Chapter 13 Techniques of Design-Oriented Analysis: The Feedback Theorem

Chapter 13 Techniques of Design-Oriented Analysis: The Feedback Theorem

从这一章开始讲负反馈Control系统和小信号建模.

13.2 The Feedback Theorem

首先介绍 Middlebrook’s Feedback Theorem

考虑下面负反馈系统

传输函数 G=uo/ui

G ( s ) = u o u i = G ∞ T 1 + T + G 0 1 1 + T G(s)=\frac{u_{o}}{u_{i}}=G_{\infty }\frac{T}{1+T}+G_{0}\frac{1}{1+T} G(s)=ui​uo​​=G∞​1+TT​+G0​1+T1​

T为Loop Gain 环路的增益

T ( s ) = u y ( s ) u x ( s ) ∣ u i = 0 T(s)=\frac{u_{y}(s)}{u_{x}(s)}\bigg|_{ui=0} T(s)=ux​(s)uy​(s)​ ​ui=0​

ideal forward gain 理想正向增益, G_inf为通过uz 消除(null) uy后, ui到uo的传输函数.

G_inf其实就是利用运放虚短和虚断来推导Vout/Vin

G ∞ ( s ) = u o ( s ) u i ( s ) ∣ u y → 0 G_{\infty }(s)=\frac{u_{o}(s)}{u_{i}(s)}\bigg|_{u_y\to 0} G∞​(s)=ui​(s)uo​(s)​ ​uy​→0​

当Loop Gain T-> inf时, G=G_inf

G0为通过uz 消除(null) ux后, ui到uo的传输函数

G 0 ( s ) = u o ( s ) u i ( s ) ∣ u x → 0 G_{0}(s)=\frac{u_{o}(s)}{u_{i}(s)}\bigg|_{u_x\to 0} G0​(s)=ui​(s)uo​(s)​ ​ux​→0​

当Loop Gain T-> 0时, G=G0

Null loop Gain Tn(s): 引入Uz来消除null uo(s)

T n ( s ) = u y ( s ) u x ( s ) ∣ u 0 → 0 T_n(s)=\frac{u_{y}(s)}{u_{x}(s)}\bigg|_{u_0\to 0} Tn​(s)=ux​(s)uy​(s)​ ​u0​→0​

T n ( s ) T ( s ) = G ∞ ( s ) G 0 ( s ) \frac{T_n (s)}{T(s)}=\frac{G_\infty (s)}{G_0(s)} T(s)Tn​(s)​=G0​(s)G∞​(s)​

13.3 Example: Op Amp PD Compensator Circuit

我们以下面负反馈op-amp为例

假设运放为单极点系统

G o p ( s ) = G o p 0 ( 1 + s ω 1 ) G_{op}(s)=\frac{G_{op0}}{(1+\frac{s}{\omega_1})} Gop​(s)=(1+ω1​s​)Gop0​​

Voltage injection模型为

Ideal forward gain: 其实就是利用运放虚短和虚断来推导Vout/Vin, 即G_inf

G ∞ ( s ) = v o u t ( s ) v i n ( s ) ∣ v y → 0 G_{\infty }(s)=\frac{v_{out}(s)}{v_{in}(s)}\bigg|_{v_y\to 0} G∞​(s)=vin​(s)vout​(s)​ ​vy​→0​

vy null to 0, 因此op输入端v-也被null to 0.

我们可以用运放的虚短和虚断特性来推导vout/vin. v- = v+ = 0即virtual ground

Loop Gain, T(s) 环路的增益.

T ( s ) = v o u t v x v − v o u t v y v − T(s)=\frac{v_{out}}{v_{x}}\frac{v{^-}}{v_{out}}\frac{v_y}{v{^-}} T(s)=vx​vout​​vout​v−​v−vy​​

前两项就是电阻电容的voltage divider传输函数, 第三项为Gop

G0为调节Vz, 从而Vx nulled to 0. 即运放输出为0

G 0 ( s ) = v o u t ( s ) v i n ( s ) ∣ v x → 0 G_{0}(s)=\frac{v_{out}(s)}{v_{in}(s)}\bigg|_{v_x\to 0} G0​(s)=vin​(s)vout​(s)​ ​vx​→0​

因此G0也是电阻电容的voltage divider传输函数

Tn为null output的loop gain

T n ( s ) = v y ( s ) v x ( s ) ∣ v o u t → 0 T_{n }(s)=\frac{v_{y}(s)}{v_{x}(s)}\bigg|_{v_{out}\to 0} Tn​(s)=vx​(s)vy​(s)​ ​vout​→0​

因此Loop Gain可推导为

T ( s ) = G 0 ( s ) T n ( s ) G ∞ ( s ) T(s)=\frac{G_{0}(s)T_{n}(s)}{G_{\infty }(s)} T(s)=G∞​(s)G0​(s)Tn​(s)​

最终Transfer Function, G= Vout/Vin

G ( s ) = v o u t v i n = G ∞ T 1 + T + G 0 1 1 + T G(s)=\frac{v_{out}}{v_{in}}=G_{\infty }\frac{T}{1+T}+G_{0}\frac{1}{1+T} G(s)=vin​vout​​=G∞​1+TT​+G0​1+T1​

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