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TOPIC 4: FREQUENCY TOPIC 4: FREQUENCY SELECTIVE CIRCUITSSELECTIVE CIRCUITS

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INTRODUCTION•Transfer Function•Frequency Selective Circuits

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TRANSFER FUNCTION

• The s-domain ratio of the Laplace transform of the output (response) to the Laplace transform of the input (source) when all initial conditions are zero.

• The transfer function depends on what is defined as the output signal.

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DEFINITION

)()(

)(

)()(

jHjH

sX

sYsH 0ICs all

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POLES AND ZEROS

• The roots of the denominator polynomial are called the poles of H(s): the values of s at which H(s) becomes infinitely large.

• The roots of the numerator polynomial are called the zeros of H(s): the values of s at which H(s) becomes zero.

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FREQUENCY RESPONSE

• The transfer function is a useful tool to compute the frequency response of a circuit (i.e. the steady state response to a varying-frequency sinusoidal source).

• The magnitude and phase of the output signal depend only on the magnitude and phase of the transfer function, H(j).

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FREQUENCY RESPONSE

• Frequency response analysis is used to analyze the effect of varying source frequency on circuit voltages and currents.

• The circuit’s response depends on:– the types of elements

– the way the elements are connected

– the impedance of the elements

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FREQUENCY SELECTIVE CIRCUITS

• Frequency selective circuits is a circuits that pass to the output only those input signals that reside in a desired range of frequencies.

• Can be constructed with the careful choice of circuit elements, their values, and their connections.

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PASSIVE FILTERS• Passband & Stopband• Cutoff Frequency• Bode plot

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PASSIVE FILTERS

• Frequency selective circuits are also called filters.

• Filters attenuate, that is weaken or lessen the effect of any input signals with frequencies outside a particular frequency.

• Called passive filters because their filtering capabilities depend only on the passive elements (i.e. R,L,C).

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PASSBAND & STOPBAND

• The signal passed from the input to the output fall within a band of frequencies called Passband.

• Frequencies not in a circuit’s passband are in its Stopband.

• Filters are categorized by the location of the passband.

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FREQUENCY RESPONSE PLOT

• One way of identifying the type of filter circuit is to examine a frequency response plot.

• Two parts: one is a graph of H(j) vs frequency. Called magnitude plot.

• The other part is a graph of (j) vs frequency. Called phase plot.

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TYPE OF FILTERS

•Low Pass Filter

•High Pass Filter

•Band Pass Filter

Band Reject Filter

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lowpasslowpass highpasshighpass

bandpassbandpass bandrejectbandreject

FILTER’S FREQUENCY RESPONSE

c c

c1 c2c2c1

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MAGNITUDE

TYPE H(0) H(∞) H(ωC)@H(ωo)

LOWPASS 1 0 1/√2

HIGHPASS 0 1 1/√2

BANDPASS 0 0 1

BANDREJECT 1 1 0

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CUT OFF FREQUENCY

• LPF and HPF have one passband and one stopband, which are defined by the cut off frequency that separates them.

• BPF passes a input signal to the output when the input frequency is within the band defined by the two cut off frequencies.

• BRF passes a input signal to the output when the input frequency is outside the band defined by the two cut off frequencies.

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CUT OFF FREQUENCY

• The cutoff frequency (fc) is the frequency either above which or below which the power output of a circuit, such as a line, amplifier, or filter, is reduced to 1/2 of the passband power; the half-power point.

• This is equivalent to a voltage (or amplitude) reduction to 70.7% of the passband, because voltage, V2 is proportional to power, P.

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CUT OFF FREQUENCY

• This happens to be close to −3 decibels, and the cutoff frequency is frequently referred to as the −3 dB point.

• Also called the knee frequency, due to a frequency response curve's physical appearance.

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CUT OFF FREQUENCY

6

max

max

max

2

12

12

1

)()(

o

iccO

V

H

H

VjHjV

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BODE PLOT• The most common way to describe the

frequency response is by so called Bode plot.

• Bode lot is a log-log plot for amplitude vs frequency and a linear-log plot for phase vs frequency.

• Many circuits (e.g. amplifiers, filters, resonators, etc.) uses Bode plot to specify their performance and characteristics.

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FILTER’S RESPONSE BODE PLOT

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LOW PASS FILTER

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LOW PASS FILTER (LPF)

• The filter preserves low frequencies while attenuating the frequencies above the cut off frequencies.

• There are two basic kinds of circuits that behave as LPFs:

a) Series RL

b) Series RC.

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(a) LPF RL CIRCUIT

sL

)(sVi

)(sVo R OUTPUT

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Transfer Function

sLR

R

V

VsH

i

o

)(

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The voltage transfer function

• To study the frequency response, substitute s=j:

LRsL

RsH

)(

)()(

LRjL

R

V

VjH

i

o

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Magnitude and Phase

RLj

LRL

RjH

1

22

tan)(

)()(

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When =0 and =

iV R

iV R

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Qualitative Analysis

• At low frequencies (L<< R): – jL is very small compared to R, and

inductor functions as a short circuit.

iVV 0

iVV 0

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Qualitative Analysis

• At high frequencies (L>> R): – jL is very large compared to R, and inductor

functions as a open circuit.

00 V

900 iVV

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LPF Frequency Response

H(j)

1.0

0

-90

c

32

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Cutoff Frequency

• At the cutoff frequency, voltage magnitude is equal to (1/2)Hmax :

2

1

)()(

22

LRL

RH C

L

RC

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Ex.

• The instrument must operate in the presence of sinusoidal noise consisting of signals from the surrounding electrical environment, whose fundamental frequency is 50 Hz- the frequency at which electric power is supplied.

• Electrocardiograph is an instrument that is used to measure the heart’s rhythmic beat. This instrument must be capable of detecting periodic signals whose frequency is about 1 Hz (the normal heart rate is 72 beats per minute).

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Ex.

i. Choose values for R and L in the series RL circuit such that the resulting circuit could be used in an electrocardiograph to filter out any noise above 10 Hz and pass the electric signals from the heart at or near 1 Hz. (choose L=100 mH)

ii. Then compute the magnitude of Vo at 1 Hz, 10 Hz, and 50 Hz to see how well the filter performs.

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Known quantities

• Inductor, L = 100 mH

• Cut off frequency, fc = 10 Hz

– therefore, c = 2fc = 20 rad/s

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Find R

28.6

)10100)(20( 3

LR c

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Find the magnitude of Vo

• Using the transfer function, the output voltage can be computed:

i

io

io

V

V

LRL

RV

VjHV

22

22

400

20

)()(

)(

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f(Hz) Vi Vo

1 1.0 0.995

10 1.0 0.707

50 1.0 0.196

io VV22 400

20)(

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(b) LPF RC CIRCUIT

iV

ov

R

C OUTPUTiV

ov

R

C

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Transfer Function

RCjRC

CjR

CjV

VjH

i

o

1

1

1

1)(

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Magnitude and Phase of H(j)

RCj

RC

RCV

VjH

i

o

1

22

tan)(

1

1)(

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• Zero frequency (=0): – the impedance of the

capacitor is infinite, and the capacitor acts as an open circuit.

– Vo and Vi are the same.

• Infinite frequency (=): – the impedance of the

capacitor is infinite, and the capacitor acts as an open circuit.

– Vo is zero.

• Frequency increasing from zero: – the impedance of the

capacitor decreases relative to the impedance of the resistor

– the source voltage divides between the resistive impedance and the capacitive impedance.

– Vo is smaller than Vi.

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Cutoff Frequency

• The voltage magnitude is equal to (1/2) Hmax at the cutoff frequency:

2

1

)1(

1)(

22

RC

RCHC

C

RCC

1

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GENERAL LPF CIRCUITS

iV

ov

R

C

sL

)(sVi

)(sVo R

c

c

ssH

)(

OUTPUT

OUTPUT

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Ex.

• For the series RC circuit of LPF:

a) Find the transfer function between the source voltage and the output voltage

b) Choose values for R and C that will yield a LPF with cutoff frequency of 3 kHz.

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a) Find the transfer function

• The magnitude of H(j):

RCjRC

V

VjH

i

o

1

1)(

22 )1()(

1)(

RC

RCV

VjH

i

o

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b) Find R & C

• R and C cannot be computed independently, so let’s choose C=1F.

• Convert the specified cutoff frequency from 3 kHz to c=2(3x10-3) rad/s.

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Calculate R

05.53

)101)(103)(2(

1

1

63

CR

c

50

HIGH PASS FILTER

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HIGH PASS FILTER

• HPF offer easy passage of a high frequency signal and difficult passage to a low frequency signal.

• Two types of HPF:– RC circuit

– RL circuit

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a) CAPACITIVE HPF

iV

ovR

C

OUTPUT

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s-Domain Circuit

)(sVi

)(sVo

R

sC1

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When =0 and =

iV R

iV R

ov

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• Zero frequency (=0): – the capacitor acts as an

open circuit, so there is no current flowing in R.

– Vo is zero.

• Infinite frequency (=): – the capacitor acts as an

short circuit and thus there is no voltage across the capacitor.

– Vo is equal to Vi.

• Frequency increasing from zero: – the impedance of the

capacitor decreases relative to the impedance of the resistor

– the source voltage divides between the resistive impedance and the capacitive impedance.

– Vo begins to increase.

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HPF Frequency Response

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Transfer Function

RCj

jjH

jsRCs

ssH

1)(

;

1)(

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Magnitude & Phase

RCj

RC

sH

1

22

tan90)(

)1()(

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Ex: INDUCTIVE HPF

• Show that the series RL circuit below also acts as a HPF.

iVL

ov

R

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Ex.

a) Derive an expression for the circuit’s transfer function

b) Use the result from (a) to determine an equation for the cutoff frequency

c) Choose values for R and L that will yield HPF with fc = 15 kHz.

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s-Domain circuit

)(sVisL

)(sVo

R

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Transfer Function

LRj

jjH

jsL

Rs

ssH

)(

;

)(

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Magnitude & Phase

H()=1 andH(0)=0 HPF

22 )()(

LR

jH

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Cutoff Frequency

LR

LR

H

LR

jH

c

c

cc

2

1

)()(

)()(

22

22

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R and L

• Choose R=500 , and convert fc to c:

mH

RLc

31.5)1015)(2(

5003

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GENERAL HPF CIRCUITS

cs

ssH

)(

)(sVisL

)(sVo

R

)(sVi

)(sVo

R

sC1

OUTPUT

OUTPUT

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REMARKS

• The components and connections for LPF and HPF are identical but, the choice of output is different.

• The filtering characteristics of a circuit depend on the definition of the output as well as circuit components, values, and connections.

• The cutoff frequency is similar whether the circuit is configured as LPF or HPF.

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BANDPASS FILTER

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Bandpass Filter

• BPF is essential for applications where a particular band or frequencies need to be filtered from a wider range of mixed signals.

• There are 3 important parameters that characterize a BPF, only two of them can be specified independently:– Center frequency (and two cutoff frequencies)

– Bandwidth

– Quality factor

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Cutoff freq. & Center freq.

• Ideal bandpass filters have two cutoff frequencies, c1 and c2, which identify the passband.

c1 and c2 are the frequencies for which the magnitude of H(j) equal (1/2).

• The center frequency, o is defined as the frequency for which a circuit’s transfer function is purely real.

• Also called as the resonant frequency.

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Bandwidth, and Quality Factor, Q

• The bandwidth, tells the width of the passband.

• The quality factor, Q is the ratio of the center frequency to the bandwidth.

• The quality factor describes the shape of the magnitude plot, independent of frequency.

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a) BPF: Series RLC

iv

ov R

CL

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At =0 and =

iv

ov R

L C

iv

ov R

CL

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• Zero frequency (=0): – the capacitor acts as

an open circuit and the inductor behaves like a short circuit, so there is no current flowing in R.

– Vo is zero.

At =0 and =

• Infinite frequency (=):– the capacitor acts as

an short circuit and the inductor behaves like an open circuit, so again there is no current flowing in R.

– Vo is zero

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Between =0 and =

• Both capacitor and inductor have finite impedances.

• Voltage supplied by the source will drop across both L and C, but some voltage will reach R.

• Note that the impedance of C is negative, whereas the impedance of L is positive.

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• At some frequency, the impedance of C and the impedance of L have equal magnitudes and opposite sign cancel out!

• Causing Vo to equal Vi

• This is happen at a special frequency, called the center frequency, o.

Quiz: In which case in BPF circuit causing Vo=Vi

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BPF Frequency Response

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Center Frequency

)(max ojHH

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s-Domain Circuit

)(sVi

)(sVo R

sC1sL

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Transfer Function

2

2

( )( )

1( ) ( )

;

( )( )

1( ) ( )

R sLH sRs sL LC

s j

Rj LH jR jL LC

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Magnitude & Phase

2

2 2 2

( )( )

1( ) ( )

( )( )

1[( ) ] [( ) ]

RLH jR jLC LRj LH j

RLC L

2

1

)1(

)(tan90)(

LC

LR

j

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Center Frequency

• For circuit’s transfer function is purely real:

LC

CjLj

o

oo

1

01

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Cutoff Frequencies

LCL

R

L

R

LCL

R

L

R

c

c

1

22

1

22

2

2

2

1

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Relationship Between Center Frequency and Cutoff

Frequencies

21 cco

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Bandwidth,

L

Rc

2 c1

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Quality Factor, Q

oQ

CR

L

CRR

LQ

o

o2

1

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Cutoff Frequencies in terms of

22

2

22

1

22

22

oc

oc

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b) BPF: Parallel RLC

iv

ov

R

C L

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GENERAL BPF CIRCUITS• BPF series RLC:

• BPF parallel RLC:

RCLC

LCRCssRCs

sH

o1,1

)1(

)()(

2

LR

LC

LCsLRs

sLR

sH

o

,1

)1()(

)()(

2

Quiz:

Derive the TF for BPF_RLC circuit?

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Remarks

• The general circuit transfer functions for both series and parallel BPF:

22)(

oss

ssH

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BANDREJECT FILTER

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Bandreject Filter

• A bandreject attenuates voltages at frequencies within the stopband, which is between c1 and c2. It passes frequencies outside the stopband.

• BRF are characterized by the same parameters as BPF:– Center freq. (and two cutoff frequencies)

– Bandwidth

– Quality Factor

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BRF: Series RLC

iv

ov

R

C

L

OUTPUT

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At =0 and =

iv

ov

R

C

L

iv

ov

R

C

L

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• Zero frequency (=0): – the capacitor acts as

an open circuit and the inductor behaves like a short circuit.

– the output voltage is defined over an effective open circuit.

– Magnitude of Vo and Vi are similar

At =0 and =

• Infinite frequency (=):– the capacitor acts as

an short circuit and the inductor behaves like an open circuit,.

– the output voltage is defined over an effective open circuit.

– Magnitude of Vo and Vi are similar

96

Between =0 and = • Both capacitor and inductor have finite

impedances of opposite signs.• As the frequency is increased from zero, the

impedance of the inductor increases and that for the capacitor decreases.

• At some frequency between the two passbands, the impedance of C and L are equal but opposite sign.

• The series combination of L and C is that short circuit so the magnitude of Vo must be zero. This is happen at a special frequency, called the center frequency, o.

97

BRF Frequency Response

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Center Frequency

• The center freq. is still defined as the frequency for which the sum of the impedances of L and C is zero.

• Only, the magnitude at the center freq. is minimum.

)(min ojHH

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s-Domain Circuit

)(sVi

)(sVo

R

sC1

sL

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Transfer Function

jLR

LC

LCjH

js

LCsLR

s

LCs

sCsLRsCsL

sH

2

2

2

2

1

1)(

;

1

1

1

1)(

101

Magnitude & Phase

2

1

)1(tan)(

LC

LR

j

2

22

2

1

1)(

LR

LC

LCjH

102

Center Frequency

• For circuit’s transfer function is purely real:

LC

CjLj

o

oo

1

01

103

Cutoff Frequencies

LC

1

L2

R

L2

R

LC

1

L2

R

L2

R

2

2c

2

1c

2nd-term of the transfer function

3rd -term of the transfer function

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Relationship Between Center Frequency and Cutoff

Frequencies

21 cco

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Bandwidth,

L

Rc

2 c1

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Quality Factor, Q

oQ

CR

L

CRR

LQ

o

o2

1

107

Cutoff Frequencies in terms of

22

2

22

1

22

22

oc

oc

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Remarks

RC

LC

LCRCs

s

LCssH

o

1

1

1

1)(

2

2

)(sVi

)(sVo

R

sC1

sL

)(sVi

)(sVoR

sC1

sL

LR

LC

LCsLR

s

LCssH

o

1

1

1)(

2

2

109

GENERAL BRF CIRCUIT

• The general circuit transfer functions for both series and parallel BRF:

22

22

)(o

o

ss

ssH