Lecture II

(1)

A. Rivetti – INFN Sezione di Torino

Lecture II

Lecture II:

•

Linear circuit theory review

• Amplifier basics

• MOS small signal model

(2)

Nodal analysis

I_s ^R¹ V_s

R₂ R₃

R₄

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

Nodal analysis

(3)

Writing nodal equations

R₂ R₄

I_s ^R¹ ^R³ V_s

v₁ v₂











 



 

 



0 0

4 2 3

2 2

1 2

2 2 1

1 1

R V R v v

R v v

R v R v v

I

s s

Nodal analysis

(4)

Writing the circuit matrix

I_s ^R¹ V_s

R₂ R₃

R₄ v₁ v₂

















































V R I v

v R

R R

R

R R

R

s s

2 4 1

4 3

2 2

1

1 1

1

Nodal analysis

(5)

Solving the circuit matrix

R R

R

4 3

2 2

1

1 1

1











R R

R

V R I R

s s

4 3

2 4

2

1 1 1 1

1









V R R

R I R

s s

4 2

2 1

1 1









 

 ¹

v

1

 

 ²

v

2

Nodal analysis

(6)

Another example

I_s

V_s

R₁

R₂

R₃ R₄

Nodal analysis

(7)

Lecture II

Lecture II:

•

(8)

Amplifier characteristic

) ( )

) ( ( )

( t a

₀

a

₁

x t a

₂

x t

² ^...

a x t

y

^ ^ ^ ^ ^ _n ⁿ

 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:

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

) ( )

( t a

₀

a

₁

x t y

^ ^

The above expression does not obey the superposition principle

Amplifier basics

(9)

Small signal model

 If a₀does not depend on the signal, we can write:

) ( )

( t a

₁

x t

y 



^

 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

(10)

Voltage amplifier

V_s(t) Ri

Rs

V_i(t) Vout

 A_V = Vout/Vi

 Input impedance high (ideally infinite)

 Output impedance small (ideally zero)

(11)

VA small signal model

A_VV_i Vout

V_s(t) R_I

R_S

V_i(t)

R_O

R_L

Note: impedances may also be complex

(12)

Current amplifier

 A_V = Iout/Ii

 Input impedance small (ideally zero)

 Output impedance high (ideally infinite)

Rs I_i(t) Ri Iout

I_s(t)

(13)

CA small signal model

Rs I_i(t) Ri

I_s(t) I_s(t) Ro I_out(t) R_L

(14)

Transconductance amplifier

V_s(t) Ri

Rs

V_i(t) Iout

 A_V = Iout/Vi

 Input impedance high (ideally infinite)

 Output impedance high (ideally infinite)

 Important: the gain is not a number

(15)

TCA small signal model

V_s(t) R_I

R_S

V_i(t) I_s(t) Ro I_out(t) R_L

(16)

Transimpedance amplifier

 A_V = Vout/Ii

 Input impedance small (ideally zero)

 Output impedance small (ideally zero)

 Note: Gain is not a number

Rs I_i(t) Ri Vout

I_s(t)

(17)

TA small signal model

Note: impedances may also be complex Rs I_i(t) Ri

I_s(t) A_VV_i Vout

R_O

R_L

(18)

Lecture II

Lecture II:

•

(19)

Simplified small signal DC model

R_S

g_m

V_GS

I_DS

= = ⁿ C_OX W

L (V_GS – V_TH) 2 n C_OX W

L I_DS

=

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

V_s(t) V_s(t) gmVs

MOS small signal DC model

(20)

Practical example

What is the equivalent small signal model of this?

W=100 m L=10 m

_nC_OX=190 A/V² V_TH=0.6 V

Vdrain=2.5 V Vgate=1.25 V

V_gate

V_drain

V_s

(21)

Gm simulation(1)

Vs=1mV pk-pk 355.7

356.7

0 1 2

time (S)

current (A)

(22)

Gm simulation (2)

Vs=250mV pk-pk 355

660

current (A)

0 1 2

time (S)

(23)

Output impedance

V_gate

V_drain

V_s

r₀

(24)

Including the output impedance

R_S

g_m

V_GS

I_DS

= = ⁿ C_OX W

L (V_GS – V_TH) 2 n C_OX W

L I_DS

=

ro 1

I_DS

=

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

V_s(t) V_s(t) gmVs ro

(25)

Bulk transconductance

g_mb

V_BS

I_DS

= = ⁿ C_OX W

L (V_GS – V_TH)

V_SB

V_TH

= g_m

2_F + V_SB

R_S

V_s(t) V_s(t) gmVs ro gmbvbs

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

(26)

Small signal DC model

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

R_S

V_s(t) V_s(t) gmVs ro gmbvbs

gm models the gate transconductance

gmb models the bulk transconductance (the bulk effect)

(27)

Some numbers…

g_m

V_GS

I_DS

= = 2 n C_OX W

L I_DS ro 1

I_DS

= I_DS = 100A, W/L=50, _nC_OX=190A/V²

=0.01V^-1 gm = 1mS ro = 1M