# RF Fundamentals: What actually creates the 90 degree coupler phase shift? Part 2

*Part 1 addresses transmission line theory. Part 3 covers the derivation of parameters, including S parameters, and conclusions derived from this analysis.*

**Coupling Factor>**

The term ζ is the *coupling factor* and is equal to the magnitude of the maximum coupling between the lines. The maximum coupling is written as maximum coupling = ζdB = 20Log(ζ) for BL = π/2. With maximum coupling comes
minimum output at the through port resulting in a maximum insertion loss.

This is also written as maximum insertion loss = 20Log(1- ζ2)1/2 for BL = π/2. These results are demonstrated in the equations derived from the previous even and odd analysis. From these equations, all the power entering and leaving the coupler can be accounted for with no power going to the isolated port. The *coupling factor* can be used to define this entirely at maximum mid band coupling. This analysis is only for single section couplers but it is useful for understanding other coupler configurations. Multi-section couplers have electrical characteristics very similar to single section couplers as long as these couplers are symmetrical and have an odd number of sections.

The central coupled line in a multi-section coupler has the highest coupling and also has nearly the same even and odd mode impedance relationships as a single section coupler. The coupled line sections add and subtract small amounts of coupling over frequency to produce broadband equal ripple characteristics.

For example, the coupling at mid-band of a 3-section coupler is C_{mid-band} ≈ ζ_{Center} - 2ζ_{Ends}. This means that the coupling at mid band is reduced by twice the coupling value of the end lines when compared to a single section coupler thus producing a more broadband equal ripple response at less coupling. This approximation is accurate for coupling less than -10 dB. For a given coupling value this means the central line must have a higher coupling value to offset the effect of the outer coupled lines.

**Isolated and Input Port Analysis**

The analysis up to this point covers the electrical characterization of the coupled and through ports for single section coupled lines only The input and isolated ports also need investigation to complete this coupled line analysis. Using the earlier equations for coupled and through lines, the voltage at the isolated port can be determined by setting z = L and the input port voltage can also be determined by setting z = 0.

The total voltage at the isolated port is 0 volts and the total voltage at the input port is V volts which is half the applied voltage of 2V volts from the combined mode voltages at the input port. This is because the input impedance is Z_{Ø}ohms and is matched to the source impedance of Z_{Ø}ohms dividing its voltage by 2. The incident even mode voltage at the isolated port is greater than the incident odd mode voltage, however the total voltage adds up to 0 volts because the even mode reflection coefficient is negative and the odd mode reflection coefficient is positive.

The sum of the incident and reflected voltages for both modes produces no net voltage at the isolated port.
V_{iE}(1+ P_{E}) " V_{iO}(1+ P_{O}) = 0 volts. The denominators are equal (1- P_{E} ^{2}e^{-jB(2L)}) = (1- P_{O}^{2}e^{-jB(2L)}) and can be factored
out of this equation.

The input port as well as all other ports impedance are matched to Z_{Ø}ohms, and this means that there can be no net reflected voltage at the input port. The voltage relationship between the reflected modes heading back to the input port is the same as the incident mode voltages going to the isolated port scaled by the negative magnitude of the reflection coefficients -|P_{E}| (|P_{E}| = |P_{O}|).

This means that there will be no net voltage generated by the reflected mode at the input port just as there will be no net voltage generated by the incident mode at the isolated port when the effect of the reflection coefficients is summed into the net voltage in both cases.