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Chapter 8 Operational Amplifier as A Black Box 8.1 General Considerations 8.2 Op-Amp-Based Circuits 8.3 Nonlinear Functions 8.4 Op-Amp Nonidealities 8.5 Design Examples 1 Chapter Outline CH8 Operational Amplifier as A Black Box 2 Basic Op Amp Vout A0 Vin1 Vin 2 Op amp is a circuit that has two inputs and one output. It amplifies the difference between the two inputs. CH8 Operational Amplifier as A Black Box 3 Inverting and Non-inverting Op Amp If the negative input is grounded, the gain is positive. If the positive input is grounded, the gain is negative. CH8 Operational Amplifier as A Black Box 4 Ideal Op Amp Infinite gain Infinite input impedance Zero output impedance Infinite speed CH8 Operational Amplifier as A Black Box 5 Virtual Short Vin1 Vin2 Vout A0 (Vin1 Vin 2 ) Vout finite, A 0 수천 (Vin1 Vin 2 ) Vout 0 A0 Due to infinite gain of op amp, the circuit forces Vin2 to be close to Vin1, thus creating a virtual short. CH8 Operational Amplifier as A Black Box 6 Unity Gain Amplifier Vout A0 (Vin Vout ) Vout A0 Vin 1 A0 CH8 Operational Amplifier as A Black Box 7 Op Amp with Supply Rails To explicitly show the supply voltages, VCC and VEE are shown. Ex) VCC=+5V, VEE=-5V; VCC=+5V, VEE=0V (general) CH8 Operational Amplifier as A Black Box 8 Noninverting Amplifier (Infinite A0) Vout R1 1 Vin R2 A noninverting amplifier returns a fraction of output signal thru a resistor divider to the negative input. With a high Ao, Vout/Vin depends only on ratio of resistors, which is very precise. CH8 Operational Amplifier as A Black Box 9 Noninverting Amplifier (Finite A0) Ex 8.3 Vin1 Vin , Vout Ao (Vin1 Vin 2 ) Vin 2 Vout R2 R1 R2 Vout Vin 1 A0 R 1 1 1 R2 R2 1 R1 R2 1 A0 R1 R2 R2 A0 R1 R1 1 1 1 1 R R 2 2 A0 The error term indicates the larger the closed-loop gain, the less accurate the circuit becomes. CH8 Operational Amplifier as A Black Box 10 Extreme Cases of R2 (Infinite A0) If R2 is zero, the loop is open and Vout /Vin is equal to the intrinsic gain of the op amp. If R2 is infinite, the circuit becomes a unity-gain amplifier and Vout /Vin becomes equal to one. CH8 Operational Amplifier as A Black Box 11 Inverting Amplifier 0 Vout Vin R1 R2 Virtual ground: + 와 –node 의 전 압 이 같아서 ground(0V)로 보인다는 말임 Vout R1 Vin R2 Infinite A0 forces the negative input to be a virtual ground. CH8 Operational Amplifier as A Black Box 12 Another View of Inverting Amplifier Inverting CH8 Operational Amplifier as A Black Box Noninverting 13 Gain Error Due to Finite A0 Vin1 0, Vout Ao (Vin1 Vin 2 ) AoV X Vin VX V Vout X R2 R1 Vout R 1 Vin R2 R1 R2 1 1 R1 1 1 A0 R2 1 R1 1 1 A R 0 2 The larger the closed loop gain, the more inaccurate the circuit is. CH8 Operational Amplifier as A Black Box 14 Complex Impedances Around the Op Amp Vout Z1 Vin Z2 The closed-loop gain is still equal to the ratio of two impedances. CH8 Operational Amplifier as A Black Box 15 Integrator Vout 1 Vin R1C1s CH8 Operational Amplifier as A Black Box Vin dV 1 C1 out Vout Vin dt R1 dt R1C1 16 Integrator with Pulse Input 1 V1 Vout Vin dt t 0 t Tb R1C1 R1C1 CH8 Operational Amplifier as A Black Box 17 Comparison of Integrator and RC Lowpass Filter The RC low-pass filter is actually a “passive” approximation to an integrator. With the RC time constant large enough, the RC filter output approaches a ramp. CH8 Operational Amplifier as A Black Box 18 Lossy Integrator Vout 1 Vin 1 1 1 R1C1s A0 A0 When finite op amp gain is considered, the integrator becomes lossy as the pole moves from the origin to 1/[(1+A0)R1C1]. It can be approximated as an RC circuit with C boosted by a factor of A0+1. CH8 Operational Amplifier as A Black Box 19 Differentiator Vout dVin R1C1 dt CH8 Operational Amplifier as A Black Box Vout R1 R1C1s 1 Vin C1s 20 Differentiator with Pulse Input Vout R1C1V1 (t ) CH8 Operational Amplifier as A Black Box 21 Comparison of Differentiator and High-Pass Filter The RC high-pass filter is actually a passive approximation to the differentiator. When the RC time constant is small enough, the RC filter approximates a differentiator. CH8 Operational Amplifier as A Black Box 22 Lossy Differentiator Vout R1C1s Vin 1 1 R1C1s A0 A0 Finite op amp gain의 경우, 미분기는 원점 zero에서 – (A0+1)/R1C1의 pole을 추가 RC HPF와 비교하여 R R/(A0+1)로 근사화 가능. CH8 Operational Amplifier as A Black Box 23 Op Amp with General Impedances Vout Z1 1 Vin Z2 This circuit cannot operate as ideal integrator or differentiator. CH8 Operational Amplifier as A Black Box 24 Voltage Adder Vout Ao Vout V1 V2 RF R1 R2 RF V1 V2 R If R1 = R2=R If Ao is infinite, X is pinned at ground, currents proportional to V1 and V2 will flow to X and then across RF to produce an output proportional to the sum of two voltages. CH8 Operational Amplifier as A Black Box 25 Precision Rectifier OP AMP 1)Vin 0 D1 순방향 FB loop(on) Vin1 Vin 2 ,VY VD,on Vout 2)Vin 0 D1 역방향 FB loop(off) VY A0 (Vin1 Vin 2 ) CH8 Operational Amplifier as A Black Box 26 Inverting Precision Rectifier (input=Vin,output=Vx) OP AMP 1)Vin 0 D1 F.B. FB loop(O) Vin1 Vin 2 0 V X 2)Vin 0 D1 R.B. FB loop(X) VX Vin ,VY A0VX CH8 Operational Amplifier as A Black Box 27 Logarithmic Amplifier VX 0 I C I s eVBE / VT VBE Vout Vout VT ln Vin R1 I S By inserting a bipolar transistor in the loop, an amplifier with logarithmic characteristic can be constructed. This is because the current to voltage conversion of a bipolar transistor is a natural logarithm. CH8 Operational Amplifier as A Black Box 28 Square-Root Amplifier VX 0 1 W nCox VGS VTH 2 2 L Vout I DS VGS Vout 2Vin VTH W nCox R1 L By replacing the bipolar transistor with a MOSFET, an amplifier with a square-root characteristic can be built. This is because the current to voltage conversion of a MOSFET is square-root. CH8 Operational Amplifier as A Black Box 29 Op Amp Nonidealities: DC Offsets by mismatch Wafer 제작과정에서 생길 수 있는 length, oxide thickness등등의 차이 CH8 Operational Amplifier as A Black Box 30 Op Amp Nonidealities: DC Offsets Offsets in an op amp that arise from input stage mismatch cause the input-output characteristic to shift in either the positive or negative direction (the plot displays positive direction). CH8 Operational Amplifier as A Black Box 31 Effects of DC Offsets Vout R1 1 Vin Vos R2 As it can be seen, the op amp amplifies the input as well as the offset, thus creating errors. CH8 Operational Amplifier as A Black Box 32 Saturation Due to DC Offsets Since the offset will be amplified just like the input signal, output of the first stage may drive the second stage into saturation. CH8 Operational Amplifier as A Black Box 33 Offset in Integrator t Vos 1 Vout Vos Vos dt Vos t R1C1 0 R1C1 시간이 지나면 출력전압이 계속 커져서 전원전압까지 올라감 CH8 Operational Amplifier as A Black Box 34 Offset in Integrator 출력전압이 계속 커지는 것을 막아줌 R2 Vout Vos 1 R1 Vout R2 1 Vin R1 R2C1s 1 A resistor can be placed in parallel with the capacitor to “absorb” the offset. However, this means the closed-loop transfer function no longer has a pole at origin. CH8 Operational Amplifier as A Black Box 35 Input Bias Current The effect of bipolar base currents can be modeled as current sources tied from the input to ground. CH8 Operational Amplifier as A Black Box 36 Effects of Input Bias Current on Noninverting Amplifier R1 Vout R2 I B 2 R1 I B 2 R2 It turns out that IB1 has no effect on the output and IB2 affects the output by producing a voltage drop across R1. CH8 Operational Amplifier as A Black Box 37 Input Bias Current Cancellation R1 Vout Vcorr 1 I B 2 R1 R2 We can cancel the effect of input bias current by inserting a correction voltage in series with the positive terminal. In order to produce a zero output, Vcorr=-IB2(R1||R2). CH8 Operational Amplifier as A Black Box 38 Correction for Variation I B1 I B 2 Current offset이 없는 경우에 위와 같이 전류가 흐른는 경우에도 출력으로 DC offset이 생기게 되는데, OP-AMP의 +node에서 바라본 저항과 –node에서 바라본 저항이 같아지도록 하면 제거된다. Since the correction voltage is dependent upon , and varies with process, we insert a parallel resistor combination in series with the positive input. As long as IB1= IB2, the correction voltage can track the variation. CH8 Operational Amplifier as A Black Box 39 Effects of Input Bias Currents on Integrator Vout 1 R1C1 I B 2 R1 dt Input bias current will be integrated by the integrator and eventually saturate the amplifier. CH8 Operational Amplifier as A Black Box 40 Integrator’s Input Bias Current Cancellation 앞과 같은 방법으로 저항 추가 By placing a resistor in series with the positive input, integrator input bias current can be cancelled. However, the output still saturates due to other effects such as input mismatch, etc. CH8 Operational Amplifier as A Black Box 41 Speed Limitation Vout A0 s s Vin1 Vin 2 1 1 Open loop Gain(feedback 회로를 연결하지 않은)의 주파수 특성 Due to internal capacitances, the gain of op amps begins to roll off. CH8 Operational Amplifier as A Black Box 42 Continued : Gain X bandwidth = constant Closed loop Gain(feedback 회로를 연결된)의 주파수 특성 Vout Vin 1 A0 Ao , A0 ( s ) R2 1 1 / 1 A0 R1 R2 Vout ( s) Vin 1 A0 ( s ) A0 R2 s R2 A0 ( s ) A 1 R1 R2 1 R1 R2 0 A0 R2 A0 1 R2 A0 R R2 R1 R2 , Gain( DC ) 1 , BW 1 A0 1 s R2 R2 R1 R2 1 A0 1 R1 R2 R2 1 A0 1 R1 R2 Gain BW A01 OP-AMP를 이용해서 만든 회로는 OPAMP의 한계를 넘어서는 특성을 절대로 보일 수 없다. 43 Bandwidth and Gain Tradeoff Having a loop around the op amp (inverting, noninverting, etc) helps to increase its bandwidth. However, it also decreases the low frequency gain. CH8 Operational Amplifier as A Black Box 44 Slew Rate of Op Amp # step 입력에 대해서 출력이 step출력이 나오지 않게 됨: OP-AMP가 가지고 있는 최대 단위 시간당 전압의 변화를 slew rate라고 함. 1)step입력에 대해서 M2가 off되고, M4가 공급해 주는 전류에 의해서, Vout이 High로 올라 간다. 출력전압은 capacitor에 공급되는 전류에 의해서 High로 올라가기 때문에, linear하게 커지는 출력전압이 나온다. 2)step입력에 대해서 M1이 off되고, M2가 빼내는 전류에 의해서, Vout이 Low로 떨어진다. 출력전압은 capacitor에서 빼내는 전류에 의해서 Low로 떨어지기 때문에, linear하게 떨어지는 출력전압이 나온다. # Slew rate한계를 가지는 OP-AMP를 사용한 회로: 출력이 단위시간당 전압의 변화가 OPAMP의 slew rate를 넘게 되는 경우에 출력 파형이 입력파형과 다르게 나옴. A) 출력 전압의 큰 변화가 생기는 경우, B) 주파수가 높은 경우 CH8 Operational Amplifier as A Black Box 45 Slew Rate of Op Amp: 전압의 큰 변화 In the linear region, when the input doubles, the output and the output slope also double. However, when the input is large, the op amp slews so the output slope is fixed by a constant current source charging a capacitor. This further limits the speed of the op amp. CH8 Operational Amplifier as A Black Box 46 Comparison of Settling with and without Slew Rate As it can be seen, the settling speed is faster without slew rate (as determined by the closed-loop time constant). CH8 Operational Amplifier as A Black Box 47 Slew Rate Limit on Sinusoidal Signals: 주파수가 높은 경우 dVout R1 V0 1 cos t dt R2 As long as the output slope is less than the slew rate, the op amp can avoid slewing. However, as operating frequency and/or amplitude is increased, the slew rate becomes insufficient and the output becomes distorted. CH8 Operational Amplifier as A Black Box 48 Maximum Op Amp Swing Vout Vmax Vmin Vmax Vmin sin t 2 2 FP SR Vmax Vmin 2 To determine the maximum frequency before op amp slews, first determine the maximum swing the op amp can have and divide the slew rate by it. CH8 Operational Amplifier as A Black Box 49 Nonzero Output Resistance A0 Rout R1 vout R1 vin R2 1 Rout A R1 0 R2 R2 In practical op amps, the output resistance is not zero. It can be seen from the closed loop gain that the nonzero output resistance increases the gain error. CH8 Operational Amplifier as A Black Box 50 Design Examples Many design problems are presented at the end of the chapter to study the effects of finite loop gain, restrictions on peak to peak swing to avoid slewing, and how to design for a certain gain error. CH8 Operational Amplifier as A Black Box 51