OPERATIONAL AMPLIFIER APPLICATIONS


 Norman Leonard
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1 OPERATIONAL AMPLIFIER APPLICATIONS 2.1 The Ideal Op Amp (Chapter 2.1) Amplifier Applications 2.2 The Inverting Configuration (Chapter 2.2) 2.3 The Noninverting Configuration (Chapter 2.3) 2.4 Difference Amplifiers (Chapter 2.4) 2.5 Integrators and Differentiators (Chapter 2.5) 2.6 Effect of Finite OpenLoop Gain and Bandwidth (Chapter 2.7) 2.7 LargeSignal Operation of Op Amp (Chapter 2.8) Filter Applications 2.8 FirstOrder and SecondOrder Filter Functions (Chapter 13.4) 2.9 The 2ndOrder LCR Resonator (Chapter 13.5) 2.10 Active Filters Based on Inductor Replacement (Chapter 13.6) 2.11 Active Filters Based on the TwoIntegratorLoop Topology (Chapter 13.7) 2.12 SingleAmplifier Biquadratic Active Filters (Chapter 13.8) 21
2 2.1 Ideal Op Amp Introduction The operational amplifier is a circuit building block of universal importance Their applications were initially in the area of analog computation and instrumentation The popularity of the opamp is due to its versatility in many electronics applications It works at levels that are quite close to their predicted theoretical performance The op amp is treated as a building block to study its terminal characteristics and its applications Circuit realization of the op amp, which is rather complex, will not be discussed in this chapter Opamp symbol and terminals Two input terminals: inverting input terminal () and noninverting input terminal (+) One output terminal DC power supplies V +, V and other terminals for frequency compensation and offset nulling Circuit symbol for op amp Op amp with dc power supplies 22
3 Ideal characteristics of op amp Differentialinput singleendedoutput amplifier Infinite input impedance i + i 0 regardless of the input voltage Zero output impedance v O A(v 2 v 1 ) regardless of the load Infinite openloop differential gain A Commonmode gain is zero Infinite bandwidth Differential and commonmode signals Two independent input signals: v 1 and v 2 Differentialmode input signal (v Id ): v Id (v 2 v 1 ) Commonmode input signal (v Icm ): v Icm (v 1 + v 2 )/2 Alternative expression of v 1 and v 2 : v 1 v Icm v Id /2 v 2 v Icm + v Id /2 Exercise 2.2 (Textbook) Exercise 2.3 (Textbook) 23
4 2.2 The Inverting Configuration The inverting closeloop configuration External components R 1 and R 2 form a close loop Output is fed back to the inverting input terminal Input signal is applied from the inverting terminal Invertingconfiguration using ideal op amp The required conditions to apply virtual short for opamp circuit: Negative feedback configuration Infinite openloop gain Closedloop gain: G v O /v I R 2 /R 1 Infinite differential gain: v + v v O /A 0 (virtual short) Infinite input impedance: i 2 i 1 v I /R 1 Zero output impedance: v O v i 1 R 2 v I (R 2 /R 1 ) Voltage gain is negative (input and output signals are out of phase) Closedloop gain depends entirely on external passive components (independent of opamp gain) Closeloop amplifier trades gain (high openloop gain) for accuracy (finite but accurate closedloop gain) 24
5 Equivalent circuit model for the inverting configuration Input impedance: R i v I /i I v I /(v I /R 1 ) R 1 For high input closedloop impedance, R 1 should be large, but is limited to provide sufficient G In general, the inverting configuration suffers from a low input impedance Output impedance: R o 0 Opencircuit voltage gain: A vo R 2 /R 1 Other circuit example for inverting configuration v v
6 Application: the weighted summer A weighted summer using the inverting configuration v 0 i + i +.. +i v + v v A weighted summer for coefficients of both signs v v + v v v Exercise 2.4 (Textbook) Exercise 2.6 (Textbook) Exercise 2.7 (Textbook) 26
7 2.3 Noninverting Configuration The noninverting closeloop configuration External components R 1 and R 2 form a close loop Output is fed back to the inverting input terminal Input signal is applied from the noninverting terminal Noninverting configuration using ideal op amp The required conditions to apply virtual short for opamp circuit: Negative feedback configuration Infinite openloop gain Closedloop gain: G v O /v I 1 + R 2 /R 1 Infinite differential gain: v + v v O /A 0 Infinite input impedance: i 2 i 1 v /R 1 v I /R 1 Zero output impedance: v O v + i 1 R 2 v I (1 + R 2 /R 1 ) Closedloop gain depends entirely on external passive components (independent of opamp gain) Closeloop amplifier trades gain (high openloop gain) for accuracy (finite but accurate closedloop gain) Equivalent circuit model for the noninverting configuration Input impedance: R i Output impedance: R o 0 (1+R 2 /R 1 )v i Opencircuit voltage gain: A vo 1 + R 2 /R 1 27
8 The voltage follower Unitygain buffer based on noninverting configuration Equivalent voltage amplifier model: Input resistance of the voltage follower R i Output resistance of the voltage follower R o 0 Opencircuit voltage gain of the voltage follower A vo 1 The closedloop gain is unity regardless of source and load It is typically used as a buffer voltage amplifier to connect a source with a high impedance to a lowimpedance load Exercise 2.9 (Textbook) 28
9 Exercise 1: Assume the op amps are ideal, find the voltage gain (v o /v i ) of the following circuits. (1) (2) 4 (3) (4) 29
10 Difference amplifier 2.4 Difference Amplifiers Amplifier circuit with an output proportional to the difference of the input voltages Ideal difference amplifier: Amplifies the differential input signal v Id Rejects the commonmode input signal v Icm Practical difference amplifier: Output voltage: A d is the differential gain A cm is the commonmode gain Commonmode rejection ratio (CMRR): CMRR 20 log A A Single opamp difference amplifier v + v v v v + v v v v v A v + A v v / 1 + / v / v 1 + / + v / 1 + / + v / 1 + / v A A 1 + / 1 + / / 1 + /
11 Superposition technique for linear timeinvariant circuit v 0 v v v 0 v 1 + v v + v v v / 1 + / + v / 1 + / v v v O1 A / 1 + / + A 1 + / 1 + / CMRR 20log / 1 + / / 1 + / v O2 The condition for difference amplifier operation: R 2 /R 1 R 4 /R 3 v O (R 2 /R 1 )(v 2 v 1 ) For simplicity, the resistances can be chosen as: R 3 R 1 and R 4 R 2 Differential input resistance R id : Differential input resistance: R id 2R 1 Large R 1 can be used to increase R id, however, R 2 has to be impractically large to maintain the required gain Gain can be adjusted by changing R 1 and R 2 simultaneously Mismatch between the resistor pairs may lead to finite A cm and CMRR 211
12 Instrumentation amplifier v A v + A v A A A A A CMRR A A 1 + Differentialmode gain can be adjusted by tuning R 1 Commonmode gain is zero Input impedance is infinite and output impedance is zero It s preferable to obtain all the required gain in the 1 st stage, leaving the 2 nd stage with a gain of 1 Mismatch between the resistor pairs may lead to finite A cm and CMRR The CMRR is enhanced by the differential gain of the first stage (CMR A d1 A d2 /A cm2 ) Exercise 2.15 (Textbook) Exercise 2.17 (Textbook) 212
13 2.5 Integrators and Differentiators Inverting configuration with general impedance R 1 and R 2 in inverting configuration can be replaced by Z 1 (s) and Z 2 (s) Transfer function (magnitude and phase) can be evaluated Inverting integrator (Miller integrator) Time domain analysis: v t V + 1 C i t dt V + 1 C v t R Frequency domain analysis: v t v t V 1 RC v t dt V V Z Z 1 src V jω V jω Z 1 Z jωrc V jω V jω 1 ωrc V jω V jω 90 Integrator frequency (ω int ), at which gain is 0 db, is the inverse of the timeconstant ω int 1/RC The capacitor acts as an opencircuit at dc (ω 0) openloop configuration at dc (infinite gain) Any tiny dc in the input could result in output saturation dt 213
14 The Miller integrator with parallel feedback resistance To prevent integrator saturation due to infinite dc gain, parallel feedback resistance is included G (db) 20 log R V Z V Z V jω V jω Z /R Z 1 + jω C V V (ω) /R 1 + s C /R 1 + (ω C) Large resistance is typically used as R F >> R Closedloop gain at dc R F /R 1 C Corner frequency (3dB frequency) 1/R F C Integrator frequency (ω int ) is still the same ω int 1/RC Closedloop gain at high frequency (ω >>1/R F C) 1/jωRC The integrator characteristics is no longer ideal 1 RC ω (log scale) 214
15 The opamp differentiator Time domain analysis i C dv t dt v t ir RC dv t dt Frequency domain analysis V V Z Z src V jω V jω Z jωrc Z V jω V jω ωrc V jω V jω 90 Differentiator timeconstant: RC Gain ( ωrc) becomes infinite at very high frequencies Highfrequency noise is magnified (generally avoided in practice) 215
16 The differentiator with series resistance To prevent magnifying highfrequency noise, series resistance R F is included G (db) 20 log R V Z V Z src 1 + s C 1 RC 1 C ω (log scale) V jω V jω Z R Z + 1/jωC jωrc 1 + jω C V V (ω) ωrc 1 + (ω C) Small resistance is typically used as R F << R Closedloop gain at infinite frequency R/R F Corner frequency (3dB frequency) 1/R F C Closedloop gain at low frequency (ω << 1/R F C ) jωrc The differentiator characteristics is no longer ideal 216
17 Exercise 2: For a Miller integrator with R 10 k and C 10 nf, a shunt resistance R F is used to suppress the dc gain. Find the minimum value of R F if a period signal with a period of 0.1 s is applied at the input. Example 2.4 (Textbook) Example 2.5 (Textbook) Exercise 2.18 (Textbook) Exercise 2.20 (Textbook) 217
18 2.6 Effect of Finite OpenLoop Gain and Bandwidth Nonideal opamp characteristics with finite gain Gain is finite and independent of frequency Openloop gain can be expressed as: A(jω) A 0 Nonideal opamp characteristics with finite gain and bandwidth Both gain and bandwidth are finite Openloop gain is typically approximated by STC form: A(jω) A 0 /(1 + jω/ω b ) At low frequencies (ω << ω b ), the openloop op amp is approximated by A(jω) A 0 At high frequencies (ω >> ω b ), the openloop op amp is approximated by A(jω) ω b A 0 /ω Unitygain bandwidth (f t ω t /2) is defined as the frequency at which A(jω t ) 1 ω t A 0 ω b 218
19 Inverting configuration using opamp with finite openloop gain Closedloop gain: i v ( v /A ) v + v /A v v A i v A v + v /A G v v Closedloop gain approaches the ideal value of R 2 /R 1 as A 0 approaches to infinite To minimize the dependence of G on openloop gain, we should have A 0 >> 1 + R 2 /R 1 The openloop gain of op amps are typically large (up to 10 5 ) Closed loop circuit analysis with ideal op amp model is generally a good approximation Input impedance: / / /A v i Output impedance: v v + v /A / v v + v G/A / 1 + G/A
20 Inverting configuration using op amp with finite openloop gain and bandwidth The openloop model: By analogy to the response of lowpass STC circuit, the openloop gain of the op amp is characterized by a singlepole model A A(s) 1 + s/ω The lowfrequency gain is A 0 and the 3dB frequency is ω b The closeloop gain: G V V A(s) The closed loop response is still a lowpass STC A 1 + s/ω / 1 + (1 + R / )/A G 1 + s (1 + R / )/ω A 1 + s/ω 1 + (1 + / )/A Gain (db) The lowfrequency gain is: The 3dB frequency is: For most cases where G / ω ω A / ω G / G 1 + (1 + / )/A ω 1 + (1 + / )/A (1 + / )/ω A 1 / A A 0 G 0 ω b ω 3dB ω t ω (rad/s) 220
21 Noninverting configuration using op amp with finite openloop gain and bandwidth The openloop op amp is also modeled by a lowpass STC The closed loop gain: G V V A(s) 1 + / 1 + (1 + R / )/A G 1 + s (1 + R / )/ω A 1 + s/ω 1 + (1 + / )/A The closed loop response is still a lowpass STC v v /A(s) The lowfrequency gain is: The 3dB frequency is: For most cases where G 1 + ω A ω A ω 1 + R R 1 + R ω G 1 + / G 1 + (1 + / )/A ω 1 + (1 + / )/A 1 + / A ω 1 + / A 221
22 Exercise 3: Consider an inverting amplifier where the openloop gain and 3dB bandwidth of the op amp are and 1 rad/s, respectively. Find the gain and bandwidth of the closeloop gain (exact and approximated values) for the following cases: R 2 /R 1 1, 100, 200, and Exercise 4: An op amp has an openloop gain of 80 db and ω t of rad/s. (1) The op amp is used in an inverting amplifier with R 2 /R Find the closeloop gain at dc and at ω 1000 rad/s. (2) Two identical inverting amplifiers with R 2 /R are cascaded. Find the closeloop gain at dc and at ω 1000 rad/s. (3) For the cascaded amplifier in (2), find the frequency at which the gain is 3 db lower than the dc gain. Exercise 2.26 (Textbook) Example 2.6 (Textbook) Exercise 2.27 (Textbook) Exercise 2.28 (Textbook) 222
23 Output voltage saturation 2.7 LargeSignal Operation of Op Amps Rated output voltage (v O,max ) specifies the maximum output voltage swing of op amp Linear amplifier operation (for required v O < v O,max ): v O (1 + R 2 /R 1 )v I Clipped output waveform (for required v O > v O,max ): v O v O,max The maximum input swing allowed for output voltage limited case: v I,max v O,max /(1 + R 2 /R 1 ) Output is typically limited by voltage in cases where R L is large Output current limits Maximum output current (i O,max ) specifies the output current limitation of op amp Linear amplifier operation (for required i O < i O,max ): v O (1 + R 2 /R 1 )v I and i L v O /R L Clipped output waveform (for required i O > i O,max ): i L i O,max i F The maximum input swing allowed for output current limited case: v I,max i O,max [R L (R 1 + R 2 )]/(1 + R 2 /R 1 ) Output is typically limited by current in cases where R L is small 223
24 Slew rate Slew rate is the maximum rate of change possible at the output: Slew rate may cause nonlinear distortion for largesignal operation SR dv dt (V/sec) Input step function Smallsignal distortion (finite BW) Largesignal distortion (SR) V V s/ω Fullpower bandwidth v t V 1 e dv dt Vω Defined as the highest frequency allowed for a unitygain buffer with a sinusoidal output at v O,max v (t) V sin (ωt) v (t) V sin (ωt) dv (t) ωv dt cos (ωt) dv (t) ωv dt < SR distortionless dv (t) dt f ω 2π ωv > SR distortion SR 2πV, v O,max v O ω M SR ω 224
25 Example 2.7 (Textbook) Exercise 2.29 (Textbook) Exercise 2.30 (Textbook) 225
26 2.8 FirstOrder and SecondOrder Filter Functions Filter Transfer Function A filter is a linear twoport network represented by the ratio of the output to input voltage Transfer function T(s) V o (s)/v i (s) Transmission: evaluating T(s) for physical frequency s jω T(jω) T(jω) e j(ω) Gain: 20 log T(jω) (db) Attenuation: 20 log T(jω) (db) Output frequency spectrum : V o (s) T(s) V i (s) Types of Filters 226
27 Filter Specification Relaxed characteristics are specified for practical filter implementations instead of the ideal case Passband edge : ω p Maximum allowed variation in passband transmission : A max Stopband edge : ω s Minimum required stopband attenuation : A min Then find a transfer function T(s) whose magnitude T(jω) meets the specifications The final process, which will be introduced in more detail, is to realize the filter circuit based on the transfer function 227
28 Filter Transfer Function The filter transfer function is written as the ratio of two polynomials: T s a s + a s + + a s + b s + + b The degree of the denominator filter order To ensure the stability of the filter N M The coefficients a i and b j are real numbers The transfer function can be factored and expressed as: T s a s z s z s z s p s p s p Zeros: z 1, z 2,, z M and (NM) zeros at infinity Poles: p 1, p 2,, p N Zeros and poles can be either a real or a complex number Complex zeros and poles must occur in conjugate pairs The poles have to be on the LHP of splane Transfer function examples: a (s + ω )(s + ω ) T s s + b s + b s + b s + b s + b a T s s + b s + b s + b s + b s + b 228
29 Cascade Filter Design Firstorder and secondorder filters can be cascaded to realize highorder filters Cascade design is one of the most popular methods for the design of active filters Cascading does not change the transfer functions of individual blocks if the output resistance is low FirstOrder Filters Bilinear transfer function T s a s + a s + b a s + a s + ω 229
30 FirstOrder Filters (Cont d) 230
31 SecondOrder Filters Biquadratic transfer function T s a s + a s + a s + b s + b a s + a s + a s + (ω /Q)s + ω It is typically used for the case of complexconjugate natural modes The natural modes (poles) are determined by ω 0 and Q: p, p ω 2Q ± jω 1 1 4Q The radial distance of the natural modes from the origin is equal to ω 0 (pole frequency) The pole quality factor Q determines the distance of the poles from the jω axis The transmission zeros are determined by the numerator coefficients Seven special secondorder filters can be realized based on the pair of natural modes Lowpass (LP) and highpass (HP) functions: The LP function has both zeros at s The HP function has both zeros at s 0 Duality between the LP and HP responses can be observed For complexconjugate natural modes: Q > 1/2 Maximally flat response: Q 1/2 The magnitude response exhibits a peak: Q >1/2 231
32 SecondOrder Filters (Cont d) 232
33 SecondOrder Filters (Cont d) Bandpass (BP) functions: The BP function has one zero at s 0 and the other at s The magnitude response peaks at the center frequency ω ω 0 The selectivity is usually measured by its 3dB bandwidth: ω, ω ω Q ± ω 2Q BW ω ω ω Q The BP filter becomes more selective as Q increases 233
34 SecondOrder Filters (Cont d) Notch functions: Transmission zeros are located on the jω axis, at the complexconjugate locations ω n No zeros at either s 0 or s The magnitude response exhibits zero transmission at ω ω n (notch frequency) Regular notch: ω n ω 0 The selectivity is usually measured by its 3dB bandwidth: ω, ω ω Q ± ω 2Q BW ω ω ω Q 234
35 SecondOrder Filters (Cont d) Lowpass notch: ω n > ω 0 Highpass notch: ω n < ω
36 SecondOrder Filters (Cont d) Allpass (AP) functions: The two transmission zeros are in the right half of the s plane, at the mirrorimage locations of the poles The magnitude response is constant over all frequencies The flat gain is equal to a 2 The frequency selectivity of AP function is in its phase response 236
37 The Resonator Natural Modes 2.9 The SecondOrder LCR Resonator Parallel Resonator Current Excitation Voltage Excitation Current Excitation V I Z 1 1/sL + sc + 1/R s/c s + (1/RC)s + (1/LC) ω 1/ LC Voltage Excitation V R (1/sC) V R 1/sC + sl 1/LC s + (1/RC)s + (1/LC) Q ω RC R C/L The LCR resonator can be excited by either current or voltage source The excitation should be applied without change the natural structure of the circuit The natural modes of the circuits are identical (will not be changed by the excitation methods) The similar characteristics also applies to series LCR resonator 237
38 Realization of Transmission Zeros Values of s at which Z 2 (s) 0 and Z 1 (s) 0 Z 2 behaves as a short Values of s at which Z 1 (s) and Z 2 (s) Z 1 behaves as an open Realization of Filter Functions LowPass Filter HighPass Filter Bandpass Filter T s V 1/LC V s + (1/RC)s + (1/LC) T s V s V s + (1/RC)s + (1/LC) T s V 1/RC s V s + (1/RC)s + (1/LC) 238
39 Notch Filter T s V s + (1/LC) V s + (1/RC)s + (1/LC) LowPass Notch Filter HighPass Notch Filter T s V C s + (1/LC ) V C s + (1/RC)s + (1/LC) T s V s + (1/CL ) V s + (1/RC)s + (1/LC) ω 1/LC > ω 1/LC ω 1/CL < ω 1/LC 239
40 2.10 SecondOrder Active Filters (Inductor Replacement) SecondOrder Active Filters by Op AmpRC Circuits Inductors are not suitable for IC implementation Use op amprc circuits to replace the inductors Secondorder filter functions based on RLC resonator The Antoniou InductanceSimulation Circuit Inductors are realized by op amprc circuits with negative feedbacks The equivalent inductance is given by Z V I sc / sl L C / 240
41 The Op AmpRC Resonator The inductor is replaced by the Antoniou circuit The pole frequency and the quality factor are given by ω 1/ C L 1/ C C / Q ω C C /C A simplified case where R 1 R 2 R 3 R 5 R and C 4 C 6 C ω 1/RC Q /R 241
42 Filter Realization LowPass Filter HighPass Filter Bandpass Filter Notch Filter 242
43 LPN Filter HPN Filter AllPass Filter 243
44 2.11 SecondOrder Active Filters (TwoIntegratorLoop) Derivation of the TwoIntegratorLoop Biquad Integrator V i V o V i V o V 1 V src ω s Highpass implementation: V KV + 1 Q ω s V ω s V T s V V Ks s + ω /Q s + ω Bandpass implementation: T s ω s Kω s T s s + ω /Q s + ω Lowpass implementation: T s ω s Kω T s s + ω /Q s + ω 244
45 Circuit Implementation (I)  KHN Biquad V + V + V + V + V + ( ω /s)v + V + ( ω /s) V + T s V V + s + Highpass transfer function: s + + R ω s + + ω 1, Q , K 2 1 Q T s V V Ks s + (ω /Q)s + ω Bandpass transfer function: T s V V Kω s s + (ω /Q)s + ω Lowpass transfer function: T s V V Kω s + (ω /Q)s + ω Notch and allpass transfer function: T s 1 V K R s ω R s + ω V s + (ω /Q)s + ω 245
46 Circuit Implementation (II) TowThomas Biquad Use an additional inverter to make all the coefficients of the summer the same sign All op amps are in singleended mode The highpass function is no longer available T s T s V V V V Kω s s + (ω /Q)s + ω Kω s + (ω /Q)s + ω An feedforward scheme can be employed T s V V C C s + 1 C 1 r s + 1 R C R s + (ω /Q)s + ω 246
47 2.12 SingleAmplifier Biquadratic Active Filters Characteristics of the SAB Circuits Only one op amp is required to implement biquad circuit Exhibit a greater dependence on the limited gain and bandwidth of the op amp More sensitive to the unavoidable tolerances in the values of resistors and capacitors Limited to less stringent filter specifications with pole Q factors less than 10 Synthesis of the SAB Circuits The 2 nd order filter is realized by a closedloop system with an op amp and a RC feedback network Steps of SAB synthesis: Synthesis of a feedback loop with a pair of complex conjugate poles characterized by ω 0 and Q Injecting the input signal in a way that realizes the desired transmission zeros Natural modes (poles) of the filter: V i V o A V A V 1 + Aβ The closedloop characteristic equation: 1 + Aβ At s 0 t s 1 A 0 The poles of the closedloop system are identical to the zeros of the RC network 247
48 RC Networks with complex transmission zeros t s V V s + s s + s 1 C C R + C C C C C t s 1 C C V V s + s s + s 1 R C C C C 1 C C 1 C C Characteristics Equation of the Filter s + s ω ω Q + ω s + s 1 C C C C 1 C C Q C C 1 C + 1 C Let C 1 C 2 C, R 3 R, R 4 R/m m 4Q RC 2Q/ω 248
49 Injection the Input Signal The method of injection the input signal into the feedback loop through the grounded nodes A component with a ground node can be connected to the input source The filter transmission zeros depends on where the input signal is injected T s V V s α/c s + s 1 C + 1 C C C 249
50 Generation of Equivalent Feedback Loops Equivalent Loop Characteristics Equation: Characteristics Equation: β t(s) 1 + At s 0 t s 1 A 0 β 1 1 t s t(s) 1 + At s 0 t s 1 A
51 Generation of Equivalent Feedback Loops (Cont d) 251
52 Supplement Operational amplifier Circuit symbol for op amp Circuit model for ideal op amp
53 Supplement Negative feedback Opposite polarity to the change of output voltage due to negative feedback As output voltage v O increase v increases differential input (v + v ) decreases output v O decrease Finite output voltage with negative feedback For a finite input voltage v I, the output voltage v O is finite Assume v O v (R 2 v I + R 1 v O )/(R 1 + R 2 ) v + v < 0 v O Assume v O v (R 2 v I + R 1 v O )/(R 1 + R 2 ) v + v > 0 v O Using an ideal op amp in a negative feedback system Output voltage is finite The voltage gain of the op amp is infinite v + v v O /A 0 (virtual short)
54 Circuit analysis technique (1) Supplement Solve the nodal voltages and branch currents sequentially Basis of the circuit analysis: ohm s law, KVL and KCL v v / i v / v + i v + + v i v + v i v / 0 0 V i v / v v i + + v virtual short
55 Supplement Circuit analysis technique (2) Solve the circuit by nodal analysis Define the nodal voltages with necessary variables Specify branch currents based on the nodal voltages KCL for current equations Solve the simultaneous current equations i v / v i v v i v / 0 0 V i v / i + i 0 v + v 0 i + i + i 0 v + v + v v 0 v + + v
56 Supplement Weighted summer for coefficients with both signs v v v v v v v v v + v v v
57 Supplement Exercise 1: Assume the op amps are ideal, find the voltage gain (v o /v i ) of the following circuits. (1) (2) 4 V V V V 2 3 (3) (4) V V V V + V V V V Exercise 2: For a Miller integrator with R 10 k and C 10 nf, a shunt resistance R F is used to suppress the dc gain. Find the minimum value of R F if a period signal with a period of 0.1 s is applied at the input. ω > 10/(R F C) R F > 16 M
58 Supplement Exercise 3: Consider an inverting amplifier where the openloop gain and 3dB bandwidth of the op amp are and 10 rad/s, respectively. Find the gain and bandwidth of the closeloop gain (exact and approximated values) for the following cases: R 2 /R 1 1, 100, 200, and / G 1 + (1 + / )/A ω 1 + (1 + / )/A (1 + / )/ω A 1 / A G / ω ω A 1 + / ω G R 2 /R G o G 0 (approx.) ω 3dB ω 3dB (approx.)
59 Supplement Exercise 4: An op amp has an openloop gain of 80 db and ω t of rad/s. (1) The op amp is used in an inverting amplifier with R 2 /R Find the closeloop gain at dc and at ω 1000 rad/s. (2) Two identical inverting amplifiers with R 2 /R are cascaded. Find the closeloop gain at dc and at ω 1000 rad/s. (3) For the cascaded amplifier in (2), find the frequency at which the gain is 3 db lower than the dc gain. (1) G(jω) 100/(1 + jω/1000); gain 100; gain 1000) (2) G(jω) 10000/(1 + jω/1000) 2 ; gain 10000; gain 1000) 5000 (3) G(jω) 10000/(1 + ω 2 / )7070 ω 643.8
60 Supplement Difference amplifier + i i i v v i v v + v i v v i v v v v i v / 1 + / v i i i i (v v )/ (v v )/ i i (v v )/ v / v v / 1 + / v v + v + v v v / v 1 + / + v / 1 + / + v / 1 + / v + v v + v + + v + v v / 1 + / v A / 1 + / + v A v + A v A 1 + / 1 + /
61 Supplement Instrumentation amplifier v v v R 2 v v v R 2 v v + v R 2 v + v v R 2
62 Supplement Difference amplifier with mismatch (1 + /2) (1 /2) A / / 2 ( + ) A 1 + / 1 + / ( + ) (1 /2) (1 + /2) For R 1 1 k, R k and 1% R 1 R 1 (1 + /2) R 3 R 1 (1 /2) R 2 R 2 (1 /2) R 4 R 2 (1 + /2) A R R A 100 A A CMRR A A 5100 (74dB) 1 v 1.1V, v 1V v 0.1V, v 1.05V v A v + A v V (desirable output 10V) 2 v 100.1V, v 100V v 0.1V, v V v A v + A v V (desirable output 10V)
63 Supplement Instrumentation amplifier with mismatch Analysis of the first stage amplifier For R 1 20 k, R 2 80 k and 1% R 22 R 2 (1 + /2) R 21 R 2 (1 /2) R 21 R 2 (1/2) v v + v v 2 v v v v 2 v v v v 2 v v v v + v 2 A v + v v 2 2 v v + 1 v 2 2 R 22 R 2 (1+/2) v 1 + v 5v v v + v 4 v v v
64 Supplement Instrumentation amplifier with mismatch Analysis of the complete instrumentation amplifier At the output of the first stage 1 + v + v A v + v v v v 1 + v v + v 2 v v v + v v A v + A v A A v + A v v A v + A v 1 + v + + v 2 v 1 + / 2 v 1 + / v + v 1 + / + v 1 + / 4 A 1 + A A 2 A A 1 + / CMRR A CMR
65 Supplement Instrumentation amplifier with mismatch For R 1 20 k, R 2 80 k, R 3 10 k, R k and 1% 1V 100V 0.802V V v A v + A v 99 k A A v + A v (1/2) 79.2 k (1+/2) 1.01 k (1/2) v + v 1 + / A A A (1+/2) 80.8 k (1/2) 0.99 k (1+/2) A A / k V 1.1V 100.1V 1.302V V V v 0.1V 0.1V v 1 + / v 0.5V 0.5V v 1.05V V v v + v V V
66 Supplement Largesignal operation V O,max 15 V, I O,max 6 ma 1.5mA 1.5mA 15V Assuming voltage limited case: v O 15 V v I v O /A v 1.5 V i O 3 ma < 6 ma 1.5mA 10 k 1.5V 1mA 1mA 10V Assuming voltage limited case: v O 15 V v I v O /A v 1.5 V i O 9 ma > 6 ma (not allowed!) Reassuming current limited case: i F + i L v I /1 k + 10v I /2 k 6 ma v I 1 V and v O 10 V 1V 5mA 2 k
67 Supplement Filter specifications: Ideal lowpass filter ω p 1 k (rad/s) Lowpass filter (spec A) ω p 1 k (rad/s) ω s 2 k (rad/s) A max 1 db A min 40 db Lowpass filter (spec B) ω p 1 k (rad/s) ω s 3 k (rad/s) A max 3 db A min 10 db 1 (0dB) 0.89 (1dB) (3dB) A max Amin Spec B Spec A (10dB) 0.01 (40dB) 0 (db) 0 ω p ω s ω s
68 Supplement Filter realization: (1) Filter spec (2) Transfer function (3) Ckt implementation Lowpass filter (spec B) ω p 1 k (rad/s) ω s 3 k (rad/s) A max 3 db A min 10 db 1 (0dB) 0.89 (1dB) (3dB) Multiple Choices T s s T jω jω(1 10 ) T ω 1k T ω 3k A max ( 3dB) ( 10dB) Multiple Choices Amin R 1 M, C 1 nf Spec B (10dB) 0 (db) 0 ω p ω s ω
69 Supplement Filter transfer functions: Examine the transfer functions T s s + 5s + 4s 10 s + 4s + 6s + 4 Filter implementations (s 1)(s j)(s + 3 j) (s + 2)(s j)(s + 1 j) T s s 6s + 10 s + 4s + 6s + 4 (s 3 + j)(s 3 j) (s + 2)(s j)(s + 1 j) T s T s s s + 4s + 6s s 2s (s 1 + j)(s 1 j) T s s + 5s + 4s 10 s + 4s + 6s + 4 (s 1)(s j)(s + 3 j) (s + 2)(s j)(s + 1 j) s 1 s + 2 s + 6s + 10 s + 2s + 2 The first term is a bilinear transfer function: 1 st order filter function The second term is a biquadratic transfer function: 2 nd order filter function Filters with high order transfer functions can be realized by cascading 1 st order and 2 nd order filters The circuits used to realize bilinear and biquadratic transfer functions will be introduced
70 Supplement Firstorder filters: T jω a jω + ω T jω a /ω a /ω 1 + jω/ω 1 + ω /ω T jω tan ω/ω T jω jωa a jω + ω 1 jω /ω T jω a 1 + ω /ω Example: 10 s T s s s + 10 T 1 R 1 R 2 10k T 2 C10nF T jω tan ω /ω R 1 R 2 10k C1nF 10 s + 10 T (s)t (s) s s + 10 T s 10 s /C s + 1/ C T s s s + 10 (/ )s s + 1/ C
71 Supplement Firstorder filters: (1) lowpass: a /ω > a ω pole < a /a (zero) (2) highpass: a /ω < a ω pole > a /a (zero) 1/sC R + 1/sC V V + V 2 V V 1 src 1 + src s 1/RC s + 1/RC s ω s + ω
72 Secondorder filters: T s T s General form for 2 nd order filter: T s For two conjugate poles: Poles: s s + 3s + 2 s s s + 2s s j a s + a s + a s + b s + b a s + a s + a s + (ω /Q)s + ω (ω /Q) 4ω < 0 Q > 0.5 p, p ω 2Q ± jω 1 1 4Q 1 s + 2 Biquadratic transfer functions: T s T s T s Supplement (1st order filter) 2 1 s + 1 j a a s + b s + b s + (ω /Q)s + ω a s s + b s + b a s s + b s + b a s s + (ω /Q)s + ω a s s + (ω /Q)s + ω 2nd order filter T s a s + a a s + a s + b s + b s + (ω /Q)s + ω T s a s + a s s + b s + b a s + a s s + (ω /Q)s + ω T s a s + a a s + a s + b s + b s + (ω /Q)s + ω T s a s + a s + a s + b s + b a s + a s + a s + (ω /Q)s + ω
73 Lowpass filter: T s Supplement a a s + b s + b s + (ω /Q)s + ω Frequency response: T jω a ω + jω(ω /Q) + ω T jω a (ω ω ) +(ωω /Q) Monotonic decrease (Q < 0.707): T jω a ω ω + ωω Q a ω Q Gain peaking (Q > 0.707): ω T jω 0 ω ω 1 1 2Q and T jω a Q ω 1 1/(4Q )
74 Highpass filter: T s a s s + b s + b Supplement a s s + (ω /Q)s + ω Frequency response: a ω T jω ω + jω(ω /Q) + ω T jω a ω (ω ω ) +(ωω /Q) Monotonic decrease (Q < 0.707): T jω a ω a Q ω ω + ωω Q Gain peaking (Q > 0.707): ω T jω 0 ω ω / 1 1 2Q and T jω a Q 1 1/(4Q )
75 Supplement Bandpass filter: T s a s s + b s + b a s s + (ω /Q)s + ω Frequency response: Center frequency: ja ω T jω ω + jω(ω /Q) + ω T jω a Q ω T jω a ω (ω ω ) +(ωω /Q) 3dB bandwidth: T jω a Q 2ω Q (ω ω ) +ω ω 2ω ω Q ω ω ±ω ω ω ω > ω Q ω ω ω ω ω ω 1 + 1/(4Q ) + ω /(2Q) ω ω < ω Q ω ω ω ω ω ω 1 + 1/(4Q ) ω /(2Q) BW ω ω ω /2Q
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