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https://f4inx.github.io/posts/IQ-quadrature-sign.html

IQ modulator and quadrature coupler sign issues.

01 Jun 2024

Introduction

IQ modulators and quadrature couplers are often used together to perform a frequency translation while rejecting the unwanted sideband. However, the documentation often does not clarify whether the quadrature signal has a +90° or -90° phase shift, and whether the IQ modulator expects a quadrature input with a +90° or -90° phase shift.

The purpose of this page is to clarify this sign issue.

Upconverter side

Here, we assume an upconverter is used. The principle of an IQ upconverter is to perform a complex frequency translation as follows:

"RF" = Re[(I + j \cdot Q) \cdot e^(j \cdot \omega_("rf") \cdot t)]

In the case of a complex cosine signal in baseband with frequency &&\omega_b&&:

I + j \cdot Q = e^(j \cdot \omega_b \cdot t) = cos(\omega_b \cdot t) + j \cdot sin(\omega_b \cdot t)

By application of the trigonometric identity &&sin(x) = cos(x - pi / 2)&&:

I + j \cdot Q = cos(\omega_b \cdot t) + j \cdot cos(\omega_b \cdot t - \pi / 2)

This clearly shows that the Q input has a phase lag compared to the I input, which is a -90° phase.

Coupler side case 1: coupled line coupler

Let’s assume that the coupled line coupler QCH-451+ from Mini-Circuits is used. The datasheet indicates which port is the quadrature output but not whether its phase is +90° or -90°.

Using ChatGPT and Plotly, the following plots were easily generated:

It’s clear from the plots than the quadrature output has a +90° phase (lead) compared to the in-phase output.

Besides clarifying the +/-90° sign issue, it is useful to check to what physical element correspond each port number. The hypothesis is that this coupler is made with an equivalent to coupled lines performed using an LC network, as pure coupled lines are obviously too long to fit in the component.

Analysis of the transmission curves enables to deduce the connexions:

This leads to the following schematic, compared to the datasheet information:

The symmetry of the schematic is coherent with the symmetry of the table given in the Mini-Circuits datasheet:

The +90 phase shift stays even at low frequencies. This is coherent with the similar behaviour of an high-pass CR network, which is part of the equivalent schematic at low frequencies.

Coupler side case 2: branch-line coupler

In this part, the ports are numbered like in Microwaves101 (https://www.microwaves101.com/encyclopedias/branchline-couplers), from which this schematic is taken:

This part was performed in two steps. First, using scikit-rf with a some help of ChatGPT, a Python script was created to produce preliminary plots and an s4p file. The second step was to reuse the plotting code from the previous section, with some help from ChatGPT to refactor the duplicate parts, to make the following plots:

Contrary to coupled lines, which have a very broadband quadrature effect, the branch-line coupler has a narrowband quadrature effect. This is still useful for narrowband applications.

It’s clear from the plots than the coupled output has a -90° phase (lag) compared to the direct output.

Conclusion

Quadrature couplers are important for making image reject mixers, but the actual phase sign is sometimes not clearly stated. According to the theory, IQ mixers needs a -90° phase (lag) for the Q input. For coupled lines couplers, the coupled output has a +90° (lead) phase compared to the uncoupled output. Conversely, for branchline couplers, the coupled output has a -90° (lag) phase compared to the uncoupled output. These points must be considered when selecting the wanted mixer side.

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