Minimum gain to sustain oscillations and emitter resistance if R_C = 2.5 kOhm

Minimum Gain to Sustain Oscillations:

In an oscillator circuit, the minimum gain required to sustain oscillations is determined by the feedback factor, which is the ratio of the output signal fed back to the input. For a transistor-based oscillator, the feedback factor is typically determined by the emitter resistor, (R_E).

The minimum gain, often denoted as (A_{min}), required for sustained oscillations is given by:

(A_{min} = \frac{1}{1 - \beta})

where (\beta) is the transistor's current gain. In this case, (\beta) is given as 50.

Substituting (\beta = 50), we get:

(A_{min} = \frac{1}{1 - 0.5} = 2)

Therefore, the minimum gain required to sustain oscillations is 2.

Emitter Resistance:

The emitter resistance, (R_E), plays a crucial role in determining the feedback factor and the stability of the oscillator circuit. The emitter resistance affects the transistor's bias point and its gain.

The value of (R_E) is typically chosen to be small enough to provide sufficient feedback for sustained oscillations but large enough to ensure that the transistor operates in the active region.

In this case, the emitter resistance is given as (R_C = 2.5) kΩ. Since (R_E) is not explicitly provided, we cannot determine its exact value. However, we can infer that (R_E) is less than (R_C) to ensure sufficient feedback.

Typically, in transistor oscillator circuits, the emitter resistance is chosen to be a fraction of the collector resistance, often ranging from 1/5 to 1/10 of (R_C). This ensures that the transistor is biased in the active region and provides enough feedback for sustained oscillations.

In summary, the minimum gain required to sustain oscillations is 2, and the emitter resistance, (R_E), should be chosen to be less than (R_C) to ensure sufficient feedback and proper biasing of the transistor.