I'm trying to figure out why my professor starts off the way he does when trying to find voltage gain. How he starts off is shown in the last picture. Any explanation would be appreciated.
To calculate voltage gain, you need to compute the value of Vo/Vs. Your circuit is divided into 2 main parts, one on the left (which has Vs) and one on the right (which has Vo). Looking at the right hand side, the value of every component is known except for Vbe which determines the magnitude of the current source. Vbe is found on the left side by using the voltage divider formula which is the first equation. The second equation gives the value of Vo from the left side of the circuit (This equation perhaps should have come first). Then it is just a matter of substituting the value of Vbe from the first equation into the second equation which results in the third equation. The fourth equation is simply the third equation with both sides divided by Vs so that the gain, Vo/Vs, is obtained as a single expression.