Part 2 addresses coupling factor, signal propagation delay, maximum coupling, and vector analysis.
Part 3 covers the derivation of parameters, including S parameters, and conclusions derived from this analysis.
In a coupler made up of parallel coupled lines there is a phase relationship between the through port and the coupled port. The electrical phase of the coupled port is +90 degrees when the through port is referenced as 0 degrees. This +90 degree phase occurs at all frequencies for which the coupler has a good match. Parallel line couplers which have this characteristic are designed symmetrically with an odd number of sections.
It is true that these couplers have 90 degree electrical length lines at the center frequency of operation, but this has nothing to do with the 90 degree phase relationship between coupled and through ports. It does have everything to do with getting maximum coupling amplitude at the center frequency. Whatever creates this phase shift must then be independent of frequency.
The explanation for this can summed up in one sentence but would take pages of derivation to prove it: "At a coupled port equal forward and reverse propagating waves combine 0 degree and 180 degree phase components generated by mode reflection coefficients to produce a 90 degree phase shift independent of frequency."
This statement now needs to be put into the correct context with transmission line theory along with even and odd mode analysis. It should now be clear that variation of line length will not "tune" the relative phase shift of the coupled port. It is only the even and odd mode impedances that can do this. This frequency independent 90 degree phase shift is a very useful property, especially for passive broadband designs.
Transmission Line Theory
A transmission line of length L with a characteristic impedance Zx is terminate at both ends with load and source impedances Z Ø along with a source voltage Vs shown in Figure 1. The forward and reverse propagating waves are shown in complex exponential form. Initial voltage division at the input produces voltage V and the reflection coefficient P determines the voltage amplitude of the reverse propagating wave.
Figure 1.
There are multiple reflections on this transmission line and the reflection coefficient at both ends is P because the terminating impedances are both Z¾. This infinite series of reflections can be analyzed by the following equations. The infinite series equations can be put into a closed form using mathematical identities. The net result is that the initial forward and reverse reflected voltage components are scaled by the constant factor 1/(1- P2e-jB(2L)) in a closed form expression.
The terminating impedances could also be connecting transmission lines with characteristic impedances and terminations equal to Z Ø. The result would be the same but in a slightly different form as follows. Vs/2 = Incident voltage wave from source. The transmitted wave amplitude into the Zx characteristic impedance line would then be (Vs/2)(2 Zx/( Zx + ZØ)) = Vs(Zx/(Zx + ZØ)) which is the original incident voltage divider result.
These results can now be applied to coupled lines using even and odd mode analysis. Each mode will first be analyzed independently using transmission line theory and then the mode voltages will be combined to produce the total coupled line response.
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