To determine the feedback factor ($\beta$) of an amplifier, we use the relationship between its voltage gain without feedback (open-loop gain) and its voltage gain with feedback (closed-loop gain). This problem deals with an amplifier where the voltage gain is reduced due to negative feedback, which is a common characteristic of feedback systems.

Amplifier Gain Concepts

In amplifier circuits, two important gain parameters are considered:

• Open-loop gain ($A$): This is the voltage gain of the amplifier circuit without any feedback applied. In this problem, the amplifier's voltage gain without feedback is given as $A = 10$.
• Closed-loop gain ($A_f$): This is the voltage gain of the amplifier when a feedback path is introduced. For a negative feedback system, the closed-loop gain is always less than the open-loop gain. Here, the voltage gain with feedback is given as $A_f = 5$.

The feedback factor ($\beta$) represents the fraction of the output signal that is fed back to the input. For negative feedback, the general formula relating these quantities is:

$A_f = \frac{A}{1 + A\beta}$

Where:

• $A_f$ = Closed-loop voltage gain
• $A$ = Open-loop voltage gain
• $\beta$ = Feedback factor

Feedback Factor Calculation

We are provided with the following values related to the amplifier's voltage gain and feedback:

• Open-loop voltage gain, $A = 10$
• Closed-loop voltage gain, $A_f = 5$

We need to find the feedback factor, $\beta$.

Let's substitute the given values into the formula for closed-loop gain:

$5 = \frac{10}{1 + 10\beta}$

Now, we will solve this equation for $\beta$ step-by-step:

Step 1: Multiply both sides by the denominator $(1 + 10\beta)$:

$5 \times (1 + 10\beta) = 10$

Step 2: Distribute the 5 on the left side of the equation:

$5 + 50\beta = 10$

Step 3: Subtract 5 from both sides of the equation to isolate the term with $\beta$:

$50\beta = 10 - 5$

$50\beta = 5$

Step 4: Divide both sides by 50 to find the value of $\beta$:

$\beta = \frac{5}{50}$

$\beta = \frac{1}{10}$

$\beta = 0.1$

Feedback Factor Result

The calculated feedback factor for the amplifier is $0.1$. This value indicates the fraction of the output signal that is fed back to the input to achieve the reduced voltage gain of 5 from an initial gain of 10.

The final answer is $\beta = 0.1$.