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

Hint: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.