hfe of transistor in active region

I've been studying NPN's and I'm trying to calculate the voltages and currents.

Consider the following transistor circuit

simulate this circuit – Schematic created using CircuitLab

with Vb, Vc and Ve the voltage at Q1's base, collecter and emitter respectively. Assume the sources and resistors are chosen in a way that Q1 is operating in forward active region, so $$V_C > V_B > V_E$$ In active region the following relation applies $$I_C = \beta I_B$$ and $$I_E = I_B + I_C$$ With network analysis we can then find $$I_C = \frac{V_{EQ}-V_{BE}}{\frac{R_B}{\beta} + R_E\frac{\beta +1}{\beta }}$$ From this point I'm not sure how to proceed; When looking up Beta (or Hfe) in the 2N2222 datasheet

or by graph from this datasheet

we see that Hfe depends on the both Ic and Vce, but in order to calculate Ic we need to know Hfe. So how do I get an exact answer for Ic (and thereby solving Ib, Ie, Vb, Vc and Ve)?

• Have a look here. It is the same datasheet from onsemi. Does it help you? – Vladimir Cravero Apr 4 '16 at 20:57
• You usually take a typical value of 100. Make an iteration with that value for β and see if all the other values fall into place and taking the value 100 was close to the operating point of the transistor. – Claudio Avi Chami Apr 4 '16 at 21:06
• @VladimirCravero Yes indeed this was what I was looking for, the Ic-Hfe graph helps! However I'm still looking for an answer for my first question; Ic and Hfe are dependent to eachother, how do I solve Ic? – user1712263 Apr 4 '16 at 23:56
• @ClaudioAviChami I'm not sure I get what you mean, will it solve Ic (thus solving the rest of the currents and voltages). Could you please elaborate it with an example, that could really help me out. – user1712263 Apr 4 '16 at 23:57
• Since you have not provided values for the resistors, I cannot give you an example. But if you take hFE = 100 and solve, you will obtain Ic and Vce. Then you can go to the graph and observe if for that Ic and Vce, hFE = 100 is a valid assumption. hFE has a big variance, that is the reason why polarization of a transistor is done in such a way as to mitigate the influence of this big hFE variance. – Claudio Avi Chami Apr 5 '16 at 3:44