The voltage gain of a common emitter amplifier with emitter resistance (R_E) is a crucial parameter in electronic circuit design. This gain indicates how much the amplifier will amplify an input signal. Let us solve this by understanding the circuit dynamics:

The voltage gain (A_v) for a common emitter amplifier with an emitter resistance can be derived from the small-signal model:

The small signal voltage gain formula is given by:

A_v = \frac{V_{out}}{V_{in}} = -\frac{\beta \cdot R_C}{R_E + r_e}

where:

• \beta is the current gain of the transistor.
• R_C is the collector resistance.
• R_E is the emitter resistance.
• r_e is the intrinsic emitter resistance, approximately given by r_e \approx \frac{26mV}{I_E}, where I_E is the emitter current.

In practical circuits with emitter degeneration (a resistor R_E is added to stabilize the gain), the effective emitter resistance seen at the base is (1 + \beta)R_E. Therefore, the correct formula for the small-signal voltage gain incorporating R_E becomes:

A_v = \frac{- \beta R_C}{r_e + (1 + \beta)R_E}

Thus, among the provided options, the correct formulation for the voltage gain of a common emitter amplifier with R_E is:

\frac{- \beta R_C}{r_e + (1 + \beta)R_E}

This expression accounts for the degenerative feedback brought by R_E, which stabilizes the gain but reduces its magnitude.