29 Multistage Amplifier Frequency Response Summary of frequency response of single stages CE CS suffers from Miller effect CC CD wideband see Section 10 5 CB CG wideband see Section 10 6 wideband means that the stage operates to near the frequency limit of the device fT How to find the Bode plot for a general multistage amplifier can t handle n poles and m zeroes analytically SPICE develop analytical tool for an important special case no zeroes exactly one dominant pole 1 2 3 n Ao V out V in 1 j 1 1 j 2 1 j n the example shows a voltage gain it could be Iout Vin or Vout Iin EECS 105 Fall 1998 Lecture 29 Finding the Dominant Pole Multiplying out the denominator Ao V out 2 n V in 1 b 1 j b 2 j b n j The coefficient b1 originates from the sum of j i factors n 1 1 1 b 1 1 2 n i 1 1 1 i Therefore if we can estimate the linear coefficient b1 in the demoninator polynomial we can estimate of the dominant pole Procedure see P R Gray and R G Meyer Analysis and Design of Analog Integrated Circuits 3rd ed Wiley 1994 pp 502 504 1 Find circuit equations with current sources driving each capacitor 2 Denominator polynomial is determinant of the matrix of coefficients 3 b1 term comes from a sum of terms each of which has the form RTj Cj where Cj is the jth capacitor and RTj is the Th venin resistance across the jth capacitor terminals with all capacitors open circuited EECS 105 Fall 1998 Lecture 29 Open Circuit Time Constants The dominant pole of the system can be estimated by n 1 1 R Tj C j b1 j 1 n j 1 1 where j RTj Cj is the open circuit time constant for capacitor Cj This technique is valuable because it estimates the contribution of each capacitor to the dominant pole frequency separately which enables the designer to understand what part of a complicated circuit is responsible for limiting the bandwidth of the amplifier EECS 105 Fall 1998 Lecture 29 Example Revisit CE Amplifier Small signal model C RS Vs r V C gmV ro roc RL Vout Apply procedure to each capacitor separately 1 C s Th venin resistance is found by inspection as the resistance across its terminals with all capacitors open circuited R T R S r R in C R T C o 2 C s Th venin resistance is not obvious must use test source and network analysis EECS 105 Fall 1998 Lecture 29 Time Constant for C Circuit for finding RT vt R in RS r v it gmv R out ro roc RL v is given by v i t R s r i t R in vo is given by v o i o R out i t g m v R out i t g m R in 1 R out vt is given by v t v o v i t 1 g m R in R out R in solving for RT vt it R T R in R out g m R in R out C o R T C R in R out g m R in R out C EECS 105 Fall 1998 Lecture 29 Estimate of Dominant Pole for CE Amplifier Estimate dominant pole as inverse of sum of open circuit time constants 1 1 R T C R T C R in C R in R out g m R in R out C inspection identical to exact analysis which also assumed 1 2 Advantage of open circuit time constants general technique Example include Ccs and estimate its effect on 1 EECS 105 Fall 1998 Lecture 29 Multistage Amplifier Frequency Response Applying the open circuit time constant technique to find the dominant pole frequency use CS CB cascode as an example V iSUP iOUT Q2 RL RS M1 Vs VBIAS vOUT V Systematic approach 1 two port small signal models for each stage not the device models 2 carefully add capacitances across the appropriate nodes of two port models which may not correspond to the familiar device configuation for some models EECS 105 Fall 1998 Lecture 29 Two Port Model for Cascode The base collector capacitor C 2 is located between the output of the CB stage the collector of Q2 and small signal ground the base of Q2 Cgd1 RS Vs Cgs1 I2 Iout Vgs1 gm1Vgs1 ro1 1 gm2 C 2 I2 o2ro2 C 2 RL Vout We have omitted Cdb1 which would be in parallel with C 2 at the output of the CS stage and Ccs2 which would be in parallel with C 2 In addition the current supply transistor will contribute additional capacitance to the output node Time constants C C gd1o gs1o R S C gs1 R in R out g m1 R in R out C gd1 1 1 where R in R S and R out r o1 g g m2 m2 Since the output resistance is only 1 gm2 the Th venin resistance for Cgd1 is not magnified i e the Miller effect is minimal C gd1o g m1 1 R S R S C gd1 R S 1 g m1 g m2 C gd1 g m2 g m2 EECS 105 Fall 1998 Lecture 29 Cascode Frequency Response cont The base emitter capacitor of Q2 has a time constant of C The base collector capacitor of Q2 has a time constant of C 2o 1 C 2 g m2 2o o2 r o2 r oc R C 2 R L C 2 L Applying the theorem the dominant pole of the cascode is approximately 1 3db C gs1o C gd1o C 2o C 2o 1 1 3db R S C gs1 R S 1 g m1 g m2 C gd1 C 2 R L C 2 g m2 EECS 105 Fall 1998 Lecture 29 Gain Bandwidth Product A useful metric of an amplifier s frequency response is the product of the lowfrequency gain Avo and the 3 dB frequency 3dB For the cascode the gain is Avo gm1RL and the gain bandwidth product is g m1 R L A vo 3dB 1 R S C gs1 R S 1 g m1 g m2 C gd1 C 2 R L C 2 g m2 If the voltage source resistance is small then g m1 R L A vo 3dB C g R C 2 m2 L 2 which has the same form as the common base gain bandwidth product and which is much greater than the Miller degraded common source EECS 105 Fall 1998 Lecture 29
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