The small-signal output resistance of a common-emitter bipolar junction transistor (BJT) amplifier can be calculated using the following steps:
Find the DC operating point: Determine the DC biasing conditions of the amplifier, including the collector current (Ic), collector voltage (Vc), and emitter current (Ie). This involves setting up the DC biasing circuit with appropriate resistors and voltage sources.
Calculate the transconductance (gm): The transconductance (gm) of the BJT is the change in collector current (ΞIc) for a small change in base-emitter voltage (ΞVbe) while keeping the collector-emitter voltage constant. It can be approximated as gm = Ic / VT, where VT is the thermal voltage (around 25 mV at room temperature).
Derive the small-signal equivalent circuit: Replace the BJT with its small-signal model. This involves replacing the BJT with its hybrid-Ο (pi) model. The small-signal output resistance (ro) is one of the components in this model.
Calculate the small-signal output resistance (ro): The small-signal output resistance (ro) is the inverse of the slope of the output characteristic curve of the BJT. It is given by ro = VA / Ic, where VA is the Early voltage of the transistor and Ic is the DC collector current.
The formula for the small-signal output resistance (ro) can also be written as:
ro = (VA + Vce) / Ic,
where Vce is the collector-emitter voltage when considering the small-signal output resistance. Vce is often approximated as Vc (the DC collector voltage) in many cases.
Keep in mind that the above calculation assumes that the output impedance of the source driving the amplifier is much smaller than the calculated small-signal output resistance of the amplifier itself. This ensures that the source impedance doesn't significantly affect the calculations.
Remember that these calculations are based on small-signal analysis, which assumes that the signal variations are small enough not to affect the DC operating point significantly.