• nodes : variables

• branches : gains

e.g. y = a ∙ x

e.g. y = 3x + 5z – 0.1y

Signal Flow Graph

x ya

x zy 53

-0.1

The value of a node is equal to the sum of all signal coming into the node.The incoming signal needs to be weighted by the branch gains.

1 21 2 31 3 41 4 51 5y a y a y a y a y

Note: y6, y7, y8 are gained up versions of y1.

r yu G11

-G2

R yz G3

1-H1

x

G1

G2

-1

N

1

An input node is a node with only out going arrows.

R, N and r are input nodes.

Parallel branches can be summed to form a single branch

fig_03_31

Series branches can be multiplied to form a single branch

r yr1 y

1

Feedback connections can be simplified into a single branch

1

G

GH

Note: the internal node E is lost!

Ry

z G3

1-H1

x

G1

G2

-1

1y

Ry

z G3

1-H1

x

G1

G2

-1

1y

z’ 1

Ry1

x

G1

G2

-1

1y

z’3

3 11

G

G HR

Ry1

x

G2

-1

1y

z’3

3 11

G

G H

y

x

G1

G2

-1

1y

z’3

3 11

G

G HR

2 3

3 1 2 31

G Gy R

G H G G

31

3 1 2 31

Gy G R

G H G G

2 3 1 3

3 1 2 3 3 1 2 3

2 3 1 3

3 1 2 3

1 1

1

G G GGy R

G H G G G H G G

G G GGR

G H G G

Overall:

Mason’s Gain Formula• A forward path: a path from input to output• Forward path gain Mk: total product of

gains along the path• A loop is a closed path in which you can

start at any point, follow the arrows, and come back to the same point

• A loop gain Li: total product of gains along a loop

• Loop i and loop j are non-touching if they do not share any nodes or branches

• The determinant Δ:

. . . .3 4

1 ...i i j i j k i j k mall i all non touching all n t all n t

pairs of loops loops loops

L L L L L L L L L L

• Δk: The determinant of the S.F.G. after removing the k-th forward path

• Mason’s Gain formula:

. . o k k

all forwardipath k

y MI O T F

y

3213..

11 GGHGLLLTN

jiall

i

Get T.F. from N to y

1 forward path: N y

M = 1

2 loops: L1 = -H1G3

L2 = -G2G3

Δ1: remove nodes N, y, and branch N y

All loops broken: Δ1 = 1

3213

11

1

11

GGHG

MM

N

y kk

0

Ry

z G3

1-H1

x

G1

G2

-1

N1

Get T.F. from R to y

2 f.p.: R x z y : M1=G2G3

R z y : M2=G1G3

2 loops:L1 = -G3H1

L2 = -G2G3

3213..

11 GGHGLLLTN

jiall

i 0

Δ1: remove M1 and compute Δ

Δ1 = 1

Δ2: remove M2 and compute Δ

Δ2 = 1

3213

3132

1 GGHG

GGGGMM

R

yH kkk

Overall:

2 3 1 3

3 1 2 3 3 1 2 3

1

1 1

G G GGy R N

G H G G G H G G

H4

H1 H2 H3

H5

H6

H7

Forward path:

M1 = H1 H2 H3

M2 = H4

Loops:

L1 = H1 H5

L2 = H2 H6

L3 = H3 H7

L4 = H4 H7 H6

H5

L1 and L3 are non-touching

Δ1: If M1 is taken out, all loops are broken.

therefore Δ1 = 1

Δ2: If M2 is taken out, the loop in the middle (L2) is still there.

therefore Δ2 = 1 – L2 = 1 – H2H6

Total T.F.:

73515674736251

6244321

6221

1

)1(

HHHHHHHHHHHHHH

HHHHHHH

HHMMMH kk

7351

5674736251

31

1

1

HHHH

HHHHHHHHHH

LLLi

Application to integrated circuits• Identify input and output variables

• At each internal “circuit node”:– Identify the equivalent resistance (impedance)– Use current into this R as a node variable– Use voltage at circuit node as another

variable

• Construct SFG for the circuit

• Use Mason’s gain formula to find I/O TF

-1/Zc

Vi+

Vo1

VBN

VoVi-

Vm

Rm Ro1

There are two input variables: Vi+, Vi-Vo is output variavle

Internal circuit nodes:

Vm, and IRm

Vo1, and IRo1VoVi-

IRm Vm IRo1 IoRm Ro1 Ro

Vi+

-gm1

-gm2

-gm4 -gm6

1/Zc

-1/Zc

-sCL

-sC1

1/Zc

This is a simplied version, by ignoring Cgds and the dynamics at the mirror node.

Vo1 Vo

IRo1 IoRo1 RoVi+ -Vi-

-gm1 -gm6

1/Zc

-1/Zc

-1/Zc

-sCL

-sC1If Rm=1/gm4

gm1 = gm2

Two forward paths: M1=gm1Ro1gm6Ro; M2=- gm1Ro1Ro/Zc

The corresponding ’s are both 1

Loops: L1= -Ro1(sC1+1/Zc); L2= -Ro(sCL+1/Zc) L3= Ro1Ro(-gm6+1/Zc)/Zc

L1 an L2 are non-touching

By MGF: TF = (M1+M2)/(1-L1-L2-L3+L1L2)= gm1Ro1Ro(gm6Zc - 1)/{Zc+Ro1(sC1Zc+1)+Ro(sCLZc+1)+ Ro1(sC1Zc+1)Ro(sCL+1/Zc) + Ro1Ro(gm6Zc-1)/Zc}

1/Zc

When s 0, if Zc inf,TF = gm1Ro1Ro(gm6Zc - 1)/{Zc+Ro1(sC1Zc+1)+Ro(sCLZc+1) + Ro1(sC1Zc+1)Ro(sCL+1/Zc) + Ro1Ro(gm6Zc-1)/Zc} gm1Ro1Rogm6If Zc = 1/sCC:

TF = gm1Ro1Ro(gm6 - sCC)/{1+sRo1(C1+CC)+sRo(CL+CC) + s2Ro1Ro (C1+CC) (CL+CC) + sCCRo1Ro(gm6-sCC)}

= gm1Ro1Ro(gm6 - sCC)/{1+sRo1(C1+CC)+sRo(CL+CC) + s2Ro1Ro (C1CL+CCCL+CCC1) + sCCRo1Rogm6}

For buffer connection, closed-loop characteristic equation:1+gm1Ro1Rogm6+s{Ro1(C1+CC)+Ro(CL+CC)+Ro1Ro(CCgm6-gm1CC)} + s2Ro1Ro (C1CL+CCCL+CCC1)=0

Closed-loop BandWidth:≈const/s-coeff≈gm1gm6/(CCgm6-gm1CC)=gm1/CC*gm6/(gm6-gm1)

For stability: gm6 >= gm1

For damping ratio >= 0.5, need b2 >a*c. That is: (CCgm6-gm1CC)2 >= gm1gm6 (C1CL+CCCL+CCC1)(CC)2/(C1CL+CCCL+CCC1) >= gm1gm6/(gm6-gm1)2

BW2 <= gm1gm6 /(C1CL+CCCL+CCC1)

Notice: Ro1 and Ro plays no role in these conditions.

Let closed-loop poles be p= -x+-jy.Step response settling time is determined by x.Ts = -ln(tol)/x, tol is settling tolerance.For fastest settling, want large x = -ln(tol)/Ts.Perform a shift of imag axis: s=z-x. Want to maximize a while z remains stable.1+gm1Ro1Rogm6+(z-x) {Ro1(C1+CC)+Ro(CL+CC)+Ro1Ro(CCgm6-gm1CC)} + (z-x)2Ro1Ro (C1CL+CCCL+CCC1)=0

gm1gm6-x(CCgm6-gm1CC)+x2(C1CL+CCCL+CCC1) +z{(CCgm6-gm1CC)-2x(C1CL+CCCL+CCC1)} +z2(C1CL+CCCL+CCC1)=0

For stability of z, x < (gm6-gm1)/2(C1CL/CC+CL+C1)

If Zc = Rz + 1/sCc:

TF = gm1Ro1Ro(sCc(Rzgm6 -1)+gm6)/ {(sCcRz+1)+Ro1(sC1(sCcRz+1)+sCc)+Ro(sCL(sCcRz+1)+sCc) - Ro1(sC1(sCcRz+1)+sCc)Ro(sCL+1/Zc) + Ro1Ro(gm6(sCcRz+1)-sCc)/Zc}

Vi+

VBN

VoVi-

Vm

Rm Ro1

1/Zc

Closed-loop:

Vo1 Vo

IRo1 IoRo1 Ro

Vi+ -Vo

-gm1 -gm6

1/Zc

-1/Zc

-1/Zc

-sCL

-sC1

Vi+ 1

-1

1/Zc

Vo1 Vo

IRo1 IoRo1 Ro

Vi+ -Vo

-gm1 -gm6

1/Zc

-1/Zc

-1/Zc

-sCL

-sC1

Vi+ 1

-1

Forward paths remain the same.Two new loops added: L4=-M1; L5=-M2These are touching with previous loop.

1-L1-L2-L3-L4-L5+L1L2, set =0:

Zc+ gm1Ro1Ro(gm6Zc - 1)+Ro1(sC1Zc+1)+Ro(sCLZc+1)-Ro1(sC1Zc+1)Ro(sCL+1/Zc) + Ro1Ro(gm6Zc-1)/Zc = 0

This is the closed-loop characteristic equation.

Zc+ gm1Ro1Ro(gm6Zc - 1)+Ro1(sC1Zc+1)+Ro(sCLZc+1)-Ro1(sC1Zc+1)Ro(sCL+1/Zc) + Ro1Ro(gm6Zc-1)/Zc = 0

If Zc= Rz + 1/sCC, The char eq becomes:L

M1 M2

M3 M4

M5

M6c

VDD

VSS

Cc

Vin- Vin+

M9

ic

VA

V1

Mb3M7

Mb1

Isupply

Mb2

Mb4

Mb5 Mb6

Vout

Vbb

Vo1 Vo

IRo1 IoRo1 RoVi+ -Vo

-gm1

-gm5

-sCL -sCC-sC1

Vi+

1

IRA RAVA

-sCC-sCA

sCC

sCC

gmc

2 21 1 1 2 5

3 4 1 5

1 1 5 1 1 3

1 2 3 4 5 1 2 2 3 1 3 1 5 1 2 3

1 1 1 5

; ( );

( ); ;

; 1 1 ( )

1

1 ( ( ) ( )

o o L C o A C

A C A o o A m mc C

m m o o A C A

o o L C A C A o o A m mc C

L sR C L sR C C L s R R C

L sR C C L sR R R g g C

M g g R R L sR C C

L L L L L L L L L L L L L L L L

s R C R C C R C C R R R g g C

21 1

21 1

3 21 1 1 1

21 5 1 1

3 21 1

1 1

)

( ( ) ( ) ( )

( ) )

( ( )( ) )

1 ( ( ))

(( )( ) )

Open Loop TF = /

o o L C o L C A C A

o A C A o A C

o o A L C C A o o A C

o o m C o o L o A L C L A

o o A L C C A C

s R C R C C R C C R C C

R C R C C R R C

s R R R C C C C C R R R C C

sR R g C s R C R C R R C C C C

s R R R C C C C C C

M

1 5 1

1

1 5

; (0)

1Dominant pole: ;

m m o o

m

o o m C C

TF g g R R

gGB

R R g C C

1 5 1

5 1 1

21 1

31 1

1

11 5 12

Buffer connection closed-loop characteristic equation:

0 1

( )( ))

( )

If we want Re(pole) < -0.5GB. Relacing by :2

0

m m o o

m o o A m mc C A

o A C o L

o o A L C

m

C

m m o

g g R R

sg R R R g g C C

s R C R C R C

s R R R C C C

gs z

C

g g R R

1

2 3 21 11 1 1 1 1 14 8

1 1 5 5 1 1 1

231 14

2 31 1 1 1 14

31 1

(1 (1 / ))

( / ) / ) /

( ( ) ( / )

/ )

( )

For stabilit

o A m A C

m o L o C A C m o o A L C

o o m m A C A m C m L C o A L

m A L C

o o L o A L C m A L

o o A L C

R g C C

g R C R C C R C g R R R C C C

zR R g g R C C g C g C C C g R C

g R C C C

z R R C C g R C C g R C C

z R R R C C C

1 2 0 3y, all coeff 0, and .a a a a

M1 M2

M3 M4

M5

M6c

VDD

VSS

Cc

Vin- Vin+

M9

ic

VA

V1

Mb3M7

Mb1

Isupply

Mb2

Mb4

Mb5 Mb6

Vout

Vbb

Vo1 Vo

IRo1 IoRo1 RoVi+ -Vo

-gm1

-gm5

-sCL -sCC-sC1

Vi+

1

IRA RAVA

-sCC-sCA

sCC

sCC

gmc+ goc

If we include a finite roc for M6c:

goc

There is one more FP: -gm1R’o1gocR’AsCCRo Its =1

It also introduces a loop: ro1 goc RA (gmc+goc).This loop is non touching with the one at Vo.

' ' '1 1 1 1 1 1

' 2 ' 22 3 5

' ' ' ' '4 1 5 6 1 1 1 1

' '1 1 5 1 1 3

/ (1 ); / (1 ) ;

( ); ( );

; ( ) / (1 )

; 1 1 (

o o o oc A A A oc A o

o L C A C A o A C

o o A m mc C o oc A mc oc o oc o oc o oc

m m o o A C

R R R g R R R g R L sR C

L sR C C L sR C C L s R R C

L sR R R g g C L R g R g g R g R g R g

M g g R R L sR C

' '

2 1 1 2 2 2 1 1

1 2 3 4 5 6 1 2 2 3 1 3 1 5 2 6 1 2 3

' ' ' ' ' '1 1 1 1 1 5

2 ' '1 1

)

; 1; | | | |

1

1 ( (1 )( ) ( ) )

( ( ) ( )

A

m o oc A C o

o oc o o o oc L C A C A o o A m mc C

o o L C o L C A

C

M g R g R sC R M M

L L L L L L L L L L L L L L L L L L L

R g s R C R R g C C R C C R R R g g C

s R C R C C R C C R

' ' 21 1

3 ' ' ' ' 21 1 1 1

21 1 5 1 1

3 21 1

1

1 1 1 5 1

(

( )

( ) )

( ( )( ) )

(1 ) 1 ( )

(( )( ) )

)(1

( )

)

1 ;

C A

o A C A o A C

o o A L C C A o o A C

o oc o o m C o o L

o o A L C C A C

o oc m m o

o A L C L o

o

A oc

C C

R C R C C R R C

s R R R C C C C C R R R C C

R g sR R g C s R C R C

s R R R C C C C C C

R

R g M

R C C C C R g

g g R R

2Most things stay the same in the TF, except one term in s in denominator

M1 M2

M3 M4

M5

M6c

VDD

VSS

Cc

Vin- Vin+

M9

ic

VA

V1

Mb3M7

Mb1

Isupply

Mb2

Mb4

Mb5 Mb6

Vout

Vbb

Vo1 Vo

IRo1 IoRo1 RoVi+ -Vo

-gm1

-gm5

-sCL -sCC-sCgd-sC1-sCgd

Vi+

1

IRA RAVA

-sCC-sCA

sCC

sCC

gmc+ goc

If we include Cgd effect for M5:

goc

sCgd

sCgd

C1 and CL should be modified just a little bit.It also introduces a loop: Ro1 Ro sCgd (-gm5+sCgd).This loop is non touching with the one at VA.In the main forward path, gm5 is replaced by gm5-sCgd.

' '1 1 1 1 3

' '2 1 1 2 2 2 1 1

1 2 3 4 5 6 1 2 2 3 1 3 1

'7 1 5

7 7

5

5 2 6 1 2

1

3

3

Additional Loop:

; 1 1 ( )

; 1; | | | |

1

(1 )

(

( )

)m o o A C A

m o oc A

o o g

m g

C o

o oc

d m g

d

d

M g R R L sR C C

M g R g R sC R M M

L L L L L L L L L L L L L L

L R R sC g

L L L L L

R

sC

L L L

g sC

g

1 5

21 1 1

3 21

1 5

2 2 2

1

1 1 5

3 2

1 1

1

1 1 5

( )

1

( ( )(1 ))

(( )( ) )

(1 ) ;

Most changes are negl

( )

o o m C

o o L o A L C L A o oc

o o A L C

o o gd m

o o gd o o gd m A C A

o o gC A C

o oc m m o o

d A C A

R R sC g

R R s C R R s C g R C C

R R s C R

sR R g C

s R C R C R R C C C C R g

s R R R C C C C C C

R g M g g R

C C

R

term in redigible, except one which introduce a RHP z ero.

Quick tips• Each circuit node makes one loop, with loop

gain = -R*sCtot

• If there is two way current injection between node A and B, it makes a loop with loop gain = +RA*RB*YAB*YBA

• These loops are non touching if the involved circuit nodes are separate.

M1 M2

M3 M4

M6

M8c

VDD

VSS

Cc2

Vin- Vin+

M5

VA

V2

M9 M7Mb1

Isupply

M8

Vo

Vbb

Cc1

V1

Mb2

1 1 1 2 7 2

2 2 2 1 2 2 1 6 1

2 2 1

1 2 1 2 1 2

1 2 2 2

Loops:

; ( )

( ); ( )

( ); ( );

, , , and ; , , and ; and are non touching.

1

oA o A C m C

C o o C m C

A A A C o o L C C

A o oA o

A o oA o o

L sRC L R R sC g sC

L sR C C L R R sC g sC

L sR C C L sR C C C

L L L L L L L L L

L L L L L L L

1 1 2 1 2

1 2 1 1 2 2 1 2 1 2

1 2 2 6 1 2 1 1

2 2 2 1 2 2 1 1

...

Two forward paths from to :

; 1 ( )

; 1 ( )

oA oA oA

A o o A o A A o A

i o

m m o A A C

m C o A A C

L L L L L L L L

L L L L L L L L L L L L L L L L L L L

V V

M g R g R sR C C sRC

M g R sC R sR C C sRC

41321

24232121

:loops Five

GGGGG

HGHGGHGG

412

3211

:paths forward Two

GGM

GGGM

4132124232121

41321

1R(s)

Y(s)

:gain Total

GGGGGHGHGGHGG

GGGGG

41321242321211

:tDeterminan

GGGGGHGHGGHGG

1

loops. no 1,path forward removingAfter

1

1

loops. no 2,path forward removingAfter

2

s

1

s

1x y

s

1b3

b2

b1

-a1

-a2

-a3

x2

x1

e

x3

Σ Σ

x3eyx

x2x1

b1

b3

b2

-a3-a2

-a1

11s

1s

1s

Example:

s

1

s

1x y

s

1b3

b2

b1

-a1

-a2

-a3

x2

x1

e

x3

Σ Σ

• Forward paths:

s

bM

s

bM

s

bM

13

22

2

33

1

• Loops:

33

3

22

2

11

s

aL

s

aL

s

aL

Determinant:

Δ1: If M1 is taken out, all loops are broken.

therefore Δ1 = 1

Δ2: If M2 is taken out, all loops are broken.

therefore Δ2 = 1

Δ3: Similarly, Δ3 = 1

33

22111

s

a

s

a

s

aL

ialli

33

221

33

221

321

1..

sa

sa

sa

sb

sb

sb

MMMMFT ii

1 10 1 1 101 100 100

2 ( 20) 2 2 ( 20)

1 1000 101 1

2 ( 20) ( 20)

s s s

s s s s s s s

s

s s s s s

4

1 1000 1 10 11* 1 100

2 ( 20) 2 ( 20) 2s s s

Gs s s s s s s

y R N

11

1

RsL Cs

1

22

1

RsL U y+

-

2R+-

Vc

I2

I1

One forward path, two loops, no non-touching loops.

1 1 2 2

1 1 1 11 ; 2L L

L s R Cs Cs L s R

21 1 2 2

1 1 11 ; 1 1M R

L s R Cs L s R

11222211

2

))((..

RsLRsLRsLRsLCs

RFT

1 1. .

1 1 2

MT F

L L

1001

2

s

s

10

( 20)s s U Y+

-

+

-

2

5s +

+

Two forward paths, three loops, no non-touching loops.1 10

1 100 ; 1 12 ( 20)

sM

s s s

1 21 100 ; 2 1

2 5

sM

s s

1 101 100 ;

2 ( 20)

1 10 1 22 ; 1 100

2 ( 20) 2 5

sL

s s s

s sL L

s s s s s