What is the correct relationship between α and β in a transistor?
A. β = α / (1 – α)
B. β = α / (1 + α)
C. β = (1 + α) / α
D. β = (1 – α)
Hint: A transistor has three main currents: the collector current, base current, and emitter current. The parameters α and β represent the ratios between certain currents in a transistor. We can derive the correct answer by expressing these ratios and comparing them.
Step-by-step Explanation:
Alpha (α) in a transistor is known as the current gain in the common-base configuration. It is defined as the ratio of the change in collector current (ΔiC) to the change in emitter current (ΔiE). The value of α is always less than 1.
Beta (β), on the other hand, is the current gain in the common-emitter configuration. It is the ratio of the change in collector current (ΔiC) to the change in base current (ΔiB).
The following relations hold:
[latex]\alpha = \frac{\Delta i_C}{\Delta i_E}[/latex]
[latex]\beta = \frac{\Delta i_C}{\Delta i_B}[/latex]
Now, we can substitute the expression for β:
[latex]\beta = \frac{\Delta i_C}{\Delta i_E} \times \frac{\Delta i_E}{\Delta i_B} = \alpha \times \frac{\Delta i_E}{\Delta i_B} \quad \text{(Equation 1)}[/latex]
Next, using the relationship between the currents:
[latex]\Delta i_B = \Delta i_E – \Delta i_C \quad \text{(Equation 2)}[/latex]
Substitute Equation (2) into Equation (1):
[latex]\beta = \alpha \times \frac{1}{1 – \frac{\Delta i_C}{\Delta i_E}} = \alpha \times \frac{1}{1 – \alpha}[/latex]
Thus, the correct formula is:
[latex]\beta = \frac{\alpha}{1 – \alpha}[/latex]
So, the correct answer is Option A.
Note: A transistor is a semiconductor device used to amplify or switch electronic signals and electrical power. It typically has at least three terminals for connecting to an external circuit. Transistors are fundamental in many modern technologies, including computers and other electronic devices.