Is the following equation wrong in this BJT DC bias analysis?
It's been enough days, I think.

Two ways to think about the problem. Let's start with the way the problem was presented at the outset:

$$\begin{align*}
I_\text{C}&amp;=\beta\: I_\text{B}+\left(\beta+1\right)\:I_\text{CO}\\\\
V_\text{BB}&amp;=I_\text{B}\: R_\text{BB}+V_{\text{BE}_\text{ON}}+\left(I_\text{B}+I_\text{C}\right)\:R_\text{E}
\end{align*}$$

Take note that this is two equations and two unknowns: \$I_\text{B}\$ and \$I_\text{C}\$. So it's already solvable as a simultaneous pair of equations.

Yet you decided to introduce this KVL on your own:

$$I_\text{B}=\frac{V_\text{BB}-V_{\text{BE}_\text{ON}}}{R_\text{BB}+\left(\beta+1\right)\:R_\text{E}}$$

That was completely unnecessary. You already had two equations and two unknowns and this "over-specified" the problem and did so, incorrectly.

Do you notice that your introduced equation for \$I_\text{B}\$ doesn't include the term, \$I_\text{CO}\$, at all? The problem is in the fact the equation you introduced for \$I_\text{B}\$ makes the assumption that \$I_\text{E}=\left(\beta+1\right)\:I_\text{B}\$. However, the reality is:

$$\begin{align*}
I_\text{E}&amp;=I_\text{B}+I_\text{C}\\\\
&amp;=I_\text{B}+\beta\: I_\text{B}+\left(\beta+1\right)\:I_\text{CO}\\\\
&amp;=\left(\beta+1\right)\:I_\text{B}+\left(\beta+1\right)\:I_\text{CO}\\\\
&amp;=\left(\beta+1\right)\left(I_\text{B}+I_\text{CO}\right)
\end{align*}$$

That's where you made your mistake.

So, you made two errors:


You didn't just do a simultaneous solution of the two equations, when you should have done so.
You introduced a new KVL equation that was incorrect, because it did not account for \$I_\text{CO}\$. (It would be correct if \$I_\text{CO}=0\$.)


The remedy is to either solve the two equations simultaneously, as you probably should have done; or else you should apply the corrected equation for \$I_\text{E}\$ when solving your added KVL equation:

$$\begin{align*}
V_\text{BB} - I_\text{B}\: R_\text{BB} - V_{\text{BE}_\text{ON}} - I_\text{E}\:R_\text{E}&amp;=0\:\text{V}\\\\
V_\text{BB}&amp;= I_\text{B}\: R_\text{BB} + V_{\text{BE}_\text{ON}}  \left(\beta+1\right)\left(I_\text{B}+I_\text{CO}\right)\:R_\text{E}\\\\
V_\text{BB}&amp;= I_\text{B}\: R_\text{BB} + V_{\text{BE}_\text{ON}} + \left(\beta+1\right)\:I_\text{B}\:R_\text{E}+\left(\beta+1\right)\:I_\text{CO}\:R_\text{E}\\\\
I_\text{B}\: \left(R_\text{BB} + \left(\beta+1\right)\:R_\text{E}\right)&amp;=V_\text{BB}- V_{\text{BE}_\text{ON}}-\left(\beta+1\right)\:I_\text{CO}\:R_\text{E} \\\\
I_\text{B}&amp;=\frac{V_\text{BB}- V_{\text{BE}_\text{ON}}-\left(\beta+1\right)\:I_\text{CO}\:R_\text{E}}{R_\text{BB} + \left(\beta+1\right)\:R_\text{E}}
\end{align*}$$

Do you note the difference, now? You cobbled up an equation for \$I_\text{B}\$ out of thin air, without taking into account the newly introduced term, \$I_\text{CO}\$. That works fine if you discount \$I_\text{CO}\approx 0\$. But if you have to include it, then you need to include it. And you didn't.

If you use this corrected value for \$I_\text{B}\$, perhaps things will work out okay.

$$\begin{align*}
I_\text{C}&amp;=\beta\: I_\text{B}+\left(\beta+1\right)\:I_\text{CO}\\\\
&amp;=\beta\: \frac{V_\text{BB}- V_{\text{BE}_\text{ON}}-\left(\beta+1\right)\:I_\text{CO}\:R_\text{E}}{R_\text{BB} + \left(\beta+1\right)\:R_\text{E}}+\left(\beta+1\right)\:I_\text{CO}\\\\
&amp;=\frac{\beta \left(V_\text{BB}- V_{\text{BE}_\text{ON}}-\left(\beta+1\right)\:I_\text{CO}\:R_\text{E}\right)+\left(\beta+1\right)\:I_\text{CO}\left(R_\text{BB} + \left(\beta+1\right)\:R_\text{E}\right)}{R_\text{BB} + \left(\beta+1\right)\:R_\text{E}}\\\\
&amp;=\beta\frac{V_\text{BB}-V_{\text{BE}_\text{ON}}}{R_\text{BB} + \left(\beta+1\right)\:R_\text{E}}+\left(\beta+1\right)\frac{\:R_\text{BB}+\left(\beta+1\right)\:R_\text{E}-\beta\:R_\text{E}}{R_\text{BB} + \left(\beta+1\right)\:R_\text{E}}\:I_\text{CO}\\\\
&amp;=\beta\frac{V_\text{BB}-V_{\text{BE}_\text{ON}}}{R_\text{BB} + \left(\beta+1\right)\:R_\text{E}}+\left(\beta+1\right)\frac{\:R_\text{BB}+R_\text{E}}{R_\text{BB} + \left(\beta+1\right)\:R_\text{E}}\:I_\text{CO}
\end{align*}$$

That's it. You have the equation, done correctly.

Of course, you could also have just solved it as a pair of simultaneous equations in the first place.