Question

A common-emitter amplifier gives an output of $3\,{\text{V}}$ for an input of $0.01\,{\text{V}}$. If $\beta$ of the transistor is $100$ and the input resistance is $1\,{\text{k}}\Omega$, then the collector resistance is:
A. $1\,{\text{k}}\Omega$
B. $3\,{\text{k}}\Omega$
C. $35\,{\text{k}}\Omega$
D. $30\,{\text{k}}\Omega$

Answer 1

We are asked to calculate the collector resistance. Here, you will need to recall the formula for amplification factor to get a relation between base and collector current. And use this relation in the formula of voltage gain to calculate the collector resistance.

Complete step by step answer:
Given, output voltage ${V_o} = 3\,{\text{V}}$.Input voltage, ${V_i} = 0.01\,{\text{V}}$.
Amplification factor of the transistor, $\beta = 100$.Input resistance, ${R_i} = 1\,{\text{k}}\Omega = 1000\Omega$.Let the collector resistance and collector current be ${R_C}$ and ${I_C}$ respectively. Let the base resistance and base current be ${R_B}$ and ${I_B}$ respectively.The amplification factor of a transistor is given by the formula,
$\beta = \dfrac{{{I_C}}}{{{I_B}}}$
Putting the value of $\beta$ we get,
$100 = \dfrac{{{I_C}}}{{{I_B}}}$
$\Rightarrow {I_C} = 100{I_B}$ (i)
The gain in voltage is written as,
${\text{Voltage}}\,{\text{gain}} = \dfrac{{{\text{Output}}\,{\text{voltage}}}}{{{\text{Input}}\,{\text{voltage}}}}$ (ii)
Putting the values of output and input voltage we get,
${\text{Voltage}}\,{\text{gain}} = \dfrac{{3\,{\text{V}}}}{{0.01\,{\text{V}}}}$
$\Rightarrow {\text{Voltage}}\,{\text{gain}} = 300$ (iii)
As the given amplifier is a common emitter amplifier so, base will act as input and collector will act as output.

The base voltage will be the input voltage which can be written as,
${\text{Input}}\,{\text{voltage}} = {I_B}{R_B}$
The collector voltage will be the output voltage which can be written as,
${\text{Output}}\,{\text{voltage}} = {I_C}{R_C}$
Putting these values of input and output voltage in equation (ii) we get,
${\text{Voltage}}\,{\text{gain}} = \dfrac{{{I_C}{R_C}}}{{{I_B}{R_B}}}$
Using equation (i) in the above equation we get,
${\text{Voltage}}\,{\text{gain}} = \dfrac{{100{I_B}{R_C}}}{{{I_B}{R_B}}}$
$\Rightarrow {\text{Voltage}}\,{\text{gain}} = \dfrac{{100{R_C}}}{{{R_B}}}$ (iv)
The input voltage will be the base voltage which means,
${R_i} = {R_B} = 1\,000\Omega$
Putting this value of ${R_B}$ in equation (iv) we get,
${\text{Voltage}}\,{\text{gain}} = \dfrac{{100{R_C}}}{{1000}}$
$\Rightarrow {\text{Voltage}}\,{\text{gain}} = \dfrac{{{R_C}}}{{10}}$ (v)
Equating equations (v) and (iii) we get,
$\dfrac{{{R_C}}}{{10}} = 300$
$\Rightarrow {R_C} = 3000\,\Omega$
$\therefore {R_C} = 3\,{\text{k}}\,\Omega$
Therefore, the collector resistance is $3\,{\text{k}}\,\Omega$.

Hence, the correct answer is option B.

Note: There are three terminals of a transistor, these are emitter, base and collector. And there can be three configurations of a transistor, these are common emitter configuration, common base configuration and common collector configuration. In the case of a common emitter transistor, an emitter is connected commonly for both input and output terminals.