Question

In a common emitter amplifier, the load resistance of the output circuit is 1000 times the resistance of the input circuit. if $\alpha = 0.98$, the voltage gain is
A.$9800$
B.$49000$
C.$9800$
D.\[4900\]

Answer 1

Hint-
Given,
${R_L} = 1000\,{R_i}$
The current gain of a transistor is denoted by the symbol $\beta $.
It is the ratio of voltage gain to resistance gain
That is,
\[\beta = \dfrac{{{V_g}}}{{{R_g}}}\]
The relationship between $\alpha $ and $\beta $ is given as
$\beta = \dfrac{\alpha }{{1 - \alpha }}$
Resistance gain can be calculated as
${R_g} = \dfrac{{{R_L}}}{{{R_i}}}$

Step by step solution:
Let ${R_L}$ be the load resistance and ${R_i}$ be the input resistance
It is given that the load resistance is thousand times the resistance of the input circuit.
Therefore, we can write
${R_L} = 1000\,{R_i}$
The value of $\alpha $ is given as
$\alpha = 0.98$
For a transistor $\alpha $ is the ratio of collector current to the emitter current.
The current gain of a transistor is denoted by the symbol $\beta $.
It is the ratio of voltage gain to resistance gain
That is,
\[\beta = \dfrac{{{V_g}}}{{{R_g}}}\].........................(1)
The relationship between $\alpha $ and $\beta $ is given as
$\beta = \dfrac{\alpha }{{1 - \alpha }}$
From this we can find the value of $\beta $ as
$
  \beta = \dfrac{{0.98}}{{1 - 0.98}} \\
   = 49 \\
 $
Resistance gain can be calculated as
${R_g} = \dfrac{{{R_L}}}{{{R_i}}}$ …………………...(2)
Where ${R_L}$ is the load resistance and \[{R_i}\] is the input resistance
Substituting value of $\beta $ and ${R_g}$in equation(1),we get
\[
  {V_g} = \beta \times {R_g} \\
   = 49 \times \dfrac{{{R_L}}}{{{R_i}}} \\
   = 49 \times \dfrac{{1000{R_i}}}{{{R_i}}} \\
   = 49000 \\
 \]

Hence the answer is option B

Note: For a transistor there are 3 different configurations -common emitter configuration, common collector configuration and common base configuration. $\alpha $ is the current gain in common base configuration. Given as the ratio of collector current to emitter current. Beta is the current gain in a common emitter amplifier. In this configuration the output current is collector current and the input current is base current. So, it is the ratio of collector current to base current. Since we are asked to find the voltage gain. We calculated current gain as a ratio of voltage gain to resistance gain.