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Lecture II Lecture II: • Linear circuit theory review • Amplifier basics • MOS small signal model A. Rivetti – INFN Sezione di Torino Nodal analysis R2 Is R1 R4 R3 Vs Nodal analysis provides a systematic and reliable method to calculate all voltages and currents in a linear circuit A. Rivetti – INFN Sezione di Torino Nodal analysis Writing nodal equations v1 Is R1 R2 v2 R3 R4 Vs v v 1 1 v2 0 I s R R 1 2 v2 v1 v2 v2 V s 0 R3 R 4 R2 A. Rivetti – INFN Sezione di Torino Nodal analysis Writing the circuit matrix R2 v1 Is 1 1 R1 R2 1 R2 R1 v2 R3 R4 Vs I s v1 R2 1 1 1 V s v2 R4 R 2 R3 R 4 1 A. Rivetti – INFN Sezione di Torino Nodal analysis Solving the circuit matrix v1 v2 1 Is V R 1 s 4 1 R 1 R 1R 1 R 2 2 2 1 4 1 1 R 2 1 1 R R R 2 2 1 3 1 1 R R R 2 2 1 1 R R 1 1 R 2 3 4 Is 2 V R s 4 A. Rivetti – INFN Sezione di Torino Nodal analysis Another example Vs R2 Is R1 R3 R4 A. Rivetti – INFN Sezione di Torino Nodal analysis Lecture II Lecture II: • Linear circuit theory review • Amplifier basics • MOS small signal model A. Rivetti – INFN Sezione di Torino Amplifier characteristic The input-output characteristic of an amplifier is usually a non-linear function Over some interval of the input signal, this function can be approximated by a polynomial: 2 n y(t ) a a x(t ) a x(t ) ... a x(t ) 0 1 2 n For narrow range of the input signal, we may write: y(t ) a a x(t ) 0 1 The above expression does not obey the superposition principle A. Rivetti – INFN Sezione di Torino Amplifier basics Small signal model If a0 does not depend on the signal, we can write: y(t ) a x(t ) 1 This is an expression that obeys the superposition principle The small signal model takes into account only variations of signals within a circuit The small signal equivalent circuit can be studied with the methods of linear circuit analysis A. Rivetti – INFN Sezione di Torino Amplifier basics Voltage amplifier Rs Vs(t) Vi(t) Ri Vout AV = Vout/Vi Input impedance high (ideally infinite) Output impedance small (ideally zero) A. Rivetti – INFN Sezione di Torino Amplifier basics VA small signal model RS Vs(t) RO Vi(t) RI AVVi RL Vout Note: impedances may also be complex A. Rivetti – INFN Sezione di Torino Amplifier basics Current amplifier Is(t) Rs Ii(t) Ri Iout AV = Iout/Ii Input impedance small (ideally zero) Output impedance high (ideally infinite) A. Rivetti – INFN Sezione di Torino Amplifier basics CA small signal model Is(t) Rs Ii(t) Ri Is(t) Ro Iout(t) RL Note: impedances may also be complex A. Rivetti – INFN Sezione di Torino Amplifier basics Transconductance amplifier Rs Vs(t) Vi(t) Ri Iout AV = Iout/Vi Input impedance high (ideally infinite) Output impedance high (ideally infinite) Important: the gain is not a number A. Rivetti – INFN Sezione di Torino Amplifier basics TCA small signal model RS Vs(t) Vi(t) RI Is(t) Ro Iout(t) RL Note: impedances may also be complex A. Rivetti – INFN Sezione di Torino Amplifier basics Transimpedance amplifier Is(t) Rs Ii(t) Ri Vout AV = Vout/Ii Input impedance small (ideally zero) Output impedance small (ideally zero) Note: Gain is not a number A. Rivetti – INFN Sezione di Torino Amplifier basics TA small signal model RO Is(t) Rs Ii(t) Ri AVVi RL Vout Note: impedances may also be complex A. Rivetti – INFN Sezione di Torino Amplifier basics Lecture II Lecture II: • Linear circuit theory review • Amplifier basics • MOS small signal model A. Rivetti – INFN Sezione di Torino Simplified small signal DC model The MOS transistor in saturation can be seen as a voltage controlled current source RS Vs(t) Vs(t) gm = IDS VGS gmVs W m C n (VGS – VTH) = OX = L 2 mn COX W IDS L A. Rivetti – INFN Sezione di Torino MOS small signal DC model Practical example What is the equivalent small signal model of this? Vdrain Vgate W=100 mm L=10 mm mnCOX=190 mA/V2 VTH=0.6 V Vdrain=2.5 V Vgate=1.25 V Vs A. Rivetti – INFN Sezione di Torino MOS small signal DC model current (mA) Gm simulation(1) 356.7 355.7 0 1 2 time (mS) Vs=1mV pk-pk A. Rivetti – INFN Sezione di Torino MOS small signal DC model current (mA) Gm simulation (2) 660 355 0 1 2 time (mS) Vs=250mV pk-pk A. Rivetti – INFN Sezione di Torino MOS small signal DC model Output impedance Vdrain r0 Vgate Vs A. Rivetti – INFN Sezione di Torino MOS small signal DC model Including the output impedance The MOS transistor in saturation can be seen as a voltage controlled current source with finite output impedance RS Vs(t) Vs(t) gm = ro = IDS VGS 1 gmVs ro W m C n (VGS – VTH) = OX = L 2 mn COX W IDS L lIDS A. Rivetti – INFN Sezione di Torino MOS small signal DC model Bulk transconductance For a more accurate model, the bulk effect must also be taken into account RS Vs(t) Vs(t) gmb = IDS VBS gmVs ro gmbvbs VTH W m (VGS – VTH) = n COX = L VSB gm g 2fF + VSB A. Rivetti – INFN Sezione di Torino MOS small signal DC model Small signal DC model The saturated MOS transistor is a voltage controlled current source with finite output impedance RS Vs(t) Vs(t) gmVs ro gmbvbs gm models the gate transconductance gmb models the bulk transconductance (the bulk effect) A. Rivetti – INFN Sezione di Torino MOS small signal DC model Some numbers… gm = IDS VGS = 2 mn COX W IDS L ro = 1 lIDS IDS = 100mA, W/L=50, mnCOX=190mA/V2 l=0.01V-1 gm = 1mS ro = 1MW A. Rivetti – INFN Sezione di Torino MOS small signal DC model