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I have seen a lot of calculations where they take GAIN as 10 or 20 instead of the original value in the datasheet.For example 2n2222 transistor

VCC = 5 ,Led(Vf = 3v).
For Ic = 20mA,Rc = [VCC-VF(LED)-VCE(SAT)]/Ic = [5-3-0.25]/20mA = 87 ohms.
Ib=Ic/Gain = 20mA/110 = 0.18mA
Rb = [Vcc-Vbe]/Ib = [5-0.83]/0.18mA = 23k ohms.

If my calculations are right,i took gain as 110 as per the datasheet.but some people take gain as 10 or 20 ... i want to know why is that ?????

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If you are using a transistor as a switch, then an easy way to insure that it switches between cutoff and saturation is to use a much lower Beta than specified in the datasheet.  In your problem, you want to use your circuit to turn on an LED.  This means you need for your 2N2222 transistor to go into saturation.  In saturation, your Vce voltage needs to be very small, approaching zero. 

In your calculations you used 0.25 volts for Vce. 

Rc = [VCC-VF(LED)-VCE(SAT)]/Ic = [5-3-0.25]/20mA = 87 ohms                                

That’s a reasonable number to use but how can you insure that it will be 0.25 volts or less?  One possible answer is, use a lower value for Beta to insure that you get your transistor into saturation.

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