Chapter 12(Analog)

8/9/2019 Chapter 12(Analog)

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Chapter 12 The Frequency Response of

Amplifiers

Section 12.1 The Miller TheoremAmplifiers were introduced before. In the previous chapters, we only talked about the

gain of an amplifier. We did not talk about the frequency of the input signal. In fact,

we deliberately ignored frequencies to simplify the discussion. In Fig. 1.1!1, we

illustrate a small signal equivalent circuit of a transistor.

Fig. 1.1!1 A small signal equivalent of a transistor

"et us point out that the above equivalent circuit is for low frequencies only. As

the frequency of the input signal gets higher, capacitors appear as shown in Fig. 1.1!

. #ote that capacitors e$ist even in low frequency cases. %ut they are not significant

then.

1!1


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Fig. 1.1! A small signal equivalent circuit with capacitors considered

With a capacitor introduced between the gate and drain terminals, it will be

helpful for us to learn the &iller 'heorem. "et us consider Fig. 1.1!(.

)a* 'he circuit for &iller 'heorem

1!


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)b* 'he equivalent circuit of the circuit in Fig. 1.1!()a* by removing Z

Fig. 1.1!( +ircuits for $plaining &iller 'heorem

As shown in Fig. 1.1!()a*, there are three nodes, 1 #,# and (# . (# is

grounded. %etween 1# and # , there is an impedance Z . It is further assumed

that kv

v=

1

. -ince we know1

vv

, we may remove Z and have the following

equation

Z

vk

Z

vkv

Z

vvi 11111

*1) === )1.1!1*

We define /Z as follows

*1)/1

k

ZZ

= )1.1!*

Finally, we have

/1

1

1Z

vi = )1.1!(*

-imilarly, we have

*1)*

11)

/

=

=

k

kZ

k

ZZ

)1.1!0*

1!(


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and/

Z

vi = )1.1!*

'he above discussion indicates that we may have an equivalent circuit for the circuit

in Fig. 1.1!()a*, as illustrated in Fig. 1.1!()b*.

'he above discussion is called &iller 'heorem.

For our amplifier, we will have kv

v

gs

ds = . For a capacitor, its corresponding

impedance is GDCjZ

1

= . 'herefore, for an amplifier, we have GDCkjZ *1)1

/1 +=

andGDC

kj

Z

=1

1

1/

. 'his means that we may have two capacitors, namely

GDCkC *1)1 += and GDCk

C

= 11 , shunting the input and output terminals of the

amplifier as shown in Fig. 1.1!0. 'hese two capacitors are often called &iller

capacitors.

Fig. 1.1!0 'he &iller capacitors

From Fig. 1.1!0, we observe the following

)1* For high frequency signals, the capacitors will become short!circuited. 'hus an

amplifier always acts as a low!pass filter.

)* 'he higher the gain, the larger the capacitor 1C . 'his means that the bandwidth

of an amplifier is smaller for a higher gain.

1!0


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In the following, we shall show e$periments to demonstrate the conclusions we

drew in the above.

Experiment 12.1-1 An Amplifier !ith a "o! #ain

In this e$periment, we used the circuit as shown in Fig. 1.1!. 'he circuit

represents a typical low gain amplifier. 'he program is shown in 'able 1.1!1 and the

result is in Fig. 1.1!2. As we can see, the amplifier is indeed a low!pass filter and its

bandwidth is quite broad.

Fig. 1.1! An amplifier with a low gain

'able 1.1!1 3rogram for $periment 1.1!1

$ample 2!1

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677 1 5 (.(6

8" 1 11 155k

.param W19u

&1 11 ( 5

1!


6/18

:nch "95.(u W9/W1/ m91

:A79/5.;u


7/18

result is shown in Fig. 1.1!@. As can be seen, the bandwidth is significantly reduced.

Fig. 1.1!? An amplifier with a high gain

'able 1.1! 3rogram for $periment 1.1!

$ 2!5

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677 1 5 6

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1!?


8/18

&0 ( 11 1

:pch "91u W9/W0/ m91

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9/18

Fig. 1.1!@ 'he gain vs frequency for the amplifier in Fig. 1.1!?

Section 12.2 The #ain-&an'!i'th (ro'uct for an Amplifier!ith Fee')ac*

In the previous section, we showed that an amplifier acts a low!pass filter. %esides,

the bandwidth decreases as the gain increases. In this section, we shall show that the

gain!bandwidth product for an amplifier with feedback is a constant.

Fig. 1.!1 shows a schematic diagram of an amplifier with feedback.

Fig. 1.!1 An amplifier with a feedback

From Fig. 1.!1, we have the following

1!;


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11/18

Experiment 12.2-1 The Enlar%in% of the &an'!i'th of an +perational

Amplifier )y ,e%atie Fee')ac*

In this e$periment, we used the circuit in Fig. 1.! as the operational amplifier.

'he open!loop program is in 'able 1.!1 and the bandwidth is shown in Fig. 1.!(.

As can be seen, the bandwidth is quite narrow because of the high gain.

Fig. 1.! 'he operational amplifier for e$periments in -ection 1.

'able 1.!1 3rogram for $periment 1.!1

open loop test

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.op

1!11


12/18

677 677D 5 (.(6

6-- 6--D 5 5.56

&1 1 1 677D 677D 3+E W95C "9C

& 1 677D 677D 3+E W95C "9C

&( ( ( 1 677D 3+E W95C "9C

&0 0 ( 677D 3+E W95C "9C

& ( 6% 6--D #+E W9155C "9C

&2 0 6%2 2 6--D #+E W9155C "9C

&? 6i! ? 6--D #+E W9155C "9C

&@ 2 6i: ? 6--D #+E W9155C "9C

&; ? 6%; 6--D 6--D #+E W9155C "9C

&15 6o 0 677D 677D 3+EW915C "9C

&11 6o 6%11 6--D 6--D #+E W95C "9C

6%IA- 6% 5 1.;6

6%IA-2 6%2 5 1.;6

6%IA-; 6%; 5 5.26

6%IA-1 6%11 5 1.?6

6% 6i! 5 1.2v

6in1 11 5 A+ 1

.A+ 7+ 155 1 155555k

6in 6i: 11 1.2v


13/18

Fig. 1.!( 'he bandwidth of the amplifier in Fig. 1.!1

We then incorporate feedback as shown in Fig. 1.!0. 'he program is in 'able

1.! and the new bandwidth is shown in Fig. 1.!. As can be seen, the bandwidth

is significantly enlarged as predicted.

Fig. 1.!0

'able 1.!

open loop test

.38>'+'

1!1(


14/18

.>3'I># 3>-'

."I% B+4mm5(v.lB ''

.C#38>'+'

.op

677 677D 5 (.(6

6-- 6--D 5 5.56

&1 1 1 677D 677D 3+E W95C "9C

& 1 677D 677D 3+E W95C "9C

&( ( ( 1 677D 3+E W95C "9C

&0 0 ( 677D 3+E W95C "9C

& ( 6% 6--D #+E W9155C "9C

&2 0 6%2 2 6--D #+E W9155C "9C

&? 6i! ? 6--D #+E W9155C "9C

&@ 2 6i: ? 6--D #+E W9155C "9C

&; ? 6%; 6--D 6--D #+E W9155C "9C

&15 6o 0 677D 677D 3+EW915C "9C

&11 6o 6%11 6--D 6--D #+E W95C "9C

6%IA- 6% 5 1.;6

6%IA-2 6%2 5 1.;6

6%IA-; 6%; 5 5.26

6%IA-1 6%11 5 1.?6

6% 6i! 5 1.2v

6in1 11 5 A+ 1

.A+ 7+ 155 1 155555k

6in 6i: 11 1.2v


15/18

Fig. 1.!

Experiment 12.2-2 Another Experiment !ith a ifferent Fee')ac* Circuit

In this e$periment, we used the same amplifier circuit as that used in $periment

1.!1. 'he feedback circuit diagram is e$actly as shown in Fig. 1.!1. 'he circuit

is shown in Fig. 1.!2, the program is in 'able 1.!( and the new bandwidth is

shown in Fig. 1.!?. As can be seen, the bandwidth is enlarged.

1!1


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Fig. 1.!2 'he circuit used for $periment 1.!

'able 1.!( 3rogram for $periment 1.!

Eigh =ain Amp


17/18

6-- 6--D 5 56

&1 1 1 677D 677D 3+E W95C "9C

& 1 677D 677D 3+E W95C "9C &( ( ( 1 677D 3+E W95C "9C

&0 0 ( 677D 3+E W95C "9C

& ( 6%? 6--D #+E W9155C

"9C

&2 0 6%@ 2 6--D #+E W9155C

"9C

&? 6i! ? 6--D #+E W9155C

"9C

&@ 2 6i: ? 6--D #+E W9155C

"9C

&; ? 6%; 6--D 6--D #+E W9155C

"9C

&15 vout 0 677D 677D 3+E W915C "9C

&11 vout 6%11 6--D 6--D #+E W95C

"9C

vin: vin 5 A+ 1 sin)1.2 5.55551 1k*

6I#! 6i! 5 1.2

6%IA-? 6%? 5 1.;6

6%IA-@ 6%@ 5 1.;6

6%IA-; 6%; 5 5.26

6%IA-1 6%11 5 1.?6

81 vout @ 15555G

8 @ 5 155G

8( @ 6i: 15G

80 vin 6i: 15G

< transient simulation


18/18

.end

Fig. 1.!? 'he bandwidth produced in $periment 1.!

1!1@

Chapter 12(Analog)

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