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6.002x CIRCUITS AND ELECTRONICS

Small Signal Circuits  

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n ( ) i

VvI

Iout

IOUT

vvfdvdv

vfv

II

⋅=

=

=

)(

n Small signal notation vA = VA + va

total operating point

small signal

Review: VS

+ – + –

vI = VI + vi

RL

vi

VI

vO = VO + vO

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interesting input signal

+ – + –

TIO Vvv −=

ov

iv

t behaves linear for small perturbations

I Graphical View Review: VS

RL

vO vi

VI

vO Vs

vI

operating point

VO

VI, VO

VI

0 VT

vo

vi

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II Mathematical view

( )L

TISO RVvKVv

2

2−−=

( )i

Vv

LTIS

Io v

RVvKV

dvdv

II

⋅⎥⎦⎤

⎢⎣⎡ −−

=

=

2

2

( ) iLTIo vRVVKv ⋅−−=

gm

related to VI constant for fixed DC bias

Review: interesting input signal

+ – + –

VS

RL

vO vi

VI

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interesting input signal

+ – + –

TIO Vvv −=

VS

RL

vO vi

VI

vO Vs

vI

new operating point

VO

VI, VO

VI

0 VT

vo

vi

How to choose the bias point aka operating point

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Demo

How to choose the bias point

vO

iDS

( ) iLTIo vRVVKv ⋅−−=gm

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III The Small Signal Circuit View We can derive small circuit equivalent models for our devices, and thereby conduct small signal analysis directly on circuits

We can replace large signal element models with small signal element models in the same circuit topology

Foundations: Section 8.2.1, and also at the end of this lecture sequence

e.g. large signal circuit model for amp +

–

VS

RL

vO

vI

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Small Signal Circuit Analysis Find operating point using DC bias inputs using large signal model. Develop small signal (linearized) models for elements. Replace original elements with small signal models to get the small signal circuit

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2

3

Analyze resulting linearized circuit…

Key: Can use superposition and other linear circuit tools with linearized circuit!

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Small Signal Circuit Analysis How to develop small signal (linearized) models for each of the elements around the operating point.

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Large signal element model

Small signal

Find operating point using DC bias inputs using large signal model.

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Voltage Sources and DC Supply VS

Large signal

DC voltage source behaves as short to small signals.

Small signal

DC current source behaves as open to small signals.

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Similarly, R small signal

large signal

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MOSFET

Large signal

2

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MOSFET 2 large signal

( )22 TGSDS VvKi −=

S

D

iDS

G

vGS -

+

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MOSFET 2 large signal

( )22 TGSDS VvKi −=

Small signal ( )2

2 TGSDS VvKi −=

S

D

vGS

iDS

G

-

+

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MOSFET small signal

2 Large signal

( )22 TGSDS VvKi −=

S

D

vGS

iDS

G

-

+

( ) gsTGSds vVVKi −=

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Amplifier example

+ –

VS

RL

vO

vI

Let us perform a small signal analysis of our MOSFET amplifier using the small signal circuit method

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Large signal model

+ –

VS

RL

vO

vI

Amplifier example: Step 1 Find operating point from large signal model

Small signal model

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Large signal model

( )22 TIDS VvKi −=

( ) LTISO RVvKVv 2

2−−=

Amplifier example: Step 2

+ –

+ – SVRL

iDS

vO

vI

Find small signal models for elements

+ –

+ – SVRL

iDS

vO

vI S

D

G

Developing the small signal model

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Large signal model

( )22 TIDS VvKi −=

( ) LTISO RVvKVv 2

2−−=

Amplifier example: Step 3

+ –

+ – SVRL

iDS

vO

vI

Replace elements in original circuit with small signal models

+ –

+ – SVRL

iDS

vO

vI S

D

G

Developing the small signal model

Large signal model

( ) iTIds vVVKi ⋅−=

Small signal model Amplifier example:

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+ – vi ids

RL

vO

( )22 TIDS VvKi −=

( ) LTISO RVvKVv 2

2−−=

+ –

+ – SVRL

iDS

vO

vI

Analyze resulting SS circuit

Large signal model

( ) iTIds vVVKi ⋅−=

Small signal model Amplifier example:

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+ – vi ids

RL

vO

( )22 TIDS VvKi −=

( ) LTISO RVvKVv 2

2−−=

+ –

+ – SVRL

iDS

vO

vI

Since small signal models are linear, our linear tools will now apply…

Analyze resulting SS circuit

Node method:

Large signal model

( ) iTIds vVVKi ⋅−=

Small signal model Amplifier example: Analyze resulting SS circuit

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( ) iLTIo vRVVKv ⋅−−=

+ – vi ids

RL

vO

( )22 TIDS VvKi −=

( ) LTISO RVvKVv 2

2−−=

+ –

+ – SVRL

iDS

vO

vI

But,  note  that  we  can  obtain  insight  into  small-­‐signal  behavior  directly  from  the  small-­‐signal  circuit,  without  having  to  do  a  large  signal  analysis  

No?ce,  first  we  need  to  find  opera?ng  point  voltages  and  currents.  Get  these  from  large  signal  analysis  

To find relationship between small signal parameters of circuit: (1) we replaced large signal device models with corresponding small signal device models in the original circuit

(2) then analyzed the resulting small signal circuit.

The Small Signal Circuit View – A Perspective

Why was this possible? 23  Also see section 8.2.1 of A&L

+ – vi ids

RL

vO

+ –

+ – SVRL

iDS

vO

vI

The Small Signal Circuit View – Foundations

But is the same equation as with small signal variables replacing total variables, so must reflect same typology as in C, except that small signal models are used.

Replace total variables with operating point variables plus small signal variables

Operating point variables themselves satisfy the same KVL, KCL equations

so, we can cancel them out, leaving

KVL, KCL applied to some circuit C yields:

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vi ids

RL

vO

+–

+– SVRL

iDS

vO

vI

+–

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Caution: