lecture 18 compensation of feedback amplifiers two-stage ...class.ece.iastate.edu/ee435/lectures/ee...

EE 435

Lecture 18

Compensation of Feedback Amplifiers

Two-Stage Op Amp Design Strategies

Simple pole calculations for 2-stage op amp

Since the poles of the 2-stage op amp must be widely

separated, a simple calculation of the poles from the

characteristic polynomial is possible.

Assume p1 and p2 are the poles and p1 << p2

D(s)=s2+a1s+a0

but

D(s)=(s+p1)(s+p2)=s2+s(p1+p2)+p1p2 s2 + p2s + p1p2

determines p2

determines p1

thus

p2=-a1 and p1 = -a0/a1

.• •

•

•

•

Revie

w f

rom

last

lectu

re .•

•

• •

•

.• • • • • Review from last lecture .• • • • •

Compensation

Compensation (alt Frequency Compensation) is the

manipulation of the poles and/or zeros of the open-loop

amplifier so that when feedback is applied, the closed-loop

amplifier will perform acceptably

Note this definition does not mention stability, positive

feedback, negative feedback, phase margin, or oscillation.

Note that acceptable performance is strictly determined by

the user in the context of the specific application


Compensation

• Often Phase Margin or Gain Margin criteria are used instead of pole Q

criteria when compensating amplifiers

(for historical reasons but must still be conversant with this approach)

• Nyquist plots are an alternative stability criteria that is used some in

the design of amplifiers

• Phase Margin and Gain Margin criteria are directly related to the

Nyquist Plots

• Compensation requirements are stongly β dependent

β (s )N (s )D (s )(s )D F B

Compensation requirements usually determined by closed-loop pole locations:

Characteristic Polynomial obtained from denominator term of basic feedback equation

1+A s β s

A s β s defined to be the “loop gain” of a feedback amplifier


Nyquist Plots

The Nyquist Plot is a plot of the Loop Gain (Aβ) versus jω in the complex plane

for - ∞ < ω < ∞

Theorem: A system is stable iff the Nyquist Plot does not encircle the point

-1+j0.

Note: If there are multiple crossings of the real axis by the Nyquist Plot,

the term encirclement requires a formal definition that will not be presented

here

Review of Basic Concepts.• • • • • Review from last lecture .• • • • •

Nyquist PlotsReview of Basic Concepts

-1+j0 Re

Im

ω = - ∞

ω = ∞ω = 0

Example

• Stable since -1+j0 is not encircled

• Useful for determining stability when few computational tools are available

• Legacy of engineers and mathematicians of pre-computer era

Aβ(jω)


Nyquist PlotsReview of Basic Concepts

-1+j0 Re

Im

-1+j0 is the image of ALL poles

The Nyquist Plot is the image of the entire imaginary axis and separates

the image complex plane into two parts

Everything outside of the Nyquist Plot is the image of the LHP

Nyquist plot can be generated with pencil and paper

Re

Im

s-plane A(s)β

FBD s = 1+A s β s

Important in the ‘30s - ‘60’s


Nyquist Plots

Review of Basic Concepts

1+j0

Im

Re

Conceptually would like to

be sure Nyquist Plot does

not get too close to -1+j0

Nyquist Plots


But identification of a suitable

neighborhood is not natural

1+j0

Im

Re




Nyquist Plots


1+j0

Im

Re




Re

Im β A(s)

Might be useful to be sure image of 45o lines do not encircle -1+j0

Nyquist Plots


What if this happened ?

1+j0

Im

Re




Re

Imβ A(s)

At least one pole would make an angle

of less than 45o wrt Im axis

Nyquist Plots


-1+j0 Re

Im

Unit Circle

Phase

Margin

Phase margin is 180o – angle of Aβ when the magnitude of Aβ =1

Nyquist Plots


-1+j0 Re

Im

Unit Circle

Gain

Margin

Gain margin is 1 – magnitude of Aβ when the angle of Aβ =180o

Nyquist and Gain-Phase Plots

-40

-30

-20

-10

0

10

20

30

40

50

60

70

Nyquist and Gain-Phase Plots convey identical information but gain-phase

plots often easier to work with

-300

-250

-200

-150

-100

-50

0

Note: The two plots do not correspond to the same system in this slide

ω = -∞

ω = ∞

ω = 0-1+j0

Im

Re

ω

Mag Phase

Mag

Phase

ω

ω

Nyquist and Gain-Phase Plots

-40

-30

-20

-10

0

10

20

30

40

50

60

70

Nyquist and Gain-Phase Plots convey identical information but gain-phase

plots often easier to work with

-300

-250

-200

-150

-100

-50

0

Aβ plots change with different values of β

Often β is independent of frequency

in this case Aβ plot is just a shifted version of A

in this case phase of Aβ is equal to the phase of A

Instead of plotting Aβ, often plot |A| and phase of A and superimpose |β| and

phase of β to get gain and phase margins

do not need to replot |A| and phase of A to assess performance with different β

ω = -∞

ω = ∞

ω = 0-1+j0

Im

Re

ω

Mag Phase

Mag

Phase

ω

ω

-40

-30

-20

-10

0

10

20

30

40

50

60

70

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Gain and Phase Margin Examples

Magnitude in d

B1β

Phase Margin

ω

ω

-180o

Angle

in d

egre

es

1s

1000T(s)

-100

-80

-60

-40

-20

0

20

40

60

80

100

-80

-60

-40

-20

0

20

40

60

80

Magnitude in d

Bω

ω

Angle

in d

egre

es


Be aware of the multiple

values of the arctan function !

Discontinuities do not exist in magnitude or phase plots

-80

-60

-40

-20

0

20

40

60

80

-300

-250

-200

-150

-100

-50

0

Magnitude in d

B 1β

Phase Margin

ω

ω

Angle

in d

egre

es


-80

-60

-40

-20

0

20

40

60

80

-300

-250

-200

-150

-100

-50

0

Magnitude in d

B

1β

Phase Margin

ω

ω

Angle

in d

egre

es


1000A s

ss+1 +1

200


Magnitude in d

BA

ngle

in

degre

es

1β

ω

Phase Margin

ω

β=.031

1000A s

ss+1 +1

200


Magnitude in d

BA

ngle

in

degre

es

1β

ω

Phase Margin

ω

Unstable !

β=.31

-40

-30

-20

-10

0

10

20

30

40

50

60

70

-300

-250

-200

-150

-100

-50

0

Magnitude in d

BA

ngle

in d

egre

es

1β

Phase Margin


ω

ω

-180o

31s

1000T(s)

-40

-30

-20

-10

0

10

20

30

40

50

60

70

-300

-250

-200

-150

-100

-50

0

Magnitude in d

BA

ngle

in d

egre

es

1βGain Margin


ω

ω

-180o

31s

1000T(s)

-40

-30

-20

-10

0

10

20

30

40

50

60

70

-300

-250

-200

-150

-100

-50

0

Magnitude in d

BA

ngle

in d

egre

es

1β

Phase Margin


ω

ω

-180o

31s

1000T(s)

-40

-30

-20

-10

0

10

20

30

40

50

60

70

-300

-250

-200

-150

-100

-50

0

Magnitude in d

BA

ngle

in d

egre

es

1β

Gain Margin


ω

ω

-180o

31s

1000T(s)

In general, the relationship between the phase margin and the

pole Q is dependent upon the order of the transfer function and on

the location of the zeros

In the special case that the open loop amplifier is second-order low-

pass, a closed form analytical relationship between pole Q and phase

margin exists and this is independent of A0 and β..

)sin(φ

)cos(φQ

M

M

24

1

M2Q

1

4Q

11cosφ

The region of interest is invariable only for 0.5 < Q < 0.7

larger Q introduces unacceptable ringing and settling

smaller Q slows the amplifier down too much

Relationship between pole Q and phase margin

Phase Margin vs Q

0

1

2

3

4

5

6

7

0 20 40 60 80 100

Phase Margin

Po

le Q

Second-order low-pass Amplifier

Phase Margin vs Q

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

40 50 60 70 80

Phase Margin

Po

le Q


Phase Margin vs Q

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

40 50 60 70 80

Phase Margin

Po

le Q


Q=.707

Q=0.5

φM ≈65o φM ≈77o

Magnitude Response of 2nd-order Lowpass Function

Q2

1

From Laker-Sansen Text

QMAX for no peaking = 7072

1.

Phase Response of 2nd-order Lowpass Function

From Laker-Sansen TextQ2

1

Step Response of 2nd-order Lowpass Function

QMAX for no overshoot = 1/2

From Laker-Sansen Text

Q2

1

From Laker-Sansen TextQ2

1

Step Response of 2nd-order Lowpass Function

Compensation Summary

• Gain and phase margin performance often strongly dependent upon architecture

• Relationship between overshoot and ringing and phase margin were developed only for 2nd-order lowpass gain characteristics and differ dramatically for higher-order structures

• Absolute gain and phase margin criteria are not robust to changes in architecture or order

• It is often difficult to correctly “break the loop” to determine the loop gain Aβ with the correct loading on the loop (will discuss this more later)

End of Lecture 18

lecture 18 compensation of feedback amplifiers two-stage ...class.ece.iastate.edu/ee435/lectures/ee...

Documents