Basic Concepts: Amplifiers and Feedback Jose Silva …ece.tamu.edu/~jose-silva-martinez/courses/ECEN620/Leture notes 2.pdf · 2 ECEN-620-2017 Jose Silva-Martinez Feedback Properties:

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Jose Silva-Martinez

Basic Concepts: Amplifiers and Feedback

Jose Silva-Martinez

Department of Electrical & Computer Engineering

Texas A&M University

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Feedback Properties: Transfer function and sensitivity function

The following non-inverted amplifier will be used as a testbed to verify the properties of

passive feedback systems.

𝐻(𝑠) =𝐴𝑉

1+𝐴𝑉(𝑅𝑖𝑛

𝑅𝑖𝑛+𝑅𝐹)

Where 𝐴𝑉 =𝑉0

𝑉+−𝑉− is the

amplifier’s gain.

The term 𝐴𝑉 (𝑅𝑖𝑛

𝑅𝑖𝑛+𝑅𝐹) corresponds to system’s loop transfer function.

Factor 𝑅𝑖𝑛

𝑅𝑖𝑛+𝑅𝐹 represents the feedback factor .

vi

+

-vo

RF

Rin

Non-Inverting Amplifier

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𝐻(𝑠) = (𝑅𝑖𝑛+𝑅𝐹

𝑅𝑖𝑛)(

1

1+ 1

𝐴𝑉(𝑅𝑖𝑛


) ≅ (𝑅𝑖𝑛+𝑅𝐹

𝑅𝑖𝑛)(1 −

1


𝑅𝑖𝑛+𝑅𝐹))

In case the loop gain 𝑨𝑽 (𝑹𝒊𝒏

𝑹𝒊𝒏+𝑹𝑭) ≫ 𝟏 , the system safely approximates the first factor,

then we called this term as the ideal system transfer function

𝐻𝑖𝑑𝑒𝑎𝑙(𝑠) = (𝑅𝑖𝑛+𝑅𝐹

𝑅𝑖𝑛) = 1 +

𝑅𝐹

𝑅𝑖𝑛

This is a very desirable result since the transfer function is a ratio of passive

elements connected through the feedback network; in fact the gain becomes

equal to 1/.

The overall (closed loop) transfer function is then low sensitive to amplifier’s gain

(AV) variations. AV is usually very sensitive to Process-Voltage-Temperature variations

Feedback Properties

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Practical single-dominant pole

amplifier gain is usually modeled as

𝐴𝑉 = (𝐴𝑉0

1+𝑠

𝜔𝑃

);

𝐴𝑉0 is amplifier DC gain and 𝜔𝑃 is

amplifier’s dominant pole.

𝐻(𝑠) = 𝐻𝑖𝑑𝑒𝑎𝑙(𝑠)

(

1

1 + 1

𝐴𝑉 (𝑅𝑖𝑛

𝑅𝑖𝑛 + 𝑅𝐹))

= 𝐻𝑖𝑑𝑒𝑎𝑙(𝑠)(1 −

1

1 + 𝐴𝑉 (𝑅𝑖𝑛

𝑅𝑖𝑛 + 𝑅𝐹))

The gain error is determined by

Feedback Properties

LG

Hsignal=H(s)/(1+LG)

Magnitude

Frequency

H(s)

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in

Fin

V

P

R

RR

A

s

GainLoopGainLoops

0

11

1

1)(

.

It is important to recognize that the error function is equal to 1/(loop gain).

Error function monotonically increases after the first pole of loop gain.

High frequency signals do not benefit from feedback as much as in-band (low

frequency) signals

Error Function

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in

Fin

0V

P

R

RR

A

s1

)s(

.

The error function is inversely proportional to amplifier’s frequency

response.

f (Hz)

20 log10(|AV(s)|)

-20 dB/decade

106 dB

unity-gain

frequency

0 dB

20 dB/decade

Error (dB)

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To maintain good accuracy, loop gain must be high enough until the

maximum frequency of interest.

For a closed loop system with a

targeted error 0, we must satisfy the

following condition:

0

in

Fin

0V

2

P

max

R

RR

A

f

f1

)s(

max is the maximum frequency of interest, often defined as system

bandwidth.

f (Hz)

0 dB

unity-gain

frequency

20 dB/decade

Error (dB)

-100 dB

fmax fP

Error Function

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For the case max /P >10, the condition for limited

overall gain error can be simplified to

0

in

Fin

0V

P

max

R

RR

A

f

f

)s(

.

If AV0 = 105 V/V (considered very large for CMOS

solutions), the error measured at different frequencies

is listed in this Table.

1+RF/Rin ωmax/ ωP Error

magnitude

10 0.01 ~ 10-4

10 1 ~1.4∙10-4

10 10 ~10-3

10 100 ~10-2

10 1000 ~10-1

100 10 ~10-2

1000 10 ~10-1

10000 10 ~100

Error Function

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Inverting Amplifier. The case of the inverting amplifier is a bit more complex than the

case of the non-inverting amplifier. Assume that the amplifier’s gain includes the effect of

the feedback network.

The voltage at the amplifier’s inverting

terminal (ERROR VOLTAGE) is:

𝑉− = (𝑅𝐹

𝑅𝑖𝑛+𝑅𝐹)𝑉𝑖 + (

𝑅𝑖𝑛

𝑅𝑖𝑛+𝑅𝐹)𝑉0

Output voltage

𝑉0 = −𝐴𝑉𝑉− = −𝐴𝑉 (𝑅𝐹

𝑅𝑖𝑛+𝑅𝐹)𝑉𝑖 − 𝐴𝑉 (

𝑅𝑖𝑛


The first term of the right most term is the so-called direct path gain

The second right hand most term is obtained in this case by breaking the loop and

grounding Vi terminal.

+

-

vo

RF

Rin

vi

Inverting Amplifiers

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It can be found that signal transfer function

is determined as

𝑽𝟎

𝑽𝒊= −(

𝑹𝑭

𝑹𝒊𝒏)(

𝟏

𝟏+𝟏

𝑨𝑽(𝑹𝒊𝒏

𝑹𝒊𝒏+𝑹𝑭)

)

Error factor is the same for both

inverting and non-inverting

configurations

+

-

vo

RF

Rin

vi

f (Hz)

20 log10(|AV(s)|)

-20 dB/decade

106 dB

unity-gain

frequency

0 dB

20 dB/decade

Error (dB)

Inverting Amplifiers

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The ERROR VOLTAGE (inverting

terminal of the amplifier) is determined by

𝑉− = (𝑅𝐹

𝑅𝑖𝑛+𝑅𝐹)𝑉𝑖 + (

𝑅𝑖𝑛


If the amplifier gain is given by AV

and it is not affected by RF, then the error

voltage is computed as

𝑉− =(

𝑅𝐹𝑅𝑖𝑛+𝑅𝐹

)

1+(𝐴𝑉)(𝑅𝑖𝑛

𝑅𝑖𝑛+𝑅𝐹)𝑉𝑖 ≅ (

𝑅𝐹𝑅𝑖𝑛

𝐴𝑉)𝑉𝑖

The larger the loop gain, the smaller the value of the error voltage is.

+

-

vo

RF

Rin

vi

Inverting Amplifier

LG

Herror=1/(1+LG)

Magnitude

Frequency

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Let us go a bit deeper: Systematic and Intuitive Analysis

Transconductance amplifiers with floating elements in feedback Network

Typically the CMOS amplifier does not have a low-impedance output stage: these

are the most practical cases since buffers

present voltage headroom issues.

The analysis of this type of amplifiers is

cumbersome, and often we get lost on

the algebra.

The results of typical analysis are not

evident and hard to be properly interpreted; results are not intuitive!

We would like to get insights and so guidelines to optimize our architecture and design

procedures.

vi

vo

Zin

Zi

ZF

ZL

+

-

Transconductance Amplifiers

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Let us first consider the case of the inverting amplifier including amplifier’s input and

output impedance.

The transimpedance amplifier’s small signal model employing a voltage controlled

current source is used here.

Using KCL at both nodes vx

and vo we find

𝑉𝑋 (1

𝑍𝑖𝑛+

1

𝑍1+

1

𝑍𝐹) −

𝑉0

𝑍𝐹=

𝑉𝑖

𝑍𝑖𝑛

𝑉𝑋 (𝑔𝑚 −1

𝑍𝐹) + 𝑉0 (

1

𝑍𝐿+

1

𝑍𝐹) = 0

Solving these equations result in 𝑉0

𝑉𝑖= −

𝑌𝑖𝑛

𝑌𝐹(1+(𝑌𝑖𝑛+𝑌1+𝑌𝐹

𝑌𝐹)(𝑌𝐿+𝑌𝐹𝑔𝑚−𝑌𝐹

))

Notice that loop gain is then given by 𝐿𝐺 = (𝑌𝐹

𝑌𝑖𝑛+𝑌1+𝑌𝐹) (𝑔𝑚− 𝑌𝐹𝑌𝐿+𝑌𝐹

)

vi

vo

Zin

Z1

ZF

vx

i1 ii

io

ZLgm vx

v+


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𝐿𝐺 = (𝑍𝑖𝑛 ∥ 𝑍1

𝑍𝑖𝑛 ∥ 𝑍1+𝑍𝐹) [(𝑔𝑚−

1𝑍𝐹) (𝑍𝐿 ∥ 𝑍𝐹)]

The first term is the feedback factor,

The 2nd factor is the amplifier’s gain.

Notice that amplifier’s transconductance gain is affected by the feedback element!

A right hand plane zero arises when ZF is capacitive; that may hurt your phase margin.

When considering capacitors

or more complex networks it

is not evident where the poles

and zeros are located.

The culprit for these

complications is the floating

element ZF.

vi

vo

Zin

Z1

ZF

vx

ZLgm vx

v+


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Modeling Bi-Directional Floating Impedances

A major issue is the mapping of circuits that have bi-directional elements connecting

different nodes.

The representation of the floating element ZF through the four components

depicted above is consistent and accurate.

The 4-element representation uses unidirectional (without floating passives)

Resulting passive components are grounded (unidirectional)

This model can be easily incorporated into the block diagram’s representation and then

solved by using Mason’s rule.

va vb

iab

Zab Zab

vb

iab

ZabZab

vava vb

iab

Zab

Floating Passives

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Design Example: Inverting Transconductance Amplifier

Converting the floating element into grounded (unidirectional) passive elements and

unidirectional voltage controlled current

sources

Using the 4-element equivalent,

and combining elements, the

amplifier’s circuit using

unidirectional elements results in

vi

vo

Zin

Z1

ZF

vx

ZLgm vx

v+

X

F

m vZ

1g

FL Z||Z

0

F

vZ

1

F1 Z||Z

inv ovinZ

Xv

Floating Passives

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Feedforward gain and loop Gain can be easily found as

𝐹𝐹 = (𝑍1||𝑍𝐹

𝑍𝑖𝑛+𝑍1||𝑍𝐹) ((𝑔𝑚 −

1

𝑍𝐹) 𝑍𝐿||𝑍𝐹) ;

𝐿𝐺 = (𝑍𝑖𝑛||𝑍1||𝑍𝐹

𝑍𝐹) ((𝑔𝑚 −

1

𝑍𝐹) 𝑍𝐿||𝑍𝐹)

The effect of the feedback impedance on the amplifier’s transconductance is

evident in this model.

Exact location of poles and zeros are now captured, and exact computation of

both loop gain and closed loop transfer function is straightforward.

X

F

m vZ

1g

FL Z||Z

0

F

vZ

1

F1 Z||Z

inv ovinZ

Xv

Floating Passives

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Unidirectional Block Diagrams and Mason Rule

An elegant yet more insightful solution for unidirectional networks employs the

Mason rule.

Unidirectional building blocks means that the output is driven by the input, but

variations at the output does not affect at all the block’s input.

Examples of these blocks are

a) Voltage controlled voltage sources, voltage controlled current sources,

current controlled voltage sources and current controlled current sources.

b) Grounded passives (resistors, capacitors and inductors)

c) Examples of non-directional elements are the transformers, and floating

impedances

The transfer function of a given linear system represented by “unidirectional

building blocks” can always be obtained by identifying loops and direct

trajectories.

Mason’s Rule

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Once the system is represented by unidirectional elements, we must:

Identify the direct trajectories(paths) from the input(s) to output

KyAn(s) from Vy to output

AmAn from Vx to output

The loops must also be

identified

Am(s)

Loops that are not touched by certain direct paths; e.g.

The loop is not touched by the direct path KyAn

The loops that are not sharing elements or nodes (un-touched loops) have to be

identified. In this case we do not find any of these loops.

Feedback Properties

S An(s)SAm(s)

KyVy

Vx Vo

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For the schematic, the following paths and loop can be identified:

i) Direct path 1 from Vy: KyAnVy

ii) Direct path 2 from Vx:

Am(s) An(s)Vx

iii) Loop1: Am(s)

iv) Loop2: An(s)

v) Loop3: Am(s)An(s)

vi) Notice that Am(s) and An(s) do not have any element in common. This is an

example of un-touched loops

MASON Rule: If the system is linear, then every single input generates an output

component that can be computed according to the following rule:

.....loopsuntouchedofproductloops1

.........loopsuntouchedandpathsdirectofproductpathsdirect

vi

vo

Mason’s Rule

S An(s)SAm(s)

KyVy

Vx Vo

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In the case of the following block diagram composed by unidirectional blocks we can obtain the

following transfer functions:

𝐻𝑜𝑥 =𝑉0

𝑉𝑥=𝐴𝑚(𝑠)𝐴𝑛(𝑠)

1−𝐴𝑚(𝑠)

𝐻𝑜𝑦 =𝑉0

𝑉𝑦=

𝐾𝑦𝐴𝑛(𝑠)−{𝐾𝑦𝐴𝑛(𝑠)}{𝐴𝑚(𝑠)}

1−𝐴𝑚(𝑠)

Once all transfer functions are identified, the overall output voltage is computed as follows:

𝑉0 = 𝐻𝑜𝑦𝑉𝑦 +𝐻𝑜𝑥𝑉𝑥

For the case of the second schematic, the transfer

functions can be computed as

𝐻𝑜𝑥 =𝐴𝑚(𝑠)𝐴𝑛(𝑠)

1−𝐴𝑚(𝑠)−𝐴𝑛(𝑠)−𝐴𝑛(𝑠)𝐴𝑚(𝑠)−𝐴𝑛(𝑠)𝐴𝑚(𝑠)

The denominator is the result of the single loop

Am(s), the single loop An(s), the loop involving Am(s)An(s) and last term is due to the

product of the two non-touching loops An(s)Am(s).

Am(s) An(s)+ +Vx

Vy Ky

loop is not touchedby Vy path

Am(s) An(s)+ +Vx

Vy Ky

Untouched loops

Mason’s Rule

S An(s)SAm(s)

KyVy

Vx Vo

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The transfer function for the second input Vy can be computed as follows:

𝐻𝑜𝑦 =𝑉0𝑉𝑦=

𝐾𝑦𝐴𝑛(𝑠) − {𝐾𝑦𝐴𝑛(𝑠)}{𝐴𝑚(𝑠)}

1 − 𝐴𝑚(𝑠)−𝐴𝑛(𝑠)−𝐴𝑛(𝑠)𝐴𝑚(𝑠)+ 𝐴𝑛(𝑠)𝐴𝑚(𝑠)

The direct path does not touch the loop determined by Am(s), then the term

{𝐾𝑦𝐴𝑛(𝑠)}{𝐴𝑚(𝑠)} arises in the numerator.

Mason’s Rule

S An(s)SAm(s)

KyVy

Vx Vo

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MASON Rule: If the system is linear, then every single input generates an output

component that can be computed according to the following rule:

.....loopsuntouchedofproductloops1

.........loopsuntouchedandpathsdirectofproductpathsdirect

vi

vo

Zeros (numerator) are fully determined by direct paths in absence of untouched

loops.

Direct paths that do not touch loops create additional terms and play relevant role

on the definition of the zeros

Poles are entirely determined by the loops

Noise shaping and sensitivity functions are quite related to the direct paths

System quality and stability are quite related to loop transfer function

Relevant remarks from Mason’s Rule

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See some examples related to Sigma-Delta Modulators

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Sensitivity Function

When dealing with multivariable functions and when some of the parameters may have

significant variations then it is useful to compute the sensitivity functions to quantify

the individual variability effects on system performance.

The sensitivity of a multivariable function H(x, y, z, …) as function of the parameter

x is defined as

∫𝐻

𝑥= (

𝑥

𝐻) (

𝑑𝐻

𝑑𝑥) =

(𝑑𝐻

𝑑𝑥)

𝐻

𝑥

or 𝑑𝐻

𝐻= ∫ (

𝑑𝑋

𝑋)

𝐻

𝑥

The sensitivity function represents the first order variation of H as function of

the parameter x, normalized by the factor H/x.

Properties of Sensitivity Function

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To get more insight, let us consider the following approximation:

∫𝑯

𝒙=

𝒅𝑯

𝑯𝒅𝒙

𝒙

≅∆𝑯

𝑯∆𝒙

𝒙

The sensitivity function then measures the variation of the normalized transfer

function ∆𝐻

𝐻 (percentage) as function of the variation of normalized parameter

∆𝑥

𝑥

(percentage).

If the sensitivity function is computed as 10, then 1% variation in parameter x will

produce a variation of 10% in the overall transfer function H.

Then, it is highly desirable to maintaining the sensitivity function of H with respect to

critical parameters lesser than 1.

For the case of large parameter variations such as gain and bandwidth of the

operational amplifier, it is highly desirable to keep the sensitivity functions well below

unity.


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Be careful with circuit enhancing devices such as the ones that make use of

negative resistors (positive feedback) to boost amplifier’s gain! Example:

Typical DC Amplifier’s gain: 𝐴𝑉𝑇 =𝑔𝑚

𝐺𝐿

Enhanced voltage gain: 𝐴𝑉𝐸 =𝑔𝑚

𝐺𝐿−𝑮𝑪= 𝐴𝑉𝑇 (

𝟏

𝟏− 𝑮𝑪𝑮𝑳

) = 𝐴𝑉𝑇 (1

1− 𝑋)

Enhancing gain factor = 1

1− 𝐺𝐶𝐺𝐿

; for more gain boosting, GC GL

Sensitivity function: ∫𝑨𝑽𝑬

𝑿= −

𝑿

𝟏− 𝑿= −

𝑮𝑪𝑮𝑳

𝟏− 𝑮𝑪𝑮𝑳

= 𝟏

𝟏− 𝑮𝑳𝑮𝑪

The design trade-off is evident! The larger the gain boosting is, the larger the amplifier

sensitivity to GC, GL tolerances is. Sweet Spots result, in most of the cases, in

very sensitive systems! Corner and Montecarlo simulations are

mandatory in these cases.


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Similar results are obtained when cancelling signal components!

Typical blocker canceling circuit: 𝐴𝑉𝑇 = 𝑔𝑚(𝑍1 − 𝑍2) = 𝑔𝑚𝑍1 (1 −𝑍2

𝑍1)

Z1 and Z2 are usually function of frequency; for noise or signal canceling at certain

frequency it is desirable to design the system such that Z1(c)= Z2(c) and Z1() Z2()

when c

: 𝐴𝑉𝑇 = 0 𝑤ℎ𝑒𝑛 𝜔 = 𝜔𝑐

Sensitivity function: ∫𝟏−𝑿

𝑿= −

𝑿

𝟏− 𝑿= −

𝒁𝟐

𝒁𝟏

𝟏− 𝒁𝟐

𝒁𝟏

= 𝟏

𝟏− 𝒁𝟏

𝒁𝟐

The signal cancelling system is very sensitive to variations on either one of the two

impedances (or components) involved in the cancellation.

If you use partial positive feedback, be sure that you are not very aggressive in your

design; e.g. 𝒁𝟐

𝒁𝟏 (= 0.8 at 𝝎 = 𝝎𝒄)

𝒈𝒎𝒁𝟏 (𝟏 −𝒁𝟐

𝒁𝟏) = 𝒈𝒎𝒁𝟏(𝟎. 𝟐) 𝒘𝒉𝒊𝒍𝒆 ∫

𝒈𝒎𝒁𝟏(𝟏−𝒁𝟐

𝒁𝟏)

𝒁𝟐

𝒁𝟏

= 𝟓

Sensitivity Function

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These results mean:

Signal attenuation by a factor of 5 (14dB) at the critical frequency, no bad

at all

If Z2/Z1 presents tolerances of 5%, then the attenuation factor will change

by 25%; worst case attenuation will be more than -11dB.

What about circuits like this: ∫𝟏+𝑿

𝑿 ?

∫

𝟏+𝑿

𝑿

= +𝑿

𝟏 + 𝑿

The sensitivity function is <1 if and only if X > 0.

Some systems like Doherty RF Power Amplifiers make use of this

property

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Non-Inverting Amplifier. Let us consider the case of the non-inverting amplifier:

𝐻(𝑠) = (𝑅𝑖𝑛+𝑅𝐹

𝑅𝑖𝑛)(

1

1+1



) = (𝑅𝑖𝑛+𝑅𝐹

𝑅𝑖𝑛) (

1

1+𝜉)

The computation of the sensitivity of H with respect to the error function 𝜉LGyields,

∫𝐻

𝜉= (

𝜉

(𝑅𝑖𝑛+𝑅𝐹𝑅𝑖𝑛

)(1

1+𝜉))(

𝑑

𝑑𝜉(𝑅𝑖𝑛+𝑅𝐹

𝑅𝑖𝑛) (

1

1+𝜉)) = −

𝜉

1+𝜉

In case 𝜉 ≪ 1, the sensitivity function can be approximated as ∫𝐻

𝜉≅ −𝜉

The error function is computed as 𝜉 = 1

𝐿𝑜𝑜𝑝 𝐺𝑎𝑖𝑛 =

1



If, we expect variations of 𝜉 in the range of 100% but we do want the effects on H(s) to be more

than 1%, then 𝜉 must be maintained under 0.01; e.g. overall loop gain > 40dB.

Main Concepts behind Feedback

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Better Linearity: Intuitively, when loop gain is large, the error function

decreases, thus better linearity figures are obtained; error voltage is the

input of your OPAMP. Assumption: the feedback network is linear!

The ERROR VOLTAGE is determined by

𝑽− =(

𝑹𝑭𝑹𝒊𝒏 + 𝑹𝑭

)

𝟏 + (𝑨𝑽) (𝑹𝒊𝒏

𝑹𝒊𝒏 + 𝑹𝑭)𝑽𝒊

≅ (

𝑹𝑭𝑹𝒊𝒏

𝑨𝑽)𝑽𝒊

If loop gain is large, then amplifier’s (true) input signal swing reduces at the same time!

Non-linearities due to AV are drastically attenuated (small signal operation mode)

+

-

vo

RF

Rin

vi

Properties of Feedback: Linearity

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General Remarks:

Entire loop transfer function is

important. Be careful when breaking the

loop; be sure the loading effects are

included in your model.

Pay special attention to the error voltage

Verror (in this specific model)

𝑉𝑒𝑟𝑟𝑜𝑟 = (1

1+𝛽 𝐻(𝑠))𝑉𝑖𝑛 = (

1

1+𝐿𝐺)𝑉𝑖𝑛; (similar to NTF shaping in S modulators)

Large loop gain reduces the error voltage

If 𝑉𝑒𝑟𝑟𝑜𝑟 ≅ 0; 𝑡ℎ𝑒𝑛 𝑉𝑖𝑛 ≅ 𝛽 𝑉𝑜𝑢𝑡 ; then 𝛽 becomes the most sensitive block of the closed

loop system. Accuracy on the absolute value of 𝛽 is critical.

Noise and non-linearities in the feedback network have direct impact on closed loop system

performance.

Since 𝑉𝑒𝑟𝑟𝑜𝑟 is very small in the range where the gain of the high-gain network is high,

then closed loop system linearity is superior.

High Gain Network

SVerror

Feedback Network

VinVout

+-

Vout


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Linearity improves if and only if LG is large.

If LG is small, then 𝑉𝑒𝑟𝑟𝑜𝑟 ≈ 𝑉𝑖𝑛 then the high-gain network has to manage the entire input

signal range; but OPAMPs are non-linear. Large signal non-linearities are quite large and

usually can not be tolerated.

𝑉𝑜𝑢𝑡 = (𝐻(𝑠)

1+𝛽 𝐻(𝑠))𝑉𝑖𝑛

𝑉𝑜𝑢𝑡 ≅ (1

𝛽)𝑉𝑖𝑛 𝑖𝑓 1 ≪ 𝛽𝐻(𝑠)

𝑉𝑜𝑢𝑡 ≅ (𝐻(𝑠))𝑉𝑖𝑛 𝑖𝑓 1 > 𝛽𝐻(𝑠)

On top of this, since this is a closed loop system 𝐿𝐺 = 𝛽 𝐻(𝑠) is critical for loop stability.

Phase margin is relevant to avoid excessive closed loop peaking.


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Root Locus provides more information on system stability. Root locus show how open-

loop poles move when the loop is closed and 𝛽 changes

How the poles move when feedback factor changes?

How stability is affected if poles and zeros move with process variations?

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Intuitive explanation on the linearity issue: Let us consider 3 cases for different frequencies

𝑉𝑒𝑟𝑟𝑜𝑟 = (1

1 + 𝐿𝐺)𝑉𝑖𝑛 = (𝐻𝑒𝑟𝑟𝑜𝑟(𝑠))𝑉𝑖𝑛

𝑉𝑜𝑢𝑡 = (𝐻(𝑠)

1 + 𝛽 𝐻(𝑠))𝑉𝑖𝑛 = (𝐻𝑠𝑖𝑔𝑛𝑎𝑙(𝑠)) 𝑉𝑖𝑛

High Gain Network

SVerror

Feedback Network

VinVout

+-

Vout


Herror=1/(1+LG)

f1 f2

f3

Magnitude

FrequencyHsignal=H(s)/(1+LG)

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Case 1: f=f1 in the graph

LG is large then error transfer

function is quite small

Closed loop system linearity is

superior!

Case 2: f=f2 in the graph is such that loop

gain is 10 (20dB)

LG is not quite large then error

transfer function is small and error

voltage increases

-1

-0.5

0

0.5

1

Vin

Verror

-1

-0.5

0

0.5

1

Vin

Verror


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Case 3: f=f3 in the graph is such that loop gain is close to unity (0dB)

LG gain is modest, then error

voltage is comparable to Vin.

Remark: error voltage is the input

of the non-linear amplifier’s first

stage!

HD3 is proportional to Verror2

Usually the magnitude of LG decreases with frequency, then high

frequency linearity is more limited than low-frequency linearity.

-1

-0.5

0

0.5

1

Vin

Verror


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𝑉𝑒 = (1

1+𝐿𝐺)𝑉𝑖𝑛 ; 𝑉𝑜𝑢𝑡 = (

∝ 𝐻(𝑠)

1+𝛽 𝐻(𝑠))𝑉𝑖𝑛

Z1

Vin

Zout

VeVout

Gm2Ve2

GmVe

Gm3Ve3

Z2

Gm2 and Gm3 are obtained from modeling the large signal transconductance gain of your amplifier. For a Diff Pair (gain compression), Gm2 =0 and Gm3 is inversely proportional the square and cubic power the overdrive voltage.

The smaller the error voltage, the better the linearity is. Keep Ve small; this happen when loop gain is

large. Fundamental observation for multi-standard applications wherein cross-products must be minimized.

Worst case linearity happen at high frequency where loop gain is limited!

What about (out-of-band) blockers? Can blockers affect in-band linearity?

How high frequency blocker effects can be minimized? Intermodulation products?

iout

Ve

Very linear

Limited

linearity

Keep Ve small!

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Although Root locus is not covered in this course due to lack

of time, it is highly recommended to be knowledgeable on this

topic.

Recommended book by Melsa and Schultz:

Very enjoyable chapters on Stability

Analysis and also on Root Locus

Methodologies.

Documents

Basic Concepts: Amplifiers and Feedback Jose Silva …ece.tamu.edu/~jose-silva-martinez/courses/ECEN620/Leture notes 2.pdf · 2 ECEN-620-2017 Jose Silva-Martinez Feedback Properties: