Chapter 9. Feedback

Contents Introduction The General Feedback Structure Some Properties of Negative Feedback The Four Basic Feedback Topologies The Feedback Voltage Amplifier (Series—Shunt) The Feedback Trans-conductance Amplifier (Series—Series) The Feedback Trans-resistance Amplifier (Shunt—Shunt) The Feedback Current Amplifier (Shunt—Series) Summary of the Feedback Analysis Method Determining the Loop Gain The Stability Problem Effect of Feedback on the Amplifier Poles Stability Study Using Bode Plots Frequency Compensation

Introduction Negative feedback의 장점 이득의 둔감화 : 증폭기 이득이 부품 값 의 변화나 온도에 둔감해짐 비선형 왜곡의 감소 잡음의 영향을 줄이기 입력 저항과 출력 저항을 제어함 증폭기의 대역폭 확대 Negative feedback의 장점은 증폭기 이득의 감소를 낳는다.

1. The General Feedback Structure β : feedback factor Aβ : Loop gain Af : closed-loop gain A : Open-loop gain 1+Aβ : Amount of feedback factor

Example 9.1

Assume that the op amp has infinite input resistance and zero output resistance. Find an expression for the feedback factor β. decrease in Af. Find the condition under which the closed-loop gain is almost entirely determined by the feedback network. (c) If the open-loop gain 104 V/V, find R2/R1 to obtain a closed-loop gain of 10 V/V. (d) What is the amount of feedback in decibels? (e) If Vs =1 V, find Vo, Vf, and Vi. (f) If A decreases by 20%, what is the corresponding decrease in Af.

2. Some Properties of Negative Feedback 2.1 Gain de-sensitivity 2.2 Bandwidth extension

2.3 Interference reduction (a) Open-loop system (b) Closed-loop system

Reducing power supply hum by feedback 정류기의 ripple (스위치가 1위치) (스위치가 2위치)

2.4 Reduction in nonlinear distortion

3. The four basic feedback topologies (a) Voltage amplifiers (series–shunt feedback) (b) Current amplifiers (shunt–series feedback) (c) Trans-conductance amplifiers(series–series feedback) (d) Trans-resistance amplifiers (shunt–shunt)

3.1 Voltage Amplifiers Examples of feedback voltage amplifiers

3.2 Current amplifiers Examples of feedback current amplifiers

Example 9.2 For the feedback current amplifier shown in the figure, find expressions for the open-loop gain A ≡ Io/Ii, the feedback factor β = If/Io, and the closed-loop gain Af=Io/Is. For simplicity, neglect the Early effect in Q1 and Q2.

3.3 Trans-conductance amplifiers Examples of feedback trans-conductance amplifiers

3.4 Trans-resistance amplifiers Examples of feedback trans-resistance amplifiers

4.1 The Feedback Voltage Amplifier (Series–Shunt) (a) The ideal series–shunt feedback amplifier (b) equivalent circuit.

amp feedback

Series–Shunt feedback solution by hybrid parameters

Shunt–Shunt feedback solution by admittance parameters

Basic amplifier Basic amplifier Basic amplifier Basic amplifier

Basic amplifier Basic amplifier Basic amplifier Basic amplifier

4.2 The Practical Case hybrid parameter

hybrid parameter + + - -

Summary of the rules for the series–shunt case Summary of the rules for finding the A circuit and β for the series–shunt case of Fig. 10.14(a).

Example 9.3 The op amp has an open loop gain μ, a differential input resistance Rid, and an output resistance ro. Find expressions for A, β, the closed-loop gain , the input resistance Rin, and the output resistance Rout. Also find numerical values, given μ = 104, Rid = 100 kΩ, ro = 1 kΩ, RL = 2kΩ, R1 = 1 kΩ, R2 = 1 MΩ, and Rs = 10 kΩ.

Example 9.4 Find voltage gain Vo/Vs, input resistance Rin, output resistance Rout. (Neglect ro).

5. The Feedback Trans-conductance Amplifier (Series–Series) 5.1 The Ideal Case

5.2 The practical case

Impedance parameters + + + - + - - -

Summary of the rules for the series–series case

Example 9.5 Determine its closed-loop gain Af = Io/Vs, the input resistance Rin, the output resistance Rout.

Example 9.6 The circuit shown (called a feedback triple) is composed of three gain stages with series–series feedback provided by the network composed of RE1, RF, and RE2. Assume that the bias circuit, which is not shown, establishes Ic1 = 0.6 mA, Ic2 = 1mA, and Ic3 = 4 mA. Also assume that for all three transistors, hfe = 100 and ro = ∞. Anticipating that the loop gain will be large, find an approximate expression and value for the closed-loop gain Af ≡ Io/Vs and hence for Ic/Vs. Also find Vo/Vs. Use feedback analysis to find A, β, Af, Vo/Vs, Rin and Rout For the calculation of Rout assume that ro of Q3 is 25 kΩ.

(a) When Aβ >>1

(b)

9.6 The Feedback Trans-resistance Amplifier (Shunt–Shunt) 6.1 The Ideal Case

6.2 The practical case

Summary of the rules for the shunt-shunt case

Example 9.7 If the loop gain is large, find an approximate expression for the closed-loop open-circuit trans-resistance of the feedback amplifier. Find the A circuit and expressions for A, Ri, and Ro. Find expressions for the loop gain, Af, Rif, Rin, Rof, and Rout. Find the values of Ri, Ro, A, β, Af, Rif, Rin, Rof and Rout for the case μ = 104 V/V, Rid = ∞, ro = 100Ω, RF=10kΩ and Rs = RL = 1kΩ. If instead of a current source Is having a source resistance Rs = 1kΩ, the amplifier is fed from a voltage source Vs having a source resistance Rs = 1kΩ, find an expression for and the value of the voltage gain Vo/Vs.

(a) When Aβ >>1

(b) (c)

Measurement of Y-parameters The computation of the parameters follows directly from the definition + + - -

9.7 The Feedback Current Amplifier (Shunt–Series) 7.1 The Ideal Case

7.2 The practical case

Example 9.8 Assume that the amplifier μ has an input resistance Rid, an open-circuit voltage gain μ, and an output resistance ro1. (a) If the loop gain is large, find an approximate expression for the closed-loop gain Af ≡ Io/Is. (b) Find the A circuit and derive expressions for A, Ri, and Ro (c) Give expressions for Aβ, Af, Rif, Rin, Rof, and Rout. (d) Find numerical values for Aβ, Af, Rif, Rin, Rof, and Rout for the following case:

(a) Aβ >> 1, Af ≈1/ β

(b)

(c)

9. Determining the Loop Gain (Aβ) Loop gain을 구하기 위해 피드백 루프를 절단한 후 그 부분에 전압원 (Vt)를 연결하고 반대쪽 절단부에 유도되는 전압(Vr)을 계산/측정하여 비율을 구하면 된다. 이 때 반대쪽 절단부는 절단 이전의 임피던스 Zt로 종단한다. 이 때 loop gain은 아래 식과 같다 다른 방법으로는 종단부를 open으로 처리하고 전달 함수를 구하여 Toc를 구하고, 종단부를 short로 처리하여 전달함수Tsc를 구한 후 아래 식을 이용하면 된다.

9.2 Equivalence of Circuits from a Feedback-Loop Point of View

10. The Stability Problem 분모가 0이 되면 unstable. 안정도 판별 기준 :

10.2 The Nyquist criterion Aβ를 주파수를 바꿔가며 복소 평면에 표시했을 때 Stable : 위상이 180도일 때 | Aβ|<1 Unstable : 위상이 180도일 때 | Aβ|>1 Figure 9.34 The Nyquist plot of an unstable amplifier.

11. Effect of Feedback on the Amplifier Poles

12. Stability Study Using Bode Plots Gain margin : Aβ의 위상이 180도일 때 절대값이 1보다 작은 정도 Phase margin : Aβ의 절대값이 1일 때 위상이 180에서 떨어져 있는 정도. 최소 45도 이상이 되어야 함. Figure 9.42 Bode plot for the loop gain Aβ illustrating the definitions of the gain and phase margins

12.3 An Alternative Approach for Investigating Stability (β의 위상은 주파수에 무관하다고 가정) (20log10A 는 고정이고 1/β를 변화시키며 차이를 확인)

13. Frequency Compensation Pole 추가

13.3 Miller compensation and pole splitting Cf가 없을 때 : Cf가 있는 경우: fp1은 더 내려가고 fp2는 더 올라감 (pole splitting)