using bode plots

8/17/2019 Using Bode Plots

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EECE 301Signals & SystemsProf. Mark Fowler

Note Set #37• C-T Systems: Using Bode Plots


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Bode Plot Ideas Can Help Visualize What Circuits Do…

1 1( )

1 1

RC H s

RCs s RC

1( )

1

H

jRC

RC Lowpass Filter

Break Point = 1/ RC

Break Point = 1/ RC


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( )1 1

RCs s H s

RCs s RC

RC Highpass Filter

c= 1/RC

RLC Bandpass Filter

2 2

2( )

2n

n n

s H s

s s

101

102

103

104

105

-60

-50

-40-30

-20

-10

0

10

CT Frequency (rad/sec)

| H ( ) | ( d B )

–172

–5828

0.1

1

3

–1000

–1000

–100+ j995

–100- j995

2

2( )

( )ns

H ss a

2( )

( )( )ns

H ss a s b

All slopes are 20 dB/decade!


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A Better Bandpass Filter? Suppose you want a BPF…. with faster “rolloff”!

1 2

Need an s2

term in the numerator!At 1 we need to change slope by –40 dB/dec… So need double pole @ 1!

At 2 we need to change slope by –40 dB/dec… So need double pole @ 2!

2

2 21 2

( )( ) ( )

Ks H s

s s

There are (at least) three ways to get this!

1 2

1 2 1 2

( )( )( ) ( )( )

K s K s H s

s s s s

1 22 2

1 2

( )( ) ( )

K s K s H s

s s

BPF RLC w/

distinct poles

21 2

2 21 2

( )( ) ( )

K s K H s

s s

HPF RLC w/

Rep. polesBPF RLC w/

repeated poles

LPF RLC w/

Rep. poles

Same exact circuits… just different choices of RLC!!!


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Looks like we could just cascade two of our RLC circuits… Here we cascade BPFs.

Although TF Theory says this will work… the problem is that the second

circuit “Loads” the first one!!!So… one approach would be to re-analyze this cascade and see if it will stillwork but with some “tweeks” on the component choices.

Our “cascade theory” onlyholds when attaching the

2nd system does not change

how the first one behaves!

Another approach is to use an op amp as a “buffer” between the stages!

It may be desirable

to add another

buffer here…

VERY large input resistance

of op amp prevents “loading”

of first stage!

VERY small output resistance

of op amp minimizes impact

on 2nd stage!

1 1 1 2 2 2

Bandpass RLC Bandpass RLC

Remember… there are two ways to choose the components here:1. Each stage has repeated poles 2. Each stage has distinct poles


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2nd Ord Highpass RLC 2nd Ord Lowpass RLC

Another way to make a better BPF:Here we must choose the components so that

each stage has repeated poles.

Although these ideas lead to workable circuits they are not necessarily the best…For one thing… they need inductors (which are big and can’t be made in an IC!)There are other forms… See this link for the form used below.

+

–

+

–

2nd Ord Highpass RC 2nd Ord Lowpass RC

http://pdfserv.maxim-ic.com/en/an/AN1795.pdf


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Design Example using Bode Plot Insight

Suppose you want to build a “treble booster” for an electric guitar.You decide that something like this might work:

100 1000

0dB

20dB)( f H

(Hz) f

100Hz 628 rad/s =

1000Hz 6283 rad/s =

Notice that we are doing our rough“design thinking” in terms of Bode

Plot approximations!!!

The A string on a guitar has a fundamental frequency of 110 Hz

The A note on 17th fret of the high-E string has a fundamental frequency

of 880 Hz

)/1(

)/1()(

)/1(

)/1()(

2

1

2

1

j

j H

s

ss H

From our Bode Plot Insight… we know we can get this from a single real pole,single real zero system… with the “zero first, then the pole”:

with: 1 = 628 rad/s

2 = 6280 rad/s


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A series combination

…has impedance Z (s) = R + 1/Cs

Note: we could get R + sL withan inductor but inductors are

generally avoided when possible

So what do we get if we could some how form a ratio of such impedances?

1

1

/1

/1

)(

)(

22

11

1

2

22

11

2

1

C sR

C sR

C

C

sC R

sC R

s Z

s Z

222

111

/1

/1 : Let

C R

C R

Aha!!! What we want!

Now, how do we get a circuit to do this? Let’s explore!

2

1

1

2

2

1

/1

/1

)(

)(

j

j

C

C

Z

Z


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Okay…how do we build a circuit that has a transfer function

that is a ratio of impedances?!

)(t vin)(t vo

i

f

R

RGain

Ratio of

resistances

Extending the analysis to include impedances we can show that:

)(t vin )(t vo

)(s Z i

)(s Z f

)(

)()(

s Z

s Z s H

i

f

Won’t affect ourmagnitude: | H ( )|

Recall the op-amp inverting amplifier!


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)(t vin)(t vo

But wait!! You then remember that op amps must always have negativefeedback at DC so putting C f here is not a good idea…

So we have to continue…

We also might not like this circuit because it might not give us a verylarge input impedance… and that might excessively “load” the circuit

that you plug into this (e.g., the guitar)

Back to the drawing board!!!

Now, you can choose the R’s & C’sto give the desired frequency points

p. 98, The Art of Electronics,Horowitz & Hill, Cambridge

Press, 1980


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Okay, then you remember there is also Non-Inverting Op-Amp circuit…

)(t vin)(t vo

Infinite input impedance

i

f

R RGain 1

We avoid the “no DC feedback”issue… but we’re not yet sure we’ll

get what we want!

( 1 / ) ( ) 1 /( ) 1

1 / 1 / 1 /

f i i f i f i

i i i i i i

R R C s R R R C s H s

R C s R C s R C s

Applying this gain formula we get:

Oh Cool!! We Get

What We want!

1

1)()(

sC R

sC R Rs H

ii

ii f

rad/s6283

1

rad/s6281

2

1

ii

i f i

C R

C R R

Set: Choose:

F1.0

5.115

i

i f

C

k Rk R

Using standard values


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Computed Frequency Response using Matlab

f (Hz)

| H ( f ) | ( d B )

Discrepancydue to use of

standardvalues


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1. Through insight gained from knowing how to do Bode plots byhand… we recognized the kind of transfer function we needed

2. Through insight gained in circuits class about impedances werecognized a key building block needed: Series R-C

3. Through insight gained in electronics class about op-amps we founda possible solution… the inverting op-amp approach

4. We then scrutinized our design for any overlooked issuesa. We discovered two problems that we needed to fix

5. We used further insight into op-amps to realize that we could fix theinput impedance issue using a non-inverting form of the op-ampcircuit

6. We didn’t give up at first sign that the inverting form might not giveus the form we want…a. Through mathematical analysis we showed that we did in fact

get what we wanted!!!!!!

Summary of Bode-Plot-Driven Design Example

using bode plots

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