Question

What are the $\alpha$ and $\beta$ parameters for a transistor? Obtain a relation between them.

Answer 1

First of all, we will individually find the parameters $\alpha$ and $\beta$ . We know that in a transistor, the emitter current is equal to the sum of base current and the collector current. In this equation we will try to bring $\alpha$ and $\beta$ . We will manipulate accordingly and obtain the relationship.

Complete step by step solution:
In the given question, we are supplied the following data:
There is a transistor given, whose two of the parameters are alpha and beta.
We are asked to define those two parameters and obtain a relation between them.

To begin with, let us discuss a bit about the two parameters alpha and beta.
A transistor 's alpha is the current gain defined as the ratio of change in collector current to change in emitter current in the common base configuration, while beta is the current gain in the ${\text{CE}}$ configuration. It is defined as a change in the current of the collector to the current of the base.The maximum alpha value is one, while the beta value varies between ${\text{50}}$ and ${\text{250}}$ . Therefore, owing to high gain, the transistor is used as an amplifier in only ${\text{CE}}$ setup.

The parameter alpha can be written as:
$\alpha = \dfrac{{{I_{\text{C}}}}}{{{I_{\text{E}}}}}$ …… (1)
Where,
$\alpha$ indicates the current gain in the common base configuration.
${I_{\text{C}}}$ indicates the current in the collector.
${I_{\text{E}}}$ indicates the current in the emitter.

Again, the parameter beta can be written as:
$\beta = \dfrac{{{I_{\text{C}}}}}{{{I_{\text{B}}}}}$ …… (2)
Where,
$\beta$ indicates the current gain in the common emitter configuration.
${I_{\text{C}}}$ indicates the current in the collector.
${I_{\text{B}}}$ indicates the current in the base.

Again, we know that in a transistor, the emitter current is equal to the sum of base current and the collector current. So, we can write mathematically as,
${I_{\text{E}}} = {I_{\text{B}}} + {I_{\text{C}}}$ …… (3)

Now, we will divide the equation by ${I_{\text{C}}}$ on both the sides and we get:
$\dfrac{{{I_{\text{E}}}}}{{{I_{\text{C}}}}} = \dfrac{{{I_{\text{B}}} + {I_{\text{C}}}}}{{{I_{\text{C}}}}} \\\ \Rightarrow \dfrac{{{I_{\text{E}}}}}{{{I_{\text{C}}}}} = \dfrac{{{I_{\text{B}}}}}{{{I_{\text{C}}}}} + 1 \\\ \Rightarrow \dfrac{1}{{\left( {\dfrac{{{I_{\text{C}}}}}{{{I_{\text{E}}}}}} \right)}} = \dfrac{1}{{\left( {\dfrac{{{I_{\text{C}}}}}{{{I_{\text{B}}}}}} \right)}} + 1 \\\ \Rightarrow \dfrac{1}{\alpha } = \dfrac{1}{\beta } + 1 \\\ \Rightarrow \dfrac{1}{\alpha } = \dfrac{{\beta + 1}}{\beta } \\\ \therefore \alpha = \dfrac{\beta }{{\beta + 1}}$

Hence, the relation between $\alpha$ and $\beta$ is $\alpha = \dfrac{\beta }{{\beta + 1}}$.

Note: It is important to remember that the maximum alpha value is one, while the beta value varies between ${\text{50}}$ and ${\text{250}}$ . Therefore, owing to high gain, the transistor is used as an amplifier in only ${\text{CE}}$ setup. Transistors are the semiconductor devices which are used to monitor the electric current flow in which a small volume of current between the collector and emitter regulates a greater current in the base leads.