For a common emitter configuration, if α and β have their usual meanings, the incorrect relationship between α and β is:

Engineering

Physics

Semiconductors

Transistors

Question

For a common emitter configuration, if α and β have their usual meanings, the incorrect relationship between α and β is:

$\frac{1}{α} = \frac{1}{β} + 1$

$α = \frac{β}{1 + β}$

$α = \frac{β}{1 - β}$

$α = \frac{β^{2}}{1 + β^{2}}$

$I_{ε} = I_{B} + I_{C}$

$\frac{I_{ε}}{I_{C}} = \frac{I_{B}}{I_{C}} + 1$

$\frac{1}{α} = \frac{1}{β} + 1$

$α = \frac{β}{1 + β}$

In a common emitter configuration of a transistor, α (alpha) and β (beta) are two important current gain parameters.

α (Alpha) is the common base current gain. It is defined as the ratio of the collector current (I_C) to the emitter current (I_E).

α = \frac{I_{C}}{I_{E}}

β (Beta) is the common emitter current gain. It is defined as the ratio of the collector current (I_C) to the base current (I_B).

β = \frac{I_{C}}{I_{B}}

The relationship between these currents is given by Kirchhoff's Current Law at the transistor terminal:

I_{E} = I_{B} + I_{C}

To find the relationship between α and β, we can substitute and rearrange these equations.

Step 1: Start with the definition of β and solve for I_B.

β = \frac{I_{C}}{I_{B}} \Rightarrow I_{B} = \frac{I_{C}}{β}

Step 2: Substitute this expression for I_B into the current law equation I_E = I_B + I_C.

I_{E} = (\frac{I_{C}}{β}) + I_{C} = I_{C} (\frac{1}{β} + 1)

Step 3: Now, use the definition of α, which is α = I_C / I_E. Substitute the expression for I_E from the previous step.

α = \frac{I_{C}}{I_{E}} = \frac{I_{C}}{I_{C} (\frac{1}{β} + 1)} = \frac{1}{\frac{1}{β} + 1}

Step 4: Simplify the complex fraction in the denominator.

α = \frac{1}{\frac{1 + β}{β}} = \frac{β}{1 + β}

This gives us the correct, standard relationship: $α = \frac{β}{1 + β}$

We can also rearrange this formula to find other correct forms. For example, taking the reciprocal:

\frac{1}{α} = \frac{1 + β}{β} = \frac{1}{β} + 1

This shows that $\frac{1}{α} = \frac{1}{β} + 1$ is also a correct relationship.

Analysis of the Options:

$\frac{1}{α} = \frac{1}{β} + 1$ is CORRECT, as derived above.
$α = \frac{β}{1 + β}$ is CORRECT, as it is the standard form.
$α = \frac{β}{1 - β}$ is INCORRECT. The sign in the denominator is wrong; it should be a plus sign.
$α = \frac{β^{2}}{1 + β^{2}}$ is INCORRECT. This is not a standard relationship between α and β.

Final Answer: The two incorrect relationships are the third and fourth options.

Detailed Solution