frequency response (i&n chap 12) - ubc ecegrecuc/253/spring14/253-p2-frequency... ·...

1

Frequency Response

(I&N Chap 12)

• Introduction & TFs

• Decibel Scale & Bode Plots

• Resonance

• Scaling

• Filter Networks

• Applications/Design

Based on slides by J. Yan

2

Network Scaling

Considering a Bode plot, there are two ways to scale.1. For magnitude scaling, the magnitude plot is shifted

up or down while the phase plot is unchanged.2. For frequency scaling, both magnitude and phase

plots are shifted left or right.

Let’s see how to change component values for resonant circuits to achieve these types of scaling.


3

Magnitude/Impedance ScalingTo scale the equivalent impedance by a factor ,

simply scale each component impedance by .

Resistors: '

1Capacitors: '

Inductors: '

For a series RLC circuit, what happen

M

M

R M

C

M

L M

K

K

Z R R K R

CZ C

sC K

Z sL L K L

= → =

= → =

= → =

0s to and with magnitude scaling?Q ω


4

Frequency ScalingTo shift the plots by a frequency factor , we wish

the impedance at the scaled frequency to

be the same as at the original frequency .

Resistors: '

1 1Capacitors: '

Induc

F

F

R

C

F

K

K

Z R R R

CZ C

j C j C K

ω ωω

ω ω

′ =

= → =

= = → =′ ′

0

tors: '

For a series RLC circuit, what happens to and with frequency scaling?

L

F

LZ j L j L L

K

Q

ω ω

ω

′ ′= = → =


5

Magnitude and Frequency Scaling

ωω ffmf

mm K

KK

CC

K

LKLRKR ==== ' ,' ,' ,'

To simultaneously scale impedance in both magnitude and frequency:

E.g., a 3rd order Butterworth filter normalized to ωc=1rad/s is shown. Scale the circuit to a cutoff frequency of 10kHz and use 15 nF capacitors.


6

Filter Networks

• Filters are used to allow a particular range of frequencies to pass through while rejecting others.

• The type of filter can readily be determined by the magnitude plot of the TF relating the filter input to the filter output.

• Filters may be classified as passive or active, depending on whether the filter has any internal sources of energy.

Filter

E.g., Low-Pass Filter


7

Common Filter NetworksWe’ll examine 4 common filters with self-explanatory names.

High-pass filter

Band-pass filter Band-rejection filter

Low-pass filter

We first focus on passive filters. We need op-amps for active ones.


8

Ideal Filter CharacteristicsQ: Consider the ideal low-pass

filter characteristic. Why do you suppose this isn’t realizable in practice? (Hint: What TF would yield this characteristic?)


9

Passive Low-Pass Filter (LPF)A simple low-pass filter is the RC series circuit where capacitor voltage is taken as the output.


10

A simple high-pass filter is the RC series circuit where resistor voltage is taken as the output.

Passive High-Pass Filter (HPF)


11

Passive Band-Pass Filter (BPF)This RLC circuit gives a band-pass

filter if vR is taken as the output.

Q: Give an intuitive reason for why both low and high frequencies are rejected.

( )2 20

,

( / ) / 4

2

Cut-off Frequencies:

lo hi

R L R L ωω

+ +=m

0

1Center Freq:

LCω = Bandwidth: hi lo

RBW

Lω ω= − =

)(

1

:

1

211

0

Cjsv

sC

v

LjR

RG

sCRLCs

sCR

sLR

R

V

VG

ωω ω −+

=

++=

++==

=

FunctionTransfer


12

Band-Rejection FilterThis RLC circuit gives a band-rejection (aka “bandstop” or

“notch”) filter if vL+vC is taken as the output.

( )2 20

,

( / ) / 4

2

Cut-off Frequencies:

lo hi

R L R L ωω

+ +=m

0

1Center Freq:

LCω =

20

21

1

1

Transfer Function:

v

V s LCG

V s LC sRC

+= =

+ +


13

Example (efts)

By now, you should realize that the RLC circuit can be used for any of our four filter types, depending on where the output is taken.


14

ExamplesDesign an RL lowpass filter that uses a 40 mH coil and has a cutoff frequency of 5

kHz.

Design an RC highpass filter that uses a 20 µF capacitor and has a cutoff frequency of 3 kHz.


15

Example: Connecting Passive FiltersWhat is the overall result of attaching the lowpass filter output to the highpass filter

input of the previous slide?


16

ExampleDetermine the center frequency and BW of these bandpass filters.


17

Passive Filter Limitations

Passive filters have some drawbacks:• The output voltage driving a load cannot be larger than the

input (i.e., gains are no greater than unity).

• Loading effects mean that they must be reanalyzed when interconnected.

• Designs sometimes require inductors which are expensive and inherently lossy.

• Component values may need to be large to achieve desired specs.

Active filters, using op-amps or transistors, can overcome these limitations.


18

Active Filter Networks

Using op-amps in filters, we can:

• achieve gains greater than unity

• “buffer” the input signal to avoid loading effects (especially helpful in “cascading”)

• design all filters with only resistors and capacitors (no need for inductors)


19

Inverting Op-Amp ConfigurationMany filters use the inverting op-amp configuration.

e.g., 1st Order Lowpass filter e.g., 1st Order Highpass filter


20

Active Band-Pass FilterOne way to design a BPF is to cascade HPF and LPF as shown below.

NB: cf slide 3.15, cascading passive filters required system reanalysis of the entire circuit. Using active filters, the design can be more modular so the poles of each stage carry into the final T.F.


21

Active Band-Rejection FilterOne way to design a notch filter is to sum HPF and LPF outputs as shown.


22

ExampleFind the voltage gain TF and identify the filter type for the circuit shown.

R2

C2

C1

R1 +

_

+

vi(t)

_

vo(t)


23

Example (Cascaded Filters)

Consider the single-stage filter above for radio tuning. You want to listen to the 100 MHz station with little interference by the 98 MHz station. By cascading filters (connect end-to-end so the output of one is the input of another), greater steepness in response can be achieved resulting in increased selectivity.

2.5kΩ10000


24

Cascading Low-Q Filters

# of Stages BW (MHz) Q Gain @98MHz

Single 2.30 43.4 53.8%

Double 1.38 72.3 28.9%

Triple 1.15 86.9 15.5%

Quadruple 0.69 144.8 8.35%

108.76

108.78

108.8

108.82

108.84

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

108.76

108.78

108.8

108.82

108.84

-80

-70

-60

-50

-40

-30

-20

-10

0

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)


Based on slides by J. Yan Slide 3.25

“Take Home Message”• Passive filters can be simple to design and easy to implement.

We could design four basic filter types using resistors, inductors and capacitors.

• Active filters utilize op-amps (or OTA) to permit gains greater than unity, isolation for better cascading effects and designs which avoid the necessity of inductors.

• Depending on the application, different filter types have been designed to optimise for passband flatness (Butterworth filters), for immediate passband-to-stopband transition (Tschebyschefffilters), and for phase response linearity (Bessel filters).

• Active filters are typically used for frequencies below 100 kHz.

frequency response (i&n chap 12) - ubc ecegrecuc/253/spring14/253-p2-frequency... ·...

Documents