Designing for Predictable Amplifier Gain Gain is hard to control Varies with operating point Non-constant gain causes distortion Gain varies from one transistor

Designing for Predictable Amplifier Gain

• Gain is hard to control• Varies with operating point• Non-constant gain causes distortion• Gain varies from one transistor to the next• Sensitive to temperature

1EEE 3308

Amplifier Gain Varies a Lot

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• Gain varies with operating point• Non-constant gain causes distortion

Input, Output, Source & LoadImpedance Variations Affect

Gain

3EEE 3308

A

v=

Vout

Vs

=Zin

Zin + ZS

Avo

ZL

Zout + ZL

Zin

Zout

Avovivi

vs

Zs

ZL vout

Input, Output, Source & LoadImpedance Variations Affect

Gain

4EEE 3308

A

v=

Vout

Vs

=Zin

Zin + ZS

Avo

ZL

Zout + ZL

Zin

Zout

Avovivi

vs

Zs

ZL vout

• Impedances vary with frequency, too.

So How Can We Possibly Design Amps That Just Work?

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• How to get gain that is stable, predictable, temperature-independent?• How to get stable biasing?• How to get desired input and output impedances?

So How Can We Possibly Design Amps That Just Work?

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• How to get gain that is stable, predictable, temperature-independent?• How to get stable biasing?• How to get desired input and output impedances?

FEEDBACK!

Classic Feedback Example:The Non-Inverting Feedback Amplifier

7EEE 3308

R2R1

vs vo

Non-Inverting Feedback Amplifier

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vo=A ⋅(vi)

R2R1

Avi

vivs

vfR2R1

vs vo

vo


9EEE 3308

vo=A ⋅ vs −vf( )

R2R1

Avi

vivs

vfR2R1

vs vo

vo


10EEE 3308

v

o=A ⋅ vs −vf( ) =A ⋅ vs −

R1

R1 +R2

vo

⎛

⎝⎜⎞

⎠⎟

R2R1

Avi

vivs

vfR2R1

vs vo

vo


11EEE 3308

v

o=A ⋅ vs −vf( ) =A ⋅ vs −

R1

R1 +R2

vo

⎛

⎝⎜⎞

⎠⎟=A ⋅vs −A ⋅

R1

R1 +R2

vo

R2R1

Avi

vivs

vfR2R1

vs vo

vo


12EEE 3308

v

o=A ⋅ vs −vf( ) =A ⋅ vs −

R1

R1 +R2

vo

⎛

⎝⎜⎞

⎠⎟=A ⋅vs −A ⋅

R1

R1 +R2

vo

v

o1+ A ⋅

R1

R1 +R2

⎛

⎝⎜⎞

⎠⎟=Avs

R2R1

Avi

vivs

vfR2R1

vs vo

vo


13EEE 3308

v

o=A ⋅ vs −vf( ) =A ⋅ vs −

R1

R1 +R2

vo

⎛

⎝⎜⎞

⎠⎟=A ⋅vs −A ⋅

R1

R1 +R2

vo

v

o1+ A ⋅

R1

R1 +R2

⎛

⎝⎜⎞

⎠⎟=Avs

vo=

Avs

1+ A ⋅R1

R1 +R2

R2R1

Avi

vivs

vfR2R1

vs vo

vo


14EEE 3308

v

o=A ⋅ vs −vf( ) =A ⋅ vs −

R1

R1 +R2

vo

⎛

⎝⎜⎞

⎠⎟=A ⋅vs −A ⋅

R1

R1 +R2

vo

v

o1+ A ⋅

R1

R1 +R2

⎛

⎝⎜⎞

⎠⎟=Avs

ACL

=vo

vs

=A

1+ A ⋅R1

R1 +R2

R2R1

Avi

vivs

vfR2R1

vs vo

vo


15EEE 3308

β =

R1

R1 +R2

A is the “open-loop gain”

is the “feedback factor”

ACL is the “closed-loop gain”

R2R1

Avi

vivs

vfR2R1

vs vo

vo


16EEE 3308

A∞ @ACL A→ ∞

=1 / β =1+R2 / R1

T = Aβ is the “loop gain”

R2R1

Avi

vivs

vfR2R1

vs vo

vo


17EEE 3308

A∞ @ACL A→ ∞

=1 / β =1+R2 / R1

R2R1

Avi

vivs

vfR2R1

vs vo

vo


18EEE 3308

A∞ @ACL A→ ∞

=1 / β =1+R2 / R1

If T is big enough, the closed-loop gain is independent of the amplifier gain A.

R2R1

Avi

vivs

vfR2R1

vs vo

vo

Feedback Analysis UsingLoop Gain and A∞

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• The A-β approach works OK for the non-inverting amp example, but it doesn’t generalize well:

- Many circuits don’t split cleanly into β and A parts- Results depend on arbitrary assumptions about amp- Some of the results are significantly wrong- Not all feedback circuits are amplifiers

Feedback Analysis UsingLoop Gain and A∞

20EEE 3308

• The A-β approach works OK for the non-inverting amp example, but it doesn’t generalize well:

- Many circuits don’t split cleanly into β and A parts- Results depend on arbitrary assumptions about amp- Some of the results are significantly wrong- Not all feedback circuits are amplifiers

• Loop gain (T) is the key parameter for feedback analysis• A∞ generalizes the ideal op amp• Combining separate analyses is design-oriented

Finding Loop Gain

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Finding Loop Gain

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Represent the amplifier by its linearized small-signal equivalent circuit.

Finding Loop Gain

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Turn off independent voltage and current sources, replacing themby their internal resistances (short for voltage sources, open for current sources).

Finding Loop Gain

24EEE 3308

R2

R1

Ri

Ro

Avovi

vi

Finding Loop Gain

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Choose a branch through which the feedback signal flows...

R2

R1

Ri

Ro

Avovi

vi

Finding Loop Gain

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R2

R1

Ri

Ro

Avovi

vi

Finding Loop Gain

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R2

R1

Ri

Ro

Avovi

vi

Finding Loop Gain

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Break the branch.

R2

R1

Ri

Ro

Avovi

vi

Finding Loop Gain

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Call the input side the x port, and the output side the y port.

x y

R2

R1

Ri

Ro

Avovi

vi

Feedback signal flow

Finding Loop Gain

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Find the resistance, call it Rix, looking into the x port with port y shorted.

y

vx

y

R2

R1

Ri

Ro

Avovi

vi

Rix

Finding Loop Gain

31EEE 3308


y

vx

y

R2

R1

Ri

Ro

Avovi

vi

(R1||Ri)

Finding Loop Gain

32EEE 3308

R1

Ri

Ro

Avovi

vi

Rix = R1 PRi( ) +R2

vx

R2


Finding Loop Gain

33EEE 3308

vy

R1

Ri

Ro

Avovi

vi

vx

R2

Place a copy of Rix across the y port.

Rix = R1 PRi( ) +R2

Finding Loop Gain

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Find the loop gain T = -vy/vx using standard amplifier analysis.

vy

R1

Ri

Ro

Avovi

vi

Rix = R1 PRi( ) +R2vx

R2

Finding Loop Gain

35EEE 3308

In this case,

vy

vx

R2

R1

Ri

Ro

Avovi

vi

T =

R1 PRi

R1 PRi( ) +R2

Avo

Rix

Rix +R0

=R1 PRi

Rix

Avo

Rix

Rix +R0

=R1 PRi

Rix +R0

Avo

Rix = R1 PRi( ) +R2

Finding Loop Gain

36EEE 3308

You get the same thing for T if you break the loop in other places.

R2

R1

Ri

Ro

Avovi

vi

Finding Loop Gain

37EEE 3308

R2

R1

Ri

Ro

Avovi

vi


Finding Loop Gain

38EEE 3308

R2

R1

Ri

Ro

Avovi

vi


Finding Loop Gain

39EEE 3308

R2

R1

Ri

Ro

Avovi

vi

T is a key property of any feedback circuit. T is independent of how youfind it. It’s independent of where any inputs may be applied or any outputsare defined.

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Finding Loop Gain: Summary

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Represent the amplifier by its linearized small-signal equivalent circuit.

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Turn off independent voltage and current sources, replacing themby their internal resistances (short for voltage sources, open for current sources).


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Choose a branch through which the feedback signal flows.

R2

R1

Ri

Ro

Avovi

vi


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Break the branch.

R2

R1

Ri

Ro

Avovi

vi


x y

45EEE 3308


y

vx

y

R2

R1

Ri

Ro

Avovi

vi

Rix


46EEE 3308

vy

R1

Ri

Ro

Avovi

vi

vx

R2

Place a copy of Rix across the y port.

Rix = R1 PRi( ) +R2


47EEE 3308

Find the loop gain T = -vy/vx.

vy

R1

Ri

Ro

Avovi

vi

Rixvx

R2


48EEE 3308

Finding A∞

R2R1

Avi

vs

vf

voRi vi

ii+

ii-

A∞ is the source-to-output gain when the controlled source gain A goes to infinity.

R2R1

vs vo

ii-

vi

49EEE 3308

Finding A∞

R2R1

Avi

vs

vf

voRi vi

ii+

ii-


If v

o=Avi is finite, then A → ∞⇒ vi =

v0

A→ 0 .

R2R1

vs vo

ii-

vi

50EEE 3308

Finding A∞

R2R1

Avi

vs

vf

voRi vi

ii+

ii-


Also, zero voltage across Ri ⇒ ii+ → 0 and ii− → 0 .

R2R1

vs vo

ii-

vi

If v


v0

A→ 0 .

51EEE 3308

Finding A∞

R2R1

Avi

vs

vf

voRi vi

ii+

ii-


v i=0 ii+ =0 ii− =0

These are equivalent to the ideal op assumptions:

R2R1

vs vo

ii-

vi

Also, zero voltage across Ri ⇒ ii+ → 0 and ii− → 0 .

If v


v0

A→ 0 .

EEE 3308 52

Finding A∞: The Ideal Op Amp Assumptions

R2R1

Avi

vs

vfR2R1

vs vovoRi vi

ii+

ii-

Ideal Op Amp Assumptions: v i=0 ii+ =0 ii− =0

vi=0 ⇒ vf =vs

ii− =0 ⇒ vf =R1

R1 +R2

vo by voltage division;

Combining, A∞ =ACL A→ ∞=

vo

Vs A→ ∞

=R1 +R2

R1

ii+

ii-

vi

53EEE 3308

Finding A∞

The A∞ approach can be applied to any feedback circuit, even when there is no op amp as such.

In general, A∞ is the overall source-to-output gain when the signal controlling the controlled source is forced to be zero because of infinite controlled-source gain.

As with the ideal op amp, assuming infinite gain leads to simpler circuit analysis.

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Putting It All Together

Once you know T and A∞ you can find the overall gain using

A

CL=

A∞

1+1 / T.

The loop gain is a measure of how close the circuit is to ideal.

Documents

Designing for Predictable Amplifier Gain Gain is hard to control Varies with operating point Non-constant gain causes distortion Gain varies from one transistor