Lectures on FREQUENCY RESPONSE

Transcription

1 University of California Berkeley College of Engineering Department of Electrical Engineering and Computer Science Robert W. Brodersen EECS40 Analog Circuit Design Lectures on FREQUENCY RESPONSE

2 Bode Plots FR- Solve impedance network transfer function. H( ω) ( ω) ( ω) V OUT V IN ( H(ω), V out (ω) & V in (ω) are phasors ) V IN ( ω) R C V OUT jωc ( ω) H( ω) H( ω) ( ω) ( ω) V OUT V IN jωc jωrc jωc R + jωc Convert H(ω) to polar coordinates, H(ω) < θ H( ω) [ H( ω) ( H ( ω) )] θ I m { H( ω) } atan R e { H( ω) } - 2 R e I M IF H( ω) N( ω) D( ω) then { H( ω) } R e { N( ω) D ( ω) } { H( ω) } I M { N( ω) D ( ω) }

3 Bode Plots (Cont.) FR-2 ( H( ω) H ( ω) ) jωrc jωrc / / + ( ωrc) 2 2 / ω 2 ω 3dB Bode Plot Magnitude H( ω) db 20 log H( ω) -3dB 20dB/Decade 6dB/Octave ω p /RC ω 3dB ω ω» ω 3dB H( ω) ω ω 3dB 2 / ω dB ω 6dB/Octave - drops by 2 every time frequency doubles

4 Bode Plots (Cont.) FR-3 H( ω) H( ω) j ω H( ω) exp ( jθ( ω) ) ω 3dB ω 3dB j ω j ω ω dB ω 2 ω 3dB j ω ω 3dB j ω ω 3dB θω ( ) I m H( ω) atan R e H( ω)

5 Bode Plots (Cont.) FR-4 Re{ H( ω) } ω ω 3dB log( ω) Im{ H( ω) } ω ω 3dB ω ω 3dB θ 0 -π/4 0.ω 3dB ω 3dB 0ω 3dB ω θω ( ) arctan ω 3dB -π/2 Linear Approximation ω arctan ω 3dB

6 Bode Plots (Cont.) FR-5 0 Phase Magnitude 90 Phase difference Magnitude Change t 0 t 0 R H( ω) jωrc C H( ω) ( H( ω) H ( ω) ) 2 /

7 Bode Plots (Cont.) FR-6 Pole Summary R H( ω) j ω ω 3dB ω 3dB RC C H( ω) db log( ω) 0dB -0dB -20dB 20dB/Decade --- ω 0 θ -π/4 -π/2 0.ω 3dB ω 3dB 0ω 3dB ω 3dB

8 Bode Plots (Cont.) 2 Poles FR-7 ν IN Ideal Unity gain Buffer R R 2 C C 2

9 Two Poles: H( ω) Bode Plots (Cont.) j ω j ω H ωp ( ω) H ωp2 ( ω) ω p ω p2 FR-8 20 log H( ω) 20 log( H ωp ( ω) H ωp2 ( ω) ) 20 log H ωp ( ω) + 20 log H ωp2 ( ω) ω p ω p2 --- ω ω 2

10 Bode Plots (Cont.) FR-9 H( ω) [ H ωp exp( jθ ωp )] [ H ωp2 exp( jθ ωp2 )] H ωp H ωp2 exp( j[ θ ωp + θ ωp2 ]) H( ω) exp( jθ( ω) ) θω ( ) θ ωp + θ ωp2 θ 0 -π/4 0.ω p ω p 0ω p 0.ω p2 ω p2 0ω p2 log( ω) -π/2-3π/4 -π

11 Bode Plots (Cont.) FR-0 ν IN R R 2 C C 2 (+) - If we have 80 degree phase shift we have a problem. The negative feedback will turn into positive feedback. ν IN (-) + Can t have positive feedback in a loop with gain >

12 θ 0 -π/4 Bode Plots (Cont.) ω p ω p2 ω p3 ω p 0.ω p 0ω p ω p log( ω) 3 poles on top of each other FR- -π/2-3π/4 -π Likely unstable circuit Can t kill gain here without adding phase shift. -5π/4-3π/2

13 Bode Plots (Cont.) FR-2 C Zero at zero frequency, pole at /RC ν IN R R R jωc jωrc jωrc jωrc + j ω RC H( ω) db ω p RC log( ω)

14 π/2 θ π/4 0 Left half plane zero H( ω) + j----- ω ω Z Right half plane zero H( ω) j----- ω ω Z Single Zero at ω z θ 0 -π/4 -π/2 θ ω arc tan ω Z 0.ω Z ω Z 0ω Z FR-3 log( ω) H( ω) db ω 0ω Z 0.ω Z ω Z log( ω) For both: H( ω) ω / ω Z

15 ν IN R Bode Plots (Cont.) Pole FR-4 C R 2 H( ω) R R + R jω( R R 2 )C C Pole, Zero ν IN H( ω) R 2 + jωr C R + R jω( R R 2 )C R R 2

16 Pole, Zero H( ω) db Bode Plots (Cont.) ω p > ω z FR-5 ω p ω z ω p < ω z H( ω) + j----- ω ω Z + j----- ω ω p

17 Capacitances FR-6 G C OX ε SiO2 t OX ff µ 2 F m 2 C GS C GD 0 CGSO 5 0 F m - S D 0 CGDO 5 0 F m - C SB C GB C DB 0 CGBO 4 0 F m - B CJ F m 2 PB φ B 0.8V

18 Sat : Capacitances (Cont.) Linear : C 2 C GS -- C OX L W + CGSO W C OX L W GS CGSO W 3 2 C C GD CGDO W C OX L W GD CGDO W 2 CJ AS C SB CJSW PS (similar for C V BS MJ V BS MJSW DB ) PB PB MJ -- (default) 2 C GB CGBO L MJSW 3 (default) FR-7 PS Perimeter of Source AS Area of Source CGBO Capacitance of gate to bulk overlap

19 Capacitances (Cont.) Layout Source Drain FR-8 W L λ 2λ 4λ 2λ 2λ (Minimum size device, W/L 2) Area of Source AS 4λ W Area of Drain AD AS Perimeter of Source PS 8λ + W

20 Capacitances (Cont.) M NMOS L2u W2u + AS4p AD4p PS6u PD6u FR-9 G S D CGSO CGDO Capacitor (in linear)

21 Miller Approximation i C FR-20 i t A C ν t A i C C d ( ) C d ( + A ) C ( + A) dt dt d dt A ν t C( + A)

22 Miller Approximation (Cont.) FR-2 C R L C g m ν gs R L R L ν OUT g m R OUT + jω[ C( + g m R OUT ) + R OUT C] C( + g m R OUT ) C MILLER ω p C( + g m R OUT )

23 Inverter FR-22 R L C GD C GD νout C GB C GS C D C G g m R L R OUT R L r o C G C GB + C GS C D ignored (not usually possible)

24 Inverter (Cont.) FR-23 Z GD ν Z G g m R L ν ν in ν Z G ν ν OUT Z GD ν OUT g m g ν m R OUT in + jω{ [ C GD ( + g m R L ) + C G ] + R L C GD } ω 2 R OUT C G C GD C GD jω C GD g m j ω ω p + j ω ω p2 jω C GD g m jω ω p ω p2 ω ω p ω p2

25 Inverter (Cont.) FR-24 ω p [ C G + C GD ( + g m R OUT ) ] + R L C GD C MILLER ω p R OUT C GD R OUT R IN ---- C G g m ω Z g m C GD H( ω) + j----- ω ω Z j ω ω p + j ω ω p2

26 H( ω) Hs ( ) + j----- ω ω Z j ω ω p + j ω ω p2 s --- s Z s - s - s p s p2 Inverter (Cont.) s Z s p s p2 jω Z jω p jω p2 FR-25 X ω p2 X g m ω p g C G C ω Z m MILLER O C GD

27 Inverter (Cont.) FR-26 R L R L ω p C MILLER C G ω p ω p C G g m ---- g m as Cgd increases

28 Inverter (Cont.) FR-27 C µ R L C D R OUT R L r o C π Case (Miller Capacitance not important) : C π, R OUT C D» ( + g m R OUT )C µ C MILLER ω p ω C p π R OUT C D

29 Inverter (Cont.) FR-28 Z OUT Z OUT R L r o jωc D r o C D R OUT jωc D R OUT jωc A ν g m Z OUT g D m R OUT jωc D ω p R OUT C D g m R OUT jωr OUT C D

30 Inverter (Cont.) FR-29 ν' in C π ' jωc π jωc π jω C π ω p C π

31 Case 2 (Large Cd) : Inverter (Cont.) FR-30 R OUT C D» ( + g m R OUT )C µ, C π ω p ω R OUT C p D ( C π + C µ ) Case 3 (Large Cµ) : ( + g m R OUT )C µ» R OUT C D, C π ω p ω p ( + g m R OUT )C µ ---- ( C π + C D ) For case 2 and 3, g m ω p C MILLER ω ZERO g m C µ X X X X O C µ large C µ 0 C µ 0 g g ω m p ω m Z ( C π + C D ) R C µ large C µ OUT C D C π

32 Source Follower FR-3 ν g C π g m ( ν g ) g mb ( ) ν OUt R S ν g ( ν + jω C π in ) + ν OUT R S ν g + g ν ( + χ) g m g m jωc π

33 Source Follower (Cont.) ( + χ) g m + jωc π ( jωc π + g m ) ν g R S FR-32 ν OUT g m R S + ( + χ) g m R S ω Z g m C π ω p C π ( A) jω C π g m jωr C π ( + χ g m R S ) IN ( + χ)g m R S A ν A g m R S + ( + χ)g m R S A g m R S + ( + χ)g m R S

34 C GD FR-33 Source Follower again with C SB C GB C GS R S C SB Small Signal : ν g C G C GD + C GB g m ( ν g ) C GS C SB R S

35 ν G Source Follower (Cont.) ν G jωc G + ( ν G ) jωc GS FR-34 ( ν G ) jωc GS g m ν G ( ) ν OUT + ν R OUT jωc SB S ν OUT g m R S jω C GS g m R S g m R jω C IN C GS R G S ( C GS + C SB ) ω 2 R + g m R S + g m R S R C GS C G + C SB ( C G + C GS ) IN S + g m R S let denominator + j ω - p + j ω p - 2 jω ω p p 2 p p 2

36 Source Follower (Cont.) FR-35 for p and p 2 widely separated, if we assume that p is the dominant pole, -» p - p 2 p R C IN C GS R G S ( C GS + C SB ) g m R S + g m R S R C IN C GS G R + g m R O ( C GS + C SB ) S where, R O ---- R S g m thus, C GS C G R + g p m R O ( C GS + C SB ) S R O [ C GS C G + C SB C G + C SB C GS ]

37 Source Follower (Cont.) FR-36 2 limiting cases, Case : C GS C G » R + g m R S O ( C GS + C SB ) ω p C GS C G g m R S Case 2 : Miller cap C GS ( A) g m R S A g m R S g m R S A g m R S + g m R S C R O ( C GS + C SB )» R IN C GS G g m R S ω p R O ( C GS + C SB )

38 Common Gate FR-37 R S + - R L C SB C GS C GD C DB assume r o R S ν S + - R L C S C D

39 S ν S R S Common Gate (Cont.) ν S jωc S + g m ν S FR-38 KCL@ g m ν S jωc D + ν OUT R L g m R L ν OUT + g m R S ( + jωr L C D ) + jω R SC S + g m R S no zeros, p R L C D p R S C S + g m R S R S ---- C S g m

40 Common Gate (Cont.) R eq C eq R L C D p R eq C eq R L C ν S R S ---- R eq C eq C S g m p R eq C eq R S ---- C S g m Since all caps go to ground, finding poles reduces to finding Req s and Ceq s at the nodes.

41 FR-40 Cascode - reduces the Miller probelm C GD2 R L ( 2) V BIAS C GS2 C D2 C GD ( ) C D C GS

42 Cascode (Cont.) FR-4 If we assume that R S is very large, then, At () : R eq R S C eq C GS + C GD ( A) A ---- g m g m p R eq C eq C eq C GS + 2 C GD At (2) : R eq R L r o large C eq C GD2 + C D2 p R eq C eq No large Miller multiplication of a capacitance!

43 Zero Value Time Constant Analysis General case of dominant pole approximation. Use this technique for complex circuits where we can t identify node with large R eq and C eq. Strategy : ) Set all caps C j 0 except for C i 2) Find resistance seen by C i 3) Calculate R i C i for all caps 4) ω 3dB R i C i FR-42

44 ZVTC with Source Follower FR-43 C GD C GB C GS R S C SB ν g C G C GD + C GB g m ( ν g ) C GS C SB R S

45 C : ZVTC with Source Follower FR-44 R C G C C G C 2 : g m ( ) R 2 R S ---- g m C 2 C SB C SB R S

46 C 3 : ZVTC with Source Follower (Cont.) ν t + - i t g m ( ν g ) C GS FR-45 do small signal analysis to find R S ν --- t R 3 seen by C GS i t R 3 R O where R O R S g m R S g m C 3 C GS hence, ω 3dB R C + R 2 C 2 + R 3 C C + G R O C + R SB R IN O g m R S CGS