漸近增益模型

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漸近增益模型（asymptotic gain model）^[1]^[2]，也稱為Rosenstark方法^[3]，是由以下漸近增益的關係來表示负反馈放大器的增益：

G=G_{\infty }\left({\frac {T}{T+1}}\right)+G_{0}\left({\frac {1}{T+1}}\right)\ ,

其中 $T$ 是輸入源無效時的返回比（在只有單一迴路，且由單輸入模組組成的系統時，等於負的环路增益），G_∞是漸近增益，G₀是直接傳遞項。此型式的增益可以提供對於電路一些直觀的直覺，而且比直接推導增益要容易計算。

圖1是漸近增益模型的方塊圖。漸近增益的關係也可以用信号流图表示，如圖二。漸近增益模型是外元素定理（英语：extra element theorem）的特例。

名詞定義

漸近增益G_∞是返回比趨近無限大時的系統增益：

G_{\infty }=G\ {\Big |}_{T\rightarrow \infty }\ ,

而直接傳遞項G₀也稱為前饋增益，是返回比為零時的系統增益：

G_{0}=G\ {\Big |}_{T\rightarrow 0}\ .

優點

此模型表示回授放大器的特性，包括負載效應以及放大器電路及回授電路的特質。
一般回授放大器的設計是使返回比T遠大於1。此時若假設直接傳遞項G₀很小（實際上多半也很小），系統的增益G會近似等於漸近增益G_∞。
漸近增益多半只是電路中被動元件的乘積，容易觀察出來。
此模型不需要先確認回授的拓樸（串聯－串聯型，串聯－分流型等），因為其分析方式是一樣的。

實現方式

使用此模型分析，可分為以下步驟：

選擇一個電路中的相依電源（英语：dependent source）。
找此電源的返回比。
將電路中的T趨近無限大，調整電路，找到增益G_∞
將電路中的T設定為為0，調整電路，找到增益 G₀。
將T, G_∞和 G₀代入漸近增益的公式中。

上述的步驟可以在SPICE中用小信號模型的手工分析求得。此作法中，已經可以找到設備的相依電源。相反的，若是用實際設備進行實驗，或是用數值產生的模型進行SPICE模擬，無法求得設備的相依電源，就需要透過其他方式來計算返回比。

和經典控制理論的關係

和經典回授控制理論中不考慮前饋項的影響（G₀），若省略前饋項，漸近增益模型的增益為

G=G_{\infty }{\frac {T}{1+T}}={\frac {G_{\infty }T}{1+{\frac {1}{G_{\infty }}}G_{\infty }T}}\ ,

在經典控制理論中，若開迴路增益用A來表示，則有回授時的增益（閉迴路增益）為：

A_{\mathrm {FB} }={\frac {A}{1+{\beta }_{\mathrm {FB} }A}}\ .

比較上述兩式，可以計算回授因素 β_FB：

\beta _{\mathrm {FB} }={\frac {1}{G_{\infty }}}\ ,

而開迴路增益為：

A=G_{\infty }\ T\ .

若其準確度足夠，上式公式是另外一個計算T的方式：計算開迴路增益以及G_∞，用此式來計算T。一般這兩項會比T容易計算。

參考資料

^ Middlebrook, RD: Design-oriented analysis of feedback amplifiers; Proc. of National Electronics Conference, Vol. XX, Oct. 1964, pp. 1–4
^ Rosenstark, Sol. Feedback amplifier principles. NY: Collier Macmillan. 1986: 15. ISBN 0-02-947810-3.
^ Palumbo, Gaetano & Salvatore Pennisi. Feedback amplifiers: theory and design. Boston/Dordrecht/London: Kluwer Academic. 2002: §3.3 pp. 69–72. ISBN 0-7923-7643-9.
^ Paul R. Gray, Hurst P J Lewis S H & Meyer RG. Analysis and design of analog integrated circuits Fourth. New York: Wiley. 2001. Figure 8.42 p. 604 [2019-09-06]. ISBN 0-471-32168-0. （原始内容存档于2009-04-28）.

外部連結

Lecture notes on the asymptotic gain model （页面存档备份，存于互联网档案馆）

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