I ran about 90 feet of direct burial 14/2 wire from an existing breaker box from a 20 amp breaker to my new breaker box also 20 amp. I had everything done but now it is tripping the breaker in the new shed and the original breaker box. It was told it was from a voltage drop because the wire is not long enough. Is there anyway to correct this without running new wire and digging everything up? I also have the option to run the original line out from a 220 breaker but don't know if that will fix it or if I need to add some form of voltage converter to the breaker so I run the additional plugs with out burning them up or causing damage. Any suggestions would be appreciated.
You are using the wrong wire
I ran about 90 feet of direct burial 14/2 wire from an existing breaker box from a 20 amp breaker to my new breaker box also 20 amp.
For 20 Amp, you normally need 12/2 wire, not 14/2. 14/2 is for 15 Amp circuits. Do you need bigger wire (e.g., 10/2 for 120 V or 10/3 for 240 V)? I'm not sure, that depends on how much of an issue there really is with the voltage drop, and also depending on what equipment you are actually using (or planning to use in the future) at the end of the cable.
14 is too small for 20A, and over that distance I think you need a gauge larger. Check the codes first, but I think you want 10-gauge.
As already stated, motors have a big inductive load at startup. That large current will cause some sag anyway, and that is exacerbated by a small gauge. Voltage drop is current times resistance, and smaller wire has higher internal resistance.