ELE205 Electronics 2Filters-2Linga Reddy CenkeramaddiDepartment of ICTUiA, GrimstadFebruary 2016

FILTERS Contd.

2

FILTERS

Varying the frequency of a sinusoidal source alters the impedance of inductors and capacitors. The careful choice of circuit elements and their connections enables us to construct circuits that pass a desired range of frequencies. Such circuits are called frequency selective circuits or filters.

The frequency range that frequencies allowed to pass from the input to the output of the circuit is called the passband.

Frequencies not in a circuits passband are in its stopband.

3

IDEAL FILTERS

Ideal low-pass filter (LPF)

|H(jw)|

w

passband Stopband1

wcIdeal high-pass filter

(HPF)

passbandStopband

wc w

|H(jw)|

1

Ideal band-pass filter (BPF)

Ideal band-reject filter (BRF)

passband

Stopband

Stopband

wc1 wc2 w

1

|H(jw)|

passband

passbandStopband

wc1 wc2 w

1

|H(jw)|

4

The input is a sinusoidal voltage source withvarying frequency. The impedance of aninductor is jL.

At low frequencies, the inductors impedance isvery small compared to that of resistorimpedance, and the inductor effectively functionsas a short circuit. vo=vi.

As the frequency increases, the impedance of theinductor increases relative to that of the resistor.The magnitude of the output voltage decreases.As , inductor approaches to open circuit.Output voltage approaches zero.

Low-Pass FILTERS

5

QUANTITATIVE ANALYSIS

LR

LR

LR

LR

jjH

ssH )()(

RL

LR

LR

jH 1

22tan)( ,)(

6

CUTOFF FREQUENCY

Cutoff frequency is the frequency at which the magnitude of the transfer function is Hmax/2.

For the series RL circuit, Hmax=1. Then,

LRjH c

LR

c

LR

c 222

1)(

At the cutoff frequency (wc) = -450.

For the example R=1, L=1H, c=1 rad/s

7

EXAMPLE

Design a series RL low-pass filter to filter out any noise above 10 Hz.

R and L cannot be specified independently to generate a value for c. Therefore, let us choose L=100 mH. Then,

28.6)10100)(10)(2( 3 LR c

iiLRLR

o VVV 2222 40020

)(

F(Hz) |Vi| |Vo|

1 1.0 0.995

10 1.0 0.707

60 1.0 0.164

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A SERIES RC CIRCUIT

1) Zero frequency =0: The impedance of thecapacitor is infinite, and the capacitor acts as anopen circuit. vo=vi.

2) Increasing frequency decreases the impedance of the capacitorrelative to the impedance of the resistor, and the source voltage dividesbetween the resistor and capacitor. The output voltage is thus smallerthan the source voltage.

3) Infinite frequency =: The impedance of the capacitor is zero, andthe capacitor acts as a short circuit. vo=0.

Based on the above analysis, the series RC circuit functions as a low-pass filter.

9

2121

1

1

)()(RC

RC

RC

RC jHs

sH

QUANTITATIVE ANALYSIS

RC

jH

jHH

c

RCc

RCc

1

)1(2

1)(

1)0(

212

1

max

10

Relating the Frequency Domain to the Time Domain

Remember the natural response of the first-order RL and RC circuits. It is an exponential with a time constant of

Compare the time constants to the cutoff frequencies for these circuits and notice that

RCRL or

c1

HIGH-PASS FILTERS

At =0, the capacitor behaves like an opencircuit. No current flows through resistor.Therefore vo=0.

As the frequency increases, the impedance of thecapacitor decreases relative to the impedance ofthe resistor and the voltage across the resistorincreases.

When the frequency of the source is infinite, thecapacitor behaves as a short circuit and vo=vi.

12

QUANTITATIVE ANALYSIS

RCj

jH

jjjH

sssH

RC

RCRC

10

212

11

tan90)(

)(

)()(

13

SERIES RL CIRCUIT

LR

jH

jHH

jH

jjjH

sssH

c

LR

c

cc

LR

LR

LR

22

max

22

)(2

1

1)(

)(

)()(

14

LOADED SERIES RL CIRCUIT

LKR K

sKssH

ss

RsH

cRRR

c

LR

RRRRR

R

sLRsLR

sLRsLR

L

L

L

L

L

L

L

L

L

L

/ , ,)(

)(

The effect of the load resistor isto reduce the passbandmagnitude by the factor K andto lower the cutoff frequency bythe same factor.

15

BANDPASS FILTERS

At =0, the capacitor behaves like an opencircuit, and the inductor behaves like a shortcircuit. The current through the circuit is zero,therefore vo=0.

At =, the capacitor behaves like a shortcircuit, and the inductor behaves like anopen circuit. The current through the circuitis zero, therefore vo=0.

Between these two frequencies, both thecapacitor and the inductor have finiteimpedances. Current will flow through thecircuit and some voltage reaches to the load.

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At some frequency, the impedance of the capacitorand the impedance of the inductor have equalmagnitudes and opposite signs; the two impedancescancel out, causing the output voltage to equal thesource voltage. This special frequency is called thecenter frequency o. On either side of wo, the outputvoltage is less than the source voltage. Note that atthe center frequency, the series combination of theinductor and capacitor appears as a short circuit andcircuit behaves as a purely resistive one.

CENTER FREQUENCY

17

QUANTITATIVE ANALYSIS

LCLRLR

ssssH

12)(

21

10

2221

tan90)(

)(

LC

LR

LR

LC

LR

j

jH

18

CENTER FREQUENCY AND CUTOFF FREQUENCIES

At the center frequency, the transfer function will be real, that means:

LCCjLj o

oo

10

1

At the cutoff frequencies, the magnitude of H(j) is Hmax/2.

1

)(

21211

1

2221max

LR

LCLCLC

LR

LC

LR

oLC

LR

oo

o

jHH

19

0/1

11

1

1

2

1

2

212221

CRLRCR

Lcc

cc

RCLR

cLR

ccLC

LR

c

c

LCLRLRc

LCLR

LR

c

12

222

12

221

It is easy to show that the center frequencyis the geometric mean of the two cutofffrequencies

21 cco

20

BANDWIDTH AND THE QUALITY FACTOR

The bandwidth of a bandpass filter is defined as the difference between the two cutoff frequencies.

LR

LCLR

LR

LCLR

LR

cc

12

2212

22

12

The quality factor is defined as the ratio of the center frequency to bandwidth 2

1

CRL

LRLCoQ

21

CUTOFF FREQUENCIES IN TERMS OF CENTER FREQUENCY, BANDWIDTH

AND QUALITY FACTOR

22

2

22

1

22

22

oc

oc

2

1

2

1

2

11

2

1

2

11

2

1

QQ

QQ

oc

oc

22

EXAMPLE

Design a series RLC bandpass filter with cutoff frequencies 1 and 10kHz.

Cutoff frequencies giveus two equations but wehave 3 parameters tochoose. Thus, we needto select a value foreither R, L, and C anduse the equations to findother values. Here, wechoose C=1F.

24.143)3514.0)(10(

)10(533.2

3514.0100010000

28.3162

mH 533.2)10()28.3162(2

11

Hz28.3162)10000)(1000(

26

3

2

12

622

21

CQLR

fffQ

CL

fff

cc

o

o

cco

23

LCRCs

RCs

sC

CL

sC

sCeq

ssH

sLsLsLsZ

12

11

1

)(

)()(

This transfer function and the transfer function for the series RLCbandpass filter are equal when R/L=1/RC. Therefore, all parameters ofthis circuit can be obtained by replacing R/L by 1/RC. Thus,

LCo1

RC1

LCRQ

2

LCRCRCcLCRCRCc

12

21

21

2

12

21

21

1

EXAMPLE

24

BANDPASS FILTER WITH PRACTICAL VOLTAGE SOURCE

+R

Vi(s)+Vo(s)

sL 1/sCRi

LCLRR LR

ssssHi 12

)(

i

oo

LRR

LC

LR

RRRjHH

LC

jHi

)(1

)(

max

2221

LCLRR

LRR

LCLRR

LRR

iic

iic

1

22

1

22

2

2

2

1

i

i

RRC

LQ

LRR

25

The addition of a nonzerosource resistance to aseries RLC bandpass filterleaves the centerfrequency unchanged butwidens the passband andreduces the passbandmagnitude.

26

Relating the Frequency Domain to the Time Domain

The natural response of a series RLC circuit is related to the neper frequency and the resonant frequency L

R2 LCo 1

o is also used in frequency domain as the center frequency. The bandwidth and the neper frequency are related by 2

The natural response of a series RLC circuit may be under-damped, overdamped,or critically damped. The transition from overdamped to criticallydamped occurswhen . The transition from an overdamped to an underdamped responseoccurs when Q=1/2. A circuit whose frequency response contains a sharp peak atthe center frequency indicates a high Q and a narrow bandwidth, will have anunderdamped natural response.

22 o

27

R=200, L=2H, C=0.5F

R=200, L=1H, C=1F

R=200, L=0.2H, C=5F

28

BANDREJECT FILTERS

++

L

R

vivo

C

++

L

R

vivo

C

++

L

R

vivo

C

At =0, the capacitor behaves as an open circuit and inductor behaves as ashort circuit. At =, these roles switch. The output is equal to the input.This RLC circuit then has two passbands, one below a lower cutofffrequency, and the other is above an upper cutoff frequency.

Between these two passbands, both the inductor and the capacitor havefinite impedances of opposite signs. Current flows through the circuit.Some voltage drops across the resistor. Thus, the output voltage is smallerthan the source voltage.

29

QUANTITATIVE ANALYSIS

LCLR

LC

sC

sC

sss

sLRsLsH

12

12

1

1

)(

21

1

2221

21

tan)(

)(

LC

LR

LR

LC

LC

j

jH

CRLQ

LR

LCo

2

1

LCLRLRcLCL

RL

Rc

12

222

12

221

30

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