I have a 7937 hooked up to a CAT3550 and we have PoE runing to that device but it keeps on power cycling. I tried to extend the power on the 3550 by issuing the following command SWITCH#power inline delay shutdown 20 initial 300. All this does is it powers up the 7937 for aprox. 20sec then it goes back to shutdown and tries to power up again.
The reason the 7937 phone does not work with the 3550 is because the 3550 only supports Cisco Pre-Standard Power whereas the 7937 phone only supports IEEE 802.3af power
PoE power snapshot
7937 Data Sheet
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