Research on Frequency Response of CE-BJT and CS-MOSFET

8/10/2019 Research on Frequency Response of CE-BJT and CS-MOSFET

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Feedback Ampli fiers: One and Two Pole cases

Negative Feedback:

a

f

_

a

f

_

There must be 180ophase shift somewhere in the loop. This is often provided by aninverting amplifier or by use of a differential amplifier.

Closed Loop Gain:1

aA

af=

+

When a >> 1, then1a

Aaf f

=

This is a very useful approximation.

The product af occurs frequently: Loop Gain or Loop Transmission T = af

a

f

a

f


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At low frequencies, the amplifier does not produce any excess phase shift. The feedbackblock is a passive network.

But, all amplifiers contain poles. Beyond some frequency there will be excess phase shiftand this will affect the stability of the closed loop system.

Frequency Response

Using negative feedback, we have chosen to exchange gain a for improved performance

Since A = 1/f, there is little variation of closed loop gain with a.Gain is determined by the passive network f

But as frequency increases, we run the possibility of

Instability Gain peaking Ringing and overshoot in the transient response

We will develop methods for evaluation and compensation of these problems.

Bandwidth of feedback ampli fiers: Single Pole case

Assume the amplifier has a frequency dependent transfer function

1

( ) ( )( )

1 ( ) 1 (

( )1 /

o

a s a sA s

a s f T s

and

aa s

s p

= =+ +

=

)

where p1is on the negative real axis of the s plane.


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With substitution, it can be shown that:

1

1( )

1 1 /[ (1o

o o

aA s

a f s p a f

=

+ + )]

We see the low frequency gain with feedback as the first term followed by a bandwidthterm. The 3dB bandwidth has been expanded by the factor 1 + aof = 1 + To.

xx

p1p1(1+To)

xx

p1p1(1+To)

S - plane

Bode PlotBode Plot

20 log |a(j)|

a(j)

0

-90

-180

20 log |A(j

)| = 20 log (1/f)

|p1| |p1(1+To)|

x

20 log |a(j)|

a(j)

0

-90

-180

20 log |A(j

)| = 20 log (1/f)

20 log |a(j)|

a(j)

0

-90

-180

20 log |A(j

)| = 20 log (1/f)

|p1| |p1(1+To)|

x


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Note that the separation between a and A, labeled as x,

20log(| ( ) |) 20log(1/ ) 20log(| ( ) |) 20log( ( ))x a j f a j f T j = = =

Therefore, a plot of T(j) in dB would be the equivalent of the plot above with thevertical scale shifted to show 1/f at 0 dB.

We see from the single pole case, the maximum excess phase shift that the amplifier canproduce is 90 degrees.

Stability condit ion:

If |T(j

)| > 1 at a frequency where

T(j

) = -180

o

,then the amplifier is unstable.

This is the opposite of the Barkhousen criteria used to judge oscillation with positivefeedback. In fact, a round trip 360 degrees (180 for the inverting amplifier at lowfrequency plus the excess 180 due to poles) will produce positive feedback andoscillations.

This is a feedback based definition. The traditional methods using T(j)

Bode Plots Nyquist diagram Root locus plots

can also be used to determine stability. I find the Bode method most useful for providingdesign insight. To see how this may work, first define what is meant by PHASEMARGIN in the context of feedback systems.


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Two pole frequency and step response. Low pass. No zeros.

Frequency Response

3 dB

bandwidth2 2 4

3 1 2 2 4 4dB n = + +

Gain

peaking 21

0.7072 1

PM

=

Research on Frequency Response of CE-BJT and CS-MOSFET

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