I know I should know about this or be able to figure this out for myself. But I am feeling lazy today.
Is there a mathematical relationship between isolation and phase and amplitude imbalance in a 3dB hybrid coupler? Or even, does high isolation guarantee better phase and amplitude balance?
In a perfect hybrid, the isn't the isolation (i.e. the return loss at input port) a function of mismatch between between the two loads? So at least for the phase and impedance mismatch between the output lines, those would be equivalent to a load mismatch.
I guess I am asking about a real, physical device and not an abstract perfect hybrid. When I look at specs, I see 25 db isolation. This means for 3db coupler directivity of 22db. This isolation comes from leakage between arms, independent of load mismatch. It would be there with perfect load. Physically, you would want coupling only in the coupling region and zero coupling outside that region. This is practically impossible.
I will get back to this when I have time and report findings. I guess ultimately I am interested in how isolation hurts quadrature.
Look into the I/Q modulation; i.e. gain and phase imbalance, and isolation. The math should be the same. I was doing a hybrid fed, orthogonal feed patch for circular polarization. The isolation between LHC and RHC uses the same charts for gain & phase imbalance for image suppression in direct conversion. Though I haven't seen one for I to Q isolation, but the DSP guys have this all figured out for their corrections. I was able to get ~30 dB suppression which was good enough for my application, but some of these radio systems need 60 dB. I thought I'd be smart and put a PA on each arm of the hybrid, and of course they were run into compression, so AM-AM and AM-PM modulate the antenna polarization. Ended up having to match gain & phase on VNA when driven into compression, and group pairs by lot codes. The AM-AM and AM-PM varied drastically between lot codes; sometime compression, sometimes expansion. Real hair puller; lesson learned.
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 or a hybrid coupler) the 16 S-parameters of the scattering matrix of the four-port reduce to 4 independent S-parameters: S11, S12, S13, 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. This results in 16 equations - one for each element of the unity matrix. 12 of these equations are not independent and are just variations of the first four. So the 16 equations reduce to four complex equations involving the four complex S-parameters: S11, S12, S13, S14.
Go to www.microwaves101.com/encyclopedias/1158...metric-coupled-lines if you would like to see the
detailed four simplified equations.
If you simply by assuming S11=0 (and by symmetry S44=0) and assume S12 and S13 are non-zero, then the phase difference between
S12 and S13 is pi/2 exactly.
All the lossless symmetric four-ports in the world must satisfy these equations. Let’s just look at one of those circuits – a two line coupler. From the even and odd mode analysis provided by Microwaves101.com, if the values of Zoo and Zoe satisfy the equation Z0*Z0=Zoo*Zoe , then S11=0 (and by symmetry S44=0) and the four equations reduce to two equations.
If you assume that S12 and S13 are non-zero, then the phase difference between S12 and S13 is pi/2 exactly.
This is true for all frequencies, for any value of coupling and for any length of the coupled lines! If you don’t believe this, then plug into AWR Microwave Office and it will compute that S11 and S44 have a return loss and isolation greater that 300dB over the frequency range from 1 to 200 GHz Then the phase difference between S12 and S13 is pi/2 exactly regardless of the length of the coupled lines. 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.
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