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So I'm currently studying Electronic Devices and Circuit Theory by Boylestad and Nashelsky (11th ed.) and I'm confused as to how the relation between \$I_{CEO}\$ and \$I_{CBO}\$ is derived. So the derivation in the book starts with the equation $$I_C = \alpha I_E + I_{CBO}\ \tag1$$ But since \$I_E = I_C + I_B\$ , $$I_C = \alpha (I_C + I_B) + I_{CBO}\ \tag2$$ Rearranging yields $$I_C = \frac{\alpha I_B}{1-\alpha} + \frac{I_{CBO}}{1-\alpha}\ \tag3$$ Then it goes on to say that \$I_C\$ when \$I_B = 0\$ is assigned the notation \$I_{CEO} = \frac{I_{CBO}}{1-\alpha} \approx \beta I_{CBO}\$ which is consistent with other sources.

The weird thing I noticed is that letting \$\alpha I_E = I_C\$ in eq. 1 produces $$I_C = I_C + I_{CBO}\ \tag4$$ and letting \$ \frac{\alpha I_B}{1-\alpha}=\beta I_B = I_C\$ in eq. 3 produces $$I_C = I_C + \frac{I_{CBO}}{1-\alpha}\ \tag5$$ which are surprisingly not the same in addition to being paradoxy/fallacious... So my question is what went wrong?

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The original equations are: $$I_C = \alpha I_E + I_{CBO} \tag{a}$$ $$I_C = \beta I_B + (\beta +1)I_{CBO} \tag{b}$$

Before equation 4, you mentioned that \$\alpha I_E = I_C\$. But this is valid only when \$I_{CBO} = 0\$ and that is what equation 4 says.

In the same way, \$I_C = \beta I_B\$ is valid only if \$I_{CBO} = 0\$ and that is what equation 5 says.

Both equation 4 and 5 are valid only under the approximation that \$I_{CBO} = 0\$.

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