitted to the termination from this point will then have an amplitude of 0 volts, regardless of the termination used at this isolated port.

Derivation of Coupler Scattering Matrix Parameters
Since all the S parameters are proportional to voltage, those S parameters involving ratios can use coupling and through port voltage ratios derived earlier for the corresponding S parameter components.

Setting ζ= (0.5)^1/2 = 0.7071 makes the coupling value -3dB at the coupler center frequency (BL= 90 degrees). The numerical S parameter values are S12= S21 = S34 = S43 = j.7071 and S14= S41 = S32 = S23 = .7071. This creates the following numerical S parameter matrix for a 3dB coupler.

Many very interesting coupler properties can be demonstrated by evaluating this S parameter matrix using different port terminations. When all port terminations are matched (ΓA = ΓB = ΓC = 0), this coupler divides the input port power (A1)² equally between coupled and through ports with a 90deg. phase difference.

This continues to be the case if just the isolated port termination (ΓB ≠ 0) is not matched because no power is delivered to this port. If the coupled port has a mismatch equal to the through port mismatch (ΓA = ΓC ≠ 0, ΓB = 0 ) then again this coupler divides the input port power (A1)² equally between coupled and through ports with a 90 deg. phase difference, but this time some of the power is reflected back. This reflected power goes entirely to the matched isolated port where it is dissipated with no further reflections. The input port does not receive any reflected power because of 180 deg. vector cancellation created by two 90deg. phase shifts within the coupler. Having just the coupled port mismatched as a pure resistance (Γ A ≠ 0) results in unequal power division and degraded input port match however the 90 deg. phase difference between the coupled and through ports is maintained.

This is because there are no voltage wave components that have been reflected back through the coupler to the coupled port to change the phase of the original instantaneous incident voltage which has a 90deg phase difference. Mismatches at the coupled port and isolated port will significantly change the 90deg phase difference along with creating unequal power division and degrading input port match. The reason some port mismatch conditions do not degrade coupler performance is because the characteristic symmetry of a 3dB coupler along with its' port isolation. This symmetry property is used in balanced amplifiers and mixers to significantly improve mismatches by canceling the reflections from these microwave components.

Conclusion
A complete description of the fundamental characteristics of a pair of matched coupled lines has been given. It has just been described using words, mathematical formulas and step by step vector analysis. The important conclusions include:

• Relative port phase shift is independent of coupler length.
• Phase shift is affected by mode impedances.
• Maximum coupling occurs at BL = π/2 radians = 90deg
• Good port match and isolation requires Z_Ø = ( Z_E Z_O)^1/2
• High isolation requires mode line phase shifts to be equal.
• Mode phase velocity and mode path length determine line phase shift of each mode.
• |ISOLATION_dB| " |COUPLING_dB| = DIRECTIVITY_dB.
• Delay = τ = - ζ θ/ ζf
• The coupled and through port paths have equal delay.
• Multiple reflections affect delay if they are large.
• Multi-section coupler line sections add and subtract coupling.
• Equal coupled and through port mismatches preserve 3dB coupler amplitude balance.
• Equal coupled and through port mismatches preserve 3dB coupler 90deg port phase relationships.

About the Author

Mr. Kaarsberg received a BSECE degree in Electrical and Computer Engineering from the University of Wisconsin Madison in 1982 and a Master's Degree in Electrical Engineering from the University of Massachusetts in 1990. He joined M/A-COM in 1988 as a Senior Electrical Engineer and was involved in the design, development, millimeter-wave components. During the 1990s he worked for ATI to develop QAM and FSK digital Radio systems and then for ITT to work on microwave and RF synthesizers. In 2000 he rejoined M/A-COM as a Principle Microwave Engineer where he is currently working on the design of broadband stripline circuits for beamformer applications and on developing new low cost high performance microwave plastic packages. He has also designed and patented multi-layer microwave boards for high-density switch matrix applications.

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