Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 9 LECTUE 9 MULTIPLESTAGE FEQUENCY ESPONSE I (EADING: GHLM 56527) Objective The objective of this presentation is:.) Develop methods for the frequency analysis of multiple stage amplifiers 2.) Illustrate by examples Outline Dominant Pole Approximation ZeroValue (Opencircuit) Time Constant Analysis Examples ShortCircuit Time Constants Examples Summary Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 92 Dominant Pole Approximation If one of the poles is significantly closer to the origin of the complex frequency plane, its magnitude is a good approximation to the 3dB frequency. Consider the following general transfer function: A(s) N(s) D(s) a a s a 2 s 2 a m s m b b s b 2 s 2 b n s n Equating denominator terms gives, b p p 2 p n n Σ i p i K s p s p 2 s p n If p << p 2, p 3, then b p and ω 3dB p b Complex frequency plane: jω splane p 3 p 2 p σ Fig. 9
Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 93 OpenCircuit Time Constant Analysis This method is suitable for finding the dominant pole of a circuit with multiple capacitors. Consider the following nport network, We may express the nodal equations as, I (g sc )V g 2 V 2 g 3 V 3 I 2 g 2 V (g 22 sc 2 )V 2 g 23 V 3 I 3 g 3 V g 32 V 2 (g 33 sc 3 )V 3 The determinant of the above can be expressed as (s) K K s K 2 s 2 K 3 s 3 K ( b s b 2 s 2 b 3 s 3 ) I V I 2 V 2 I 3 V 3 C C 2 C 3 nport network where K (C i ) for all i Fig. 97 Consider now the K which involves a single capacitor and is given as K h sc h 2 sc 2 h 3 sc 3 The h i terms can be evaluated by expanding the determinant about each row: First row: (s) (g sc ) g 2 2 g 3 3 h (C i, i ) Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 94 OpenCircuit Time Constant Analysis Continued Second row: (s) g 2 2 (g 22 sc 2 ) 22 g 23 23 h 2 22 (C i, i 2) Third row: (s) g 3 3 g 32 32 (g 33 sc ) 33 h 3 33 (C i, i 3) K C (C i, i ) C 2 22 (C i, i 2) C 3 33 (C i, i 3) Finally, b K K (C i, i ) C 22(C i, i 2) C 2 33(C i, i 3) C 3 If we realize that the drivingpoint impedance of the ith port are expressed as V i I i (s) then o (C i, i ), 2o 22(C i, i 2), 3o 33(C i, i 3), are the drivingpoint impedances at ports, 2, 3, with all capacitors set equal to zero. b o C 2o C 2 3o C 3 where io C i are called the opencircuit time constants. If there are no dominant zeros, then the dominant pole, p, is given as ω 3dB p b o C 2o C 2 3o C 3 Σ n ( io C i ) i
Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 95 Example A more exact model of the commonemitter BJT is shown below. Using the opencircuit time constant approach, find an expression for the 3dB frequency. Solution The procedure involves finding the opencircuit time constants. io (Corresponds to C i ): v in I rb r π v r b C x Ci C f g m v L v out Fig. 93 io r π (r b I ) I r π io fo (Corresponds to C f ): r π (r b I ) ( g m V ) L V r π (r b I ) io io L g m io L fo io L g m io L I r b r π fo, xo V g m V L Fig. 94 Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 96 Example Continued xo (Corresponds to C x ): (r π r b ) I ( g m V ) L r b xo V [ I r π /( I r b r π )] (r π r b ) I L g m L [(r π r b ) I ] I r π V g m V L xo (r π r b ) I L g m [ I r π /( I r b r π )] L Fig. 95 ω 3dB io C i fo C f xo C x ω 3dB r π (r b I )C i ( io L g m io L )C f {(r π r b ) I L g m L [ I r π /( I r b r π )]}C x ω 3dB r π (r b I )C i [g m L (r π I )](C f C x ) if r b Let I kω, r b 2Ω, I C ma, β o, f T 4MHz (I C ma), C µ.5pf, C x, L 5kΩ, and find the 3dB frequency. r π β o g m (26)2.6kΩ, τ T 2πf 398ps C π g m τ T C µ 5.3pF.5pF 4.8pF T ω 3dB 9 2.6.2(4.8) (5/26)(2.6 ).5 9 2.269.4 2.3x6 rad/s (.95MHz)
Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 97 Example 2 Cascade VoltageAmplifier Frequency esponse Calculate the 3dB frequency of the cascade voltage amplifier shown which has the following parameters: I kω L kω L2 5kΩ C gs 5pF C gs2 pf C gd C gd2 pf C bd C bd2 2pF g m 3mA/V g m2 6mA/V V DD V DD C gd C gd2 v in L M M2 v out V in C gs V g m V C bd Finding the opencircuit time constants: C gs : gs I kω C gs2 : gs2 L kω C gd : (Use the model to the right) I ( g m V ) L I ( g m I ) L gd I L g m I L 2kΩ 3kΩ 32kΩ L C gs2 V 2 g m V 2 V g m V V L2 out C bd2 Fig. 96 L Fig. 97 Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 98 Example 2 Continued C gd2 : (Can use the results of C gd ) gd2 L L2 g m2 L L2 kω 5kΩ 3kΩ 35kΩ C bd : bd L kω C bd2 : bd2 L2 5kΩ ω 3dB ΣT gs C gs gs2 C gs2 gd C gd gd2 C gd2 bd C bd bd2 C bd2 9 5 32 35 25 2 9 85.227x6 rad/s f 3dB 95kHz Computer simulation gives poles at 25kHz, 4.2MHz, and 39.98MHz and two zeros at 477MHz and 955MHz. How important is it for the circuit to have a dominant pole for the opencircuit time constant approach? For a circuit with two identical poles, the 3dB frequency is ω 3dB ω x 2 ).64ω x The opencircuit time constant approach gives ω 3dB ω x /2.5ω x 22% error and is pessimistic
Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 99 Example 3 Cascode VoltageAmplifier Frequency esponse Calculate the 3dB frequency of the cascade voltage amplifier shown which has the following parameters: I kω L2 kω C gs 5pF C gs2 pf C gd C gd2 pf C bd C bd2 2pF g m 3mA/V g m2 6mA/V 5kΩ V DD L M2 M v out v in V GG2 Finding the opencircuit time constants: C gs : gs I kω C gd : gd? I ( g m V ) S2 I ( g m I ) S2 V in C gd g m V 2 G DS2 D2 V C C V 2 C 2 L2 V out gs g m V G2 S C C bd C gs2 C 2 C bd2 C gd2 Fig. 98 S2 g m V 2 gd I t I S2 g m I S2 I S2 (g m I ) V V 2 L2 g m V Fig. 99 Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 9 Example 3 Continued But what is S2? Using the model shown, we see that g m2 L2 S2 L2 g m2 Note that unless > L2 that S2 is greater than /g m. r ds2 L2 gd I (g m I ) g m2 kω3.2kω 6.2kΩ C : S2 L2 g m2.2kω ω 3dB ΣT gs C gs gd C gd C 2 C 2 C 2 : L2 kω gs C gs gd C gd (C bd C gs2 ) 2 (C bd2 C gd2 ) 9 56.2.2 2 3 9 64.8 5.43x6 rad/s f 3dB 2.46MHz Note that the Miller effect, gd, is less in the cascode amplifier (6.2kΩ compared with 32kΩ). S2 g m2 L2 Fig. 9
Lecture 9 Multiple Stage Frequency esponse I (/7/2) Page 9 SUMMAY Developed the background for the opencircuit time constant analysis Good for amplifiers with multiple capacitors Works well if one of the poles is dominant, okay if not (pessimistic approx.) Illustrated the opencircuit time constant analysis Cascaded MOSFET amplifier Cascode MOSFET amplifier The input impedance to the cascoding stage depends on what is connected to the output of the cascoding stage. S2 L2 g m2 We will continue the multiple amplifier analysis techniques in the following lecture.