nodes : variables branches : gains e.g.y = a ∙ x e.g.y = 3x + 5z – 0.1y Signal Flow Graph xy a xz y

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

ry uG1G1 1 -G 2 Ry z G3G3 1 -H 1 x G1G1 G2G2 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 y r1y 1 Feedback connections can be simplified into a single branch Note: the internal node E is lost!

R y z G3G3 1 -H 1 x G1G1 G2G2 1 y R y z G3G3 1 -H 1 x G1G1 G2G2 1 y z’ 1 R y1 x G1G1 G2G2 1 y z’ R

R y1 x G2G2 1 y z’ y x G1G1 G2G2 1 y z’ R Overall:

Mason’s Gain Formula A forward path: a path from input to output Forward path gain M k : 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 L i : 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: Δ k : The determinant of the S.F.G. after removing the k-th forward path Mason’s Gain formula:

Get T.F. from N to y 1 forward path:Ny M = 1 2 loops:L 1 = -H 1 G 3 L 2 = -G 2 G 3 Δ 1 : remove nodes N, y, and branch N y All loops broken:  Δ 1 = 1 0 R y z G3G3 1 -H 1 x G1G1 G2G2 N 1

Get T.F. from R to y 2 f.p.:Rxzy :M 1 =G 2 G 3 Rzy :M 2 =G 1 G 3 2 loops:L 1 = -G 3 H 1 L 2 = -G 2 G 3 0

Δ 1 : remove M 1 and compute Δ Δ 1 = 1 Δ 2 : remove M 2 and compute Δ Δ 2 = 1 Overall:

H4H4 H1H1 H2H2 H3H3 H5H5 H6H6 H7H7 Forward path: M 1 = H 1 H 2 H 3 M 2 = H 4 Loops: L 1 = H 1 H 5 L 2 = H 2 H 6 L 3 = H 3 H 7 L 4 = H 4 H 7 H 6 H 5 L 1 and L 3 are non-touching

Δ 1 : If M 1 is taken out, all loops are broken. therefore Δ 1 = 1 Δ 2 : If M 2 is taken out, the loop in the middle (L 2 ) is still there. therefore Δ 2 = 1 – L 2 = 1 – H 2 H 6 Total T.F.:

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/Z c V i+ V o1 V BN VoVo V i- VmVm RmRm R o1 There are two input variables: Vi+, Vi- Vo is output variavle Internal circuit nodes: Vm, and I Rm Vo1, and I Ro1 VoVo V i- I Rm VmVm I Ro1 IoIo RmRm R o1 RoRo V i+ -g m1 -g m2 -g m4 -g m6 1/Z c -1/Z c -sC L -sC 1 1/Z c This is a simplied version, by ignoring Cgds and the dynamics at the mirror node.

V o1 VoVo I Ro1 IoIo R o1 RoRo V i+ -V i- -g m1 -g m6 1/Z c -1/Z c -sC L -sC 1 If R m =1/g m4 g m1 = g m2 Two forward paths: M1=g m1 R o1 g m6 R o ; M2=- g m1 R o1 R o /Z c The corresponding  ’s are both 1 Loops: L1= -R o1 (sC 1 +1/Z c ); L2= -R o (sC L +1/Z c ) L3= R o1 R o (-g m6 +1/Z c )/Z c L1 an L2 are non-touching By MGF: TF = (M1+M2)/(1-L1-L2-L3+L1L2) = g m1 R o1 R o (g m6 Z c - 1)/{Z c +R o1 (sC 1 Z c +1)+R o (sC L Z c +1) + R o1 (sC 1 Z c +1)R o (sC L +1/Z c ) + R o1 R o (g m6 Z c -1)/Z c } 1/Z c

When s  0, if Zc  inf, TF = g m1 R o1 R o (g m6 Z c - 1)/{Z c +R o1 (sC 1 Z c +1)+R o (sC L Z c +1) + R o1 (sC 1 Z c +1)R o (sC L +1/Z c ) + R o1 R o (g m6 Z c -1)/Z c }  g m1 R o1 R o g m6 If Zc = 1/sC C : TF = g m1 R o1 R o (g m6 - sC C )/{1+sR o1 (C 1 +C C )+sR o (C L +C C ) + s 2 R o1 R o (C 1 +C C ) (C L +C C ) + sC C R o1 R o (g m6 -sC C )} = g m1 R o1 R o (g m6 - sC C )/{1+sR o1 (C 1 +C C )+sR o (C L +C C ) + s 2 R o1 R o (C 1 C L +C C C L +C C C 1 ) + sC C R o1 R o g m6 } For buffer connection, closed-loop characteristic equation: 1+g m1 R o1 R o g m6 +s{R o1 (C 1 +C C )+R o (C L +C C )+R o1 R o (C C g m6 -g m1 C C )} + s 2 R o1 R o (C 1 C L +C C C L +C C C 1 )=0

Closed-loop BandWidth: ≈const/s-coeff≈g m1 g m6 /(C C g m6 -g m1 C C )=g m1 /C C* g m6 /(g m6 -g m1 ) For stability: g m6 >= g m1 For damping ratio >= 0.5, need b 2 >a*c. That is: (C C g m6 -g m1 C C ) 2 >= g m1 g m6 (C 1 C L +C C C L +C C C 1 ) (C C ) 2 /(C 1 C L +C C C L +C C C 1 ) >= g m1 g m6 /(g m6 -g m1 ) 2  BW 2 <= g m1 g m6 /(C 1 C L +C C C L +C C C 1 ) Notice: R o1 and R o 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+g m1 R o1 R o g m6 +(z-x) {R o1 (C 1 +C C )+R o (C L +C C )+R o1 R o (C C g m6 - g m1 C C )} + (z-x) 2 R o1 R o (C 1 C L +C C C L +C C C 1 )=0 g m1 g m6 -x(C C g m6 -g m1 C C )+x 2 (C 1 C L +C C C L +C C C 1 ) + z{(C C g m6 -g m1 C C )-2x(C 1 C L +C C C L +C C C 1 )} + z 2 (C 1 C L +C C C L +C C C 1 )=0 For stability of z, x < (g m6 -g m1 )/2(C 1 C L /C C +C L +C 1 )

If Zc = Rz + 1/sCc: TF = g m1 R o1 R o (sC c (R z g m6 -1)+g m6 )/ {(sC c R z +1)+R o1 (sC 1 (sC c R z +1)+sC c )+R o (sC L (sC c R z +1)+sC c ) - R o1 (sC 1 (sC c R z +1)+sC c )R o (sC L +1/Z c ) + R o1 R o (g m6 (sC c R z +1)-sC c )/Z c }

V i+ V BN VoVo V i- VmVm RmRm R o1 1/Z c Closed-loop: V o1 VoVo I Ro1 IoIo R o1 RoRo V i+ -V o -g m1 -g m6 1/Z c -1/Z c -sC L -sC 1 V i+ 1

1/Z c V o1 VoVo I Ro1 IoIo R o1 RoRo V i+ -V o -g m1 -g m6 1/Z c -1/Z c -sC L -sC 1 V i+ 1 Forward paths remain the same. Two new loops added: L4=-M1; L5=-M2 These are touching with previous loop.  1-L1-L2-L3-L4-L5+L1L2, set  =0: Z c + g m1 R o1 R o (g m6 Z c - 1)+R o1 (sC 1 Z c +1)+R o (sC L Z c +1)- R o1 (sC 1 Z c +1)R o (sC L +1/Z c ) + R o1 R o (g m6 Z c -1)/Z c = 0 This is the closed-loop characteristic equation.

Z c + g m1 R o1 R o (g m6 Z c - 1)+R o1 (sC 1 Z c +1)+R o (sC L Z c +1)- R o1 (sC 1 Z c +1)R o (sC L +1/Z c ) + R o1 R o (g m6 Z c -1)/Z c = 0 If Z c = R z + 1/sC C, The char eq becomes:L

V o1 VoVo I Ro1 IoIo R o1 RoRo V i+ -V o -g m1 -g m5 -sC L -sC C -sC 1 V i+ 1 I RA RARA VAVA -sC C -sC A sC C g mc

V o1 VoVo I Ro1 IoIo R o1 RoRo V i+ -V o -g m1 -g m5 -sC L -sC C -sC 1 V i+ 1 I RA RARA VAVA -sC C -sC A sC C g mc + g oc If we include a finite r oc for M 6c : g oc There is one more FP: - g m1 R’ o1 g oc R’ A sC C R o Its  =1

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

V o1 VoVo I Ro1 IoIo R o1 RoRo V i+ -V o -g m1 -g m5 -sC L -sC C -sC gd -sC 1 -sC gd V i+ 1 I RA RARA VAVA -sC C -sC A sC C g mc + g oc If we include C gd effect for M 5 : g oc sC gd

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.

Quick tips Each circuit node makes one loop, with loop gain = -R*sC tot If there is two way current injection between node A and B, it makes a loop with loop gain = +R A *R B *Y A  B *Y B  A These loops are non touching if the involved circuit nodes are separate.

x y b3b3 b2b2 b1b1 -a 1 -a 2 -a 3 x2x2 x1x1 e x3x3 Σ Σ x3x3 e yx x2x2 x1x1 b1b1 b3b3 b2b2 -a 2 -a s 1 s 1 s

Example: x y b3b3 b2b2 b1b1 -a 1 -a 2 -a 3 x2x2 x1x1 e x3x3 Σ Σ Forward paths:Loops:

Determinant: Δ 1 : If M 1 is taken out, all loops are broken. therefore Δ 1 = 1 Δ 2 : If M 2 is taken out, all loops are broken. therefore Δ 2 = 1 Δ 3 : Similarly, Δ 3 = 1

U y VcVc I2I2 I1I1 One forward path, two loops, no non-touching loops.

U Y Two forward paths, three loops, no non-touching loops.