Somebody knows as to calculate the base current so that the tansistor operates in cut and saturation? and since I make to caculte the resistance in the base so that it operates in cut and saturation? I am using a transistor npn 2n2222

 

The datasheet shows you Collector-Emitter Saturation Voltage at two examples of collector current. The base current is one-tenth the collector current to make the transistor saturate very well. The max base-emitter voltage is also shown.

A transistor is cutoff when it doesn't have base current.

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Uncle $crooge

 

for 2n2222 which is the gain or the HF, in the table is values 35-50-75-100-30-50 minimum, and maximum 300 I do not understand as of all to use, I thought that in cut and saturation hfe was fixed.

 

The datasheet shows that the hFE is between 30 to 300 depending on the particular transistor (they are not all exactly the same), its temperature and on its amount of collector current. Your circuit needs a transistor with enough hFE for your design so the transistor can conduct between saturation and cutoff. If your design provides more base current when needed than the hFE number then the transistor can saturate very well.
Usually a voltage divider and an emitter resistor are used to properly bias the base of a transistor that has an hFE between 30 to 300, and it can operate well at different temperatures.

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Uncle $crooge

 

What beta got to do with saturation nothing except it must be there to saturate the transistor and that depends on ic more base current for higher ic once it saturates increasing Ic will take out of it and viceversa. if you look at the load line of the 2n2222 you can readily see that. once saturate all you have is 2 diodes in parallel and current can and will flow either way. to saturate that sucker figure on a forced beta of 10 an 2n3050 power transistor you better figure for a forced beta of 5 for hi amps. look at load line of the transistor it shows easely