Impact of Negative Feedback on Amplifier Properties

Key Impacts of Negative Feedback on Amplifiers

Negative feedback significantly alters the operational characteristics of an amplifier, generally improving performance at the cost of reduced gain. The primary impacts are categorized as follows:

• Provides Gain Stability: It minimizes fluctuations in gain caused by external or internal factors.

• Reduces Non-Linear Distortion: It corrects waveforms distorted by the non-linearities of active devices.

• Reduces Noise: Noise introduced within the amplifier circuit is suppressed.

• Increases Bandwidth / Improved Frequency Response: It extends the range over which the amplifier operates effectively.

• Increased Input Impedance: Specific feedback configurations help raise the resistance seen by the input source.

• Reduced Output Impedance: Negative feedback can lower the output resistance, improving the amplifier's ability to drive loads.

Stabilization of Gain and Desensitivity

In practical applications, the transfer gain of an amplifier is not naturally constant. It fluctuates due to changes in operating points, temperature, and component aging. Negative feedback is introduced to mitigate this lack of stability.

• Mathematical Representation of Gain Stability:

  • Let $A$ be the open-loop gain.

  • Let $\beta$ be the feedback factor.

  • The gain with feedback is defined as:     $A_f = \frac{A}{1 + \beta A}$

• Derivation using the Quotient Rule ($u/v$ rule):     To determine how sensitive $A_f$ is to changes in $A$, we differentiate with respect to $A$:     $\frac{dA_f}{dA} = \frac{(1 + \beta A) \cdot 1 - A \times \beta}{(1 + \beta A)^2}$     $\frac{dA_f}{dA} = \frac{1}{(1 + \beta A)^2}$

• Fractional Change in Amplification:     To find the relative stability, we divide both sides by $A_f$:     $\frac{dA_f}{A_f} = \frac{dA}{(1 + \beta A)^2 A_f}$     Since $A_f = \frac{A}{1 + \beta A}$, we substitute and simplify:     $\frac{dA_f}{A_f} = \frac{dA}{(1 + \beta A)^2} \times \frac{1 + \beta A}{A}$     $\frac{dA_f}{A_f} = \frac{dA}{A} \times \frac{1}{1 + \beta A}$

• Definitions and Implications:

  • $\frac{dA_f}{A_f}$: Fractional change in amplification with feedback.

  • $\frac{dA}{A}$: Fractional change in amplification without feedback.

  • The change in gain with feedback is less than the change in gain without feedback by a factor of $(1 + \beta A)$.

  • Sensitivity: The ratio of the fractional change in gain with feedback to the fractional change in gain without feedback is called the Sensitivity of the Transfer Gain:     $\text{Sensitivity} = \frac{dA_f / A_f}{dA / A} = \frac{1}{1 + \beta A}$

  • Desensitivity ($D$): The reciprocal of sensitivity, defined as:     $D = 1 + \beta A$

  • Stability increases as desensitivity increases.

  • High Loop Gain Limit: If \beta A >> 1, then:     $A_f \approx \frac{A}{\beta A} = \frac{1}{\beta}$     In this state, the gain is entirely dependent on the feedback network components (which can be high-precision resistors) and is independent of the active amplifier itself.

Specific Feedback Stabilization Mechanisms

Different feedback configurations stabilize different types of transfer gain:

• Voltage Series Feedback: Stabilizes the voltage gain ($A_{vf} \approx \frac{1}{\beta}$).

• Current Series Feedback: Stabilizes the transconductance gain ($G_{Mf} \approx \frac{1}{\beta}$).

• Voltage Shunt Feedback: Stabilizes the transresistance gain ($R_{Mf} \approx \frac{1}{\beta}$).

• Current Shunt Feedback: Stabilizes the current gain ($A_{If} \approx \frac{1}{\beta}$).

Reduction in Distortion

Frequency Distortion

• Gain with negative feedback is $A_f = \frac{A}{1 + \beta A}$.

• When \beta A >> 1, the gain becomes $A_f \approx \frac{1}{\beta}$.

• If the feedback network is purely resistive, the overall gain becomes independent of frequency, even if the open-loop gain $A$ is frequency-dependent.

• Practically, this considerably reduces the frequency distortion caused by variations in internal amplifier gain across different frequencies.

Non-linear Distortion

• Variables:

  • $A$: Open-loop gain.

  • $A_f$: Closed-loop gain.

  • $D$: Distortion without feedback.

  • $D_f$: Distortion with feedback.

• Derivation:     With feedback, a portion of the distortion $\beta D_f$ is fed back and amplified by the amplifier, resulting in a signal $A \beta D_f$.     The net distortion with feedback is the difference between the initial distortion and the fed-back amplified distortion (since negative feedback is in phase opposition):     $D_f = D - A \beta D_f$     $D_f + A \beta D_f = D$     $D_f (1 + \beta A) = D$     $D_f = \frac{D}{1 + \beta A}$

• Caveat: Distortion cancellation only occurs if the amplifier itself introduces the distortion. If the original input signal was already distorted, the feedback system will not remove that distortion.

Reduction in Noise

Noise reduction follows the same logic as non-linear distortion reduction. Noise introduced by the amplifier is suppressed by the desensitivity factor.

• Formula:     $N_f = \frac{N}{1 + \beta A}$

• Caveat: This reduction applies only to noise introduced within the amplifier circuit. If the input signal itself is noisy, no noise cancellation will take place.

Frequency Response and Bandwidth

Frequency response tracks the variation in the amplitude level of the output signal as the frequency changes. The standard bandwidth is defined by the lower and upper cut-off frequencies where the gain drops by $3\,dB$ from the maximum (mid-band) level.

• Baseline Equations:

  • $A_{fmid} = \frac{A_{mid}}{1 + \beta A_{mid}}$

  • $A_{flow} = \frac{A_{low}}{1 + \beta A_{low}}$

  • $A_{fhigh} = \frac{A_{high}}{1 + \beta A_{high}}$

Impact on Lower Cut-off Frequency ($f_L$)

• The relation between low-frequency gain and mid-frequency gain is:     $A_{low} = \frac{A_{mid}}{1 - j(\frac{f_L}{f})}$

• Substituting into the feedback equation and dividing the numerator and denominator by $(1 + A_{mid} \beta)$, we obtain the lower cut-off frequency with feedback ($f_{Lf}$):     $A_{flow} = \frac{A_{fmid}}{1 - j(\frac{f_{Lf}}{f})}$     Where:     $f_{Lf} = \frac{f_L}{1 + A_{mid} \beta}$

• Conclusion: The lower cut-off frequency is reduced by the factor $(1 + A_{mid} \beta)$, improving the low-frequency response.

Impact on Upper Cut-off Frequency ($f_H$)

• Following a similar derivation for the high-frequency range:     $A_{fhigh} = \frac{A_{fmid}}{1 - j(\frac{f}{(1 + A_{mid} \beta) f_H})}$     Where:     $f_{Hf} = f_H(1 + A_{mid} \beta)$

• Conclusion: The upper cut-off frequency is increased by the factor $(1 + A_{mid} \beta)$, improving the high-frequency response.

Effect on Bandwidth ($BW$)

• Standard Bandwidth: $BW = f_H - f_L$

• Bandwidth with Feedback: $BW_f = f_{Hf} - f_{Lf}$

• Since $f_{Hf}$ increases and $f_{Lf}$ decreases:     (f_{Hf} - f_{Lf}) > (f_H - f_L)     $BW_f = BW(1 + \beta A_{mid}$

• Therefore, the bandwidth of an amplifier with feedback is greater than that without feedback.

Impact on Input Resistance ($R_{if}$)

The effect on input resistance depends on how the feedback signal is mixed with the input signal:

• Series Input Mixing: If the feedback signal is added to the input in series with the applied voltage (regardless of the sampling method), it increases the input resistance (R_{if} > R_i). This occurs because the feedback voltage opposes the source voltage, leading to a smaller input current $I_i$ than would be present without feedback.

• Shunt Input Mixing: If the feedback signal is added to the input in shunt (parallel) with the applied voltage, it decreases the input resistance (R_{if} < R_i). In this configuration, the input current $I_s = I_i + I_f$. The total current drawn from the source is increased, thereby lowering the effective input resistance.