chapter 7- frequency response

8/12/2019 Chapter 7- Frequency Response

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Chapter 7

Frequency Response


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Amplifier Frequency Response


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1-3

Electronic Circuits

Amplifier Frequency Response

In linear amplifier analyses, coupling and bypasscapacitors act as short circuits to the signal voltagesand open circuits to dc voltages.Capacitors do not change instantaneously from a shortcircuit to an open circuit as the frequency approacheszero.Internal capacitances in BJT and FET affect thefrequency response.

All amplifier gain factors are functions of signalfrequency.Includes voltage, current, transconductance, andtransresistance


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1-4

Electronic Circuits

Amplifier gain versus frequency

(f < f L ) (f > f H )

Gain decreases as the frequencydecreases because of coupling and

bypass capacitor effects


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1-6

Electronic Circuits

(f < f L ) (f > f H )

Coupling and bypass capacitors act asshort circuits, and stray and transistorcapacitances act as open circuits



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1-7

Electronic Circuits

(f < f L ) (f > f H )

Gain at f = f L and at f = f H is 3dB

Bandwidth f BW = f H - f L


Gain at f = f L and at f = f H is 3dB

Bandwidth f BW = f H - f L


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1-8

Electronic Circuits

System Transfer Functions

The frequency response of a circuit is usuallydetermined by using the complex frequency s (s = j =

j2 f ).Each capacitor is represented by its compleximpedance, 1/sC .Each inductor is represented by its compleximpedance, sL .

Name of function Expression

Voltage transfer function T(s) = V o(s) / V i(s)

Current transfer function I o(s) / I i(s)

Transresistance function V o(s) / I i(s)

Transconductance function I o(s) / V i(s)


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1-9

Electronic Circuits

s -Domain Analysis

A transfer function in the s -domain

K = constantz = zeroes

p = poles

s = z i , the transfer function is zeros = p i , the transfer function is infinite

n

m p s p s p s

z s z s z s K sT

21

21


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1-10

Electronic Circuits

s -Domain Analysis

Simple transfer functions

2

2

2

11

1

11

s

s K sT

s K sT


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1-11

Electronic Circuits

s -Domain Analysis

Series coupling capacitor circuit

+

-

R sCs

Vo

R pVi

s s

K C R R s

C R R s

R R

R

sV sV

C R R s

C sR

sV sV

sC R R

R

sV sV

s p s

s p s

p s

p

i

o

s p s

s p

i

o

s p s

p

i

o

11)()(

1)()(

1)()(

2


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1-12

Electronic Circuits

s -Domain Analysis

Parallel load capacitor circuit

+

-

R s

C p

Vo

R pVi

s K

C R R s R R

R

sV sV

C R R

R R s

R R

R

sV sV

sC V

RV

RV V

p p s p s

p

i

o

p p s

p s p s

p

i

o

p

o

p

o

s

io

11

11

)()(

1

1)()(

01

1

KCL at output node


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1-13

Electronic Circuits

Bode Plots

+

-

R sCs

Vo

R pVi

2

22

2

21

2)(

1)(

1)()()()(

11)()(

s

s

p s

p

s

s

p s

p

s s

p s p

io

s p s

s p s

p s

p

i

o

f

f R R

R jf T

R R

R jT

j j

R R R

jV jV jT sT

s s

K C R R s

C R R s

R R

R

sV sV


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1-14

Electronic Circuits

Bode Plots

+

-

R sCs

Vo

R pVi

2101010

210

21log202log20log20)(

21

2log20)(

s s p s

pdB

s

s

p s

p

dB

f f R R

R jf T

f

f

R R

R jf T


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1-15

Electronic Circuits

Bode Plots

2101010 21log202log20log20)( s s p s

pdB f f R R

R jf T


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1-16

Electronic Circuits

Bode Plots

p s

p

R R

R10log20

This term is a constant thus independent of frequency[R p /(R s+R p)] is less than unity thus dB value is less thanzero


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1-17

Electronic Circuits

Bode Plots

s f 2log20 10

When f = 1/2 s , 20log 10 (1) = 0


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1-18

Electronic Circuits

Bode Plots

The slopes in Bode plot magnitudes are described inunits of either dB/octave or dB/decade

An octave means that frequency is increased by afactor of two

A decade implies that the frequency is increased by afactor of 10The value of the function 20log 10 (2 f s ) increases by afactor of 6dB for every factor of 2 increases in

frequency, thus a slope is considered 6dB/octaveThe value of the function 20log 10 (2 f s ) increases by afactor of 20dB for every factor of 10 increases infrequency, thus a slope is considered 20dB/decade


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1-20

Electronic Circuits

Bode Plots

For f >> 1/2 s , the 2 nd and 3 rd term cancel For f


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1-21

Electronic Circuits

Bode Plots

The series capacitor C s is a coupling capacitor between theinput and output signals At a high enough frequency, C s acts as a short circuit, andthe output voltage isV o = [R p /(R s+R p )]V i

For very low frequencies, the impedance of C s increasesand approaches as an open circuit, and the output voltageapproaches zeroThis circuit is called a high-pass network

The high-frequency signals are passed through to the output

+

-

R sCs

Vo

R pVi


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1-22

Electronic Circuits

Bode Plot of The Phase Function

j Ke jB A

K =

.

22 B A

)/(tan 1 A B


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1-23

Electronic Circuits


3213

2

1

3

2

1

3

21

3

21)(

21

2)(

212)(

1)()(

)(

j j

j j

j s

j s j

p s

p

s

s

p s

p

s

s

p s

p

i

o

e K

K K e K e K

e K jf T

e f j

e f je

R R

R jf T

f j f j

R R R jf T

j j

R R

R

jV jV

jT


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1-24

Electronic Circuits


3213

2

1

3

2

1

3

21

3

2

1)(

21

2)(

j

j

j j

j s

j s j

p s

p

e K

K K

e K

e K e K jf T

e f j

e f je

R R

R jf T

The first term is a positive real quantity, the phase is 1 = 0The second term is purely imaginary, the phase is 2 =

90 oThe third term is complex, the phase is 3 = tan -1 (2 f s ) The net phase is = 90- tan -1 (2 f s)


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1-25

Electronic Circuits


so f 2tan90 1


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1-26

Electronic Circuits

Bode Plots

2

22

1

21

1)(

1

1)(

11

)()(

)()(

11

11

)()(

p p s

p

p p s

p

p p s

p

io

p p s p s

p

i

o

f R R

R jf T

R R

R jT

j R R

R

jV jV

jT sT

s K

C R R s R R

R

sV sV

+

-

R s

C p

Vo

R pVi


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1-27

Electronic Circuits

Bode Plots

2101010

210

21log201log20log20)(

21

1log20)(

p p s

pdB

p p s

pdB

f R R

R jf T

f R R

R jf T

+

-

R s

C p

Vo

R pVi


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1-28

Electronic Circuits

Bode Plots

2101010 21log201log20log20)( p p s

pdB f R R

R jf T


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1-29

Electronic Circuits

Bode Plots

The parallel capacitor C p is a load or parasitic capacitance At low frequency, C p acts as an open circuit, and the outputvoltage is

V o = [R p /(R s+R p )]V i At high frequency, the impedance of C p decreases andapproaches as a short circuit, and the output voltage is zeroThis circuit is called a low-pass network

The low-frequency signals are passed through to the output

+

-

R s

C p

Vo

R pVi


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1-30

Electronic Circuits


321

3

2

1

3

2

1

3

21

3

21)(

21)(

211)(

11

)()(

)(

j j

j j

j p

j j

p s

p

p p s

p

p p s

p

i

o

e K

K K e K e K

e K jf T

e f j

ee

R R

R jf T

f j R R R jf T

j R R

R

jV jV

jT


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1-31

Electronic Circuits


3213

2

1

3

2

1

3

21

3

21)(

21)(

j j

j j

j p

j j

p s

p

e K

K K

e K

e K e K jf T

e f je

e R R

R jf T

The first term is a positive real quantity, the phase is 1 = 0The second term is a positive real quantity, the phase

is 2 = 0

The third term is complex, the phase is 3 = tan -1 (2 f p ) The net phase is = - tan -1 (2 f p)


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1-32

Electronic Circuits


s f 2tan 1


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1-33

Electronic Circuits

Short-Circuit and Open-CircuitTime Constant

Vo

+

-

R s

C pR pVi

Cs

p s s

p

p s

p p s

p

i

o

s sC

C

R R

R R R

R

sV sV

11

1)()(

Both C S and C P presentKCL at output node


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1-34

Electronic Circuits


C s affects the low frequency response and C p affects thehigh frequency response At low frequencies, load capacitor is treated as an opencircuitEquivalent resistor seen by C s is found by setting all

independent sources equal to zero. Thus the effectiveresistance is (R s+R p )

The time constant associated with C s is s = (R s+R p )C sSince C p was made an open circuit, thus s is called anopen-circuit time constant

Vo

+

-

R s

C pR pVi

Cs


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1-35

Electronic Circuits


At high frequencies, coupling capacitor is treated as ashort circuit

Effective resistance seen by C p is (R s ||R p ) The time constant associated with C p is p = (R s ||R p )C pSince C s was made a short circuit, thus p is called ashort-circuit time constant

Vo

+

-

R s

C pR pVi

Cs


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1-36

Electronic Circuits


The lower corner or 3dB frequency is at the low end ofthe frequency scale, is a function of the open-circuittime constantf L = 1/2 s

The upper corner or 3dB frequency is at the high endof the frequency scale, is a function of the short-circuittime constantf H = 1/2 p

Midband range or bandwidthf BW = f H - f L


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Electronic Circuits



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Frequency Response:Transistor Amplifiers withCircuit Capacitors


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1-39

Electronic Circuits

(a) Common-emitter circuit with coupling capacitor and(b) small-signal equivalent circuit

Coupling Capacitor Effects

Small signal outputresistance r o is infinite

r o >> R C, r o >> R E

This circuit is a high-pass network

At high frequencies, C C acts as a short circuitand the input signal iscoupled through thetransistor to the output

At low frequencies, theimpedance of C C becomes large and theoutput is zero


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1-40

Electronic Circuits

Current-Voltage Analysis

Coupling CapacitorEffects

r g

Rr I V

R

r I V

Rr I g I r I

RV g I r I V

m

E b

inib

b

E bmbb

E mbbin

1


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1-43

Electronic Circuits


i

gs

gs

o

i

ov V

V

V V

V V

A

Set all independent source 0, thus V i=0, V gs =0, g mVgs =0.Resistance seen by C C is (R D+R L)Corner frequency f L=1/2 s


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1-44

Electronic Circuits

(a) Emitter-follower circuit with output coupling capacitor and(b) small-signal equivalent circuit

Coupling Capacitor Effects

CC1 is very large and acts as a short circuit to the input signal The equivalent resistance seen by C C2 is [R o+R L]


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Electronic Circuits



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Electronic Circuits

Review: OutputImpedance


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1-47Problem-Solving Technique

chapter 7- frequency response

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