Two-Port Power Gain

Difinition

3 Gains

$$ \begin{gather} \begin{aligned} Power \ Gain \ G_p &= \frac{ P_{L} }{ P_{in} } \\ Avaiable \ Gain \ G_A &= \frac{ P_{out} }{ P_{S} } \\ Transducer \ Gain \ G_T &= \frac{ P_L }{ P_S } \end{aligned} \end{gather} $$

Circuit Analysis

Reflection Coefficients

1. $\Gamma_L$ 与 $\Gamma_{out}$, $\Gamma_{in}$与$\Gamma_{S}$

$$ \begin{gather} \begin{aligned} \Gamma_L &= \frac{ Z_L - Z_0 }{ Z_L + Z_0 } = \frac{ V_2^+ }{ V_2^- } = \frac{ 1 }{ \Gamma_{out} } \\ \Gamma_S &= \frac{ Z_S - Z_0 }{ Z_S + Z_0 } = \frac{ V_1^+ }{ V_1^- } = \frac{ 1 }{ \Gamma_{in} } \end{aligned} \end{gather} $$

2. $\Gamma_L$ 与 $\Gamma_{in}$

$$ \begin{gather} \begin{aligned} V_1^- &= S_{11} V_1^+ + S_{12} V_2^+ = S_{11} V_1^+ S_{12} \Gamma_L V_2^- \\ V_2^- &= S_{22} V_2^+ + S_{21} V_1^+ = S_{21} V_1^+ + S_{22} \Gamma_L V_2^- \\ \therefore V_2^- &= \frac{ S_{21} V_1^+ }{ 1 - S_{22} \Gamma_L } \\ V_1^- &= (S_{11} + \frac{ S_{12} S_{21} \Gamma_L }{ 1 - S_{22} \Gamma_L }) V_1^+ \\ \therefore \Gamma_{in} &= S_{11} + \frac{ S_{12} S_{21} \Gamma_L }{ 1 -S_{22} \Gamma_L } \end{aligned} \end{gather} $$

3. $\Gamma_S$ 与 $\Gamma_{out}$, 同理可得

$$ \begin{gather} \begin{aligned} \Gamma_{out} &= S_{22} + \frac{ S_{12} S_{21} \Gamma_S}{ 1- S_{11} \Gamma_S} \end{aligned} \end{gather} $$

Powers

1. $V_1^+$ 和 $V_S$

$$ \begin{gather} \begin{aligned} V_1 &= \frac{ Z_{in} }{ Z_{in} + Z_S } V_S \\ &= V_1^+(1 + \Gamma_{in}) \\ Z_{in} &= Z_0 \frac{ 1 + \Gamma_{in} }{ 1 - \Gamma_{in} } \\ \therefore V_1^+ &= \frac{ V_S }{ 2 } \frac{ 1 - \Gamma_S }{ 1 - \Gamma_S \Gamma_{in} } \\ \end{aligned} \end{gather} $$

2. Powers

• $P_{in}$

$$ \begin{gather} \begin{aligned} P_{in} &= \frac{ |V_1|^2 }{ 2Z_{in} } \\ &= \frac{ |V_1^+|^2 }{ 2 Z_0 } (1- |\Gamma_{in}|^2) \\ &= \frac{ |V_s|^2 }{ 8 Z_0 } \frac{ (1- \Gamma_S)^2 ( 1- | \Gamma_{in}|^2) }{ (1 - \Gamma_S \Gamma_{in})^2 } \end{aligned} \end{gather} $$

• 利用$V_2^- = \dfrac{ S_{21} }{ 1 - S_{22} \Gamma_L } V_1^+$获取$P_L$

$$ \begin{gather} \begin{aligned} P_L &= \frac{ |V_2^-|^2 }{ 2Z_0 } (1 - | \Gamma_L|^2) \\ &= \frac{ |V_s|^2 }{ 8 Z_0 } \frac{ (1- \Gamma_S)^2 ( 1- | \Gamma_{L}|^2) }{ (1 - \Gamma_S \Gamma_{in})^2 } \frac{ |S_{21}|^2 }{ (1 - S_{22} \Gamma_L)^2 } \end{aligned} \end{gather} $$

• $P_{avs}$, 即Maximum Power Avaiable from the source, 阻抗匹配时的功率

$$ \begin{gather} \begin{aligned} P_{avs} &= P_{in} ({\Gamma_{in} = \Gamma_S^*}) = \frac{ |V_S|^2 }{ 8Z_0 } \frac{ (1- \Gamma_S)^2 }{ (1- |\Gamma_S|^2) } \end{aligned} \end{gather} $$

• $P_{avn}$

$$ \begin{gather} \begin{aligned} P_{avn} &= P_L ({\Gamma_{out} = \Gamma_L^*}) = \frac{ |V_S|^2 }{ 8Z_0 } \frac{ (1- \Gamma_S)^2 }{ (1- |\Gamma_{out}|^2) } \frac{ |S_{21}|^2 }{ |1 - S_{11} \Gamma_S|^2 } \end{aligned} \end{gather} $$

Gains

• $G_{TU}$是Unilateral Gain, 此时, $S_{12} = 0 \Longrightarrow \Gamma_{in} = S_{11}$

$$ \begin{gather} \begin{aligned} G_P &= \frac{ P_L }{ P_{in} } = \frac{ |S_{21}|^2 ( 1- | \Gamma_L|^2) }{ (1 - S_{22} \Gamma_L)^2 (1 - | \Gamma_{in}|^2) } \\ G_A &= \frac{ P_{out} }{ P_{avs} } = \frac{ |S_{21}|^2 ( 1- | \Gamma_S|^2) }{ (1 - S_{11} \Gamma_S)^2 (1 - | \Gamma_{out}|^2) } \\ G_T &= \frac{ P_{out} }{ P_{in} } = \frac{ |S_{21}|^2 ( 1- | \Gamma_S|^2) ( 1- | \Gamma_L|^2) }{ | 1 - \Gamma_S \Gamma_{in}|^2| 1 - S_{22} \Gamma_L|^2 } \\ G_{TU} &= \frac{ P_{out} }{ P_{in} } = \frac{ |S_{21}|^2 ( 1- | \Gamma_S|^2) ( 1- | \Gamma_L|^2) }{ | 1 - \Gamma_S S_{11}|^2| 1 - S_{22} \Gamma_L|^2 } \\ \end{aligned} \end{gather} $$

2-Port Power Gain in a Amplifier Model

$$ \begin{gather} \begin{aligned} G_T &= G_S G_0 G_L \\ G_S &= \frac{ 1 - | \Gamma_S|^2 }{ |1 - \Gamma_S S_{11}|^2 } \\ G_0 &= |S_{21}|^2 \\ G_L &= \frac{ 1 - | \Gamma_L|^2 }{ | 1 - S_{22} \Gamma_L|^2} \end{aligned} \end{gather} $$

Stability

Definition

$$ \begin{gather} \begin{aligned} \mathcal{Re} \lbrace Z_{in} \rbrace \ge 0 \\ |Z_0 \frac{ 1 - \Gamma_{in} }{ 1 + \Gamma_{in} }| \ge 0 \\ \therefore | \Gamma_{in}| < 1 \\ | \Gamma_{out}| < 1 \end{aligned} \end{gather} $$

Stability Circle

Input Stability Circle

• 将$\Gamma_{in}, \Gamma_{out}$的表达式代入$(12)$, 得到

$$ \begin{gather} \begin{aligned} | \Gamma_L - \frac{ (S_{22} - \Delta S_{11}’)’ }{ |S_{22} - | \Delta|^2 } | &= | \frac{ S_{12} S_{21} }{ |S_{22}|^2 - | \Delta|^2 } | \\ C_L &= \frac{ (S_{22} - \Delta S_{11}’)’ }{ |S_{22} - | \Delta|^2} \\ R_L &= | \frac{ S_{12} S_{21} }{ |S_{22}|^2 - | \Delta|^2 } | \end{aligned} \end{gather} $$

Output Stability Circle

1. 同理可得

$$ \begin{gather} \begin{aligned} | \Gamma_S - \frac{ (S_{11} - \Delta S_{22}’)’ }{ |S_{11} - | \Delta|^2 } | &= | \frac{ S_{12} S_{21} }{ |S_{11}|^2 - | \Delta|^2 } | \\ C_L &= \frac{ (S_{11} - \Delta S_{22}’)’ }{ |S_{11} - | \Delta|^2} \\ R_L &= | \frac{ S_{12} S_{21} }{ |S_{11}|^2 - | \Delta|^2 } | \end{aligned} \end{gather} $$

2. Output Stability Circles

Test for Unconditionally Stable

Rollet Condition

1. S参数

$$ \begin{gather} \begin{aligned} K &= \frac{ 1 - |S_{11}|^2 - |S_{22}|^2 + | \Delta|^2 }{ 2 |S_{12} S_{21}| } > 1 \\ |\Delta| &< 1 \end{aligned} \end{gather} $$

2. 导抗参数

$$ \begin{gather} \begin{aligned} K &= \frac{ 2 \mathcal{Re} (W_{11})\mathcal{Re} ( W_{22}) - \mathcal{Re} ( W_{12} W_{21})}{ | W_{12} W_{21} |} \end{aligned} \end{gather} $$

mu Test

$$ \begin{gather} \begin{aligned} \mu = \frac{ 1 - |S_{11}|^2 }{ |S_{22} - \Delta S_{11}’| + |S_{12} S_{21}| } > 1 \end{aligned} \end{gather} $$

Potential Unstable Factors for MOSFET

Inner Feedback Capacitance

Y Parameter

$$ \begin{gather} \begin{aligned} [Y] &= \begin{bmatrix} j \omega C_{gd} + \dfrac{ j \omega C_{gs} }{ 1 + j \omega R_{gs} C_{gs} }& - j \omega C_{gd} \\ \dfrac{ g_m }{ 1 + j \omega R_{gs} C_{gs} } - j \omega C_{gd} & \dfrac{ 1 }{ R_{ds} } + j \omega ( C_{ds} + C_{gd}) \\ \end{bmatrix} \\ Y_{in} &= \frac{ j \omega C_{gs} }{ 1 + j \omega R_{gs} C_{gs} } \Big[ 1 + g_m R_{ds} \frac{ 1 - j \dfrac{ \omega }{ \omega_T } ( 1 + j \omega R_{gs} C_{gs}) }{ 1 + j \omega R_{ds} C_{ds} \Big( 1 + \dfrac{ C_{gd} }{ C_{ds} } + \dfrac{ B_L }{ \omega C_{ds} } \Big)} \Big] \end{aligned} \end{gather} $$

Stability Factor

• 将Y参数带入导抗参数表达的$K$中

$$ \begin{gather} \begin{aligned} K &= \Big[ 1 + \frac{ 2 }{ g_m R_{ds} } (1 + \frac{ C_{gs} }{ C_{gd} })\Big] \frac{ \omega R_{gs} C_{gs} }{ \sqrt{ 1 + ( \omega R_{gs} C_{gs})^2 } } \\ \underset{ \omega \longrightarrow \infin }{ lim } K &= \Big[ 1 + \frac{ 2 }{ g_m R_{ds} } (1 + \frac{ C_{gs} }{ C_{gd} })\Big] \\ \underset{ \omega \longrightarrow 0}{ lim } K &= 0 \end{aligned} \end{gather} $$

稳定频带边界

• Power MOSFET 经验条件: $g_m R_{ds} = \dfrac{ 10 }{ 30 }, \dfrac{ C_{gd} }{ C_{gs} } = \dfrac{ 0.1 }{ 0.2 }$

$$ \begin{gather} \begin{aligned} f_{p1} &\approx \frac{ 10^{-6} }{ 2 \pi R_{gs} C_{gs}} \\ f_{p2} &= \frac{ 1 }{ 4 \pi R_{gs} C_{gs} } \frac{ g_m R_{ds}}{ \sqrt{ 1 + \dfrac{ C_{gs} }{ C_{gd} } }} \frac{ 1 }{ \sqrt{ 1 + \dfrac{ C_{gs} }{ C_{gd} } + g_m R_{ds} } } &\approx \frac{ 1 }{ 4\pi R_{gs} C_{gs} } \end{aligned} \end{gather} $$

Parasitic Source Inductance

• $\kappa = \dfrac{ \omega_T L_S }{ R_{gs} }$

Single-Stage Amplifier Design

Design for Maximum Gain

Maximum Gain & Max Stable Gain

• 当$\Gamma_{in} = \Gamma_S^, \Gamma_{out} = \Gamma_L^$时，会出现最大增益

$$ \begin{gather} \begin{aligned} G_{Tmax} &= \frac{ |S_{21}|^2 (1 - | \Gamma_L|^2) }{ (1- | \Gamma_S|^2) (1- S_{22} \Gamma_L)^2 } \\ &= \frac{ |S_{21}| }{ |S_{12}| } ( K - \sqrt{ K^2 - 1 }) \\ G_{msg} &= \frac{ |S_{21}| }{ |S_{12}| }, K = 1 \end{aligned} \end{gather} $$

对应的反射系数

• 根据反射系数公式，结合共轭匹配，可以得到

$$ \begin{gather} \begin{aligned} \Gamma_S &= \frac{ B_1 \pm \sqrt{ |B_1|^2 - 4|C_1|^2 } }{ 2C_1 } \\ \Gamma_L &= \frac{ B_2 \pm \sqrt{ |B_2|^2 - 4|C_2|^2 } }{ 2C_2 } \\ B_1 &= 1 + |S_{11}|^2 - |S_{22}|^2 - | \Delta|^2 \\ B_2 &= 1 - |S_{11}|^2 + |S_{22}|^2 - | \Delta|^2 \\ C_1 &= S_{11} - \Delta S_{22}^* \\ C_2 &= S_{22} - \Delta S_{11}^* \end{aligned} \end{gather} $$

• 以上方程只在$K>1$时有解

Gains in Unilateral Case

$$ \begin{gather} \begin{aligned} S_{12} &= 0 \\ \Gamma_{in} &= S_{11}’, \Gamma_{out} = S_{22}’ \\ G_{TUmax} &= \frac{ |S_{21}|^2 }{ (1 - |S_{11}|^2)(1 - |S_{22}|^2) } \end{aligned} \end{gather} $$

Constant Gain Circle & Desgin for Specific Gain

Gain Factor

1. 当$\Gamma_S = S_{11}^, \Gamma_L = S_{22}^$时，增益取得最大值

$$ \begin{gather} \begin{aligned} G_S &= \frac{ 1 - |\Gamma_s|^2 }{ | 1- S_{11} \Gamma_S|^2 } \\ G_L &= \frac{ 1 - |\Gamma_L|^2 }{ | 1- S_{22} \Gamma_L|^2 } \\ G_{Smax} &= \frac{ 1 }{ 1 - |S_{11}|^2 } \\ G_{Lmax} &= \frac{ 1 }{ 1 - |S_{22}|^2 } \end{aligned} \end{gather} $$

2. Gain Factor

$$ \begin{gather} \begin{aligned} g_s &= \frac{ G_S }{ G_{S \ max} } \\ g_L &= \frac{ G_L }{ G_{L \ max} } \end{aligned} \end{gather} $$

Constant Gain Circles

$$ \begin{gather} \begin{aligned} C_S &= \frac{ g_s S_{11}^* }{ 1 - (1 - g_s)|S_{11}|^2 } \\ R_S &= \frac{ \sqrt{ 1 - g_s } ( 1 - |S_{11}|^2)}{ 1 - (1 - g_s) |S_{11}|^2 } \\ C_L &= \frac{ g_s S_{22}^* }{ 1 - (1 - g_L)|S_{22}|^2 } \\ R_L &= \frac{ \sqrt{ 1 - g_L } ( 1 - |S_{22}|^2)}{ 1 - (1 - g_L) |S_{22}|^2 } \end{aligned} \end{gather} $$

Low Noise Amplifier Design

Noise Figure Parameter

1. Noise Figure of a Transistor

$$ \begin{gather} \begin{aligned} F &= F_{min} + \frac{ R_N }{ G_S } |Y_S - Y_{opt}|^2 \end{aligned} \end{gather} $$

2. Define Noise Figure Parameter

$$ \begin{gather} \begin{aligned} Y_S &= \frac{ 1 }{ Z_S } \frac{ 1 - \Gamma_S }{ 1 + \Gamma_S } \\ Y_{opt} &= \frac{ 1 }{ Z_{opt} } \frac{ 1 - \Gamma_{opt} }{ 1 + \Gamma_{opt} } \\ \therefore F &= F_{min} + \frac{ 4R_N }{ Z_0 } \frac{ | \Gamma_S - \Gamma_{opt}|^2 }{ (1 - | \Gamma_S|^2) | 1 + \Gamma_{opt}|^2 } \\ N &= \frac{ | \Gamma_S - \Gamma_{opt}|^2 }{ 1 - | \Gamma_S|^2 } \\ &= \frac{ F - F_{min} }{ 4R_N/Z_0 } | 1 + \Gamma_{opt}|^2 \\ \end{aligned} \end{gather} $$

3. Noise Figure Center

$$ \begin{gather} \begin{aligned} | \Gamma_S - C_F| &= |R_F| \\ C_F &= \frac{ \Gamma_{opt} }{ N + 1} \\ R_F &= \frac{ \sqrt{ N(N+1 - | \Gamma_{opt}|^2) } }{ N + 1 } \end{aligned} \end{gather} $$

Low Noise MOSFET Amplifier

Broadband Transistor Amplifier Design

增加带宽的做法

Balanced Amplifiers

$$ \begin{gather} \begin{aligned} S_{21} &= - \frac{ j }{ 2 }( G_A + G_B) \\ S_{11} &= \frac{ 1 }{ 2 } ( \Gamma_A - \Gamma_B ) \end{aligned} \end{gather} $$

Distributed Amplifier

Differential Amplifier

Single-Ended & Differential Amplifier (Symbols)

Balun Lines

Differential Amplifier Circuit