Linear vs Reciprical

[Twinpeaksr]

As I work to finish my masters (RF And Digital Logic), I had a question come up:

If a device is reciprocal, is it always linear?

If a device is linear is it always reciprocal?

I posed this to the professor, he could not think of any RF device that would meet either reciprocity or linearity but not both.

Can anyone think of a situation for which this would not be true? Maybe I can name the law after me (my guess is someone else has already take credit for this and I just can't find it in the 900pages of my reference book).

Thanks!

 

[QuixoticOne]

Well Kirchoff's laws would imply a certain amount of
reciprocity:
http://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws

I guess the answer depends on what you really mean by reciprocity in the context of a multi-port circuit.

A lumped directional coupler isn't truly linear in its operation, and it may or may not be reciprocal depending on whether you're accounting for symmetries in the way its ports operate since it isn't going to be reciprocal between port 1 and port 2, since S21 is high but then for waves coming backwards S12 isn't the same as they'll flow into port 3.

You could say a 'magic tee' (which is a purely passive and therefore linear assuming it is involving nothing but empty space and metal waveguide walls) directional coupler is the same way -- reciprocity is easy to analyze, but it certainly isn't as simple as saying it is reciprocal in any given 2-port sense.

It's certainly possible to contrive circuits that will have real world losses that vary depending on the direction of travel of waves in the circuit (q.v. directional coupling), so if you took the real world circuit as a connectorized 2-port black box it would look like a linear time invariant network to a good approximation, but there would be some loss or possibly differing frqeuency dependent behavior depending on which direction one looks at signal flow in them due to directional type of effects.

I guess it just depends on where linear circuit theory breaks down and where you have to start taking into account more full applications of Maxwell's laws and real world materials physics which are outside of the simplifications circuit theory imposes.

Also a classical theoretical op-amp with feedback set up as an instrumentation amp is a good approximation of a non-reciprocal device. Signals appearing at its imput are copied to its output with perfect fidelity, whereas signals appearing at the output are not impactful of the signal terminations at the input nodes. I suppose it's really just a well terminated input, a few buffer amplifiers cascaded, and an output amplifier, with some implication of strong directional coupling involved in the overall design. It can be as linear in transfer function from input to output as much as one wants to design it to be within any given frequency and amplitude range. But it's almost meaningless to ask about its reciprocity by definition.

 

[Twinpeaksr]

Thanks for the insight, much insight and a great point, Where do you define theory vs reality. I do not practice RF in my daily job (more on the digital side) I do find the field interesting and appreciate the detail! It will take me a little bit to digest all you provided (as you may have guessed I am working almost completely in the theoretical real right now).

Thanks!

 

[f95toli]

There must be plenty of devices that are reciprocal (in the sense that they hava a symmetric S-matrix) but not linear.
Any device where S21 /S12 depends on the power would be an obvious example; and that means just about any real passive device.
A superconducting thin film band pass filter would be one rather extreme example, the kinetic inductance etc in these devices makes them slightly non-linear with a S21/S12 that depends on the power (which is, as it happens, what I was measuring today).

 

[Twinpeaksr]

I see my over look, my original comment was for isotropic homogeneous material devices.

Thanks for all the info!