with each input terminal. Normally, Zom is best represented by a parallel resistance and capacitance of 2RcM (which is >>RN)and CM/2. The dc bias currents at the input are represented by I and Ig current sources that would equal the input base currents if a differential bipolar transistor were used as the input stage of the op amp, or the input gate currents if FETs were used. The fact that the two transistors of the input stage of the op amp may not be perfectly balanced is represented by an equivalent input offset voltage source, Vos, in series with the e input. The smallest signal that can be amplified is always limited by the inherent random noise internal to the op amp itself. In Fig. 27. 3 the noise effects are represented by an equivalent input voltage source(ENv), which when multiplied by the gain of the op amp would equal the total output noise present if the inputs to the op amp were shorted. In a similar fashion, if the inputs to the op amp were open circuited, the total output noise ould equal the sum of the noise due to the equivalent input current sources(ENi* and ENI), each mult by their respective current gain to the output. Because noise is a random variable, this summation must be accomplished in a squared fashion, i. e, Eo(rms volt /Hz)=(ENV)'A+(ENI )2 AR+(ENT)A12 (27.6) Typically, the correlation(C) between the ENV and ENI sources is low, so the assumption of C=0 can be made. For the basic circuits of Fig. 27. 2(a)or(b), if the signal source v, is shorted then the output voltage due to he nonideal effects would be(using the model of Fig 27.3) 。=|Vos++A哪‖1+ +IBR (27.7) CMRR PSRR R provided that the loop gain(also called loop transmission in many texts)is related by the inequality R I R1+/A(s) 7.8 Inherent in Eq.(27. 8)is the usual condition that R, < ZIN and ZcM. If a resistor R, were in series with the noninverting input terminal, then a corresponding term must be added to the right hand side of Eq (27.7)of value -IB R2(R+ Re)/RI. On manufacturers' data sheets the individual values of Ib and I are not stated instead the average input bias current and offset current are specified as B+rB I offset=I*-IBl The output noise effects can be obtained using the model of Fig. 27.3 along with the circuits of Fig. 27. 2 as EOut (rms volts/Hz)=E +ef +(env+ e2X R1 27.10) (ENI"'RF+(ENI*)R2 where it is assumed that a resistor R2 is also in series with the noninverting input of either Fig. 27. 2(a)or(b). The thermal noise(often called Johnson or Nyquist noise)due to the resistors Ri, R2, and R is given by(in rms volt/Hz) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC with each input terminal. Normally, ZCM is best represented by a parallel resistance and capacitance of 2RCM (which is >> RIN ) and CC M /2. The dc bias currents at the input are represented by IB + and IB – current sources that would equal the input base currents if a differential bipolar transistor were used as the input stage of the op amp, or the input gate currents if FETs were used. The fact that the two transistors of the input stage of the op amp may not be perfectly balanced is represented by an equivalent input offset voltage source, VOS , in series with the input. The smallest signal that can be amplified is always limited by the inherent random noise internal to the op amp itself. In Fig. 27.3 the noise effects are represented by an equivalent input voltage source (ENV), which when multiplied by the gain of the op amp would equal the total output noise present if the inputs to the op amp were shorted. In a similar fashion, if the inputs to the op amp were open circuited, the total output noise would equal the sum of the noise due to the equivalent input current sources (ENI+ and ENI–), each multiplied by their respective current gain to the output. Because noise is a random variable, this summation must be accomplished in a squared fashion, i.e., (27.6) Typically, the correlation (C) between the ENV and ENI sources is low, so the assumption of C ª 0 can be made. For the basic circuits of Fig. 27.2(a) or (b), if the signal source vI is shorted then the output voltage due to the nonideal effects would be (using the model of Fig. 27.3) (27.7) provided that the loop gain (also called loop transmission in many texts) is related by the inequality (27.8) Inherent in Eq. (27.8) is the usual condition that R1 << ZIN and ZCM . If a resistor R2 were in series with the noninverting input terminal, then a corresponding term must be added to the right hand side of Eq. (27.7) of value –IB + R2 (R1 + RF )/R1. On manufacturers’ data sheets the individual values of IB + and IB – are not stated; instead the average input bias current and offset current are specified as (27.9) The output noise effects can be obtained using the model of Fig. 27.3 along with the circuits of Fig. 27.2 as (27.10) where it is assumed that a resistor R2 is also in series with the noninverting input of either Fig. 27.2(a) or (b). The thermal noise (often called Johnson or Nyquist noise) due to the resistors R1 , R2 , and RF is given by (in rms volt2 /Hz) E AA A O vI 2 22 2 1 2 2 12 2 rms volt /Hz ENV ENI ENI 2 ( ) =+ + + - () () () v V V V R R o OS I R CM F =+ + B F Ê Ë Á ˆ ¯ ˜ + Ê Ë Á ˆ ¯ ˜ + - CMRR PSRR D supply 1 1 R R R A s F 1 1 1 + Ê Ë Á ˆ ¯ ˜ ( ) >> I I I I II B B B = B B + = - + - + - 2 ; offset * * E E R R E E R R R R R R F F F F F out rms volts /Hz ENV ENI ENI 2 2 1 2 1 2 2 2 2 2 1 2 22 2 2 2 1 2 1 1 ( ) () () () = Ê Ë Á ˆ ¯ ˜ ++ + ¥ + Ê Ë Á ˆ ¯ ˜ ++ + Ê Ë Á ˆ ¯ ˜ - +
E- 4kT R ES= 4kTR (27.11) E2 =4kTR where k is Boltzmanns constant and T is absolute temperature(Kelvin). To obtain the total output noise, one lust multiply the Eout expression of Eq (27. 10)by the noise bandwidth of the circuit, which typically is equal to /2 times the -3 dB signal bandwidth, for a single-pole response system [ Kennedy, 1988] SPICE Computer Models The use of op amps can be considerably simplified by computer-aided analysis using the program SPICE. SPICE originated with the University of California, Berkeley, in 1975 [Nagel, 1975], although more recent user-friendly commercial versions are now available such as HSPICE, HPSPICE, IS-SPICE, PSPICe, and ZSPiCe, to mention a few of those most widely used A simple macromodel for a near-ideal op amp could be simply stated with the SPICe subcircuit file(* indicates a comment that is not processed by the file) SUBCKT IDEALOA 123 "A near-ideal op amp: (1)is noninv,(2)is inv, and(3)is output. RIN 12 1E12 El(3,0)(1,2)1E8 ENDS IDEALOA (27.12) The circuit model for IDEALOA would appear as in Fig. 27. 4(a). A more complete model, but not including nonideal offset effects, could be constructed for the 741 op amp as the subcircuit file OA741, shown in Fig.274(b) SUBCKT OA741126 *A linear model for the 741 op amp: (1)is noninv,(2)is inv, and (6)is output. RiN= 2MEG, AOL =200,000, ROUT=75 ohm, Dominant open-loop pole at 5 Hz, gain-bandwidth product *is 1 mhz RIN 12 2MEG E(3,0)(1,2)2E5 Rl34100K C1 400.318UF: RI X CI= 5HZPOLE E2(5,0)(4,0)1.0 ROUT 75 ENDS OA741 (27.13) The most widely used op amp macromodel that includes dc offset effects is the Boyle model Boyle et al, 1974]. Most op amp manufacturers use this model, usually with additions to add more poles(and perha zeroes). The various resistor and capacitor values, as well as transistor, and current and voltage generator, values are intimately related to the specifications of the op amp, as shown earlier in the nonideal model of Fig. 27.3 The appropriate equations are too involved to list here; instead, the interested reader is referred to the article by Boyle in the listed references. The Boyle model does not accurately model noise effects, nor does it fully gA more circuits-oriented approach to modeling op amps can be obtained if the input transistors are removed nodel psrr and cmrr effects and a model formed by using passive components along with both fixed and dependent voltage and current sources. Such a model is shown in Fig. 27.5. This model not only includes all the basic nonideal effects of the op amp, allowing for multiple poles and zeroes, but can also accurately include ENV and ENI noise effects e 2000 by CRC Press LLC
© 2000 by CRC Press LLC (27.11) where k is Boltzmann’s constant and T is absolute temperature (°Kelvin). To obtain the total output noise, one must multiply the E2 out expression of Eq. (27.10) by the noise bandwidth of the circuit, which typically is equal to p/2 times the –3 dB signal bandwidth, for a single-pole response system [Kennedy, 1988]. SPICE Computer Models The use of op amps can be considerably simplified by computer-aided analysis using the program SPICE. SPICE originated with the University of California, Berkeley, in 1975 [Nagel, 1975], although more recent user-friendly commercial versions are now available such as HSPICE, HPSPICE, IS-SPICE, PSPICE, and ZSPICE, to mention a few of those most widely used. A simple macromodel for a near-ideal op amp could be simply stated with the SPICE subcircuit file (* indicates a comment that is not processed by the file) .SUBCKT IDEALOA 1 2 3 *A near-ideal op amp: (1) is noninv, (2) is inv, and (3) is output. RIN 1 2 1E12 E1 (3, 0) (1, 2) 1E8 .ENDS IDEALOA (27.12) The circuit model for IDEALOA would appear as in Fig. 27.4(a). A more complete model, but not including nonideal offset effects, could be constructed for the 741 op amp as the subcircuit file OA741, shown in Fig. 27.4(b). .SUBCKT OA741 1 2 6 *A linear model for the 741 op amp: (1) is noninv, (2) is inv, and *(6) is output. RIN = 2MEG, AOL = 200,000, ROUT = 75 ohm, *Dominant open - loop pole at 5 Hz, gain - bandwidth product *is 1 MHz. RIN 1 2 2MEG E1 (3, 0) (1, 2) 2E5 R1 3 4 100K C1 4 0 0.318UF ; R1 2 C1 = 5HZPOLE E2 (5, 0) (4, 0) 1.0 ROUT 5 6 75 .ENDS OA741 (27.13) The most widely used op amp macromodel that includes dc offset effects is the Boyle model [Boyle et al., 1974]. Most op amp manufacturers use this model, usually with additions to add more poles (and perhaps zeroes). The various resistor and capacitor values, as well as transistor, and current and voltage generator, values are intimately related to the specifications of the op amp, as shown earlier in the nonideal model of Fig. 27.3. The appropriate equations are too involved to list here; instead, the interested reader is referred to the article by Boyle in the listed references. The Boyle model does not accurately model noise effects, nor does it fully model PSRR and CMRR effects. A more circuits-oriented approach to modeling op amps can be obtained if the input transistors are removed and a model formed by using passive components along with both fixed and dependent voltage and current sources. Such a model is shown in Fig. 27.5. This model not only includes all the basic nonideal effects of the op amp, allowing for multiple poles and zeroes, but can also accurately include ENV and ENI noise effects. E kT R E kT R EF F kT R 1 2 1 2 2 2 2 4 4 4 = = =