Introduction
Characteristic of Feedback Circuit
Gain Sensitivity
$$ \begin{equation} \frac{ Y }{ X } = \begin{cases} \dfrac{ A }{ 1 + A \beta } \\ \dfrac{ 1 }{ \beta } & A \beta \gg 1 \end{cases} \end{equation} $$
| $A \beta$ | Loop gain |
|---|---|
| $A$ | 越大，电路就越准确 |
Band Width
$$ \begin{aligned} A_v &= \frac{ A(s) }{ 1 + A(s) \beta } \\ &= \frac{ \dfrac{ A_0 }{ 1 + A_0 \beta } }{ 1 + \dfrac{ s }{ (1 + A \beta) \omega_0} } \end{aligned} $$
| $3dB$带宽 | 增加了$A \beta$ |
|---|---|
| 增益 | 减小 |
Non-Linear Reduction
Types of Amplifiers
Feedback Structure
V-V Feedback
Circuit
Impedence
- Output Impedence
$$ \begin{aligned} V_F &= \beta V_X \\ V_e &= - \beta V_X \\ V_M &= - A_0 \beta V_X \\ R_{out} &= \frac{ V_X - V_M }{ I_X } \end{aligned} $$
$$ \begin{equation} \frac{ V_X }{ I_X } = \frac{ R_{out} }{ 1 + A \beta} \end{equation} $$
- Input Impedence
$$ \begin{aligned} V_e &= I_X R_{in} \\ V_F &= A_0 \beta V_e \\ V_e &= V_X - V_F \end{aligned} $$
$$ \begin{equation} \frac{ V_X }{ I_X } = (1 + A_0 \beta) R_{in} \end{equation} $$
Other Feedback Strcuture & Summary
| Types | $V_{in}$ | $V_{out}$ |
|---|---|---|
| $V-V$(Voltage) | $( 1 + A \beta) R_{in}$ | $\dfrac{ R_{out}}{ 1 + A \beta }$ |
| $V-I$(Transconductance) | $( 1 + A \beta) R_{in}$ | $( 1 + A \beta) R_{out}$ |
| $I-V$(Transresistance) | $\dfrac{ R_{in}}{ 1 + A \beta }$ | $\dfrac{ R_{out}}{ 1 + A \beta }$ |
| $I-I$(Current) | $\dfrac{ R_{in}}{ 1 + A \beta }$ | $( 1 + A \beta) R_{out}$ |
Feedback Influecing Noise
$$ \begin{aligned} V_{out} &= (V_{in} - \beta V_{out} + V_n) A \\ \therefore V_{out} &= \frac{ A }{ 1 + A \beta } ( V_{in} + V_n) \end{aligned} $$