Noise in Microwave Circuit

Noise Type

Internal Noise

External Noise

Noise Power & Equivalent Noise Temperature

Thermal Noise in Resistor

• 使用Rayleigh-Jeans Approximation

$$ \begin{gather} \begin{aligned} V_n &= \sqrt{ \frac{ 4hfBR }{ e^{hf/kT} - 1} } \\ hf &\ll kT \Longrightarrow e^{hf/kT} - 1 \approx hf/kT \\ \therefore V_n &\approx \sqrt{ 4kTBR } \end{aligned} \end{gather} $$

Noise Power

• 计算传递到Load上的噪声能量

$$ \begin{gather} \begin{aligned} P_n &= ( \frac{ V_n }{ 2R })^2 R = kTB \\ \underset{ B \longrightarrow 0}{ lim } P_n &= 0 \\ \underset{ T \longrightarrow 0}{ lim } P_n &= 0 \\ \underset{ B \longrightarrow \infin }{ lim } P_n &= ? \end{aligned} \end{gather} $$

• 当$B \longrightarrow \infin$或者$T \longrightarrow \infin$， $V_n$必须使用原公式计算

Equivalent Noise Temperature

1. Definition

• 对于产生White Noise的设备，其噪声都可以等效成等效噪声温度下的电阻

$$ \begin{gather} \begin{aligned} T_e &= \frac{ N_o }{ kB } \end{aligned} \end{gather} $$

2. Equivalent Noise Temperature for a Noisy Amplifier

$$ \begin{gather} \begin{aligned} T_e &= \frac{ N_o }{ GkB } \end{aligned} \end{gather} $$

Excess Noise Ratio

• 对于一个分离设备而言

$$ \begin{gather} \begin{aligned} ENR(dB) &= 10 lg( \frac{ N_g - N_o }{ N_o }) \\ &= 10 lg( \frac{ T_g - T_o }{ T_o }) \end{aligned} \end{gather} $$

• 以上定义在一个系统中可能有所不同

Measurement of Noise Temperature (Y-Factor Method)

• 只有在$T_1 \gg T_2$时，该方法才是适用的

$$ \begin{gather} \begin{aligned} N_1 &= GkB(T_1 + T_e) \\ N_2 &= GkB(T_2 + T_e) \\ Y &= \frac{ N_1 }{ N_2 } \\ &= \frac{ T_1 + T_e }{ T_2 + T_e } > 1\\ \therefore T_e &= \frac{ T-1 - YT_2 }{ Y -1 } \end{aligned} \end{gather} $$

Noise Figure

Definition

在阻抗匹配，且温度$T = 290K$的情况下输入信噪比和输出信噪比的比值

$$ \begin{gather} \begin{aligned} F &= \frac{ S_i/N_i }{ S_o/N_o } \\ \end{aligned} \end{gather} $$

Noise Figure in Impedance Matched Cases

NF of a common 2-Port Network

$$ \begin{gather} \begin{aligned} N_i &= kT_0B \\ N_o &= GkB(T_0 + T_e) \\ S_o &= GS_i \\ F &= 1 + \frac{ T_e }{ T_0 } \ge 1 \\ T_e &= T_0(F-1) \end{aligned} \end{gather} $$

NF of a Lossy Line / Attenuator

• 定义Lossy Factor $L = \dfrac{ 1 }{ G }$

$$ \begin{gather} \begin{aligned} N_o &= kTB \\ &= GkTB + GN_{line} \\ \therefore N_{line} &= \frac{ 1 - G }{ G } kT B \\ \therefore T_e &= \frac{ N_{line} }{ kB } \\ &= \frac{ 1 - G }{ G } T \\ &= (L-1) T \\ F_{line} &= 1 + \frac{ (L-1) T }{ T_0 } \end{aligned} \end{gather} $$

• 当$T = T_0$时, $F_{line} = L$

Noise of a Cascaded System

1. Equivalent Noise Temperature & NF

$$ \begin{gather} \begin{aligned} N_1 &= G_1(N_i + N_{e1}) \\ &= G_1kB(T_0 + T_{e1}) \\ N_2 &= G_2(N_1 + N_{e2}) \\ &= G_1G_2 kB(T_0 + T_{e1}) + G_2 kBT_{e2} \\ &= G_1G_2kB( T_0 + T_{e1} + \frac{ 1 }{ G_1 } T_{e2}) \\ \therefore T_{cas} &= T_{e1} + \frac{ 1 }{ G_1 } T_{e2} \\ F &= F_1 + \frac{ 1 }{ G_1 } (F_2 - 1) \end{aligned} \end{gather} $$

2. Generalization

$$ \begin{gather} \begin{aligned} T_{cas} &= T_{e1} + \frac{ 1 }{ G_1} T_{e2} + \frac{ 1 }{ G_1 G_2 } T_{e3} + … \\ F &= F_1 + \frac{ 1 }{ G_1 } (F_2 - 1) + \frac{ 1 }{ G_1G_2 } (F_3 -1) + … \end{aligned} \end{gather} $$

Noise Figure in Mismatched Cases

• Lossy Factor $L$ 是能量损失

$$ \begin{gather} \begin{aligned} S &= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \frac{ e^{-j \beta l} }{ \sqrt{ L } } \\ \therefore \Gamma_{out} &= S_{22} + \frac{ S_{12} S_{21} \Gamma_S}{ 1 - S_{11} \Gamma_S} \\ &= \frac{ \Gamma_S }{ L }e^{-2j \beta l} \\ \therefore G_{21} &= \frac{ |S_{21}|^2 ( 1- | \Gamma_S|^2) }{ |1 - S_{11} \Gamma_S|^2 ( 1- | \Gamma_{out}|^2)} \\ &= \frac{ L ( 1 - | \Gamma_S|^2) }{ L^2 - | \Gamma_S|^2 } \\ T_e &= \frac{ 1 -G_{21} }{ G_{21} } T \\ &= \frac{ (L-1)(L + | \Gamma_S|^2) }{ L ( 1 - | \Gamma_S|^2) } T \end{aligned} \end{gather} $$

Noise Figure for a Mismatched Amplifier

$$ \begin{gather} \begin{aligned} N_o &= kT_0 GB( 1 - | \Gamma|^2) + kT_0 ( F - 1) GB \end{aligned} \end{gather} $$

Nonlinear Distortion

Causes & Output Signal

• 对输出信号进行Taylor’s Series展开

$$ \begin{gather} \begin{aligned} v_o &= a_0 + a_1 v_i + a_2 v_i^2 + … \\ a_0 &= v_o(0) & DC \\ a_1 &= \frac{ dV_o(0) }{ dt } & Linear \ Output \\ a_2 &= \frac{ dV^2(0) }{ dt^2 } & Squared \ Output \end{aligned} \end{gather} $$

Gain Compression

Output Voltage Expressions

$$ \begin{gather} \begin{aligned} v_o &= a_0 + a_1 V_0 \cos ( \omega_0 t) + a_2 V_0^2 cos^2( \omega_0 t) + a_3 V_0^3cos^3( \omega_0 t) \\ &= (a_0 + \frac{ 1 }{ 2 } a_2 V_0^2) + (a_1V_0 + \frac{ 3 }{ 4 } a_3 V_0^3) \cos(\omega_0 t) + \frac{ 1 }{ 2 } a_2V_0^2 cos(2 \omega_0 t) + \frac{ 1 }{ 4 } a_3V_0^3 cos(3 \omega_0 t) \end{aligned} \end{gather} $$

Gain at Center Frequency

$$ \begin{gather} \begin{aligned} G_v &= \frac{ v_o^{( \omega_0)} }{ v_i^{( \omega_0)} } \\ &= \frac{ a_1 V_0 + \frac{ 3 }{ 4 } a_3 V_0^3 }{ V_0 } \\ &= a_1 + \frac{ 3 }{ 4 } a_3 V_0^3 \end{aligned} \end{gather} $$

1dB Compression Point

$$ \begin{gather} \begin{aligned} OP_{1dB} &= IP_{1dB} + G - 1dB \end{aligned} \end{gather} $$

Harmonic & Intermodulation Distortion

Analysis of Two-Tone Input Signal

1. 公式推导

$$ \begin{gather} \begin{aligned} v_i &= V_0 \Big[ \cos ( \omega_1 t) + cos ( \omega_2 t)\Big] \\ v_o &= a_0 + a_1 V_0 ( \cos ( \omega_1 t) + \cos ( \omega_2 t)) + a_2 V_0^2 ( \cos \omega_1 t + \cos( \omega_2 t))^2 \\ & + a_3 V_0^3(\cos ( \omega_1 t) + \cos( \omega_2 t))^3 + …\\ &= \Big[ a_0 + a_2 V_0^2 \Big] \\ &+ \Big[ a_1V_0 + \frac{ 9 }{ 4 } a_3 V_0^3 \Big] \cos( \omega_1 t) + \Big[ a_1 V_0 + \frac{ 9 }{ 4 } a_3 V_0^3 \Big] \cos ( \omega_2 t) \\ & + \frac{ 1 }{ 2 } a_2 V_0^2cos(2 \omega_1 t) + \frac{ 1 }{ 2 } a_2V_0^2 \cos( 2 \omega_2 t) \\ &+ a_2 V_0^2 \cos \Big[ ( \omega_1 - \omega_2) t\Big] + a_2V_0^2 \cos \Big[ ( \omega_1 + \omega_2) t\Big] \\ &+ \frac{ 1 }{ 4 } a_3V_0^3 \cos( 3 \omega_1 t) + \frac{ 1 }{ 4 } a_3V_0^3 \cos( 3 \omega_2 t) \\ &+ \frac{ 3 }{ 4 } a_3 V_0^3 \Big[ \cos(2 \omega_1 - \omega_2) t \Big] + \frac{ 3 }{ 4 } a_3 V_0^3 \Big[ \cos(2 \omega_1 + \omega_2) t \Big] \\ &+ \frac{ 3 }{ 4 } a_3 V_0^3 \Big[ \cos(2 \omega_2 - \omega_1) t \Big] + \frac{ 3 }{ 4 } a_3 V_0^3 \Big[ \cos(2 \omega_2 + \omega_1) t \Big] \end{aligned} \end{gather} $$

2. 频率分布

Third-Order Intercept Point

1. $OIP_3$求解

• $P_{\omega_1}$指的是$\omega_1$(Desired Frequency)处的功率

$$ \begin{gather} \begin{aligned} P_{ \omega_1} &\approx \frac{ 1 }{ 2 } a_1^2 V_0^2 \\ P_{2 \omega_1 - \omega_2} &= \frac{ 1 }{ 2 } ( \frac{ 3 }{ 4 }a_3 V_0^3 )^2 \\ &= \frac{ 9 }{ 32 } a_3^2 V_0^6 \\ P_{ \omega_1} &= P_{2 \omega_1 - \omega_2} \\ \therefore V_{IP} &= \sqrt{ \frac{ 4a_1 }{ 3a_3 } } \\ \therefore OIP_3 &= \frac{ 2a_1^3 }{ 3a_3 } \end{aligned} \end{gather} $$

Intercept Point of a Cascaded System

$$ \begin{gather} \begin{aligned} P’{2\omega_1 - \omega_2} &= \frac{ P{ \omega_1}’^3 }{ (OIP_3’)^2 } \\ \therefore V’{2 \omega_1 - \omega_2}&= \sqrt{ P’{2 \omega_1 - \omega_2} Z_0 } \\ &= \frac{ \sqrt{ P_{ \omega_1}’^3 \cdot Z_0 } }{ OIP_3’ } \\ V’’{2 \omega_1 - \omega_2} &= \sqrt{ G_2 } V’{2 \omega_1 - \omega_2} + \frac{ \sqrt{ P_{ \omega_1}’^3 \cdot Z_0 } }{ OIP_3’ } \\ P_{2 \omega_1 - \omega_2}’’&= \Big[ \frac{ 1 }{ G_2 (OIP_3’) } + \frac{ 1 }{ OIP_3’’ } \Big]^2 P_{ \omega_1}’’^3 \\ &= \frac{ (P_{ \omega_1}’’)^3 }{ (OIP_3’’)^2 } \end{aligned} \end{gather} $$

• Intercept Point of a Cascaded System

$$ \begin{gather} \begin{aligned} OIP_3 = \begin{cases} \Big[ \dfrac{ 1 }{ G_2 (OIP_3’) } + \dfrac{ 1 }{ OIP_3’’ } \Big]^{-1} & Coherent \\ \Big[ \dfrac{ 1 }{ (G_2 (OIP_3’))^2 } + \dfrac{ 1 }{ (OIP_3’’)^2} \Big]^{-1/2} & Incoherent \end{cases}
\end{aligned} \end{gather} $$

Dynamic Range

Linear Dynamic Range

$$ \begin{gather} \begin{aligned} LDR(dB) &= OP_{1dB} - N_o \end{aligned} \end{gather} $$

Spurious-Free Dynamic Range

$$ \begin{gather} \begin{aligned} SFDR &= \frac{ P_{ \omega_1} }{ P_{2 \omega_1 - \omega_2} } \\ &= \Big( \dfrac{ OIP_3 }{ N_o } \Big)^{2/3} \\ SFDR(dB) &= \frac{ 2 }{ 3 } (OIP_3 - N_o) \end{aligned} \end{gather} $$