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## Feedback

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**Feedback**Section 8.1**Topics**• General Feedback • Examples of Feedback Circuits • Bandwidth Extension • Gain Sensitivity • Input and Output Impedance • Types of Amplifiers**General Feedback System**H(s)=Feedforward network, represents an amplifier, “Open-loop” transfer function, a.k.a. A G(s)=Feedback network, usually frequency independent, a.k.a, β. Y(s)/X(s)=“Closed-loop” transfer function**Behavior of a well-designed Negative Feedback System**The error term is minimized! An accurate copy of the input. The input of H(s) is a virtual ground because the error term is minimized!**Elements of Feedback System**• Feed forward amplifier • Means of Sensing the Output • Feedback network • Means of Generating Feedback Error C2 senses the output voltage, converts it to a current feedback signal, which is added tot eh current produced by Vin through C1.**Common-Source Stage**Poor definition of the gain (gmro): Both gm and ro depend vary with process and temperature.**Gain Desensitization Example**Assumption: The frequency is low enough that C2 does not load the output node and CGS behaves as an open circuit. gmro is sufficiently large Bias of the gate is not shown! Results: If C1 and C2 are made of the same material, then Process and temperature variations do not change C1/C2.**Gain Desensitization**(Taylor series expansion) The closed loop gain is approximately, 1/β. If Aβ is sufficiently large, Y/X is relatively insensitive to variations of A. Aβ is called the loop gain.**Calculation of Loop Gain**The input is set to 0. The negative of the transfer function is the loop gain. VF=-βAVt**Loop Gain Calculation Example**The loading of C2 is neglected!**Common Gate Loop Gain Example**Feed forward: M1 and RD Feedback: R1 and R2 R1+R2>>RD M1 operates as a subtractor. (Loop gain)**Bandwidth Modification**Gain is reduced by 1+βAo. Bandwidth is increased by 1+βAo**Achieving High gain and High Bandwidth**1. Apply feedback to Improve speed of each amplifier 2. Cascade to improve gain!**Input Impedance Modification Example**Loop is opened!**Calculation of Loop Gain**(loop gain)**Feedback Mechanism**(Feed forward amplifier) Subtraction occurs in the current domain at the Input terminal.**Output Impedance Modification**Senses Vout Return a current**Calculation of Output Resistance at Low Frequencies**(Loop gain)**Realistic Model**Transimpedance (Voltage) Transimpedance Transconductance Current amplifier More than one model is possible.**Examples of Four Types of Amplifiers**(Suffers from large output impedacne)**Voltage Sensing**R1+R2 must be large enough so that A1 does feel the effect of the resistive divider. How do you measure voltage across a port? Place a voltmeter in parallel Use a voltmeter of high input impedance so It does not disturb the circuit.**Current Sensing**(Current Meter resistance) Current is sensed by measuring voltage across r. (Implementation) RS is ideally small.**Return Voltage to the Input**(Single-ended Implementation) (Differential pair implementation)**Return Current to the Input**(KCL) (KCL) (Use RF large enough to approximate current Source)**Example**(Sensing Achieves via R1 and R2) Differential pair performs voltage subtraction.**Input Impedance of Ideal Feedback Networks**(Voltage Sensing) (Current Sensing)**Output Impedance of Ideal Feedback Networks**(Return Voltage) (Return Current)**Polarity of Feedback**• Assume the input signal goes up or down • Follow the change through the forward amplifier and the feedback network • Determine whether the returned quantity opposes or enhances the original effect produced by the input change.