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Coaxial cable often has a characteristic impedance of 50  and is connected to 50  loads. If I have a 50  load but only have ribbon cable transmission line which has a characteristic impedance of 300 . What length of ribbon cable will give maximum power transfer into the 50  load.

When you say "ribbon cable"

do you mean this:

http://en.wikipedia.org/wiki/Twin-lead

which has a characteristic impedance of 300R or this:

http://www.ece.unm.edu/~jimp/650/slides/ribbon_cables.pdf

which has between 80R and 100R characteristic impedance.

http://arcarc.xmission.com/PDF_Electronics/Transmission%20Lines.pdf

http://en.wikipedia.org/wiki/Transmission_line

If you want to connect a 300R xmission line to a 50R ximission line - presumably terminated by or directly to - a 50R load, without losing most of the signal, then you need some form of impedance matching network.

http://en.wikipedia.org/wiki/Impedance_matching

http://electronicdesign.com/communications/back-basics-impedance-matching-part-1

http://electronicdesign.com/communications/back-basics-impedance-matching-part-2

http://electronicdesign.com/communications/back-basics-impedance-matching-part-3

You have said nothing about what frequency you wish to operate at.

For a fixed frequency this may be an LC network or a transformer (which may be implemented in a number of ways).

It is possible to implement a narrow band match (i.e. a match at a fixed frequency) using carefully chosen lengths of xmission lines. This is not a good technique for broadband matching.

For a broadband impedance match then you need a transformer or an active circuit i.e. a buffer amplifier of some sort.

The Maximum Power-transfer Theorem applies to any system in which a Generator is driving a Load. The requirement is that the Impedance presented by the Load must be exactly equal the the output impedance of the Source. When feeding the Power via a cable this output impedance is transformed by the feeder IF THERE IS A STANDING WAVE ON THAT FEEDER. ... and such is set up by failing to create a match between Load and characteristic impedance of the Feeder. However you treat your cable it will not accept max power from the driving source (a transmitter ?) and whatever power it does accept it can only deliver it if the match between cable and load is exact. The best possible result is obtained when the P.A. is correctly tuned but you cannot bodge into good health a system that starts life as a sloppy design. In addition the standing-wave could destroy your output valve OR would destroy your P.A. semiconductor.

However it is possible to transpose the cable impedance by inserting a "transformer" - a length of cable which has the necessary length (in terms of wavelength) and the necessary impedance.The catch lies in that last requirement. You might have to manufacture that bit o.f cable Ken Green