lecture 12 - frequency response

7/28/2019 Lecture 12 - Frequency Response

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EE 21 Fundamentals of Electronics2

ndSem, A.Y. 2012-2013

AAMSumalde, JPARamoso

BJT and JFET Frequency Response

Introduction

Logarithmic function

axba bx log,

Common logarithm: ax 10log

Natural logarithm: aay e lnlog

Basic properties of logarithms

1. 01log10

2. baba101010

loglog/.log

3. ana n 1010 loglog

Use of log scales

- Expands the range of graphing

Reading a typical log plot

-basically, each major division is a factor of 10.

Decibels

- Decibels are always a relative measure

dBm a decibel measure where the reference power is

1mW

i.e. mWPAdBm 1/log10 10

Typical decibel values and corresponding voltage gains

Recall that these values are computed using

iOdBVVAv /log20

- Numerically, the use of dB allows us to represent

10x increase in gains by only 20dB per 10x gain.

Hence the term 20dB/decade (another relation,

6dB/octave, will be illustrated later

General Frequency Considerations


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- Analysis in the previous chapter (small-signal

analysis) assumed midfrequency spectrum

- At low frequencies, coupling and bypass

capacitors have increased reactances and affects

the response of the system.

- At high frequencies, the frequency-dependent

parameters of the small-signal equivalent circuit

(i.e. ) and stray capacitances will limit response

of the system as well.

Typical gain v. frequency response curves

(see additional handout; page 546 Boylestad 10th

ed)

- For the typical RC-coupled amplifier, thecapacitances Cc, Cs, and CEaffect the low-

frequency response

Corner Frequencies

- Also known as cutoff, band, break, orhalf-power

frequencies or points

- Magnitude of the gain is equal or close to the

midband value, which is designated at 0.707Avmid.

Ro

VAv

Ro

VoP imid

Omid

22||

At the half-power points,

midOOhpf

midOhpf

imidOhpf

PP

Ro

ViAvP

Ro

VAvP

5.0

)(5.0

|707.0|

2

2

Bandwidth: f2

f1, where f2 and f1 are the two half-powerfrequencies.

Normalization Process

- Done to obtain a dB vs. fplot

- Values in the gain axis are divided by the midband

(hence all values are in reference to that gain)

Normalized frequency response curve

Phase Response

- In small-signal analysis, the accepted phase shift

for a typical common emitter amplifier is 180o

.- However, the complete phase response is

different when considering the lower and higher

frequencies

Low-frequency analysis: Bode Plot

- RC combinations formed by Cc, CE and Cs and

the network resistive parameters determine the

cutoff frequency

RC combination in consideration:

- At low frequencies, C approaches an open circuit

- At high frequencies, C approaches a short circuit

During the frequencies between low and high, the

following graph shows the response


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where f1 is a cutoff frequency.

Considering the RC circuit above,

c

io

XR

RVV

With the boldface notation indicating both

magnitude and phase.

The magnitude is given by

22

C

i

XR

RVVo

At resonance, where Xc=R,

ViViVo 707.02

1

Hence, at the half-power points, Xc = R

This frequency, f1, can be calculated as,

RCf

RCf

Xc

2

1

2

1

1

1

Note also that we can represent this in terms of angular

frequency (rad/s):

RC

11

In terms of logarithms,

dBGv 32/1log20 10

The voltage divider equation can be manipulated to get an

expression for the voltage gain, as follows:

CjXR

R

Vi

VoAv

RXcj /1

1

=

fCRj

2

11

1

And sinceCR

f2

11 ,

ffjAv

/1

1

1

In magnitude and phase form,

anglepha se

f

f

f

fVi

VoAv

/

11

magnitude

2

1

tan,

1

1

When f=f1,

dBAv 3707.02

1

)1(1

1||

2

Going back to the logarithmic form

magnitude)/(1

1log20

2

1

10)(

ffAv dB

This equation is then expanded

2

1

2

110)( 1log20

f

fAv dB


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=

2

110 1log10

f

f.

For frequencies f >> f1, or (f1/f)2>>1, the equation can be

simplified to

2

110)( log10

f

fAv dB or

,`log20 1)(

f

fAv dB where f


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Low-frequency response BJT amplifier

- Above is a sample circuit for analyzing the LFR of

a BJT amplifier.

- The effect of capacitors Cs, Cc, and CE are

identified by matching them with the RC

combination as seen by these capacitors.

CS:

- From this block diagram, the low cutoff frequency

defined by Cs is given by the equation

SiSLS

CRRf

2

1

- By voltage divider:

-

Si

iS

midRR

RVVi

, similar to SSA derivation

At f=fLS, Vi= 0.707Vimid

- The equivalent resistance seen by CS must be

determined:

Localized AC equivalent for Rias seen by Cs:

ei rRRR //// 21

And the voltage input (combining magnitude and phase)

can be calculated using the voltage divider rule

S

ii

jXcRRssR 1

V

V

Cc:

- from this block diagram, the total series

impedance is now Ro + RL and

CLLC

CRRof

2

1

- ignoring Cs and CE, the output voltage Vo will be

0.707 of the midband value at fLC.

Localized AC equivalent for effect of Cc

OCO rRR //

R1//R2


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CE :

- again it is necessary to determine the resistance

network seen by CE, at which

Ee

LECR

f2

1

Localized AC equivalent for Re:

Where

21 ////' RRRR SS

And

eEe r

RsRR

'//

Quantitative illustration of CEeffect on gain

Ee

C

Rr

RAv

The gain is heavily altered by the presence of RE

in the equation; depending on the frequency, the bypasscapacitor CE may either short out RE or otherwise.

Summary: Low Frequency Response (BJT)

- At midband frequency level, short-circuit

equivalents are inserted for the capacitors

- The highestlow-frequency cutoff (depending on

Cc, CE, or CS) will have the greatest impact.

- Near LF cutoff values may interact and alter

frequency response curves by compounding thebode plot slopes

Example: For the loaded voltage divider circuit (as

refreshed below),

Cs = 10uF, CE = 20uF, CC = 1uFRS= 1k, R1= 40k, R2= 10k, RE= 2k, RC= 4k,

RL= 2.2k, = 100, ro = infinite, and Vcc = 20.

a. Determine the lower cutoff frequency

b. Sketch the frequency response using a bode plot.


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Low-frequency response JFET amplifier

- Analysis follows in the same manner as that of the

BJT configuration

- Above is a sample circuit for analyzing the LFR of

a BJT amplifier.

- The effect of capacitors Cs, Cc, and CE are again

identified by matching them with the RC

combination as seen by these capacitors

CG :

- The cutoff frequency as determined by the gate

capacitance CG is given by

GiLG

CRRsigf

2

1

Where

GiRR

Note: Typically, RG >> Rsig (remember that RG is usually in

the order of M). This also allows us to have a low fLG

even with a small CG.

Cc

- The cutoff frequency is given by the equation

COLG

CRRf

L

2

1

Where, based on the circuit given,

dDO rRR //

Cs

- The cutoff frequency is given by the equation

Seq

LSCR

f2

1

Where

LDddmSS

eqRRrrgR

RR

///11

But for very large rd, this equation is simplified to

m

Seqg

RR1

//


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Example: Consider the loaded JFET self-bias amplifier

below:

Given the following parameters:

CG = 0.01F Cc = 0.5F CS = 2F

Rsig = 10k RG= 1M, RD= 4.7k

RS= 1k RL= 2.2k

IDSS = 8mA VP = - 4 V rd = infinite

VDD= 20 V

Sketch the frequency response using a bode plot.

Miller Effect Capacitance

- In the high frequency region, the capacitive

elements of importance are the interelectrode

(between-terminals) capacitances and the wiring

capacitances (between leads of the network)

- Capacitors controlling the LFR are replaced with

equivalent shorts at these high frequencies.

Miller Input Capacitance

- Forinvertingamplifiers, input and output

capacitance is increased by a capacitance level

sensitive to (a) interelectrode capacitance b/w

input and output and (b) the gain Av.

- This feedback capacitance is denoted by C f.

Applying KCL:

21 IIIi

by ohms law:

i

i

i

ii

R

VI

Z

VI 1,

And

Cf

iv

Cf

ivi

Cf

oi

X

VA

X

VAV

X

VVI

12

Substituting to the KCL equation:

Cf

iv

i

i

i

i

X

VA

R

V

Z

V 1

And

vCf AXRiZi

1/

111

Simplifying the quantity Xcf/(1-Av)

CM

C

vfv

CfX

ACA

X

M

1

1

1

Finally,

CMXRiZi

111

We now define the Miller Input Capacitance as

fMi CAvC 1


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Illustration of the Miller effect capacitance:

Notes:

- The above equivalent circuit shows the

representation of the equation

CMXRiZi

111

- The effect of CM is a parallel combination thatwould be important esp. when frequency

increases

- Other interelectrode capacitances will simply be

added to CM in parallel with Ri.

Miller Output Capacitance

- Miller effect would also increase the level of

output capacitance

Applying KCL:

21 IIIo

CfX

ViVoI

Ro

VoI

21 ,

- We assume that the resistance Ro is sufficiently

large enough to ignore the I1 term, thus:

Cf

iOO

X

VVI

Since Vi = Vo/Av,

CfCf

OX

AvVo

X

Av

VoVo

I

11

And

CfX

Av

Vo

Io /11

We rearrange this equation as such:

MoCMo

f

Cf

CAvCAv

X

Io

Vo

1

/11

1

/11

Thus, we define the Miller Output Capacitance as

fMo C

Av

C

1

1

Typically,Av >> 1 , so

fMo CC

Again, this has the same capacitor-in-parallel effect as

that of the miller input capacitance.


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High-Frequency Response BJT Amplifier

At the high-frequency end, the cutoff frequency is

determined by:

a. Network capacitance (parasitic and introduced)b. Frequency dependence of(orhfe)

Network Parameters

- The above RC network is the configuration of

concern for the high-frequency response (note its

difference with the low-frequency response!)

Derivation similar to the low-frequency response corner

frequencies will yield

2/11

ffjAv

- Again, note its difference from the low-frequency

equation!

Resulting frequency plot

Note the asymptotes of the bode plot (the 0-dB line and

the -6dB/octave or -20dB/decade line).

- The above circuit is a typical loaded + internal source voltage divider with the parasitic capacitances (Cbe ,Cce and

Cbc) and the wiring capacitances (Cwi and Cwo)

- Cc, CS and CE are assumed shorted at the high frequencies


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Equivalent AC circuit with capacitances included

- The input capacitance Ci includes the input wiring capacitance Cwi, transition capacitance Cbe and the input

miller capacitance CMi

- The output capacitance Co includes the output wiring capacitance Cwo, parasitic capacitance Cce and the output

miller capacitance CMo.

- In general, the largest capacitance is Cbe and the smallest being Cce.

The -3dB frequency is defined by the equations

iTh

HiCR

f12

1

and

oTh

HoCR

f22

1

,

Where the thevenin resistances RTH1 and RTH2 pertain to the TECs of the input and output circuit, as shown:

Clearly,

iSTH RRRRR ////// 211 ,

where Ri is the total resistance at the base (for this particular example, Ri = re)

, also,

bcbeWiMibeWii CAvCCCCCC 1


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Notes:

- At high frequencies, XCi will decrease,

consequently reducing the total impedance of

the combination

ii XCRRR ////// 21 ,

- This in turn reduces the voltage across Ci, the

current Ib , and finally ,the gain of the system

For the output network,

oLCTH rRRR ////2

And

MoceWoO CCCC

- Similarly, Xco decreases as frequency increases

thus the total impedance of the output parallelbranches is reduced (thus Vo 0).

- Both fHiand fHowill define a -6dB/octave (or -

20dB/decade) asymptote

hfeorVariation

Beta varies with frequency as defined by the following relationship:

ffj

mid

/1

(note: and hfe are interchangeable)

The quantity fis determined by a set of parameters in the hybrid pi model(aka Giacoletto Model)

Where rbb 'base contact, base bulk, and base spreading resistance

Base contactactual connection to the base

Base bulk resistance from external terminal to the active region of the transistor

Base spreading resistance actual resistance within the active base region

rb'e , rceand rbc resistances between the indicated terminals (BCE) in the active region

Cbcand Cbe capacitances between the terminals (Cbc transition, Cbe diffusion)


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Replacing some of the quantities with more familiar terms:

In terms of the circuit parameters,

u

hCCr

fffe

2

1or

Or, since

emidrr

Then

uemid

hCCrr

fffe

2

1or

Since the equation involves the resistance re, then fis a function of the bias configuration.


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hfe(beta)and hfb(alpha)versus frequency in the high-frequency region

Observations:

- From the equation off , the value ofwill drop from its midband value upon hitting f , similar to how the gaindrops after hitting a cutoff frequency associated with its RC network

- The above graph compares a common-emitter connection (with parameter) and a common-base connection

(with parameter).

- From the figure, the common-base configuration displays improved high-frequency characteristics (for) as

compared to the common-emitter connection.

- The miller effect capacitance is ABSENT at a common-base configuration. (why?)

- With these characteristics, common-base high frequency parameters are often given instead of common-emitter

configurations

Conversion between fand f

1ff


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Gain-bandwidth product

Defined by the condition

1/1

ffj

mid

So that

dB

ffj

mid

dB01log20

/1log20

We define a frequency Tf at which dB =0dB (see above figure). We compute the magnitude of at the

condition point ffT as

1

//12

ffff T

mid

T

mid

And so we have

product)bandwidth-(gain

BW

midT ff

And

mid

Tff

We can now obtain an expression for Tf which mirrors the form of the other cutoff frequency equations:

uemidmidT

CCrf

2

1

= ueT

CCrf

2

1


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Example: Consider the BJT configuration shown below:

Given the following parameters:

RS = 1k , R1 = 40k, R2 = 10k, RE = 2k, RC = 4k, RL = 2.2k

CS = 10 uF, CC = 1uF, CE = 20uF

= 100, ro = infinite, Vcc=20V

C(Cbe) = 36pF, Cu(Cbc) = 4pF, CCE = 1pF, Cwi = 6pF, and Cwo = 8pF

a. Determine the frequencies HoHi ff and

b. Find the frequencies Tff and

c. Sketch the complete frequency response by combining the low-frequency response previously obtained and the

results of (a) and (b).


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High-Frequency Response JFET Amplifier

- Analysis proceeds in a similar nature as the BJT amplifier, but without the variation

- Miller effect capacitance still comes into play

Equivalent Circuit with parasitic and wiring capacitances

AC Equivalent (with capacitors of interest)

From the circuit, we can obtain a similar input TEC / output TEC as we did in the BJT amplifier:


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From the input circuit,

iTh

HiCR

f12

1

And

GsigTh RRR //1

Also,

MigsWii CCCC

With the miller input capacitance defined by

gdMi CAvC 1

For the output circuit,

oTh

HoCR

f22

1

With

dLDTh rRRR ////2

And the equivalent output capacitance

ModsWo CCCCo

The miller output capacitance is given by

gdMo CAv

C

1

1

It must be noted that unlike the BJT amplifier, we did not quickly generalize the miller output capacitance; voltage

gains of JFET networks are relatively small, not exactly >>1 unlike BJT gains.


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Example: Consider the JFET circuit below:

The parameters of the circuit are as follows:

CG = 0.01uF, CC = 0.5uF CS = 2uF

Rsig = 10k, RG = 1M, RD = 4.7k, RS = 1k, RL = 2.2k

IDSS = 8mA, VP = - 4 V, rd = infinite VDD = 20 V

Cgd = 2pF Cgs = 4pF Cds = 0.5pF CWi = 5pF CWo = 6pF

Determine the high-cutoff frequencies for the network and sketch the full frequency response.

Multistage Frequency Effects

- Additional stages introduce their own low- and high-cutoff frequencies.

- The highest low-cutoff and lowest high-cutoff frequencies determine the overall frequency response of the

system.

- Note the compounding of slopes occurring when a nearby cutoff frequency is hit.

For identical stages, the low- and high-cutoff frequencies are calculated via the following equations:

12'

/1

11

n

ff (low-cutoff frequencies)

2/12 12' ff n (high-cutoff frequencies)


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Effect of identical stages

Table of values of the quantity 12/1 n

Reference: Boylestad and Nashelsky, Electronic Devices and Circuit Theory. 7thand 10

theditions.

lecture 12 - frequency response

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