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I am reading Electronics and Devices by Boylestad and it says:

$$ \begin{align} I_E &= I_C + I_B &(1)\\ I_C &= \alpha I_E + I_{CO} &(2) \\ I_E &= \alpha I_E + I_{CO} + I_B &(3) \end{align} $$ Is equation 3 valid for all regions of operation?

because if equation 3 is valid near cutoff region, if \$I_B\$ is decreased to zero, either \$\alpha\$ has to increase or \$I_{CO}\$ should increase. Also is there any relationship between \$I_B\$, \$\alpha\$ and \$I_{CO}\$? How to connect the concept that, if there is no base current, no collector current flows in the transistor?

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    \$\begingroup\$ Hi ibsen, I fixed your equations. This site supports mathjax. You can have a look at the link or just see the edited answer to understand the basic concepts. \$\endgroup\$ – Vladimir Cravero Aug 9 '14 at 13:58
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Equation 3 is totally derivable from eq2 and eq1 so the question really is should equation 2 be valid in all regions of operation.

In detail, equation 2 is saying collector current comprises the (large normally) fraction of emitter current that flows through the collector + leakage current through the collector-base reversed junction.

And, as far as I remember this is valid under in all regions of operation that I can think of.

This page might be useful.

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"How to connect the concept that, if there is no base current, no collector current flows in the transistor?"

The well-known equation Ie=Ic+Ib must not be misunderstood. It simply says that the current Ie (caused by the voltage Vbe) is split into two parts: A very small current Ib and a much larger current Ic. With other words: You must not ask "What happens with Ic when Ib=0 ?"because such a question assumes that Ib would be the cause of Ic (which is not the case).

In general, such an equation gives the mathematical relation beween some quantities (in this case: currents) only, but it doesn`t tell you anything about cause and effect.

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