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 ¯ ˜ - +