Monolithic Amplifier Circuits: Operational Amplifiers Chapter 1 Jón Tómas Guðmundsson tumi@hi.is 1. Week Fall 2010 1
Introduction Operational amplifiers (op amps) are an integral part of many analog and mixed-signal systems The op amp is a high-gain differential amplifier Modern op amp design involves a trade-off between parameters such as voltage gain, input impedance, speed and power dissipation 2
Introduction The term operational amplifier (op amp) was coined in the 1940s That is well before the invention of the transistor and the integrated circuit They were first realized by vacuum tubes to create integrators and differentiators etc. referred to as analog computers it performed mathematical operations thus operational amplifier 3
Introduction The operational amplifier can be looked at as a black box having two inputs and one output The op amp is a circuit that amplifies the difference between the two inputs V out = A 0 (V in1 V in2 ) 4
Introduction Ideal op amps have infinite voltage gain, A 0 = infinite input impedance, R in = zero output resistance, R 0 = 0 infinite speed zero output voltage when input voltage is zero infinite bandwidth + v 2 - i + - 2 v 1 R i i 1 + - + - R o A(v - v ) 2 1 Equivalent circuit of an op amp i o + v o - infinite slew rate 5
Introduction Actual op amps are not ideal finite voltage gain, A 0 finite bandwidth, bw finite input impedance, R in + v 2 i + 2 R i i 1 + - + - R o A(v - v ) 2 1 i o + v o finite input capacitance, C in finite output resistance, R 0 - v 1 - - generates noise input bias currents input offset currents and voltages finite output swing and all of these effects are temperature dependent 6 Equivalent circuit of an op amp
Introduction The main objective of this course is to understand what causes the non-idealities and to develop design strategies to get around them The first step towards analyzing and designing op amps is to understand the transistor technologies upon which they are based 7
Introduction The very high gain of the op amp leads to an important observation The difference between V in1 and V in2 is always small V in1 V in2 = V out A 0 8
Unity-gain buffer If the voltage gain of the op amp were infinite V in1 = V in2 = V in and V out = V in 9
Unity-gain buffer For a finite gain V out = A 0 (V in1 V in2 ) = A 0 (V in V out ) or V out V in = A 0 1 + A 0 The gain approaches unity as A 0 becomes large 10
Non-inverting amplifier Non-inverting amplifier consists of an op amp and a voltage divider R 2 V in2 = V out = 1 + R 1 V in1 R 1 + R 2 R 2 If the op amp has finite gain V out V in = where A 0 is the open loop gain and V out /V in is the closed loop gain A 0 1 + R 2 R 1 +R 2 A 0 11
Inverting amplifier Now node X bears zero potential referred to as virtual ground so 0 V out R 1 This is the inverting amplifier = V in R 2 V out V in = R 1 R 2 12
Inverting amplifier If we assume a finite gain V out V in = 1 ( ) R 2 R 1 + 1 A 0 1 + R 2 R 1 = Example 1.1 13
Integrator With an ideal op amp V out V in = 1 C 1 s R 1 = 1 R 1 C 1 s so the circuit operates as an integrator (and low pass filter) 14
Integrator Also or V in dv out = C 1 R 1 dt V out = 1 RC 1 V in dt 15
Differentiator Here V out = R 1 V 1 = R 1 C 1 s in C 1 s which acts as a differentiator and a high pass filter Similarly or C 1 dv in dt = V out R 1 V out = R 1 C 1 dv in dt 16
Precision rectifier Recall that simple rectifier circuits suffer from a dead zone due to the finite voltage required to turn on a diode (0.7 V) This drawback prevents the use of simple diode circuits in high precision applications A solution to this problem is to place a diode around an op amp for a precision rectifier to rectify very small signals A diode D 1 is placed in the feedback loop 17
Precision rectifier If V in = 0 the op amp raises V Y to approximately V D1,on so turning D 1 barely on with little current so V X 0 If V in becomes slightly positive V Y rises further so that current flowing through D 1 and R 1 yields V out V in If V in becomes negative D 1 turns off 18
Logarithmic amplifier Here also V out = V BE so V BE = V T ln V in/r 1 I s V out = V T ln V in I s R 1 Logarithmic amplifiers are useful in applications where the input signal level may vary by a large fraction 19
Square-root amplifier Here V in R 1 = 1 2 µ nc ox W L (V GS V TH ) 2 and since V GS = V out V out = 2V in µ n C ox W L R 1 V TH 20
Op amp Nonidealities In practice op amps suffer from imperfections that may influence circuit performance significantly In op amp design the trade-offs between parameters requires a multi-dimensional compromise If the speed is critical while the gain error is not we choose a topology that favors the former 21
DC offsets Input offset voltage is the voltage that must be applied to the two input terminals to null the output So far we have assumed that V out = 0 if V in1 = V in2 In reality, a zero input difference may not give a zero output The internal circuit of the op amp may have random asymmetries from fabrication E.g. the bipolar transistor sensing the input may have different base-emitter voltages 22
Input bias current Input bias current is the average of the currents that flow into the inverting and non-inverting terminals of an op amp I B = I B1 + I B2 2 Op amps implemented in bipolar technology draw a base current from each input They are small ( 0.1 1µA) but may create inaccuracies in some circuits 23
Speed limitations The high frequency behavior of op amps plays a critical role in many applications In reatility, the internal capacitances of the op amp degrade the performance at high frequency The open loop gain begins to fall as the operational frequency exceeds f 1 24
Speed limitations This gain-roll off can be approximated by a first order model where V out V in1 V in2 (s) = A 0 1 + s ω 1 ω 1 = 2πf 1 The small-signal bandwidth is often defined as the unity-gain frequency f u f u can exceed 1 GHz for modern CMOS op amps 25
Speed limitations Note that for low frequencies s/ω 1 1 the gain is equal to A 0 At very high frequencies s/ω 1 1 and the gain of the op amp falls to unity at ω u = A 0 ω 1 26
Speed limitations For the non-inverting amplifier V out V in (s) = 1 + R 2 A 0 1+ s ω 1 R 1 +R 2 + A 0 1+ s ω 1 = A 0 s ω 1 + R 2 R 1 +R 2 A 0 + 1 The system is of first order and the pole of the transfer function is given by ( ) R 2 ω p,closed = 1 + A 0 ω 1 R 1 + R 2 27
Speed limitations The bandwidth of the closed-loop circuit is significantly higher than that of the op amp itself This improvement is at the cost of a reduced gain, from A 0 to 1 + R 2 A 0 (R 1 + R 2 ) = Example 1.2 28
Slew rate Slew rate is defined as the maximum rate of change of the output voltage per unit of time SR = dv out [V/µs] dt maximum It indicates how rapidly the output of an op amp can change in response to changes in the input frequency 29
Slew rate In reality the output first rises with a constant slope (i.e. as a ramp) and eventually settles as in the linear case Slewing is a non-linear phenomena 30
Slew rate If and V in (t) = V 0 sinωt ( V out (t) = V 0 1 + R ) 1 sinωt R 2 dv out dt ( = V 0 1 + R ) 1 ω cosωt R 2 The output exhibits a maximum slope of V 0 ω(1 + R 1/R 2 ) and the op amp must have SR with higher value to avoid slewing and ω FP = is the full-power bandwidth and is a measure of the large-signal speed of the op amp 31 SR V max V min 2
Slew rate = Example 1.3 32
Operational Amplifiers in circuits Operational amplifiers are everywhere in practical circuits 33
Operational Amplifiers in circuits The operational amplifier 741 has been a workhorse in circuits for decades 34
Operational Amplifiers in circuits The operational amplifiers are often included in application specific integrated circuits (ASIC) Here we see a CMOS op amp on the die level 35
Further reading This discussion is based on Chapter 8 in Razavi (2008) and section 9.1. of Razavi (2001). A nice summary of op amp parameters is also found in chapter 2 of Gayakward (2000). References Gayakward, R. A. (2000). Op-amps and Linear Integrated Circuits (4 ed.). Upper Saddle River, NJ: Prentice Hall. Razavi, B. (2001). Design of Analog CMOS Integrated Circuits. New York, NY: McGraw-Hill. Razavi, B. (2008). Fundamentals of Microelectronics. Hoboken, NJ: John Wiley & Sons. 36