Amplifier Negative Feedback Gain Explained

Amplifiers are often used with feedback circuits to improve their performance characteristics. Negative feedback is a technique where a portion of the output signal is fed back to the input in a way that opposes the original input signal. This opposition helps to stabilize the amplifier's gain, reduce distortion, decrease noise, and influence the input and output impedances.

The gain of an amplifier without feedback is called the open-loop gain, usually denoted by \( A \). When negative feedback is applied, the gain changes, and this new gain is called the closed-loop gain, denoted by \( A_f \). The amount of signal fed back is determined by the feedback network or circuit, characterized by the feedback factor, often denoted by \( \beta \) or, as used in the options, \( M \).

Calculating Amplifier Gain with Negative Feedback

The formula for the closed-loop gain \( A_f \) of an amplifier with negative feedback is derived from analyzing the signals within the feedback loop. Let's consider the input signal \( V_{in} \), the output signal \( V_{out} \), and the feedback signal \( V_f \). The input to the amplifier's core (which has open-loop gain \( A \)) is the difference between the input signal and the feedback signal:

\( V_{id} = V_{in} - V_f \)

The output voltage is related to the input difference voltage by the open-loop gain:

\( V_{out} = A \cdot V_{id} = A \cdot (V_{in} - V_f) \)

The feedback voltage \( V_f \) is a fraction of the output voltage, determined by the feedback factor \( M \):

\( V_f = M \cdot V_{out} \)

Substituting the expression for \( V_f \) into the equation for \( V_{out} \):

\( V_{out} = A \cdot (V_{in} - M \cdot V_{out}) \)

Now, let's rearrange this equation to find the ratio of \( V_{out} \) to \( V_{in} \), which is the closed-loop gain \( A_f \):

\( V_{out} = A \cdot V_{in} - A \cdot M \cdot V_{out} \)

\( V_{out} + A \cdot M \cdot V_{out} = A \cdot V_{in} \)

\( V_{out} (1 + A \cdot M) = A \cdot V_{in} \)

\( \frac{V_{out}}{V_{in}} = \frac{A}{1 + A \cdot M} \)

Thus, the closed-loop gain \( A_f \) is:

\( A_f = \frac{A}{1 + AM} \)

In this formula, \( AM \) is known as the loop gain. For negative feedback to be effective and stable, the loop gain \( AM \) is usually much greater than 1, and the feedback is negative (indicated by the + sign in the denominator, derived from \( V_{in} - V_f \)).

Comparing Options for Amplifier Gain Formula

Let's examine the given options for the negative feedback amplifier gain \( A_f \):

1. \( Af = (1 + Am)/A \)
2. \( Af = (1 - Am)/A \)
3. \( Af = A/(1 - AM) \)
4. \( Af = A/(1 + AM) \)

Comparing these options to our derived formula \( A_f = \frac{A}{1 + AM} \), we can see which one matches the standard formula for negative feedback gain.

• Option 1 and 2 do not match the standard form.
• Option 3, \( A_f = A/(1 - AM) \), is the formula for positive feedback, where the feedback signal adds to the input signal, potentially leading to oscillation.
• Option 4, \( A_f = A/(1 + AM) \), exactly matches the derived formula for negative feedback gain, where \( A \) is the open-loop gain and \( M \) is the feedback factor.

Conclusion on Negative Feedback Gain Formula

Based on the derivation and standard amplifier theory, the correct formula for the gain \( A_f \) of an amplifier with negative feedback is \( A_f = \frac{A}{1 + AM} \), where \( A \) is the open-loop gain and \( M \) is the feedback factor. This corresponds to Option 4.

Additional Information on Amplifier Feedback Circuits

Negative feedback is a fundamental concept in amplifier design due to its significant benefits:

• Gain Stability: The closed-loop gain becomes less dependent on the variations in the open-loop gain \( A \), which can change with temperature or component aging. If \( AM \gg 1 \), then \( A_f \approx \frac{A}{AM} = \frac{1}{M} \), meaning the gain is primarily determined by the stable feedback network \( M \).
• Reduced Distortion: Negative feedback tends to linearize the amplifier's operation, reducing non-linear distortion.
• Increased Bandwidth: Negative feedback typically increases the amplifier's bandwidth. The gain-bandwidth product often remains constant, so reducing the gain with feedback increases the bandwidth.
• Reduced Noise: Noise generated within the amplifier circuit is reduced relative to the signal level.
• Modified Input and Output Impedances: Depending on how the feedback is sampled and mixed (voltage or current, series or shunt), the input and output impedances are modified. Series mixing increases input impedance, shunt mixing decreases it. Voltage sampling decreases output impedance, current sampling increases it.

Understanding the gain formula \( A_f = \frac{A}{1 + AM} \) is crucial for analyzing and designing stable and high-performance amplifier circuits using negative feedback.