Feedback System Analysis

This article intends to deal with the analysis of feedback systems.

General Consideration

A negative feedback system is as follows

$Y(s)=H(s)[X(s)-G(s)Y(s)]$

Reorganize the items, we have

$\frac{Y(s)}{X(s)}=\frac{H(s)}{1+G(s)H(s)}$

$H(s)$ is called the open-loop gain, while $H(s)G(s)$ is called the loop gain, and $Y(s)/X(s)$ is called the closed-loop transfer function. The loop gain can be obtained by

Set the main input to zero
Break the loop at some point
Inject a test signal in the correct direction
Follow the loop and obtain the value that returns to the break point
The negative of the transfer function is the loop gain

Assume $H(s)=A, G(s)=\beta$ , we have

$\frac{Y}{X}=\frac{A}{1+\beta A}\approx\frac{1}{\beta}\left(1-\frac{1}{\beta A}\right)$

Thus, for negative-feedback system, as long as $\beta A$ is large enough, the closed loop gain is determined by the feedback coefficient.

Bandwidth: for one pole gain $A(s)=\frac{A_0}{1+s/\omega_0}$ , the transfer function is $Y(s)/X(s)=\frac{A_0/(1+\beta A_0)}{1+s/[(1+\beta A_0)\omega_0]}$ , and thus the bandwidth is extended by the loop gain, leaving the GBW unchanged.
Nonlinearity: reduced

Types of Amplifiers

Different Types of Amplifiers

Corresponding implementation

Amplifier Implementation

Sensing and Return Mechanism

Input and Output Impedance

Voltage-Voltage Feedback

For the output impedance

Voltage-Voltage Feedback, Output Impedance

Assume that feedback network draws no current

$Z_{out}=\frac{R_{out}}{1+\beta A_0}$

For the input impedance

Voltage-Voltage Feedback, Input Impedance

$Z_{in}=R_{in}(1+\beta A_0)$

Current-Voltage Feedback

$\frac{I_{out}}{V_{in}}=\frac{G_m}{1+G_mR_F}$

For the output impedance

Current-Voltage Feedback, Output Impedance

$Z_{out}=R_{out}(1+G_mR_F)$

For the input impedance

Current-Voltage Feedback, Input Impedance

$Z_{in}=R_{in}(1+G_mR_F)$

Voltage-Current Feedback

$\frac{V_{out}}{I_{in}}=\frac{R_0}{1+g_{mF}R_0}$

Voltage-Current Feedback, Impedance Calculation

For the output impedance

$Z_{out}=\frac{R_{out}}{1+g_{mF}R_0}$

For the input impedance

$Z_{in}=\frac{R_{in}}{1+g_{mF}R_0}$

This topology has application in fiber optic receivers, where light is converted to electrical current through a reverse-biased diode. A transimpedance amplifier (TIA) is adopted to convert this current signal into voltage signal.

Current-Current Feedback

$\frac{I_{out}}{I_{in}}=\frac{A_I}{1+\beta A_I}$

For the output impedance

$Z_{out}=R_{out}(1+\beta A_I)$

For the input impedance

$Z_{in}=\frac{R_{in}}{1+\beta A_I}$

Limitations of the Simple Analysis

What is loading effect?

The non-ideal input and output impedance of the feedback network could degrade the open-loop gain.

Feedback Analysis Difficulties

Possible methods

Possible Analysis Methods

Two Port Models

Z, Y, H, and G Models

$\left\{ \begin{aligned} V_1=Z_{11}I_1+Z_{12}I_2\\ V_2=Z_{21}I_1+Z_{22}I_2 \end{aligned} \right.$

$\left\{ \begin{aligned} I_1=Y_{11}V_1+Y_{12}V_2\\ I_2=Y_{21}V_1+Y_{22}V_2 \end{aligned} \right.$

$\left\{ \begin{aligned} V_1=H_{11}I_1+H_{12}V_2\\ I_2=H_{21}I_1+H_{22}V_2 \end{aligned} \right.$

$\left\{ \begin{aligned} I_1=G_{11}V_1+G_{12}I_2\\ V_2=G_{21}V_1+G_{22}I_2 \end{aligned} \right.$

Voltage-Voltage Feedback

Voltage-Voltage Feedback, using G Model

We have

$\begin{aligned} \frac{V_{out}}{V_{in}}&=\frac{A_0}{(1+\frac{g_{22}}{Z_{in}})(1+g_{11}Z_{out})+g_{21}A_0}\\ &=\frac{A_{v,open}}{1+g_{21}A_{v,open}} \end{aligned}\\ A_{v,open}=\frac{A_0}{(1+\frac{g_{22}}{Z_{in}})(1+g_{11}Z_{out})}\\ g_{11}=\frac{I_1}{V_1}\bigg|_{I2=0}\\ g_{22}=\frac{V_1}{I_1}\bigg|_{V1=0}$

For that matter, the $A_{v,open}$ should be calculated using the configuration

Voltage-Voltage Feedback, Open-Loop Gain Calculation

Summary

To calculate the gain with the loading effect, the procedure is as follows:

Open the loop with proper loading and calculate the open-loop gain $A_{OL}$
Determine the feedback ratio $\beta$
Calculate the closed-loop gain with $A_{OL}/(1+\beta A_{OL})$

Summary of Open-Loop Gain Calculation

Bode's Analysis

$\left\{ \begin{aligned} V_{out}&=AV_{in}+BI_1\\ V_1&=CV_{in}+DI_1 \end{aligned} \right.$

Assume $I_1=g_mV_1$ , we have

$\frac{V_{out}}{V_{in}}=A+\frac{g_mBC}{1-g_mD}$

The return ratio (RR) is defined as $-g_mD$ .

The procedure

Disable the transistor and by setting $g_m=0$ and obtain A and C
Set input to zero and calculate B and D
Solve the $A_v$ with above equations

Blackman's Theorem

$\left\{ \begin{aligned} V_{in}&=AI_{in}+BI_1\\ V_1&=CI_{in}+DI_1 \end{aligned} \right.$

It follows that

$\frac{V_{out}}{I_{in}}=A+\frac{g_mBC}{1-g_mD}$

If we define two quantities

$T_{oc}=-g_m\frac{V_1}{I_1}\bigg|_{I_{in}=0}=-g_mD\\ T_{sc}=-g_m\frac{V_1}{I_1}\bigg|_{V_{in}=0}=-g_m\frac{AD-BC}{A}$

And therefore

$Z_{in}=\frac{V_{out}}{I_{in}}=A\frac{1+T_{sc}}{1+T_{oc}}$

Stability Consideration

Consider again the closed-loop transfer function

$\frac{Y(s)}{X(s)}=\frac{H(s)}{1+G(s)H(s)}$

If $G(s)H(s)=-1$ , the gain goes to infinity. This is the condition of instability, which can be equivalently expressed as

$|\beta H(j\omega_1)|=1\\ \angle\beta H(j\omega_1)=-180^\circ$

which are called “Barkhausen’s Criteria.” We introduce several notions:

Gain Crossover Frequency: The frequency when open-loop gain reduces to 1 (0 dB), denoted as GX.
Phase Crossover Frequency: The frequency when the phase of open-loop gain reduces to $-180^\circ$ , denoted as PX.
Phase Margin: Defined as $180^\circ+\angle\beta H (f=GX)$ , denoted as PM.

If $\text{PM}=60^\circ$ , we have

$\left|\frac{Y(s)}{X(s)}\right|=\frac{H(s)}{\left|1+\exp(-j\frac{2\pi}{3})\right|}=H(s)=\frac{1}{G(s)}$

There will be negligible frequency peaking.