[Circuit] Noise Factor & Noise Figure

電路知識：Noise Factor & Noise Figure

簡介：Noise Factor & Noise Figure 概念

Noise voltage

V_{n} = \sqrt{4 k T R B}

k = 波茲曼常數

1.38 \cdot 10^{- 23} J o u l e s / K e l v i n

T = Temperature in Kelvin

K = 273 + ° C e l s i u s

R = 電阻阻值

Ω

B = noise 頻寬 (Hz)

因熱攪動導致導體內部的電子達到平衡狀態時的電子雜訊，故與所施加電壓無關
等效電路如下

By Sbyrnes321 - Own work, CC0, Link

計算範例
一個 100kΩ 的電阻，在常溫 27℃ 加上一個 1MHz 頻寬的 noise
請問 Noise voltage 為何？

\begin{aligned} V_{n} & = \sqrt{4 k T R B} \\ = \sqrt{4 \cdot 1.38 \cdot 10^{- 23} \cdot (273 + 27) \cdot 100 \cdot 10^{3} \cdot 1 \cdot 10^{6}} \\ = 40.7 u V RMS \end{aligned}

Noise Power

\begin{aligned} P & = k T B \\ P_{d B m} & = - 174 d B m + 10 l o g (B) (a t 300 K) \end{aligned}

k = 波茲曼常數

1.38 \cdot 10^{- 23} J o u l e s / K e l v i n

T = Temperature in Kelvin

K = 273 + ° C e l s i u s

B = noise 頻寬 (Hz)

如下，因電阻匹配可得到最大 Noise Power，故令

R_{L o a d = R}

\begin{aligned} P_{L o a d} & = \frac{1}{2} \cdot \frac{V_{n}^{2}}{2 R} \\ = \frac{4 k T R B}{4 R} \\ = k T B \end{aligned}

常溫 27℃ 下

\begin{aligned} P_{d B m} & = 10 l o g \frac{k T B}{1 m W} \\ = 10 l o g (\frac{1.38 \cdot 10^{- 23} \cdot 300 \cdot 1 H z}{1 \cdot 10^{- 3}}) + 10 l o g (B) \\ = - 173.829 d B m + 10 l o g (B) \\ \approx - 174 d B m + 10 l o g (B) \end{aligned}

計算範例
Ex. 1: 計算在常溫 27℃ 下，當 GSM 使用頻寬為 200kHz 時，請問其底噪是多少？

- 174 d B M + 10 l o g (200 \cdot 10^{3}) = - 121 d B m

Ex. 2: 計算在常溫 27℃ 下，當 SSB receiver 的頻寬為 2.4 kHz，請問其底噪是多少？

- 174 d B M + 10 l o g (2.4 \cdot 10^{3}) = - 140 d B m

Noise Factor

F = \frac{S N R_{i}}{S N R_{o}} = \frac{N_{o}}{N_{i} G} = 1 + \frac{N_{a}}{N_{i} G}

Noise Figure

N F = 10 l o g (F)

其實就是 Output Noise 經過 AMP 後，增加了多少 AMP 內部的 Noise

By Fvultier - Using Inkscape, CC BY-SA 4.0, Link

\begin{aligned} F & = \frac{S N R_{i}}{S N R_{o}} \\ = \frac{\frac{S_{i}}{N_{i}}}{\frac{S_{o}}{N_{o}}} \\ = \frac{\frac{S_{i}}{N_{i}}}{\frac{S_{i} G}{N_{o}}} \\ = \frac{N_{o}}{N_{i} G} \\ = \frac{N_{i} G + N_{a}}{N_{i} G} \\ = 1 + \frac{N_{a}}{N_{i} G} \end{aligned}

計算範例
請問經過此 AMP 後，Output Noise 為多少？

- 174 d B m + 10 d B + 3 d B = - 161 d B m

Cascaded Noise Factor

F_{t o t a l} = \frac{S N R_{i}}{S N R_{o}} = F_{1} + \frac{F_{2} - 1}{G_{1}} + \frac{F_{3} - 1}{G_{1} G_{2}}

By Fvultier - Using Inkscape, CC BY-SA 4.0, Link

\begin{aligned} F_{t o t a l} & = \frac{S N R_{i}}{S N R_{o}} \\ = \frac{\frac{S_{i}}{N_{i}}}{\frac{S_{o}}{N_{o}}} \\ = \frac{\frac{S_{i}}{N_{i}}}{\frac{S_{i} G_{1} G_{2} G_{3}}{N_{i} G_{1} G_{2} G_{3} + N_{a 1} G_{2} G_{3} + N_{a 2} G_{3} + N_{a 3}}} \\ = \frac{N_{i} G_{1} G_{2} G_{3} + N_{a 1} G_{2} G_{3} + N_{a 2} G_{3} + N_{a 3}}{N_{i} G_{1} G_{2} G_{3}} \\ = 1 + \frac{N_{a 1}}{N_{i} G_{1}} + \frac{N_{a 2}}{N_{i} G_{1} G_{2}} + \frac{N_{a 3}}{N_{i} G_{1} G_{2} G_{3}} \\ = \underset{F_{1}}{\underset{⏟}{1 + \frac{N_{a 1}}{N_{i} G_{1}}}} + \frac{1}{G_{1}} \underset{F_{2} - 1}{\underset{⏟}{\frac{N_{a 2}}{N_{i} G_{2}}}} + \frac{1}{G_{1} G_{2}} \underset{F_{3} - 1}{\underset{⏟}{\frac{N_{a 3}}{N_{i} G_{3}}}} \\ = F_{1} + \frac{F_{2} - 1}{G_{1}} + \frac{F_{3} - 1}{G_{1} G_{2}} \end{aligned}

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