A. Rivetti – INFN Sezione di Torino Lecture II Lecture II: Linear circuit theory review Amplifier...

A. Rivetti – INFN Sezione di Torino

Lecture II

Lecture II:

• Linear circuit theory review• Amplifier basics• MOS small signal model


Nodal analysis

Is VsR1

R2

R3

R4

Nodal analysis provides a systematic and reliable method to calculateall voltages and currents in a linear circuit

Nodal analysis


Writing nodal equations

R2 R4

Is VsR1 R3

v1 v2

0

0

4

2

3

2

2

12

2

21

1

1

RVv

Rv

Rvv

Rvv

RvI

s

s

Nodal analysis


Writing the circuit matrix

Is VsR1

R2

R3

R4v1 v2

RV

I

v

v

RRRR

RRRs

s

42

1

4322

221

1111

111

Nodal analysis


Solving the circuit matrix

RRRR

RRR

4322

221

1111

111

RRRRV

RI

s

s

4324

2

1 111

1

RV

R

IRRs

s

42

21

2 1

11

1

1v 2

2v

Nodal analysis


Another example

Is

Vs

R1

R2

R3R4

Nodal analysis


Lecture II

Lecture II:



Amplifier characteristic

)()()()( ...2

210 txatxatxaaty n

n

The input-output characteristic of an amplifier is usually a non-linearfunction Over some interval of the input signal, this function can be approximated by a polynomial:

For narrow range of the input signal, we may write:

)()(10

txaaty

The above expression does not obey the superposition principle

Amplifier basics


Small signal model

If a0 does not depend on the signal, we can write:

)()(1

txaty

This is an expression that obeys the superposition principle

The small signal model takes into account only variations of signalswithin a circuit The small signal equivalent circuit can be studied with the methodsof linear circuit analysis

Amplifier basics


Voltage amplifier

Vs(t) Ri

Rs

VoutVi(t)

AV = Vout/Vi Input impedance high (ideally infinite) Output impedance small (ideally zero)

Amplifier basics


VA small signal model

AVViVoutVs(t) RI

RS

Vi(t)

RO

RL

Note: impedances may also be complex

Amplifier basics


Current amplifier

AV = Iout/Ii Input impedance small (ideally zero) Output impedance high (ideally infinite)

RiRs IoutIi(t)Is(t)

Amplifier basics


CA small signal model


Amplifier basics

RiRs Ii(t)Is(t) RLIout(t)RoIs(t)


Transconductance amplifier

Vs(t) Ri

Rs

IoutVi(t)

AV = Iout/Vi Input impedance high (ideally infinite) Output impedance high (ideally infinite) Important: the gain is not a number

Amplifier basics


TCA small signal model


Amplifier basics

Vs(t) RI

RS

Vi(t) RLIout(t)RoIs(t)


Transimpedance amplifier

AV = Vout/Ii Input impedance small (ideally zero) Output impedance small (ideally zero) Note: Gain is not a number

Amplifier basics

RiRs VoutIi(t)Is(t)


TA small signal model


RiRs Ii(t)Is(t) AVViVout

RO

RL

Amplifier basics


Lecture II

Lecture II:



Simplified small signal DC model

RS

gm VGS

IDS= = n COX

W

L(VGS – VTH) 2 n COX

WL

IDS=

The MOS transistor in saturation can be seen as a voltage controlledcurrent source

Vs(t) Vs(t) gmVs

MOS small signal DC model


Practical example

What is the equivalent small signal model of this?

W=100 mL=10 mnCOX=190 A/V2

VTH=0.6 VVdrain=2.5 VVgate=1.25 V


Vgate

Vdrain

Vs


Gm simulation(1)


Vs=1mV pk-pk

355.7

356.7

0 1 2time (S)

curr

ent (

A)


Gm simulation (2)


Vs=250mV pk-pk

355

660

curr

ent (

A)

0 1 2time (S)


Output impedance


Vgate

Vdrain

Vs

r0


Including the output impedance

RS

gm VGS

IDS= = n COX

W

L(VGS – VTH) 2 n COX

WL

IDS=

ro1

IDS

=

The MOS transistor in saturation can be seen as a voltage controlledcurrent source with finite output impedance

Vs(t) Vs(t) rogmVs



Bulk transconductance

gmb VBS

IDS= = n COX

W

L(VGS – VTH)

VSB

VTH

=gm

2F + VSB


RS

Vs(t) Vs(t) rogmVs gmbvbs

For a more accurate model, the bulk effect must also be taken into account


Small signal DC model

The saturated MOS transistor is a voltage controlled current source with finite output impedance

RS

Vs(t) Vs(t) rogmVs gmbvbs

gm models the gate transconductancegmb models the bulk transconductance (the bulk effect)



Some numbers…

gm VGS

IDS= = 2 n COX

WL

IDSro

1

IDS

=

IDS = 100A, W/L=50, nCOX=190A/V2

=0.01V-1

gm = 1mSro = 1M


A. Rivetti – INFN Sezione di Torino Lecture II Lecture II: Linear circuit theory review Amplifier...

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