To calculate the small-signal input resistance of a common-emitter BJT amplifier, you can follow these steps:
Replace the BJT with its small-signal equivalent circuit: In the small-signal model, the BJT is represented by its small-signal parameters, such as the transconductance (gm) and the output conductance (go). The input resistance is determined by the transistors' base-emitter junction, so you can use the small-signal equivalent circuit for that purpose.
Disconnect the signal source and apply a test signal: To determine the input resistance, you need to apply a small-signal input voltage to the base of the transistor. Disconnect the DC biasing components and connect a small AC voltage source (voltage signal source) to the base of the transistor.
Find the small-signal input current: Calculate the small-signal input current (ib) flowing into the base terminal of the transistor due to the test signal.
Calculate the input resistance: Finally, divide the small-signal input voltage (vbe) by the small-signal input current (ib) to obtain the small-signal input resistance (rin).
Mathematically, the small-signal input resistance (rin) can be calculated as follows:
rin = Δvbe / Δib
Where:
Δvbe is the change in the base-emitter voltage due to the small-signal input voltage.
Δib is the corresponding change in the base current due to the small-signal input current.
Typically, you may encounter cases where the small-signal model of a BJT is given with gm and go, and you can use these parameters to find the input resistance. The input resistance (rin) is related to the transconductance (gm) as follows:
rin ≈ 1 / gm
Keep in mind that these calculations assume that the transistor is operating in its active region and that the small-signal model is valid for the given frequency range of interest. For accurate results, it's essential to use the small-signal parameters provided in the datasheet or derived from the device characteristics at the operating point of the amplifier.