Question 6.SP.4: Utilize the r-parameter equivalent circuit of Fig. 6-10(b) t......
Utilize the r-parameter equivalent circuit of Fig. 6-10(b) to find the voltage gain ratio A_v = v_L/v_i for the CE amplifier circuit of Fig. 3-10.
The small-signal equivalent circuit for the amplifier is drawn in Fig. 6-11. After finding the Thévenin equivalent for the network to the left of terminals B, E, we may write
v_{b e} = {\frac{R_{B}}{R_{B} + R_{i}}}\,v_{i} + {\frac{R_{B}R_{i}}{R_{B} + R_{i}}}\,i_{b} (1)
Ohm’s law at the output requires that
v_{ce} = v_{L} = {\frac{R_{C}R_{L}}{R_{C} + R_{L}}}\,i_{c} (2)
Applying KVL around the B, E, mesh and around the C, E mesh while noting that i_e = i_c + i_b yields, respectively,
v_{b e} = -r_{b}i_{b} – r_{e}i_{e} = -(r_{b} + r_{e})i_{b} – r_{e}i_{c} (3)
and v_{c e} = -r_{e}i_{e} + r_{m}i_{b} – (1 – \alpha)r_{c}i_{c} = -(r_{e} – r_{m})i_{b} – [(1 – \alpha)r_{c} + r_{e}]i_{c} (4)
Equating (1) to (3) and (2) to (4) allows formulation of the system of linear equations
from which, by Cramer’s rule, i_{c} = \Delta_2/\Delta, where
Then A_{v} = {\frac{v_{L}}{v_{i}}} = {\frac{(R_{L}||R_{C})i_{C}}{v_{i}}} = {\frac{R_{L}R_{C}}{R_{L} + R_{C}}}{\frac{r_{e} – r_{m}}{\Delta}}
