# 单调谐回路谐振放大器

1. $\dot{I}_{o1}=y_{fe}\dot{V}_{i1}$ 是晶体管放大作用的等效电流源
2. $g_{o1}, C_{o1}$ 为晶体管的输出导纳
3. $G_p=\frac{1}{R_p}$ 回路本身的损耗
4. $Y_L=g_{i2}+j\omega C_{i2}$ 负载本身的导纳（一般是下一级晶体管的输入导纳，所以用下标 $_i$）

## 电压增益

$\dot{A}_v=\frac{\dot{V}_{o1}}{\dot{V}_{i1}}=\frac{-y_{fe}}{y_{oe}+Y_L'}\cdot\frac{p_1^2}{p_1^2}=-\frac{p_1^2 y_{fe}}{Y'}$

$\dot{A}_v=\frac{V_{i2}}{\dot{V}_{i1}}=\frac{(p_2/p_1)\dot{V}_{o1}}{\dot{V}_{i1}}=-\frac{p_1p_2y_{fe}}{Y'}$

$\dot{A}_{v0}=-\frac{p_1p_2y_{fe}}{Y'}=-\frac{p_1p_2y_{fe}}{G_p'}=-\frac{p_1p_2y_{fe}}{G_p+g_{o1}'+g_{i2}'}\\ =-\frac{p_1p_2y_{fe}}{G_p+p_1^2g_{o1}+p_2^2g_{i2}}$

$p_2^2g_{i2}=p_1^2g_{o1}+G_p=\frac{G_p'}{2}$

$p_2^2g_{i2}=p_1^2g_{o1}=\frac{G_p'}{2}\\ \Rightarrow p_1=\sqrt{\frac{G_p'}{2g_{o1}}},\;p_2=\sqrt{\frac{G_p'}{2g_{i2}}}\\ \Rightarrow (\dot{A}_{v0})_\text{max}=-\frac{y_{fe}}{2\sqrt{g_{o1}g_{i2}}}$

（👇有损电压增益在功率增益部分👇）

## 功率增益

$A_{p0}=\frac{P_o}{P_i}\\ 其中，P_i=V_{i1}^2g_{i1}（指信号源）, P_o=V_{ab}^2 p_2^2g_{i2}=\left(\frac{p_1|y_{fe}|V_{i1}}{G_p'}\right)^2p_2^2g_{i2}\\ 故有，A_{p0}=\frac{P_o}{P_i}=\frac{p_1^2p_2^2g_{i2}|y_{fe}|^2}{g_{i1}(G_p')^2}=A_{v0}^2\frac{g_{i2}}{g_{i1}}$

$(A_{p0})_\max=\frac{|y_{fe}|^2}{4g_{o1}g_{i2}}$

$K_1=\frac{无损输出功率P_1}{有损输出功率P_1'}$ $P_1（P_1'）=V_{ab}^2 p_2^2g_{i2}= \begin{cases} \left( \dfrac{I_o}{p_1^2g_{o1}+p_2^2g_{i2}} \right)^2(p_2^2g_{i2})\Rightarrow 无损\\ \left( \dfrac{I_o}{p_1^2g_{o1}+p_2^2g_{i2}+G_p} \right)^2(p_2^2g_{i2})\Rightarrow 有损 \end{cases}$

$无载：Q_0=\frac{1}{G_p\omega_0L}\Longrightarrow G_p=\frac{1}{Q_0\omega_0L}\\ 有载：Q_L=\frac{1}{(p_1^2g_{o1}+p_2^2g_{i2}+G_p)\omega_0L}\\ \Rightarrow p_1^2g_{o1}+p_2^2g_{i2}=\frac{1}{Q_L\omega_0L}-G_p\\ =\frac{1}{\omega_0L}\left(\frac{1}{Q_L}-\frac{1}{Q_0}\right)$

\begin{align} K_1=\frac{P_1}{P_1'}&=\left( \frac{p_1^2g_{o1}+p_2^2g_{i2}+G_p}{p_1^2g_{o1}+p_2^2g_{i2}} \right)^2\\ &=\left[ \frac{\frac{1}{Q_L\omega_0L}}{\frac{1}{Q_L\omega_0L}\left(\frac{1}{Q_L}-\frac{1}{Q_0}\right)} \right]\\ &=\left( \frac{1}{1-\frac{Q_L}{Q_0}} \right)^2 \end{align}\\

$K_1(\text{dB})=2\times10\lg \left[ 1/\left( 1-\frac{Q_L}{Q_0} \right)\right]$

$(A_{p0})_\max=\frac{|y_{fe}|^2}{4g_{o1}g_{i2}}\cdot\left( 1-\frac{Q_L}{Q_0} \right)^2$

$(A_{v0})_\text{max}=-\frac{|y_{fe}|}{2\sqrt{g_{o1}g_{i2}}}\cdot\left( 1-\frac{Q_L}{Q_0} \right)$

## 通频带

\begin{align} \dot{A}_v&=-\frac{p_1p_2y_{fe}}{Y'}\\ \dot{A}_{v0}&=-\frac{p_1p_2y_{fe}}{G_p+p_1^2g_{o1}+p_2^2g_{i2}}\\ \frac{\dot{A}_v}{\dot{A}_{v0}}&=\frac{G_p'}{Y'}=G_p'Z' \end{align}\\

$z'=\frac{1}{G_p'+j\left( \omega C_\Sigma-\dfrac{1}{\omega L_1} \right)}=\frac{1}{G_p'\left( 1+j\dfrac{2Q_L\Delta f}{f_0} \right)}$

$\frac{A_v}{A_{v0}}=\frac{1}{\sqrt{1+\left(\dfrac{2Q_L\Delta f}{f_0}\right)^2}}\\ 2\Delta f_{0.7}=\frac{f_0}{Q_L}$

## 矩形系数

$K_{r0.1}=\frac{2\Delta f_0.1}{2\Delta f_{0.7}}=\sqrt{10^2-1}\approx9.95$

## 带宽增益积

$G_p'=\frac{\omega_0 C_\Sigma}{Q_L}=\frac{2\pi f_0C_\Sigma}{f_0/2 \Delta f_{0.7}}=4\pi C_\Sigma \Delta f_{0.7}$

$\dot{A}_{v0}=-\frac{p_1p_2y_{fe}}{G'}=-\frac{p_1p_2y_{fe}}{4\pi C_\Sigma \Delta f_{0.7}}$

# 多级单调谐回路谐振放大器

## 增益

$\frac{A_v}{A_{v0}}=\frac{1}{\sqrt{1+(\frac{2Q_L\Delta f}{f_0})^2}}$

m级：

$\frac{A_m}{A_{m0}}=\frac{1}{\left[1+(\frac{2Q_L\Delta f}{f_0})^2\right]^{m/2}}$

## 通频带

$2\Delta f_{0.7}=\frac{f_0}{Q_L}$

$(2\Delta f_{0.7})_m=\sqrt{2^{1/m}-1}\frac{f_0}{Q_L}=\sqrt{2^{1/m-1}}\cdot 2\Delta f_{0.7}$

## 矩形系数

$K_{r 0.1}=\frac{2\Delta f_{0.1}}{2\Delta f_{0.7}}$

$\frac{\sqrt{100-1}}{\sqrt{2-1}}\approx9.95$

$\frac{\sqrt{100^{1/m}-1}}{\sqrt{2^{1/m}-1}}\approx3.75(m=3)$