Offset Analysis

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declare([beta,beta_square,gmp,gmn,Lp,Ln,Vic,VTn,Wp,Wn,I_TAIL],[constant,real,scalar])$
assume(beta>0)$
assume(beta_square>0)$
assume(gmp>0)$
assume(gmn>0)$
assume(Lp>0)$
assume(Ln>0)$
assume(Vic>0)$
assume(VTn>0)$
assume(Wp>0)$
assume(Wn>0)$
assume(I_TAIL>0)$

Differential Pair with Resistive Load

No Mismatch

See "Analysis and Design of Analog Integrated Circuits", 4th Ed., by P. R. Gray, P. J. Hurst, S. H. Lewis, R. G. Meyer, Wiley, 2001, p. 219-221

Large-signal solution Channel-length modulation neglected

Solution:solve([
I_TAIL=I_1+I_2,
I_1=1/2*beta*(Vgs_1-VT)^2,
Vgs_1=Vg_1-Vs_12,
Vg_1=Vic+Vid/2,
I_2=1/2*beta*(Vgs_2-VT)^2,
Vgs_2=Vg_2-Vs_12,
Vg_2=Vic-Vid/2,
Vds_1=VDD-RD*I_1-Vs_12,
Vds_2=VDD-RD*I_2-Vs_12,
Vout=Vds_1-Vds_2],
[I_1,I_2,Vg_1,Vg_2,Vgs_1,Vgs_2,Vds_1,Vds_2,Vs_12,Vout]);
ev(Vout,Solution);
[[I1=−β Vid 4 ITAIL−β Vid2−2 ITAIL4,I2=β Vid 4 ITAIL−β Vid2+2 ITAIL4,Vg1=Vid+2 Vic2,Vg2=−Vid−2 Vic2,Vgs1=−β 4 ITAIL−β Vid2−β Vid−2 β VT2 β,Vgs2=−β 4 ITAIL−β Vid2+β Vid−2 β VT2 β,Vds1=β (β RD Vid−2) 4 ITAIL−β Vid2+4 β VT+4 β VDD−2 ITAIL β RD−4 Vic β4 β,Vds2=−β (β RD Vid+2) 4 ITAIL−β Vid2−4 β VT−4 β VDD+2 ITAIL β RD+4 Vic β4 β,Vs12=β 4 ITAIL−β Vid2−2 β VT+2 Vic β2 β,Vout=β RD Vid 4 ITAIL−β Vid22],[I1=β Vid 4 ITAIL−β Vid2+2 ITAIL4,I2=−β Vid 4 ITAIL−β Vid2−2 ITAIL4,Vg1=Vid+2 Vic2,Vg2=−Vid−2 Vic2,Vgs1=β 4 ITAIL−β Vid2+β Vid+2 β VT2 β,Vgs2=β 4 ITAIL−β Vid2−β Vid+2 β VT2 β,Vds1=−β (β RD Vid−2) 4 ITAIL−β Vid2−4 β VT−4 β VDD+2 ITAIL β RD+4 Vic β4 β,Vds2=β (β RD Vid+2) 4 ITAIL−β Vid2+4 β VT+4 β VDD−2 ITAIL β RD−4 Vic β4 β,Vs12=−β 4 ITAIL−β Vid2+2 β VT−2 Vic β2 β,Vout=−β RD Vid 4 ITAIL−β Vid22]]\left[ \left[ I_{1}=-{{\sqrt{\beta}\,\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}-2\,\mathit{I}_{TAIL}}\over{4}} , I_{2}={{\sqrt{\beta}\,\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}+2\,\mathit{I}_{TAIL}}\over{4}} , \mathit{Vg}_{1}={{\mathit{Vid}+2\,\mathit{Vic}}\over{2}} , \mathit{Vg}_{2}=-{{\mathit{Vid}-2\,\mathit{Vic}}\over{2}} , \mathit{Vgs}_{1}=-{{\sqrt{\beta}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}-\beta\,\mathit{Vid}-2\,\beta\,\mathit{VT}}\over{2\,\beta}} , \mathit{Vgs}_{2}=-{{\sqrt{\beta}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}+\beta\,\mathit{Vid}-2\,\beta\,\mathit{VT}}\over{2\,\beta}} , \mathit{Vds}_{1}={{\sqrt{\beta}\,\left(\beta\,\mathit{RD}\,\mathit{Vid}-2\right)\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}+4\,\beta\,\mathit{VT}+4\,\beta\,\mathit{VDD}-2\,\mathit{I}_{TAIL}\,\beta\,\mathit{RD}-4\,\mathit{Vic}\,\beta}\over{4\,\beta}} , \mathit{Vds}_{2}=-{{\sqrt{\beta}\,\left(\beta\,\mathit{RD}\,\mathit{Vid}+2\right)\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}-4\,\beta\,\mathit{VT}-4\,\beta\,\mathit{VDD}+2\,\mathit{I}_{TAIL}\,\beta\,\mathit{RD}+4\,\mathit{Vic}\,\beta}\over{4\,\beta}} , \mathit{Vs}_{12}={{\sqrt{\beta}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}-2\,\beta\,\mathit{VT}+2\,\mathit{Vic}\,\beta}\over{2\,\beta}} , \mathit{Vout}={{\sqrt{\beta}\,\mathit{RD}\,\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}}\over{2}} \right] , \left[ I_{1}={{\sqrt{\beta}\,\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}+2\,\mathit{I}_{TAIL}}\over{4}} , I_{2}=-{{\sqrt{\beta}\,\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}-2\,\mathit{I}_{TAIL}}\over{4}} , \mathit{Vg}_{1}={{\mathit{Vid}+2\,\mathit{Vic}}\over{2}} , \mathit{Vg}_{2}=-{{\mathit{Vid}-2\,\mathit{Vic}}\over{2}} , \mathit{Vgs}_{1}={{\sqrt{\beta}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}+\beta\,\mathit{Vid}+2\,\beta\,\mathit{VT}}\over{2\,\beta}} , \mathit{Vgs}_{2}={{\sqrt{\beta}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}-\beta\,\mathit{Vid}+2\,\beta\,\mathit{VT}}\over{2\,\beta}} , \mathit{Vds}_{1}=-{{\sqrt{\beta}\,\left(\beta\,\mathit{RD}\,\mathit{Vid}-2\right)\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}-4\,\beta\,\mathit{VT}-4\,\beta\,\mathit{VDD}+2\,\mathit{I}_{TAIL}\,\beta\,\mathit{RD}+4\,\mathit{Vic}\,\beta}\over{4\,\beta}} , \mathit{Vds}_{2}={{\sqrt{\beta}\,\left(\beta\,\mathit{RD}\,\mathit{Vid}+2\right)\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}+4\,\beta\,\mathit{VT}+4\,\beta\,\mathit{VDD}-2\,\mathit{I}_{TAIL}\,\beta\,\mathit{RD}-4\,\mathit{Vic}\,\beta}\over{4\,\beta}} , \mathit{Vs}_{12}=-{{\sqrt{\beta}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}+2\,\beta\,\mathit{VT}-2\,\mathit{Vic}\,\beta}\over{2\,\beta}} , \mathit{Vout}=-{{\sqrt{\beta}\,\mathit{RD}\,\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}}\over{2}} \right] \right]β RD Vid 4 ITAIL−β Vid22{{\sqrt{\beta}\,\mathit{RD}\,\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}-\beta\,\mathit{Vid}^2}}\over{2}}
taylor(-(sqrt(beta)*Vid*sqrt(4*I_TAIL-beta*Vid^2)*RD)/2,Vid,0,7);
coeff(%,Vid,1);
ev(%,I_TAIL=gmn^2/beta);
−ITAIL β RD Vid+ITAIL β3 RD Vid38 ITAIL+β5 RD Vid5128 ITAIL ITAIL+β7 RD Vid71024 ITAIL ITAIL2+⋯-\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\beta}\,\mathit{RD}\,\mathit{Vid}+{{\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\beta}^3\,\mathit{RD}\,\mathit{Vid}^3}\over{8\,\mathit{I}_{TAIL}}}+{{\sqrt{\beta}^5\,\mathit{RD}\,\mathit{Vid}^5}\over{128\,\sqrt{\mathit{I}_{TAIL}}\,\mathit{I}_{TAIL}}}+{{\sqrt{\beta}^7\,\mathit{RD}\,\mathit{Vid}^7}\over{1024\,\sqrt{\mathit{I}_{TAIL}}\,\mathit{I}_{TAIL}^2}}+\cdots−ITAIL β RD-\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\beta}\,\mathit{RD}−gmn RD-\mathit{gmn}\,\mathit{RD}

Differential Pair with Active Load

No Mismatch

See "Analysis and Design of Analog Integrated Circuits", 4th Ed., by P. R. Gray, P. J. Hurst, S. H. Lewis, R. G. Meyer, Wiley, 2001, p. 287-299

Large-signal solution Channel-length modulation neglected

Solution:solve([
I_TAIL=I_1+I_2,
I_1=1/2*betan*(Vgs_1-VTn)^2,
Vgs_1=Vg_1-Vs_12,
Vg_1=Vic+Vid/2,
I_2=1/2*betan*(Vgs_2-VTn)^2,
Vgs_2=Vg_2-Vs_12,
Vg_2=Vic-Vid/2,
I_3=-I_1,
Vgs_3=sqrt(2*(I_3)/betap)+VTp,
Vgs_4=Vgs_3,
I_4=1/2*betap*(Vgs_4-VTp)^2,
Iout=-(I_2+I_4)],
[Vgs_1,Vgs_2,Vgs_3,Vgs_4,Vg_1,Vg_2,Vs_12,Iout,I_1,I_2,I_3,I_4]);
[[Vgs1=−4 ITAIL betan−Vid2 betan2+(−Vid−2 VTn) betan2 betan,Vgs2=−4 ITAIL betan−Vid2 betan2+(Vid−2 VTn) betan2 betan,Vgs3=2 Vid 4 ITAIL betan−Vid2 betan2−2 ITAILbetap+2 VTp2,Vgs4=2 Vid 4 ITAIL betan−Vid2 betan2−2 ITAILbetap+2 VTp2,Vg1=Vid+2 Vic2,Vg2=−Vid−2 Vic2,Vs12=4 ITAIL betan−Vid2 betan2+(2 Vic−2 VTn) betan2 betan,Iout=−Vid 4 ITAIL betan−Vid2 betan22,I1=−Vid 4 ITAIL betan−Vid2 betan2−2 ITAIL4,I2=Vid 4 ITAIL betan−Vid2 betan2+2 ITAIL4,I3=Vid 4 ITAIL betan−Vid2 betan2−2 ITAIL4,I4=Vid 4 ITAIL betan−Vid2 betan2−2 ITAIL4],[Vgs1=4 ITAIL betan−Vid2 betan2+(Vid+2 VTn) betan2 betan,Vgs2=4 ITAIL betan−Vid2 betan2+(2 VTn−Vid) betan2 betan,Vgs3=2 −Vid 4 ITAIL betan−Vid2 betan2+2 ITAILbetap+2 VTp2,Vgs4=2 −Vid 4 ITAIL betan−Vid2 betan2+2 ITAILbetap+2 VTp2,Vg1=Vid+2 Vic2,Vg2=−Vid−2 Vic2,Vs12=−4 ITAIL betan−Vid2 betan2+(2 VTn−2 Vic) betan2 betan,Iout=Vid 4 ITAIL betan−Vid2 betan22,I1=Vid 4 ITAIL betan−Vid2 betan2+2 ITAIL4,I2=−Vid 4 ITAIL betan−Vid2 betan2−2 ITAIL4,I3=−Vid 4 ITAIL betan−Vid2 betan2+2 ITAIL4,I4=−Vid 4 ITAIL betan−Vid2 betan2+2 ITAIL4]]\left[ \left[ \mathit{Vgs}_{1}=-{{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+\left(-\mathit{Vid}-2\,\mathit{VTn}\right)\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Vgs}_{2}=-{{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+\left(\mathit{Vid}-2\,\mathit{VTn}\right)\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Vgs}_{3}={{\sqrt{2}\,\sqrt{{{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}-2\,\mathit{I}_{TAIL}}\over{\mathit{betap}}}}+2\,\mathit{VTp}}\over{2}} , \mathit{Vgs}_{4}={{\sqrt{2}\,\sqrt{{{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}-2\,\mathit{I}_{TAIL}}\over{\mathit{betap}}}}+2\,\mathit{VTp}}\over{2}} , \mathit{Vg}_{1}={{\mathit{Vid}+2\,\mathit{Vic}}\over{2}} , \mathit{Vg}_{2}=-{{\mathit{Vid}-2\,\mathit{Vic}}\over{2}} , \mathit{Vs}_{12}={{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+\left(2\,\mathit{Vic}-2\,\mathit{VTn}\right)\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Iout}=-{{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}}\over{2}} , I_{1}=-{{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}-2\,\mathit{I}_{TAIL}}\over{4}} , I_{2}={{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+2\,\mathit{I}_{TAIL}}\over{4}} , I_{3}={{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}-2\,\mathit{I}_{TAIL}}\over{4}} , I_{4}={{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}-2\,\mathit{I}_{TAIL}}\over{4}} \right] , \left[ \mathit{Vgs}_{1}={{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+\left(\mathit{Vid}+2\,\mathit{VTn}\right)\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Vgs}_{2}={{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+\left(2\,\mathit{VTn}-\mathit{Vid}\right)\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Vgs}_{3}={{\sqrt{2}\,\sqrt{-{{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+2\,\mathit{I}_{TAIL}}\over{\mathit{betap}}}}+2\,\mathit{VTp}}\over{2}} , \mathit{Vgs}_{4}={{\sqrt{2}\,\sqrt{-{{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+2\,\mathit{I}_{TAIL}}\over{\mathit{betap}}}}+2\,\mathit{VTp}}\over{2}} , \mathit{Vg}_{1}={{\mathit{Vid}+2\,\mathit{Vic}}\over{2}} , \mathit{Vg}_{2}=-{{\mathit{Vid}-2\,\mathit{Vic}}\over{2}} , \mathit{Vs}_{12}=-{{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+\left(2\,\mathit{VTn}-2\,\mathit{Vic}\right)\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Iout}={{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}}\over{2}} , I_{1}={{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+2\,\mathit{I}_{TAIL}}\over{4}} , I_{2}=-{{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}-2\,\mathit{I}_{TAIL}}\over{4}} , I_{3}=-{{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+2\,\mathit{I}_{TAIL}}\over{4}} , I_{4}=-{{\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{Vid}^2\,\mathit{betan}^2}+2\,\mathit{I}_{TAIL}}\over{4}} \right] \right]
ratsimp(ev((sqrt(betan)*Vid*sqrt(4*I_TAIL-betan*Vid^2))/2,
betan=beta_square*Wn/Ln));
taylor(%,Vid,0,3);
coeff(%,Vid,1);
Gm:ev(%,I_TAIL=gmn^2/(beta_square*Wn/Ln));
Wn βsquare Vid 4 ITAIL Ln−Wn βsquare Vid22 Ln{{\sqrt{\mathit{Wn}}\,\sqrt{\beta_{square}}\,\mathit{Vid}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{Ln}-\mathit{Wn}\,\beta_{square}\,\mathit{Vid}^2}}\over{2\,\mathit{Ln}}}ITAIL Ln Wn βsquare VidLn−ITAIL Wn3 βsquare3 Vid38 Ln ITAIL Ln+⋯{{\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\mathit{Ln}}\,\sqrt{\mathit{Wn}}\,\sqrt{\beta_{square}}\,\mathit{Vid}}\over{\mathit{Ln}}}-{{\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\mathit{Wn}}^3\,\sqrt{\beta_{square}}^3\,\mathit{Vid}^3}\over{8\,\sqrt{\mathit{Ln}}\,\mathit{I}_{TAIL}\,\mathit{Ln}}}+\cdotsITAIL Ln Wn βsquareLn{{\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\mathit{Ln}}\,\sqrt{\mathit{Wn}}\,\sqrt{\beta_{square}}}\over{\mathit{Ln}}}gmn\mathit{gmn}

With mismatch

A differential NMOST input pair (W/L=50/2 micron) is loaded with a PMOST current mirror (W/L=36/1 micron) in a process with A_VT,N = A_VT,P = 6 mV micron and the ratio between the beta square (for W/L = 1) is N/P = 3. Calculate the input referred mismatch.

delta_VTn contribution

Solution:solve([
I_TAIL=I_1+I_2,
I_1=1/2*betan*(Vgs_1-VTn)^2,
Vgs_1=Vg_1-Vs_12,
Vg_1=Vic,
I_2=1/2*betan*(Vgs_2-(VTn+delta_VTn))^2,
Vgs_2=Vg_2-Vs_12,
Vg_2=Vic,
I_3=-I_1,
Vgs_3=sqrt(2*(I_3)/betap)+VTp,
Vgs_4=Vgs_3,
I_4=1/2*betap*(Vgs_4-VTp)^2,
Iout=-(I_2+I_4)],
[Vgs_1,Vgs_2,Vgs_3,Vgs_4,Vg_1,Vg_2,Vs_12,Iout,I_1,I_2,I_3,I_4]);
[[Vgs1=−4 ITAIL betan−betan2 δVTn2−betan δVTn−2 VTn betan2 betan,Vgs2=−4 ITAIL betan−betan2 δVTn2−betan δVTn−2 VTn betan2 betan,Vgs3=2 δVTn 4 ITAIL betan−betan2 δVTn2−2 ITAILbetap+2 VTp2,Vgs4=2 δVTn 4 ITAIL betan−betan2 δVTn2−2 ITAILbetap+2 VTp2,Vg1=Vic,Vg2=Vic,Vs12=4 ITAIL betan−betan2 δVTn2−betan δVTn+(2 Vic−2 VTn) betan2 betan,Iout=−δVTn 4 ITAIL betan−betan2 δVTn22,I1=−δVTn 4 ITAIL betan−betan2 δVTn2−2 ITAIL4,I2=δVTn 4 ITAIL betan−betan2 δVTn2+2 ITAIL4,I3=δVTn 4 ITAIL betan−betan2 δVTn2−2 ITAIL4,I4=δVTn 4 ITAIL betan−betan2 δVTn2−2 ITAIL4],[Vgs1=4 ITAIL betan−betan2 δVTn2+betan δVTn+2 VTn betan2 betan,Vgs2=4 ITAIL betan−betan2 δVTn2+betan δVTn+2 VTn betan2 betan,Vgs3=2 −δVTn 4 ITAIL betan−betan2 δVTn2+2 ITAILbetap+2 VTp2,Vgs4=2 −δVTn 4 ITAIL betan−betan2 δVTn2+2 ITAILbetap+2 VTp2,Vg1=Vic,Vg2=Vic,Vs12=−4 ITAIL betan−betan2 δVTn2+betan δVTn+(2 VTn−2 Vic) betan2 betan,Iout=δVTn 4 ITAIL betan−betan2 δVTn22,I1=δVTn 4 ITAIL betan−betan2 δVTn2+2 ITAIL4,I2=−δVTn 4 ITAIL betan−betan2 δVTn2−2 ITAIL4,I3=−δVTn 4 ITAIL betan−betan2 δVTn2+2 ITAIL4,I4=−δVTn 4 ITAIL betan−betan2 δVTn2+2 ITAIL4]]\left[ \left[ \mathit{Vgs}_{1}=-{{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}-\mathit{betan}\,\delta_{VTn}-2\,\mathit{VTn}\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Vgs}_{2}=-{{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}-\mathit{betan}\,\delta_{VTn}-2\,\mathit{VTn}\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Vgs}_{3}={{\sqrt{2}\,\sqrt{{{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}-2\,\mathit{I}_{TAIL}}\over{\mathit{betap}}}}+2\,\mathit{VTp}}\over{2}} , \mathit{Vgs}_{4}={{\sqrt{2}\,\sqrt{{{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}-2\,\mathit{I}_{TAIL}}\over{\mathit{betap}}}}+2\,\mathit{VTp}}\over{2}} , \mathit{Vg}_{1}=\mathit{Vic} , \mathit{Vg}_{2}=\mathit{Vic} , \mathit{Vs}_{12}={{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}-\mathit{betan}\,\delta_{VTn}+\left(2\,\mathit{Vic}-2\,\mathit{VTn}\right)\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Iout}=-{{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}}\over{2}} , I_{1}=-{{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}-2\,\mathit{I}_{TAIL}}\over{4}} , I_{2}={{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}+2\,\mathit{I}_{TAIL}}\over{4}} , I_{3}={{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}-2\,\mathit{I}_{TAIL}}\over{4}} , I_{4}={{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}-2\,\mathit{I}_{TAIL}}\over{4}} \right] , \left[ \mathit{Vgs}_{1}={{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}+\mathit{betan}\,\delta_{VTn}+2\,\mathit{VTn}\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Vgs}_{2}={{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}+\mathit{betan}\,\delta_{VTn}+2\,\mathit{VTn}\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Vgs}_{3}={{\sqrt{2}\,\sqrt{-{{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}+2\,\mathit{I}_{TAIL}}\over{\mathit{betap}}}}+2\,\mathit{VTp}}\over{2}} , \mathit{Vgs}_{4}={{\sqrt{2}\,\sqrt{-{{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}+2\,\mathit{I}_{TAIL}}\over{\mathit{betap}}}}+2\,\mathit{VTp}}\over{2}} , \mathit{Vg}_{1}=\mathit{Vic} , \mathit{Vg}_{2}=\mathit{Vic} , \mathit{Vs}_{12}=-{{\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}+\mathit{betan}\,\delta_{VTn}+\left(2\,\mathit{VTn}-2\,\mathit{Vic}\right)\,\mathit{betan}}\over{2\,\mathit{betan}}} , \mathit{Iout}={{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}}\over{2}} , I_{1}={{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}+2\,\mathit{I}_{TAIL}}\over{4}} , I_{2}=-{{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}-2\,\mathit{I}_{TAIL}}\over{4}} , I_{3}=-{{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}+2\,\mathit{I}_{TAIL}}\over{4}} , I_{4}=-{{\delta_{VTn}\,\sqrt{4\,\mathit{I}_{TAIL}\,\mathit{betan}-\mathit{betan}^2\,\delta_{VTn}^2}+2\,\mathit{I}_{TAIL}}\over{4}} \right] \right]

Calculation of IOSout and VOSin

IOSout:ev((sqrt(betan)*delta_VTn*sqrt(4*I_TAIL-betan*delta_VTn^2))/2,
betan=beta_square*Wn/Ln,
I_TAIL=1/2*beta_square*Wn/Ln*(Vic-VTn)^2+1/2*beta_square*Wn/Ln*(Vic-(VTn+delta_VTn))^2);
Wn βsquare δVTn 4 (Wn βsquare (−δVTn+Vic−VTn)22 Ln+(Vic−VTn)2 Wn βsquare2 Ln)−Wn βsquare δVTn2Ln2 Ln{{\sqrt{\mathit{Wn}}\,\sqrt{\beta_{square}}\,\delta_{VTn}\,\sqrt{4\,\left({{\mathit{Wn}\,\beta_{square}\,\left(-\delta_{VTn}+\mathit{Vic}-\mathit{VTn}\right)^2}\over{2\,\mathit{Ln}}}+{{\left(\mathit{Vic}-\mathit{VTn}\right)^2\,\mathit{Wn}\,\beta_{square}}\over{2\,\mathit{Ln}}}\right)-{{\mathit{Wn}\,\beta_{square}\,\delta_{VTn}^2}\over{\mathit{Ln}}}}}\over{2\,\sqrt{\mathit{Ln}}}}
VOSin:taylor(ev(IOSout/(beta_square*Wn/Ln*(Vic-VTn))),delta_VTn,0,2);
δVTn−δVTn22 Vic−2 VTn+⋯\delta_{VTn}-{{\delta_{VTn}^2}\over{2\,\mathit{Vic}-2\,\mathit{VTn}}}+\cdots
VOSin:taylor(ev(IOSout/(beta_square*Wn/Ln*(Vic-VTn))),delta_VTn,0,1);
+δVTn+⋯+\delta_{VTn}+\cdots

delta_VTp

Solution:solve([
I_TAIL=I_1+I_2,
I_1=1/2*betan*(Vgs_1-VTn)^2,
Vgs_1=Vg_1-Vs_12,
Vg_1=Vic,
I_2=1/2*betan*(Vgs_2-VTn)^2,
Vgs_2=Vg_2-Vs_12,
Vg_2=Vic,
I_3=-I_1,
Vgs_3=sqrt(2*(I_3)/betap)+VTp,
Vgs_4=Vgs_3,
I_4=1/2*betap*(Vgs_4-(VTp+delta_VTp))^2,
Iout=-(I_2+I_4)],
[Vgs_1,Vgs_2,Vgs_3,Vgs_4,Vg_1,Vg_2,Vs_12,Iout,I_1,I_2,I_3,I_4]);
[[Vgs1=−ITAIL betan−VTn betanbetan,Vgs2=−ITAIL betan−VTn betanbetan,Vgs3=VTp betap+ITAIL i betapbetap,Vgs4=VTp betap+ITAIL i betapbetap,Vg1=Vic,Vg2=Vic,Vs12=(Vic−VTn) betan+ITAIL betanbetan,Iout=2 ITAIL i betap δVTp−betap δVTp22,I1=ITAIL2,I2=ITAIL2,I3=−ITAIL2,I4=−−betap δVTp2+2 ITAIL i betap δVTp+ITAIL2],[Vgs1=VTn betan+ITAIL betanbetan,Vgs2=VTn betan+ITAIL betanbetan,Vgs3=VTp betap+ITAIL i betapbetap,Vgs4=VTp betap+ITAIL i betapbetap,Vg1=Vic,Vg2=Vic,Vs12=−(VTn−Vic) betan+ITAIL betanbetan,Iout=2 ITAIL i betap δVTp−betap δVTp22,I1=ITAIL2,I2=ITAIL2,I3=−ITAIL2,I4=−−betap δVTp2+2 ITAIL i betap δVTp+ITAIL2]]\left[ \left[ \mathit{Vgs}_{1}=-{{\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\mathit{betan}}-\mathit{VTn}\,\mathit{betan}}\over{\mathit{betan}}} , \mathit{Vgs}_{2}=-{{\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\mathit{betan}}-\mathit{VTn}\,\mathit{betan}}\over{\mathit{betan}}} , \mathit{Vgs}_{3}={{\mathit{VTp}\,\mathit{betap}+\sqrt{\mathit{I}_{TAIL}}\,i\,\sqrt{\mathit{betap}}}\over{\mathit{betap}}} , \mathit{Vgs}_{4}={{\mathit{VTp}\,\mathit{betap}+\sqrt{\mathit{I}_{TAIL}}\,i\,\sqrt{\mathit{betap}}}\over{\mathit{betap}}} , \mathit{Vg}_{1}=\mathit{Vic} , \mathit{Vg}_{2}=\mathit{Vic} , \mathit{Vs}_{12}={{\left(\mathit{Vic}-\mathit{VTn}\right)\,\mathit{betan}+\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\mathit{betan}}}\over{\mathit{betan}}} , \mathit{Iout}={{2\,\sqrt{\mathit{I}_{TAIL}}\,i\,\sqrt{\mathit{betap}}\,\delta_{VTp}-\mathit{betap}\,\delta_{VTp}^2}\over{2}} , I_{1}={{\mathit{I}_{TAIL}}\over{2}} , I_{2}={{\mathit{I}_{TAIL}}\over{2}} , I_{3}=-{{\mathit{I}_{TAIL}}\over{2}} , I_{4}=-{{-\mathit{betap}\,\delta_{VTp}^2+2\,\sqrt{\mathit{I}_{TAIL}}\,i\,\sqrt{\mathit{betap}}\,\delta_{VTp}+\mathit{I}_{TAIL}}\over{2}} \right] , \left[ \mathit{Vgs}_{1}={{\mathit{VTn}\,\mathit{betan}+\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\mathit{betan}}}\over{\mathit{betan}}} , \mathit{Vgs}_{2}={{\mathit{VTn}\,\mathit{betan}+\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\mathit{betan}}}\over{\mathit{betan}}} , \mathit{Vgs}_{3}={{\mathit{VTp}\,\mathit{betap}+\sqrt{\mathit{I}_{TAIL}}\,i\,\sqrt{\mathit{betap}}}\over{\mathit{betap}}} , \mathit{Vgs}_{4}={{\mathit{VTp}\,\mathit{betap}+\sqrt{\mathit{I}_{TAIL}}\,i\,\sqrt{\mathit{betap}}}\over{\mathit{betap}}} , \mathit{Vg}_{1}=\mathit{Vic} , \mathit{Vg}_{2}=\mathit{Vic} , \mathit{Vs}_{12}=-{{\left(\mathit{VTn}-\mathit{Vic}\right)\,\mathit{betan}+\sqrt{\mathit{I}_{TAIL}}\,\sqrt{\mathit{betan}}}\over{\mathit{betan}}} , \mathit{Iout}={{2\,\sqrt{\mathit{I}_{TAIL}}\,i\,\sqrt{\mathit{betap}}\,\delta_{VTp}-\mathit{betap}\,\delta_{VTp}^2}\over{2}} , I_{1}={{\mathit{I}_{TAIL}}\over{2}} , I_{2}={{\mathit{I}_{TAIL}}\over{2}} , I_{3}=-{{\mathit{I}_{TAIL}}\over{2}} , I_{4}=-{{-\mathit{betap}\,\delta_{VTp}^2+2\,\sqrt{\mathit{I}_{TAIL}}\,i\,\sqrt{\mathit{betap}}\,\delta_{VTp}+\mathit{I}_{TAIL}}\over{2}} \right] \right]

Calculation of IOSout and VOSin

IOSout:ratsimp(ev(-(betap*delta_VTp^2+2*sqrt(I_TAIL)*sqrt(betap)*delta_VTp)/2,
betap=1/3*beta_square*Wp/Lp,
I_TAIL=beta_square*Wn/Ln*(Vic-VTn)^2));
−3 Ln Lp Wp βsquare δVTp2+6 Wn Wp Lp βsquare ∣Vic−VTn∣ δVTp2 332 Ln Lp32-{{\sqrt{3}\,\sqrt{\mathit{Ln}}\,\sqrt{\mathit{Lp}}\,\mathit{Wp}\,\beta_{square}\,\delta_{VTp}^2+6\,\sqrt{\mathit{Wn}}\,\sqrt{\mathit{Wp}}\,\mathit{Lp}\,\beta_{square}\,\left| \mathit{Vic}-\mathit{VTn}\right| \,\delta_{VTp}}\over{2\,3^{{{3}\over{2}}}\,\sqrt{\mathit{Ln}}\,\mathit{Lp}^{{{3}\over{2}}}}}
VOSin:ratsimp(taylor(ev(IOSout/(beta_square*Wn/Ln*(Vic-VTn))),delta_VTp,0,2));
−3 Lp Wn Ln Wp δVTp2+6 Ln Wp Lp Wn ∣Vic−VTn∣ δVTpLp Wn32 (2 332 Lp Vic−2 332 Lp VTn)-{{\sqrt{3}\,\sqrt{\mathit{Lp}}\,\sqrt{\mathit{Wn}}\,\mathit{Ln}\,\mathit{Wp}\,\delta_{VTp}^2+6\,\sqrt{\mathit{Ln}}\,\sqrt{\mathit{Wp}}\,\mathit{Lp}\,\mathit{Wn}\,\left| \mathit{Vic}-\mathit{VTn}\right| \,\delta_{VTp}}\over{\sqrt{\mathit{Lp}}\,\mathit{Wn}^{{{3}\over{2}}}\,\left(2\,3^{{{3}\over{2}}}\,\mathit{Lp}\,\mathit{Vic}-2\,3^{{{3}\over{2}}}\,\mathit{Lp}\,\mathit{VTn}\right)}}
VOSin:ratsimp(taylor(ev(IOSout/(beta_square*Wn/Ln*(Vic-VTn))),delta_VTp,0,1));
−Ln Wp ∣Vic−VTn∣ δVTpLp Wn (3 Vic−3 VTn)-{{\sqrt{\mathit{Ln}}\,\sqrt{\mathit{Wp}}\,\left| \mathit{Vic}-\mathit{VTn}\right| \,\delta_{VTp}}\over{\sqrt{\mathit{Lp}}\,\sqrt{\mathit{Wn}}\,\left(\sqrt{3}\,\mathit{Vic}-\sqrt{3}\,\mathit{VTn}\right)}}

Summing up the offset variances and

VOSin_total:sqrt(delta_VTn^2+(-(sqrt(Ln)*sqrt(Wp)*delta_VTp)/(sqrt(Lp)*sqrt(Wn)*sqrt(3)))^2);
Ln Wp δVTp23 Lp Wn+δVTn2\sqrt{{{\mathit{Ln}\,\mathit{Wp}\,\delta_{VTp}^2}\over{3\,\mathit{Lp}\,\mathit{Wn}}}+\delta_{VTn}^2}
VOSin:ratsimp(ev(VOSin_total,
delta_VTp=AVT/sqrt(Wp*Lp),
delta_VTn=AVT/sqrt(Wn*Ln)));
float(ev(%,Wn=50e-6,Ln=2e-6,Wp=36e-6,Lp=1e-6,AVT=6e-9));
3 Lp2+Ln2 ∣AVT∣3 Ln Lp Wn{{\sqrt{3\,\mathit{Lp}^2+\mathit{Ln}^2}\,\left| \mathit{AVT}\right| }\over{\sqrt{3}\,\sqrt{\mathit{Ln}}\,\mathit{Lp}\,\sqrt{\mathit{Wn}}}}9.165151389911681×10−49.165151389911681 \times 10^{-4}

The answer is 0.92 mV