Derive the expression for the voltage gain of a transistor amplifier in CE configuration in terms of the load resistance RL, current gain βa, and input resistance. Explain why input and output voltages are in opposite phase.

Applying Kirchhoff's law to the output loop, we get Vcc = VCE + ICRC.. (1)
Same way the input loop gives VBB = VBE + IBRB.. (2)
When Vi is no+2no VBE + Vi = VBE + IBIB + ΔIB(RB + ViVi = ΔIB(RB + ri) = rΔIB
The change in IB causes a change in IC. We define a parameter Bac Bac = ΔICΔIB = iClB (current gain A)
The change in IC due to a change in IB can cause a change in VCE and the voltage drop across the resistor Ri because VCC is fixed. These changes can be given by eqn 1 as
ΔVCC = ΔVCE + RCΔIC = 0
ΔVCE = RCΔIC
The change in VCE is the output voltage Vo
V0 = ΔVCE = −BacRLΔI
Voltage gain AV = V0Vi = ΔCErΔIB
Av = −βAC × RLr
−ve sign mass output voltage is opposite with phase with the input voltage