OPERATIONAL AMPLIFIER APPLICATIONS

Transcription

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 Non-inverting Configuration (Chapter 2.3) 2.4 Difference Amplifiers (Chapter 2.4) 2.5 Integrators and Differentiators (Chapter 2.5) 2.6 Effect of Finite Open-Loop Gain and Bandwidth (Chapter 2.7) 2.7 Large-Signal Operation of Op Amp (Chapter 2.8) Filter Applications 2.8 First-Order and Second-Order Filter Functions (Chapter 13.4) 2.9 The 2nd-Order LCR Resonator (Chapter 13.5) 2.10 Active Filters Based on Inductor Replacement (Chapter 13.6) 2.11 Active Filters Based on the Two-Integrator-Loop Topology (Chapter 13.7) 2.12 Single-Amplifier Biquadratic Active Filters (Chapter 13.8) 2-1

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 op-amp 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 Op-amp symbol and terminals Two input terminals: inverting input terminal () and non-inverting 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 2-2

3 Ideal characteristics of op amp Differential-input single-ended-output 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 open-loop differential gain A Common-mode gain is zero Infinite bandwidth Differential and common-mode signals Two independent input signals: v 1 and v 2 Differential-mode input signal (v Id ): v Id (v 2 v 1 ) Common-mode 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) 2-3

4 2.2 The Inverting Configuration The inverting close-loop 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 Inverting-configuration using ideal op amp The required conditions to apply virtual short for op-amp circuit: Negative feedback configuration Infinite open-loop gain Closed-loop 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) Closed-loop gain depends entirely on external passive components (independent of op-amp gain) Close-loop amplifier trades gain (high open-loop gain) for accuracy (finite but accurate closed-loop gain) 2-4

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 closed-loop 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 Open-circuit 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) 2-6

7 2.3 Non-inverting Configuration The non-inverting close-loop 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 non-inverting terminal Non-inverting configuration using ideal op amp The required conditions to apply virtual short for op-amp circuit: Negative feedback configuration Infinite open-loop gain Closed-loop 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 ) Closed-loop gain depends entirely on external passive components (independent of op-amp gain) Close-loop amplifier trades gain (high open-loop gain) for accuracy (finite but accurate closed-loop gain) Equivalent circuit model for the non-inverting configuration Input impedance: R i Output impedance: R o 0 (1+R 2 /R 1 )v i Open-circuit voltage gain: A vo 1 + R 2 /R 1 2-7

8 The voltage follower Unity-gain buffer based on non-inverting configuration Equivalent voltage amplifier model: Input resistance of the voltage follower R i Output resistance of the voltage follower R o 0 Open-circuit voltage gain of the voltage follower A vo 1 The closed-loop 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) 2-8

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) 2-9

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 common-mode input signal v Icm Practical difference amplifier: Output voltage: A d is the differential gain A cm is the common-mode gain Common-mode rejection ratio (CMRR): CMRR 20 log A A Single op-amp 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 time-invariant 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 2-11

12 Instrumentation amplifier v A v + A v A A A A A CMRR A A 1 + Differential-mode gain can be adjusted by tuning R 1 Common-mode 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) 2-12

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 time-constant ω int 1/RC The capacitor acts as an open-circuit at dc (ω 0) open-loop configuration at dc (infinite gain) Any tiny dc in the input could result in output saturation dt 2-13

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 Closed-loop 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 Closed-loop gain at high frequency (ω >>1/R F C) 1/jωRC The integrator characteristics is no longer ideal 1 RC ω (log scale) 2-14

15 The op-amp 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 time-constant: RC Gain ( ωrc) becomes infinite at very high frequencies High-frequency noise is magnified (generally avoided in practice) 2-15

16 The differentiator with series resistance To prevent magnifying high-frequency 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 Closed-loop gain at infinite frequency R/R F Corner frequency (3dB frequency) 1/R F C Closed-loop gain at low frequency (ω << 1/R F C ) jωrc The differentiator characteristics is no longer ideal 2-16

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) 2-17

18 2.6 Effect of Finite Open-Loop Gain and Bandwidth Non-ideal op-amp characteristics with finite gain Gain is finite and independent of frequency Open-loop gain can be expressed as: A(jω) A 0 Non-ideal op-amp characteristics with finite gain and bandwidth Both gain and bandwidth are finite Open-loop gain is typically approximated by STC form: A(jω) A 0 /(1 + jω/ω b ) At low frequencies (ω << ω b ), the open-loop op amp is approximated by A(jω) A 0 At high frequencies (ω >> ω b ), the open-loop op amp is approximated by A(jω) ω b A 0 /ω Unity-gain bandwidth (f t ω t /2) is defined as the frequency at which A(jω t ) 1 ω t A 0 ω b 2-18

19 Inverting configuration using op-amp with finite open-loop gain Closed-loop gain: i v ( v /A ) v + v /A v v A i v A v + v /A G v v Closed-loop gain approaches the ideal value of R 2 /R 1 as A 0 approaches to infinite To minimize the dependence of G on open-loop gain, we should have A 0 >> 1 + R 2 /R 1 The open-loop 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 open-loop gain and bandwidth The open-loop model: By analogy to the response of low-pass STC circuit, the open-loop gain of the op amp is characterized by a single-pole model A A(s) 1 + s/ω The low-frequency gain is A 0 and the 3-dB frequency is ω b The close-loop gain: G V V A(s) The closed loop response is still a low-pass 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 low-frequency gain is: The 3-dB 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) 2-20

21 Non-inverting configuration using op amp with finite open-loop gain and bandwidth The open-loop op amp is also modeled by a low-pass 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 low-pass STC v v /A(s) The low-frequency gain is: The 3-dB 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 2-21

22 Exercise 3: Consider an inverting amplifier where the open-loop gain and 3-dB 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 open-loop 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 close-loop gain at dc and at ω 1000 rad/s. (2) Two identical inverting amplifiers with R 2 /R are cascaded. Find the close-loop 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) 2-22

23 Output voltage saturation 2.7 Large-Signal 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 2-23

24 Slew rate Slew rate is the maximum rate of change possible at the output: Slew rate may cause non-linear distortion for large-signal operation SR dv dt (V/sec) Input step function Small-signal distortion (finite BW) Large-signal distortion (SR) V V s/ω Full-power bandwidth v t V 1 e dv dt Vω Defined as the highest frequency allowed for a unity-gain 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 2-24

25 Example 2.7 (Textbook) Exercise 2.29 (Textbook) Exercise 2.30 (Textbook) 2-25

26 2.8 First-Order and Second-Order Filter Functions Filter Transfer Function A filter is a linear two-port 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 2-26

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 2-27

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 s-plane 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 2-28

29 Cascade Filter Design First-order and second-order filters can be cascaded to realize high-order 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 First-Order Filters Bilinear transfer function T s a s + a s + b a s + a s + ω 2-29

30 First-Order Filters (Cont d) 2-30

31 Second-Order 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 complex-conjugate 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 second-order filters can be realized based on the pair of natural modes Low-pass (LP) and high-pass (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 complex-conjugate natural modes: Q > 1/2 Maximally flat response: Q 1/2 The magnitude response exhibits a peak: Q >1/2 2-31

32 Second-Order Filters (Cont d) 2-32

33 Second-Order Filters (Cont d) Band-pass (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 3-dB bandwidth: ω, ω ω Q ± ω 2Q BW ω ω ω Q The BP filter becomes more selective as Q increases 2-33

34 Second-Order Filters (Cont d) Notch functions: Transmission zeros are located on the jω axis, at the complex-conjugate 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 3-dB bandwidth: ω, ω ω Q ± ω 2Q BW ω ω ω Q 2-34

35 Second-Order Filters (Cont d) Low-pass notch: ω n > ω 0 High-pass notch: ω n < ω

36 Second-Order Filters (Cont d) All-pass (AP) functions: The two transmission zeros are in the right half of the s plane, at the mirror-image 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 2-36

37 The Resonator Natural Modes 2.9 The Second-Order 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 2-37

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 Low-Pass Filter High-Pass 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) 2-38

39 Notch Filter T s V s + (1/LC) V s + (1/RC)s + (1/LC) Low-Pass Notch Filter High-Pass 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 2-39

40 2.10 Second-Order Active Filters (Inductor Replacement) Second-Order Active Filters by Op Amp-RC Circuits Inductors are not suitable for IC implementation Use op amp-rc circuits to replace the inductors Second-order filter functions based on RLC resonator The Antoniou Inductance-Simulation Circuit Inductors are realized by op amp-rc circuits with negative feedbacks The equivalent inductance is given by Z V I sc / sl L C / 2-40

41 The Op Amp-RC 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 2-41

42 Filter Realization Low-Pass Filter High-Pass Filter Bandpass Filter Notch Filter 2-42

43 LPN Filter HPN Filter All-Pass Filter 2-43

44 2.11 Second-Order Active Filters (Two-Integrator-Loop) Derivation of the Two-Integrator-Loop Biquad Integrator V i V o V i V o V 1 V src ω s High-pass implementation: V KV + 1 Q ω s V ω s V T s V V Ks s + ω /Q s + ω Band-pass implementation: T s ω s Kω s T s s + ω /Q s + ω Low-pass implementation: T s ω s Kω T s s + ω /Q s + ω 2-44

45 Circuit Implementation (I) - KHN Biquad V + V + V + V + V + ( ω /s)v + V + ( ω /s) V + T s V V + s + High-pass transfer function: s + + R ω s + + ω 1, Q , K 2 1 Q T s V V Ks s + (ω /Q)s + ω Band-pass transfer function: T s V V Kω s s + (ω /Q)s + ω Low-pass transfer function: T s V V Kω s + (ω /Q)s + ω Notch and all-pass transfer function: T s 1 V K R s ω R s + ω V s + (ω /Q)s + ω 2-45

46 Circuit Implementation (II) Tow-Thomas Biquad Use an additional inverter to make all the coefficients of the summer the same sign All op amps are in single-ended mode The high-pass 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 2-46

47 2.12 Single-Amplifier 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 closed-loop 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 closed-loop characteristic equation: 1 + Aβ At s 0 t s 1 A 0 The poles of the closed-loop system are identical to the zeros of the RC network 2-47

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 2-48

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 2-49

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) 2-51

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 open-loop gain and 3-dB bandwidth of the op amp are and 10 rad/s, respectively. Find the gain and bandwidth of the close-loop 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 open-loop 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 close-loop gain at dc and at ω 1000 rad/s. (2) Two identical inverting amplifiers with R 2 /R are cascaded. Find the close-loop 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 (@dc) -100; gain (@ω 1000) (2) G(jω) 10000/(1 + jω/1000) 2 ; gain (@dc) 10000; gain (@ω 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 Large-signal 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 low-pass filter ω p 1 k (rad/s) Low-pass filter (spec A) ω p 1 k (rad/s) ω s 2 k (rad/s) A max 1 db A min 40 db Low-pass 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 Low-pass 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 First-order 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 First-order filters: (1) low-pass: a /ω > a ω pole < a /a (zero) (2) high-pass: 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 Second-order 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 Low-pass 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 High-pass 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 Band-pass 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) 3-dB 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