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This seems to be hardly any difference. But it provides us with a quick look at the modifications, too. Given the relatively minor differences, it seems like the earlier, earlier, and less complex equation wasn't too bad, after all.

This seems to be hardly any difference. But it provides us with a quick look at the modifications, too. Given the relatively minor differences, it seems like the earlier, easier, and less complex equation wasn't too bad, after all.

1. How good are the above reductions (simplifications) and should we consider doing a somewhat more detailed analysis to figure out how much error there is in making those reductions? The answer to this lies in the assumptions themselves. The active mode assumption we simply have to accept for an amplifier that operates well. So there was no risk from that. The \$V_{BE}\$ assumption varies only by about \$60\:\textrm{mV}\$ per decade change in collector current. So that isn't a biggy, either. So most of the answer here turns on \$\beta\$. If we are using "larger" values of \$\beta\$ then our reductionist approach is pretty good. But for "smaller" values, not so good. So the sum-up here is that in the case where \$\beta\$ is larger, then \$V_C\$ is very stable and also our reduction is sound; and in the case where \$\beta\$ is smaller, then \$V_C\$ is less stable and our reduction is no longer as sound. Keeping in mind that we may still ignore the Early Effect, and temperature, and variation of \$V_{BE}\$ with collector current, it may still be worthwhile doing some extra math (full KCL, with solutions) to see how that changes the situation (and if so, by how much.)
2. Another is to ask ourselves, "What is the real question we are asking?" Is it, "How much does \$V_C\$ vary with a change of '1' in \$\beta\$?" Or is it instead, "How much does \$V_C\$ vary with a change of '1%' in \$\beta\$?" These are actually different questions, despite seeming similar. But what do we really want to know? Chances are, it's more about the percent-variation. We probably are not as interested in comparing what happens to \$V_C\$ when \$\beta\$ from 300 to 301 vs from 90 to 91, as much as we are interested in comparing what happens when there is a 10% change, regardless of starting value. And we actually haven't even expressed how to address either question, yet.

This seems to be hardly any difference. But it provides us with a quick look at the modifications, too.

$$\begin{align*} \frac{\frac{\textrm{d} V_C}{V_C}}{\frac{\textrm{d} \beta}{\beta}}&=\frac{\beta}{V_C}\cdot\frac{\textrm{d} V_C}{\textrm{d} \beta}\\\\ &=-\frac{R_B \: R_C\:\left(V_{CC}-V_{BE}\right)}{\left(\frac{\beta+1}{\beta}\left(R_C+R_E\right)+\frac{R_B}{\beta}\right)\left(\frac{\beta+1}{\beta}\left(V_{CC} R_E+V_{BE} R_C\right)+V_{CC}\frac{R_B}{\beta}\right)} \end{align*}$$

This answers the following question: "What percent change in \$V_C\$ would occur for a given percent change in \$\beta\$?"

1. How good are the above reductions (simplifications) and should we consider doing a somewhat more detailed analysis to figure out how much error there is in making those reductions? The answer to this lies in the assumptions themselves. The active mode assumption we simply have to accept for an amplifier that operates well. So there was no risk from that. The \$V_{BE}\$ assumption varies only by about \$60\:\textrm{mV}\$ per decade change in collector current. So that isn't a biggy, either. So most of the answer here turns on \$\beta\$. If we are using "larger" values of \$\beta\$ then our reductionist approach is pretty good. But for "smaller" values, not so good. So the sum-up here is that in the case where \$\beta\$ is larger, then \$V_C\$ is very stable and also our reduction is sound; and in the case where \$\beta\$ is smaller, then \$V_C\$ is less stable and our reduction is no longer as sound. Keeping in mind that we may still ignore the Early Effect, and temperature, and variation of \$V_{BE}\$ with collector current, it may still be worthwhile doing some extra math (full KCL, with solutions) to see how that changes the situation (and if so, by how much.)
2. Another is to ask ourselves, "What is the real question we are asking?" Is it, "How much does \$V_C\$ vary with a change of '1' in \$\beta\$?" Or is it instead, "How much does \$V_C\$ vary with a change of '1%' in \$\beta\$?" These are actually different questions, despite seeming similar. But what do we really want to know? Chances are, it's more about the percent-variation. We probably are not as interested in comparing what happens to \$V_C\$ when \$\beta\$ changes from 300 to 301 vs changes from 90 to 91 (by 1 in each case), as much as we are interested in comparing what happens when there is a 10% change, regardless of starting value. And we actually haven't even expressed how to address either question, yet.

This seems to be hardly any difference. But it provides us with a quick look at the modifications, too. Given the relatively minor differences, it seems like the earlier, earlier, and less complex equation wasn't too bad, after all.

$$\begin{align*} \frac{\left[\frac{\textrm{d} V_C}{V_C}\right]}{\left[\frac{\textrm{d} \beta}{\beta}\right]}&=\frac{\beta}{V_C}\cdot\frac{\textrm{d} V_C}{\textrm{d} \beta}\\\\ &=-\frac{R_B \: R_C\:\left(V_{CC}-V_{BE}\right)}{\left(\frac{\beta+1}{\beta}\left(R_C+R_E\right)+\frac{R_B}{\beta}\right)\left(\frac{\beta+1}{\beta}\left(V_{CC} R_E+V_{BE} R_C\right)+V_{CC}\frac{R_B}{\beta}\right)} \end{align*}$$

Note that there is factor inserted here that scales the differential equation that would apply for a "change by 1" to make this a unitless comparison of % changes, instead. This form of equation answers the following question: "What percent change in \$V_C\$ would occur for a given percent change in \$\beta\$?"

And with the values you provided with the schematic, I find that the answer is approximately -0.32. Which means that you'd get about a 10% change in \$V_C\$ for a 31% change in \$\beta\$. Since one BJT might vary that much in \$\beta\$, from one to another, this suggests that you probably should not expect to see more than about a 10% change in \$V_C\$ when swapping BJTs into the socket from a bag of them (discounting temperature effects, the Early Effect, and variations in \$V_{BE}\$.)

And with the values you provided with the schematic, I find that the answer is approximately -0.32. Which means that you'd get about a 10% downward change in \$V_C\$ for a 31% upward change in \$\beta\$. Since one BJT might vary that much in \$\beta\$, from one to another, this suggests that you probably should not expect to see more than about a 10% change in \$V_C\$ when swapping BJTs into the socket from a bag of them (discounting temperature effects, the Early Effect, and variations in \$V_{BE}\$.)