Q.Define the current gain α_DC and β_DC for a transistor. Obtain the relation between them.

A transistor amplifies an alternating signal by superimposing a small sinusoidal AC input voltage on its steady DC bias in the common emitter configuration: the resulting small variations in base current $I_B$ produce proportionally much larger variations in collector current $I_C$, and hence in the output voltage across the collector load resistor $R_L$, with a coupling capacitor blocking the DC component so only the amplified AC variation reaches the output. From the output-loop equation $V_{CC}=V_{CE}+I_CR_L$ (with $V_{CC}$ fixed), any change in $I_C$ must be matched by an equal and opposite change in $V_{CE}$: $\Delta V_{CE}=-R_L\Delta I_C=-\beta_{AC}R_L\Delta I_B$, where the AC current gain $\beta_{AC}=\Delta I_C/\Delta I_B$ is numerically close to $\beta_{DC}$ for normal operating conditions. …