• Author / Uploaded
• Rafee Mohammad

• Categories
• Filtro electrónico
• Filtro de paso bajo
• Amplificador
• Filtro (Procesamiento de señal)
• Ancho de banda (Procesamiento de señal)

Citation preview

Back ground This project design paper gives the brief design steps of low thermal sensitivity notch filter of center frequency 50Hz and 4Hz(-3db)bandwidth that lock the center frequency with 50Hz + 0.1Hz. Introduction Filters are electric circuits, that remove unwanted frequency components from the signal, to enhance wanted ones, or both. Depending on the type of signal processed, filters can be classified as Analog and Digital filters. Analog filters can further be divided into 1.

Passive filters:- are built of passive components such as resistors, inductors and

capacitors. In these filters the maximum gain that can be attained is unity and they are applicable in all frequency ranges. 2. Active filters:- are built of operational amplifiers that employ both positive and negative feedback. These filters are more efficient than passive filters because they are able to filter out or attenuate undesired frequencies much better and can also increase the gain. Active filters are applicable in the lower frequency range (1 Hz to 1 MHz). At frequencies greater than 1MHz active components gain bandwidth product droops quickly. There are also other categories of filters .Low pass filter(high cut filter) that passes lowfrequency signals but attenuates signals with frequencies higher than the cutoff frequency. While a high pass filter(low cut filter) is an LTI filter that passes high frequencies well but attenuates frequencies lower than the cutoff frequency. Band pass filter designed to transmit a particular band of frequencies (bandwidth frequencies)while excluding those of higher or lower frequencies. On the other hand all pass or phase shift filter is a signal processing filter that passes all frequencies equally, but changes the phase relationship between various frequencies, A bandstop filter (band-rejection filter) in signal processing is a filter that passes frequencies unaltered,

1

but attenuates those in a specific range to very low levels. It is the opposite of a

band-pass

filter. The above figure is an ideal band reject filter with corner frequencies wL and wH\. The main aim of this project is the design and analysis of 50 Hz active notch filter. A notch filter is a band-stop filter that passes all frequencies except those in a stop band centered on a center frequency. Notch filter is sometimes known as band limit filter, T-notch filter ,band elimination filter or band-reject filter. The amplitude response of a notch filter is flat at all frequencies except for the stop band on either side of the center frequency. The standard reference points for the roll-offs on each side of the stop band are the points where the amplitude has decreased by 3dB, or 70.7% of its original amplitude. The -3dB points and -20dB points are determined by the size of the stop band in relation to the center frequency, in other words the Q of the filter. The Q knowledgebase item will have additional information, but it is hard to talk about the roll-off points of a notch filter without defining the Q, which is the center frequency divided by the bandwidth. Many people think that the higher the Q, the deeper the notch, This is not true, The depth of the notch depends on the matching of components. The Q affects only the location of the -3dB points, the stopband width. A notch filter has transfer function of the form H(s) =

k(s2 +b2 ) w s2 + o s+wo 2 Q

Depending on the relationship of the pole frequencies wp and the zero frequencies wo notch filters are divided into standard notch, low pass notch and high pass notch. If the zero frequency is equal to the pole frequency standard notch exists. A low pass notch occurs when the zero frequency is greater than the pole frequency. In practical sense this means the filter response below wp will be greater than the response at w0. These results 2

in an elliptic low pass filter.A high pass notch filter occurs when the zero frequency is less than the pole frequency. This means that the response below wp will be less than the response above w0. In this project Twine T networks are used for the realization of the designed filter. Twine T network The original twine-T filter, shown in the figure below is passive RC network with a quality factor of Q=0.25.To increase Q , the passive filter is implemented into the feedback loop of an amplifier, thus turning into an active band-reject filter.

Fig. active twine T network The network consists of two RC T networks. The first T network consists of C1,C2 and R3 and it is a high pass filter. That is, the lower frequency ranges are prevented from reaching the input of the op amp due to the high reactance of C1 and C2. On the other hand, the higher frequencies can pass and reach the op amp easily because the reactance of C1 and C2 is low at higher frequencies. The second T network is made up of R1, R2 and C3 and forms a low pass filter. Here the lower frequencies find C3's high reactance to be essentially open, so they pass to the op amp input. On the other hand, the higher frequencies are essentially shorted to the ground by the low reactance of C3. Therefore Both the low and high frequencies get through the (+) input of the op amp and pass through the output. If the cutoff frequencies of the two T networks do not overlap there is frequency (fo) that results in a net voltage of 0 at the (+) terminal of the op amp. The figure given bellow is an active notch filter. The detail analysis of this network is given in appendix І. 3

Application of notch filter .-Used in live sound reproduction (Public Address systems also known as PA systems ) -Instrument amplifier (especially amplifiers of preamplifiers for acoustic instruments such as acoustic guitar, mandolin, bass instrument amplifier ,etc.) -To reduce or prevent feedback, while having little noticeable effect on the rest of the frequency spectrum. -The FM Notch filter is designed to trap out single FM broadcast station energy from cable, MATV, television, or minor FM translator receiver applications. -Cable TV “notch filters” that blank out frequencies corresponding to two consecutive channel on the FCC channel spectrum. -TV notch filters:-are used to remove unwanted carrier signals from TV channels, such as the annoying “beep-beep-beep”. But, they can also be used to remove entire channels. -Mainly notch filters are used for power supply interference filtering which are 50/60Hz filters.

4

Design procedures 1.

Design brief:-This part includes the analysis and identification of the necessary

parameters required for the design of the network. This includes the pass band frequencies, stop band frequencies and pass band and stop band attenuations. 2.

Transfer function approximation:-This part includes the approximation of the

network function which can satisfy the given specifications. The approximation was done first by calculating the low pass equivalent of the notch filter then applying frequency transformation techniques the low pass prototype was transformed to a notch filter. Different approximation functions such as butterworth, elliptic, inverse chebyshev ,Bessel Thomson or chebyshev І approximations were the candidates of the transfer function approximation in our project , chebyshev І was choose for its sharp stop band characteristics. 3. Synthesis(Realization):-The realization part was done using the coefficient matching technique. Twine-T network with known transfer function was selected, and matched with the approximated transfer function to give the required component values. 4.

Analysis:- the maximum power drop across each component was messured using

the setup in multism 10.and the sensitivity analysis was also performed using multism 10. Methods of component Selection Resistors and capacitors are probably the most common and well known of all electrical components. Applying a resistor and capacitors to a circuit normally requires  calculating the nominal values needed in the circuit application (resistance, power rating, etc.) then  developing acceptable tolerances for the resistor that ensure it will function properly in all extremes of the application. The first design consideration is relatively simple, typically based on straight forward theory and linear calculations. The second design task can be more difficult 5

because resistors and capacitors have several characteristics that alter their value when used in a practical circuit. . Resistors and capacitors are often designated as "precision" or "power". Precision are designed for applications where tight tolerance and stability are primary considerations. They generally have restricted operating temperature limits and power dissipation ratings. Power are designed for the applications where tight tolerance and be quite stable, but their design emphasis is to optimize power dissipation. generally they have extended operating temperature limits

6

Network Design Given Fc =50Hz=314.16 rad/sec Bw=4Hz=25.13 rad/sec Wp1=48Hz=301.44 rad/sec Wp2=52Hz326.56 rad/sec αp=3dB assumptions αs=25 dB ws1=49Hz=307.72 rad/sec ws2=51.02Hz=320.28 rad/sec Approximation First find the equivalent low pass filter prototype Let the pass band frequency of the lowpass prototype be wp=1 rad/sec −1 wp1 wp2 −(ws2 )2 −1 ) ws2

The stop band frequency is given by ws= (bw −1 48∗52−(51)2 −1 ) 51

 Ws=( 4

 Ws=2 rad/sec Calculation for order The Chebycheve I approximation is used to approximate the transfer function that fulfill the specifications. 7

αp ⁄10 -1=0.995

ε2=10

The order for Chebycheve I approximation is given by 10αs⁄10 −1 ε2 ws cosh−1 ( ) wp

cosh−1 √

n=

=

1025⁄10 −1 0.9952 2 cosh−1 ( ) 1

cosh−1 √

=2.771

 n=3 A

2

(T(w))2 =1+ε2max Cn2 (w) Where Cn2(w)=cos(n cos-1(w)) For n=3 Cn becomes Cn(w)=cos(3 cos-1(w))

expanding this expression gives

Cn(w)=4w3-3w  Cn2(w)=16w6-24w4+9w2  1+ε2Cn2(w)=1+0.995(16w6-24w4+9w2) =15.92w6-23.88w4+8.955w2+1 2

A

max  (T(w))2=15.92w6 −23.88w 4 +8.955w2 +1

A

substituting w=-js gives

2

max  (T(s))2=15.92s6−23.88s 4 +8.955s2 +1

Solving the denominator and taking the left hand side poles k

T(s)=(s+0.2986i)(s+0.1493+0.9038i)(s+0.1493−0.9038i) k

 T(s)=s3 +0.597s2 +0.02835s+0.25059 If the Dc gain=α T(s)|s=0=α k

 α=0.25059  k=0.25059α 0.25059α

 T(s)=s3 +0.597s2 +0.02835s+0.25059 Select α=31.645 7.92997958

 T(s)=s3 +0.597s2 +0.02835s+0.25059 8

Applying frequency transformation technique to transform T(s) to notch filter substitute Bw∗S

S=s2 +w

0

where w0=central frequency, Bw=band width

2

Applying frequency scaling by wo gives Bw s w0 2 s +1

S=

0.08s

= s2 +1

7.92997958(s2 +1)3

 Tn(s)=0.2506s6 +0.0743s5 +0.7556s4+0.1490s3+0.7556s2 +0.0743s+0.2506  Tn(s)=

7.92997958(s2 +1)(s2 +1)(s2 +1) (s2 +0.2678s+1)(s2 +0.0148s+1.09)(s2 +0.0136s+0.9174)

This can be written as a product of three functions 1.8661(s2 +1)

T1(s)=s2 +0.2678s+1 => Active notch filter 2.155(s2 +1)

T2(s)= s2 +0.148s+1.09 => High pass notch 1.97192268(s2 +1)

T3(s)= s2 +0.136s+0.9174 => low pass notch Figure: magnitude and phase plot of T(s) Bode Diagram 0

Magnitude (dB)

-50 -100 -150 -200 -250 -300 720

Phase (deg)

540 360 180 0 -1

10

0

10

Frequency (rad/sec)

1

10

9

Component Selection 1.Active notch R/2 C

C

R

R R1

2C

R2

1 ) (CR)2 1 2 s2 +2k( k −1)s+ (CR)2

μ(s2 +

H(s)=

k(s2 +1)

T(s)=s2 +0.2678s+1 To find the value of C and R First we equate the value of the constant in T(s) and H(s) 1

⟹1=(CR)2 1

⟹1=(CR) 1

If we assume R=2kΩ then c=2kΩ=0.5mF When we denormalizes it by ω0 =50*pi ,then C=1.5915µF =1.6 µF 1

=> R=(1.6)(2∗π∗50) =1.98kΩ To find the value of k we equate the coefficient of s in T(s) and H(s) i.e

2(2-k)= 0.2678 10

4-2k=0.2678 4−0.2678

k=

2

k=1.8661 Using the value of k we can determine R1 and R2 since k is the gain R

k=1+R1 2

R

1.8661-1=R1 2

R

0.8661=R1 2

R1=0.8661R2 If we assume the value of R2=2 kΩ R1=1.74kΩ 2 Low Pass Notch k/k+1 C

C/k

1

k

c

k+1 k

c R2

k(s2 +1)

T(s)=s2 +.0136s+0.9174 H(s) =

μ 1 (s2 + 2 ) (1+αk+α) C 2−μ ( +α)(1+k)s 1 k + + s2 c(1+αk+α) c2 (1+αk+α)

11

To find The value of C we equate the zero frequency of T(s) and H(s) 1

1=C2 If we assume the value of C= 1F And demoralizing it by ω0 =50*pi , C= 3.18mF The closest standard value is C=3.3mF To find the value of  we let k=1.96 kΩ and Nornalizeing C with the new standard value C=1.03672558F Then equating the value of the constant in T(s) and H(s) 1

i.e

C2 (1+αk+α) 1

C2 ∗.9174

=(1 + αk + α)

1 2∗.9174

 =C

=0.9174

−1

k+1

=

1.417617157∗10−2

=

1.417617157∗10−2

k+1

1.96 kΩ +1

=7.229*10-6  C =7.855µF When we denormalizes C= 24nF C⁄ =1.68µF , the closes standard value is 1.6 µF k k k+1

=.9994 , the closes standard value is 1Ω

k+1 k

C = 3.3 mF

12

We can get the value of µ by equate the coefficient of s in T(s) and H(s) i.e .0136=

µ=2-(

(

2−μ k

+α)(1+k)

c(1+αk+α)

.0136C(1+(k+1)) (1+k)

− )

µ=1.99987679 R

µ=1+R1 2

R

.99987679=R1 2

If we assume R 2 =3.6kΩ then R1 =3.6kΩ

High Pass notch k/k+1 

/k

1

k

∝

𝑘+1

R1

𝑘

R2

H(s)= 1

k(s2 + 12 ) 1

k

α 1+k 1+ki 2

1

µ

( + +1)+ ( + − )s +s2 α α k R k α2 R R

k(s2 +1)

T(s)=s2 +.04148s+1.09 To find The value of  we equate the zero frequency of T(s) and H(s) 1 α2

=1  α = 1F

13

When we denormalize α = 3.183mF ≈ 3.3mF α=1.03672558F

Normalizing by wo

To find the value of k we equate the denominator of the constant of T(s) and H(s) 1



1

1

 (R+ 1 R

+

1+k R

k

k

(R+

α2

k

R

+ 1)=1.09

+ 1) = 1.09 ∗ α2

R

=0.17153719

R

=0. 17153719

K=0.17153719R -1 Let K= 1.96KΩ Then R =11.4.32KΩ The closest standard value R=11.5KΩ α(k+1)

= 3.3mF

k k

= 1Ω

k+1

To find µ we equate the coefficient of s in T(s) and H(s) i.e.

1+k α



(

2−μ k

1

+ R ) = .0148

0.0148α− 1 1+k

 k(

R

.01489α 1+k

2−μ

=

k 1

− R) =2-µ

.01489α

 2 − k(

1+k

1

− R) =µ

 2-(-0.155) =µ  µ=2.155 R

 µ =1+R1 2

14

 R1=1.155R2 assume R2 =20Ω then R1 =23.2Ω The value of R2 and R1 are selected in ohms to minimize the Dc offset. OP AMP selections Let the circuit be operator until fmax=10Mhz fvG =Av*f =6.45*10MHz =64.5MHz Slew Rate (min)=π* fmax* V0 (max) =3.14*10MHz*350mv =10.9v/µs When we look at the data sheet THS4011 satisfies the above conditions therefore we select it

Component tolerance calculation

15

The assumption made while calculating the component tolerance is, If we keep the center frequency of each section of the network to vary only by one third of 0.1Hz the overall variation of the central frequency would be only 0.1Hz. The following steps calculate the tolerance of each section with the above assumption. Low pass notch 1

Wo=√c2 (1+α(K+1)) Where C=3.3mF,K=1.96K,α=7.229*10-3 1

 W0=√(3.3∗10−3 )2 (1+7.299∗10−6 )(1.96∗103 =300.905rad/sec +1)) 1

 W0+∆wo=√(c−∆c)2 (1+α((K+∆k)+1) ∆wo=

0.2∗2∏

=0.2094, wo+∆wo=301.1144

3

Let Tc=2%, c-∆c=3.3*10-3(1-0.02)=3.234*10-3 1

 (301.114*3.234*10-3)2=(1+α(K(1+T

k )+1)

 1.0269029-1=α(k(1+T k ) +1)  0. 0269029=7.2999*10-3(1.96 *103)(1_+Tk)+1)  3.7214*10-3=1.96*103(1+TK)+1  1.898169=(1- TK)  0.898169=TK This is the maximum value of the tolerance that we can use for the resistor K. So Choose the tolerance for K, Tk=2%. High pass notch 1

1+k

W0=∝2 √

R

+ 1 where a=3.3*10-3F R=11.5*103 K=1.96*103 Wo=327.85rad/sec Wo=.20944rad/sec

 Wo=327.85 rad/sec 16

.1∗2π

 Wo=

3

=.2094

 Wo+Wo=301.1144 Let Tc=2% , c-C=3.3*10-3(1-.02)=3.234*10-3 1

(301.114*3.234*10-3)2=1+a (k(1+T

k )+1)

=> 1.0269029-1= (k(1+Tk)+1) => 0.269029=7.229*10-3(1.96*103(1+Tk)+1) => 3.7214*103=1.96*103(1+Tk)+1 => 1.898169=(1-Tk) => 0.898169=Tk This is the maximum value of the tolerance that we can use for the resistor K. So Choose the tolerance for K, Tk=2% Therefore Tc=2% Tk=2% High pass notch 1

1+k

W0=a2 √

R

Where a=3.3*10-3F

+1

R=11.5*103 K=1.96*103 Wo =327.85rad/sec ∆Wo=0.20944rad/sec  Wo=327.85rad.sec 1

1+(k+∆k)+1

Wo+∆Wo=a−∆a √

R+∆R

Let To=2%=0.02 Tk=2%=0.02  a -∆a=3.234 *10-3  k+∆k= 1.9992*103  Wo+∆wo=328.05944 1+1.9992∗103

 (328.05944*3.234 *10-3)=√

𝑅+∆𝑅

+1 17



9.961615∗1.993∗103 11.5∗103

=1-TR

 0.37987 = T𝑅 This is the maximum tolerance SO we can choose TR=1% Active notch 𝟏

Where R=2*103

Wo=𝑹𝑪

C=1.6*10-6 Wo+ Wo=312.5+0.20944 Let Tc=2%=0.02 1

=> 312.70944 =0.0032(1+𝑅

𝑅 )(1−𝑅𝑅 )

=> 1.000670208=(1 + 𝑅𝑅 )(1 − 𝑅𝑅 ) =>

1.000670208 0.98

=(1 + 𝑅𝑅 )

=>𝑇𝑅 =0.0219 This is the maximum tolerance that can be used for R therefore choose TR=2%

Besides the above tolerance calculation to minimize the variation of components with temperature more temperature insensitive components such as ceramic capacitors and resistors are used. - To build the active notch filter we cascade the above filters ( i.e the high pass, the low pass, and the active ) notch together with the input buffers .

18

R11 C9

5 3.3mF

XFG1

8

V5 12 V

C8 1Ω 1 1.6uF 4

1.96kΩ

7

THS4011CD 4

R16 11.5kΩ

C10 3.3mF

R14

6

7

THS4011CD 2

1.6uF

17 R7 15

6

4

V6 12 V

U3

R8

1Ω

7 3

1.96kΩ

THS4011CD 2

19

23.2Ω

16

V2 12 V

C7

C6

0

24nF

3.3mF

10

12

R15

3 2

20Ω

3

1Ω

C4

13 3.3mF

U5

3

2

0

C5

7

V3 12 V

R6

9

U6

R12 6 R13 1Ω

V1 12 V

6

4

3.6kΩ R9 11 R10 V4 3.6kΩ 12 V 0

14 V9 12 V

XBP1

28 IN

OUT

U2 7

0 3

THS4011CD 2

4

6

C2

24 1.6uF

27 R2

C1

1kΩ

XSC1 Ext Trig +

18

_

1.6uF

26

R3 23 2kΩ

29

+

7

R4 3.74kΩ

25

3

THS4011CD 22 2

C3 3.3uF

B

A

U1

2kΩ

V10 12 V

V7 12 V

R1

4

6

R5

21 V8 2kΩ 12 V

0

19

_

+

_

Simulation results The simulation was performed using software (Multism10) and the simulation set up was given in the previous figure From the simulation the following points can be noticed -The circuit can be operated until maximum frequency of 10.39MHz -3dB frequency. -The maximum attenuation occurs at 48.97Hz and this is equal to -50.237dB. The Attenuation at 50Hz is equal to -37.905dB and the attenuation at the stop band Frequency ws1=49Hz is -50.005dB and at ws2=51Hz is -18.82dB. -The maximum input voltage that results in undistorted output voltage is 50mV peak. -The gain of the circuit is 6.37. The simulation result Circuit simulated with the non standard components is given as follows -The voltage level at the central frequency W0=50Hz=-61.115db -The -3db frequencies are wo=51.918Hz and Wo=47.85Hz. -The maximum output voltage level is at 2.455MHz =27.09db -The phase response of the circuit is non linear specially around the central frequency and at high frequencies.

20

Figure The magnitude response of the non standard component circuit

Figure The. phase response of the non standard component circuit The frequency response of the circuit can be obtained from it’s bode plot and this is given in the following figure.

21

Figure: The magnitude response of the standard component circuit

Figure: The. phase response of the standard component circuit

22

Analysis The maximum power drop across each component is measured from the simulation set up at frequency of 1Khz and input voltage of 50mv(the maximum input voltage level) and is tabulated as follows.

High pass notch

low pass

Active

notch

notch

power

power

Component

Dissipation

Component power Dissipation

component

Dissipation

R12

1.247mvw

R7

5.772mW

R2

10.98µW

R13

650nvW

C5

45.375µW

C2

8.94nW

C4

96PW

R8

2.88µW

R3

10.9µW

C8

1.6pW

C6

19.6pw

C4

51.53nw

R11

1.683m

C4

0W

R1

6.27µW

R16

111.03n

R6

13.429µW

C3

61.4nw

R14

63.6µ

R9

1.623µW

R4

63µW

R15

137.3µ

R10

1.638µW

R5

3.27µW

c10

1.02µW

C7

0W

Sensitivity The sensitivity of with respect to each component was simulated in multism 10 and the resulting plot is given below.

Legend

23

Magnitude sensitivity for the low pass part

phase sensitivity for the low pass notch part Legend

Magnitude sensitivity for the high pass notch section

24

Phase sensitivity for the high pass notch part

25

Legend

Magnitude sensitivity of the active notch part

Phase sensitivity of the active notch part

From the above plots we can observe that the magnitude of the output voltage varies very little with the change in the capacitor values due to ambient conditions. But the phase changes significantly. This implies that the filter does not good phase response and Ait can not be used for phase sensitive applications.

26

Bill of materials

Quantity Description

RefDes

Type

Price

Manufacturer Suzhou Yangjie Electron

2 RESISTOR, 1Ω 1%

R11, R12

Ceramic

$0.50

Co..Ltd Suzhou Yangjie

1 RESISTOR, 1.96kΩ 1%

R13

Ceramic

$0.50

Electronics Co..Ltd. Suzhou Yangjie Electron

1 RESISTOR, 20Ω 0.5%

R14

Ceramic

$0.50

Co..Ltd Suzhou Yangjie Electron

1 RESISTOR, 23.2Ω 1%

R15

Ceramic

$0.50

CAP_ELECTROLIT, 3.3mF 3 2%

Suzhou Yangjie Electron C9, C5, C6

Fan capacitor

$0.50

CAP_ELECTROLIT, 3.3mF 1 2%

Co..Ltd

Co..Ltd Suzhou Yangjie Electron

C10

Fan capacitor

$0.50

Co..Ltd

V6, V3, V4, V1, V2, V7, V8, V9, V10, 10 DC_POWER, 12 V

V5 Suzhou Yangjie Electron

1 RESISTOR, 11.5kΩ 1%

R16

Ceramic

$0.50

Co..Ltd Suzhou Yangjie Electron

2 RESISTOR, 1Ω 0.5%

R6, R7

Ceramic

$0.50

Co..Ltd Suzhou Yangjie Electron

1 RESISTOR, 1.96kΩ 0.5%

R8

Ceramic

$0.50

Co..Ltd Suzhou Yangjie Electron

2 RESISTOR, 3.6kΩ 1% CAP_ELECTROLIT, 1.6uF 4 2%

R9, R10

Ceramic

$0.50

C4, C1, C2, C8

Suzhou Yangjie Electron Ceramic

$0.50

CAP_ELECTROLIT, 24nF 1 2%

Co..Ltd

Co..Ltd Suzhou Yangjie Electron

C7

Ceramic

$0.50

Co..Ltd 27

Suzhou Yangjie Electron $0.50

Co..Ltd Suzhou Yangjie Electron

1 RESISTOR, 1kΩ 0.5%

R1

Ceramic

$0.50

Co..Ltd Suzhou Yangjie Electron

2 RESISTOR, 2kΩ 0.1%

R2, R3

Ceramic

$0.50

Co..Ltd Suzhou Yangjie Electron

1 RESISTOR, 3.74kΩ 1%

R4

Ceramic

$0.50

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1 RESISTOR, 2kΩ 1%

R5

Ceramic

$0.50

CAP_ELECTROLIT, 3.3uF

Co..Ltd Suzhou Yangjie Electron

1 2%

C3

5 op amp THS4011

u1,u2,u3,u4,u5

Ceramic

$0.50

Co..Ltd

$1

Texas Instruments

28

Conclusion And Suggestion In conclusion we decide to use twin-t notch filters with standard components that wound reduce the sensitivity of the system while still maintaining the simplicity of the design process. We hoped that this design will be as close as possible to ideal filter. As we all know the goal of any filter design is to design it in such a way that it will be as close as possible to the ideal filter. We accomplished this by increasing the order of the filter. The disadvantage of increasing the order of the filter is that it will become more costly because more components are needed for higher order filter. High order filter could be realized by breaking it in smaller order sections that are easier to design and cascading this sections with the use of buffer. In our project the given specification was accomplished to satisfactory live. But it was not satisfied 100% some design factors limit the satisfaction of the the design brief. One of the factors that limit us was the use of standard components. Each components calculated from the design has to be approximated to standard components this resulted in slight shift of the central frequency from the required value. Another limitation is the biquad sections used to design the filter has gain limitation. the gain of each sections has to be set according to the calculated values from the design ,it was not possible to set this gains at will. Active Notch filter can be extremely helpful when used properly, however it is not right for every system . in correct placement of the notch can cause instability and a notch filter puts extra overhead on the cpu of the controllers. They are applicable many areas such as :-in live sound reproduction -In Instrument amplifier -In FM broadcast station energy from cable, MATV, television, - In Cable TV “notch filters” that blank out Frequencies corresponding to two consecutive channel on the FCC channel spectrum. 29

-In TV notch filters -In power supply interference filtering which are 50/60Hz filters After designing each section for the resistor and capacitor values ,the high order filter circuit could then be implemented by connecting all sections together .the design in this report has successfully approximated the ideal notch filter using a sixth order chebysheve filter.

30

Appendix Chebyshev approximation Chebyshev approximation approximate the constant value unity through out 0s(ω)= 10log[1+2Tn2 (𝜔 𝑠 ) 𝑝

=> = √ 10.1𝛼𝑝− 1 31

𝜔

=>s(ω)= 10log[1+2 cosh2(ncosh-1(𝜔 𝑠 )] 𝑝

𝜔

=>10.1s -1=2cosh2(ncosh-1(𝜔 𝑠 ) 𝑝

10.1𝛼𝑝 −1

=>√

2

𝜔

=cosh(ncosh-1(𝜔 𝑠 ) 𝑝

10.1𝛼𝑝 −1

=>cosh-1√

=>n=

2

10 cosh−1√

𝜔

= ncosh-1(𝜔 𝑠 ) 𝑝

.1𝛼𝑝 −1 2

𝜔 cosh−1( 𝑠 ) 𝜔𝑝

Table used to find Tn (ω) for various values of n Tn (ω)

N 0

1

1

Ω

2

2 ω2-1

3

4 ω3-3 ω

4

8 ω4-8 ω2+1

5

16 ω5-20 ω3+5 ω

6

32 ω6-48 ω4- 18 ω2 -1

7

64 ω7-112 ω6+56 ω3-7 ω

8

128 ω8-256 ω6+160 ω4-32 ω3+1

9

256 ω9-576 ω7+476 ω5-120 ω3+9 ω

10

512 ω10-1280 ω8+1120 ω6-400 ω4+50 ω2-1

32

Frequency Transformation Most approximation methods use to find the transfer function of low pass filter therefore we transform low pass transfer function to required filter using the following Form :-Low pass to high Pass………………. S→ S/ωo :-Low pass to Stop Pass………………. S→ BWS/S2+1 :-Low pass to Band Pass………………. S→S2+1/ BW

Active notch analysis 𝑘 𝑘+1

C

b

C

a

i 1

R 𝛼

𝑘+1 𝑘

R2

R1

kcL at node a 2 (𝑣 − 𝑣𝑎 ) + (𝑣𝑖 – 𝑣𝑎 )𝑐𝑠 = (𝑣𝑎 − 𝑣𝑖𝑛 )𝑐𝑠 𝑅 𝑜 2 𝑅

−2

𝑣0 +( 𝑅 − 2𝑐𝑠)𝑣𝑎 + 𝑐𝑠𝑣𝑖 + 𝑐𝑠𝑣𝑖𝑛 = 0

𝑣𝑎 =

2 𝑣 +𝑐𝑠𝑣𝑖 +𝑐𝑠𝑣𝑖𝑛 𝑅 𝑜 2 ( +2𝑐𝑠) 𝑅

33

𝑣𝑎 =

2 𝑣 2+2𝑐𝑠𝑅 𝑜

𝑐𝑠𝑅

𝑐𝑠𝑅

+ 2+2𝑐𝑠𝑅 𝑣𝑖 + 2+2𝑐𝑠𝑅 … … … …

eq(1)

Kcl at node b 𝑣𝑖𝑛 −𝑏 𝑅

= 2𝑐𝑠 𝑣𝑏 +

𝑣𝑏 −𝑣𝑖 𝑅 𝑣𝑖𝑛 𝑅

−2

1

+ ( 𝑅 − 2𝐶𝑆) 𝑉𝐵 + 𝑅 𝑉𝑖 = 0 𝑣𝑖𝑛 1 + 𝑣𝑖 𝑅 𝑅 2 + 2𝑐𝑠 𝑅

𝑣𝑏 =

𝑣

𝑣

𝑖𝑛 𝑖 = 2+2𝑐𝑠𝑅 + 2+2𝑐𝑠𝑅 … … … ….

eq(2)

Kcl at node → 𝑖 𝑣𝑏 − 𝑣𝑖 = 𝑐𝑠(𝑣𝑖 − 𝑣𝑎 ) 𝑅 𝑣𝑏 = 𝑐𝑠𝑅𝑣𝑖 + 𝑣𝑖 − 𝑐𝑠𝑅𝑣𝑎 𝑣𝑏 = (𝑐𝑠𝑅 + 1)𝑣𝑖 − 𝑐𝑠𝑅𝑣𝑎 ……………………eq(3)

Equating equation 3 and 2

=>(CSR +1 )𝑣𝑖 − 𝑐𝑠𝑅𝑣𝑎 = -CSR𝑣𝑎 =

−𝑣𝑖𝑛 2+2𝑐𝑠𝑅

𝑣𝑎 =

𝑣𝑖𝑛 2+2𝐶𝑆𝑅

𝑣

𝑖 + 2+2𝐶𝑆𝑅

1

+ (2+2𝑐𝑠𝑅 − 𝑐𝑠𝑅 − 1)𝑣𝑖

−𝑣𝑖𝑛 1 1 − ( − 𝑐𝑠𝑅 − 1)𝑣𝑖 (𝑐𝑠𝑅)(2 + 2𝑐𝑠𝑅) 𝑐𝑠𝑅 2 + 2𝑐𝑠𝑅

𝑣𝑎 =

−𝑣𝑖𝑛 (𝑐𝑠𝑅)(2+2𝑐𝑠𝑅)

1

1

+ (− (𝑐𝑠𝑅)(2+2𝑐𝑠𝑅) + 1 + 𝑐𝑠𝑅)𝑣𝑖 ……………….. eq(4)

Equating equation 1 and 4

34

2 𝑐𝑠𝑅𝑣𝑖 𝑐𝑠𝑅 𝑣𝑜 + + 𝑣 2 + 2𝑐𝑠𝑅 2 + 2𝑐𝑠𝑅 2 + 2𝑐𝑠𝑅 𝑖𝑛 −𝑣𝑖𝑛 −1 1 = +( +1+ )𝑣 (𝑐𝑠𝑅)(2 + 2𝑐𝑠𝑅) (𝑐𝑠𝑅)(2 + 2𝑐𝑠𝑅) 𝑐𝑠𝑅 𝑖 𝑐𝑠𝑅 2 −(2𝑐𝑠𝑅)𝑣𝑜 − (𝑐𝑠𝑅 2 )𝑣𝑖 − 𝑣𝑖 + (𝑐𝑠𝑅)(2 + 2𝑐𝑠𝑅)𝑣𝑖 + (2 + 2𝑐𝑠𝑅)𝑣𝑖 )𝑣𝑖𝑛 = (𝑐𝑠𝑅)(2 + 2𝑐𝑠𝑅) 𝑐𝑠𝑅)(2 + 2𝑐𝑠𝑅) (𝑐𝑠𝑅)2 + 1)𝑣𝑖𝑛 = −2𝑐𝑠𝑅𝑣𝑜 − (𝑐𝑠𝑅)2 + 1 − (𝑐𝑠𝑅)(2 + 2𝑐𝑠𝑅) − (2 + 2𝑐𝑠𝑅)𝑣𝑖 =-2csR𝑣𝑜 − (𝑐𝑠𝑅)2 + 1 − 2𝑐𝑠𝑅 − 2𝑐𝑠𝑅 2 − 2 − 2𝑐𝑠𝑅)𝑣𝑖 =-2csR𝑣0 − (−(𝑐𝑠𝑅)2 − 1 − 4𝑐𝑠𝑅)𝑣𝑖 =-2csr𝑣0 + (𝑐𝑠𝑅)2 + 1 + 4𝑐𝑠𝑅)𝐾𝑣0

where K=

1 (1+

𝑅𝑓 ) 𝑅𝑖

=(-2csR +(𝑐𝑠𝑅)2 + 1 + 4𝑐𝑠𝑅)𝐾)𝑣𝑜 =(k(𝑐𝑠𝑅)2 + 4𝑐𝑠𝑅𝑘 − 2𝑐𝑠𝑅 + 𝑘)𝑣𝑜 𝑣𝑖𝑛( 𝑠𝑐𝑅)2 +1)=(𝑘(𝑐𝑠𝑅2 + 2𝑐𝑠𝑅(2𝑘−1)+ 𝑘)𝑣𝑜 𝑣𝑜 𝑐𝑠𝑅 2 + 1 => = 𝑣𝑖𝑛 𝑘(𝑐𝑠𝑅)2 + 2𝑠𝑅(2𝑘 − 1)𝑠 + 𝑘 𝑐𝑅 2 𝑠2 +1

𝑣

=>𝑣 𝑜 = 𝑖𝑛

𝑣0 𝑣𝑖𝑛

𝑣0 𝑣𝑖𝑛

𝑣𝑜 𝑣𝑖𝑛

=

𝑘(𝑐𝑅 2 𝑠2 +2𝑐𝑅(2𝑘−1)𝑠+𝑘

=

1 𝑐𝑅2 2(2𝑘−1) 𝑘𝑠2 + 𝑠 𝑐𝑅

=

1 2 1 (𝑠 + 2 ) 𝑘 𝑐𝑅 2(2𝑘−1)𝑠 1 𝑠2 + + 2 𝑘𝑐𝑅 𝑐𝑅

=

𝑠2 +

𝐺0 (𝑠2 + 𝑠2 +2𝐺𝑜 (

𝐺0 (𝑠2 +

𝑘

+ 𝑐𝑅2

1 ) 𝑐𝑅2

2 1 −1)𝑠+ 2 𝐺0 𝑠𝑅

Where G0=1/k

1 ) 𝑐𝑅2

𝑠2 +2(2−𝐺0 )𝑠+

1 𝑐𝑅2

35

High pass notch analysis

Kcl at node a (𝑣𝑖𝑛 − 𝑣𝑎 )𝑠𝑎 + (𝑣𝑖 − 𝑣𝑎 ) 𝑣𝑖𝑛 𝑠𝑎 − (𝑠𝑎 +

𝑠𝑎 𝑘

+ (𝑣𝑜 − 𝑣𝑎 )

𝑘+1 𝑘

=0

𝑠𝑎 𝑘 + 1 𝑠𝑎 𝑘+1 + ) 𝑣𝑎 + 𝑣𝑖 + 𝑣𝑜 = 0 𝑘 𝑘 𝑘 𝑘

𝑠𝑎 𝑘+1 +( ) 𝜇) 𝑣𝑖 𝑘 𝑘 = 𝑣𝑎 1 (𝑠𝑎 + 1) (1 + ) 𝑘

𝑣𝑖𝑛 𝑠𝑎 + (

……………. 𝑒𝑞1

Kcl at node B 𝑣𝑖𝑛 − 𝑣𝑏 =

𝑣𝑖𝑛 +

𝑣𝑏 =

𝑣𝑏 − 𝑣𝑖 𝑠𝑎 𝑘 + 1 + 𝑣𝑏 𝑘 𝑘

(𝑘 + 1) 𝑣𝑖 1 = 𝑣𝑏 (1 + + 𝑠𝑎 ) 𝑘 𝑘 𝑘

𝑣𝑖𝑛 + 𝑣𝑖 ⁄𝑘 … … … … … . 𝑒𝑞2 1 (1 + ) (1 + 𝑠𝑎) 𝑘 36

Kcl at node i 𝑣𝑏 − 𝑣𝑖 𝑣𝑖 𝑠𝑎 = + (𝑣𝑖 − 𝑣𝑎 ) 𝑘 𝑟 𝑘 𝑣𝑏 𝑘 1 𝑠𝑎 𝑠𝑎 = 𝑣𝑖 ( + + ) − 𝑣𝑎 𝑘 𝑟 𝑘 𝑘 𝑘 𝑘

=>𝑣𝑏 = 𝑣𝑖 (𝑟 + 1 + 𝑠𝑎) − 𝑠𝑎 𝑣𝑎 … … . . 𝑒𝑞3 Equating Eq2 and eq3 𝑣𝑖𝑛 + 𝑣𝑖 ⁄𝑘 𝑘 = 𝑣𝑖 ( + 1 + 𝑠𝑎) − 𝑠𝑎 𝑣𝑎 1 𝑟 1 + )(1 + 𝑠𝑎) 𝑘 𝑣𝑖𝑛 1 (1 + ) (1 + 𝑠𝑎) 𝑘

+ 𝑣𝑖 (

1

𝑘 − ( + 1 + 𝑠𝑎) = −𝑠𝑎 𝑣𝑎 1 𝑟 𝑘 (1 + ) (1 + 𝑠𝑎) 𝑘

𝑘 + 1 + 𝑠𝑎 −𝑣𝑖𝑛 1 => 𝑣𝑎 = + 𝑣𝑖 ( 𝑟 − … … … . 𝑒𝑞4 1 1 𝑠𝑎 𝑠𝑎 (1 + ) (1 + 𝑠𝑎) 𝑠𝑎𝑘 (1 + ) (1 + 𝑠𝑎) 𝑘 𝑘

Equating eq1 and eq4 𝑠𝑎)2 +1

𝑣𝑖𝑛 = 𝑣𝑖 (

1 𝑘

𝑠𝑎(1+ )(1+𝑠𝑎)

𝑘 +1+𝑠𝑎 𝑟

𝑠𝑎

−

1 1 𝑘

𝑠𝑎𝑘(1+ )(1+𝑠𝑎)

𝑘

=𝑣𝑖 (𝑟

=𝑣𝑖 (

=𝑣𝑖 (

−

𝑠𝑎 𝑘

𝑘+1 ) 𝑘 1 (𝑠𝑎+1)(1+ 𝑘

( +𝜇(

)

𝑠𝑎 𝜇 𝑘 𝑘

+1+𝑠𝑎)(𝑘+1)(1+𝑠𝑎)−1−𝑠𝑎 𝑘( + (𝑘+1) 1 𝑘

𝑠𝑎 𝑘(1+ )(1+𝑠𝑎) 𝑘 𝑘 𝑟 𝑟

𝑠𝑎( + +1+2𝑠𝑎+(𝑠𝑎)2 (𝑘+1)−1−(𝑠𝑎)2 −𝑠𝑎(𝑘+1)𝜇 1 𝑘

𝑠𝑎 𝑘(1+ )(1+𝑠𝑎) 𝑘 𝑘2 𝑘 +𝑘)+𝑎(1+𝑘)( +2−𝜇)𝑠+𝑘(𝑠𝑎)2 𝑟 𝑟 𝑟 1 𝑠𝑎 𝑘(1+ )(1+𝑠𝑎) 𝑘

( +

1 𝑘 𝑟 𝑟

1 𝑘 𝑘 𝑟 1 𝑘(1+ )(1+𝑠𝑎) 𝑘

𝑘( + +1)+𝑘𝑎(1+ )( +2−𝜇)𝑠+𝑘(𝑠𝑎)2

=𝑣𝑖 (

𝑠𝑎

)

37

(𝑠𝑎)2 +1

𝑣

=>𝑣 𝑖 =

1 𝑘 𝑟 𝑟

1 𝑘

𝑘 𝑟

( + +1)+𝑎(1+ )( +2−𝜇)𝑠+(𝑠𝑎)2

𝑖𝑛

𝑣𝑜 𝜇(𝑠𝑎)2 + 1) = 𝑣𝑖𝑛 (1 + 𝑘 + 1) + 𝑎 (1 + 1) (𝑘 + 2 − 𝜇) 𝑠 + (𝑠𝑎)2 𝑟 𝑟 𝑘 𝑟 1

𝜇(𝑠2 + 2 ) 𝑎

=1

1 𝑘 𝑘+1 1 2 𝜇 ( + +1)+ ( + – )𝑠+(𝑠)2 𝑎 𝑟 𝑘 𝑘 𝑎2 𝑟 𝑟 1

𝜇(𝑠2 + 2 ) 𝑎

=1

1 𝑘 𝑘+1 1 2−𝜇 ( + +1)+ ( + )𝑠+𝑠2 𝑟 𝑟 𝑘 𝑎2 𝑟 𝑟

Low Pass Notch analysis K/K+1 C

C/k

a

i 1 C

R

𝑘+1

C

𝐾

R1 R2

kCl at node A (𝑣𝑖𝑛 − 𝑣𝑎) 𝑠𝑐 + (𝑣0 − 𝑣𝑎 ) Sc𝑣𝑖𝑛 − (𝑠𝑐 +

𝑘+1 𝑘

𝑐𝑠

𝑘+1 𝑘

+ 𝑘 ) 𝑣𝑎 +

𝑐𝑠

+ (𝑣𝑖 − 𝑣𝑎 ) 𝑘 = 0 𝑘+1 𝑘

𝑣0 +

𝑐𝑠 𝑘

𝑣𝑖 = 0

𝑘+1 𝑐𝑠 𝑣0 + 𝑣𝑖 𝑘 𝑘 = 𝑣𝑎 1 1 𝑠𝑐 (1 + ) + (1 + ) 𝑘 𝑘

𝑠𝑐𝑣𝑖𝑛 +

38

𝑣𝑎 =

𝑠𝑐𝑘𝑣𝑖𝑛 +((𝑘+1)𝜇+𝑐𝑠)𝑣𝑖 1 𝑘

𝑘(𝑠𝑐+1)(1+ )

𝑅

……………………𝑒q(1) where 𝜇 =1+𝑅1 2

Kcl at node B 𝑣𝑖𝑛 − 𝑣𝑏 = -𝑣𝑏 −

𝑣𝑏 𝑘

−

𝑘+1 𝑣𝑏 − 𝑣𝑖 𝑠𝑐 𝑣𝑏 + 𝑘 𝑘 𝑘+1 𝑘

𝑠𝑐 𝑣𝑏 = −𝑣𝑖𝑛 −

𝑣𝑖 𝑘

1 𝑘+1 𝑣𝑖 𝑣𝑏 (1 + ) + 𝑠𝑐) = 𝑣𝑖𝑛 + 𝑘 𝑘 𝑘

𝑣𝑏 =

𝑣 𝑣𝑖𝑛 + 𝑖

𝑘 1 (1+ )(1+𝑠𝑐) 𝑘

𝑘𝑣 +𝑣

𝑖𝑛 𝑖 = (𝑘+1)(𝑠𝑐+1) ………………..eq(2)

Kcl at node i 𝑣𝑏 −𝑣𝑖 𝑘

𝑐𝑠

=(𝑉𝑖 − 𝑣𝑎 ) 𝑘 + 𝑣𝑖 𝛼𝑐𝑠 𝑣𝑏 1 𝑐𝑠 = 𝑣𝑖 ( + + 𝛼𝑐𝑠) − 𝑣𝑎𝑐𝑠 𝑘 𝑘 𝑘 𝑘

𝑣𝑏 = 𝑣𝑖 (1 + 𝑐𝑠 + 𝛼𝑐𝑠𝑘) − 𝑣𝑎 𝑐𝑠 ………………..eq(3) Equating eq(2) and eq(3)

𝑘 𝑣 +𝑣

𝑖𝑛 𝑖 =>𝑣𝑖 (1 + (𝑐 + 𝛼𝑐𝑘)𝑠) − 𝑣𝑎 𝑐𝑠 = (𝑘+1)(𝑠𝑐+1

𝑘𝑣

1

𝑖𝑛 -𝑣𝑎 𝑐𝑠 = (𝑘+1)(𝑠𝑐+1 + 𝑣𝑖 ((𝑘+1)(𝑠𝑐+1) − (1 + (𝑐 + 𝛼𝑐𝑘)𝑠)

𝑣𝑎 = −𝑘𝑣

−𝑘𝑣𝑖𝑛 1 + (𝑐 + 𝛼𝑐𝑘)𝑠 1 + 𝑣𝑖 ( − 𝑠𝑐(𝑘 + 1)(𝑠𝑐 + 1) 𝑠𝑐(𝑘 + 1)(𝑠𝑐 + 1) 𝑠𝑐

𝑖𝑛 𝑣𝑎 =𝑠𝑐(𝑘+1)(𝑠𝑐+1) + 𝑣𝑖 (

(𝑘+1)𝑠𝑐+1)+(𝑐+𝛼𝑐𝑘)𝑠(𝑘+1)(𝑠𝑐+1)−1 𝑠𝑐(𝑘+1)(𝑠𝑐+1)

)…………..eq(4) 39

Equating eq(4) and eq(3) 𝑠𝑐𝑘 𝑣

𝑘+1)𝜇+𝑐𝑠)𝑣

−𝑘𝑣

𝑖𝑛 𝑖𝑛 =>(𝑠𝑐+1)(𝑘+1) + (𝑠𝑐+1)(𝑘+1)𝑖 = 𝑠𝑐(𝑘+1)(𝑠𝑐+1) + 𝑣𝑖 (

𝑠𝑐𝑘

𝑠𝑐(𝑘+1)(𝑠𝑐+1)

)

(𝑘+1)(𝑠𝑐+1)+(𝑐+𝛼𝑐𝑘)𝑠(𝑘+1)(𝑠𝑐+1)−1−

𝑘

((𝑠𝑐+1)(𝑘+1) + 𝑠𝑐(𝑘+1)(𝑠𝑐+1))𝑣𝑖𝑛 = 𝑣𝑖 ( (𝑠𝑐)2 𝑘+𝑘 )𝑣 (𝑠𝑐+1)(𝑘+1)𝑠𝑐 𝑖𝑛

(𝑘+1)(𝑠𝑐+1)+(𝑐+𝛼𝑐𝑘)𝑠

𝑠𝑐(𝑘+1)(𝑠𝑐+1)

(𝑘+1)(𝑠𝑐+1)(1+(𝑐+𝛼𝑐𝑘)𝑠)−1−𝑠𝑐((𝑘+1)𝜇+𝑐𝑠)

= 𝑣𝑖 (

𝑠𝑐(𝑘+1)(𝑠𝑐+1)

((𝑘+1)𝜇+𝑐𝑠)

) − (𝑠𝑐+1)(𝑘+1)

)

𝑣𝑖 (𝑠𝑐)2 𝑘 + 𝑘 = 𝑣𝑖𝑛 𝑘 + 𝑐((2 − 𝜇) + 𝛼𝑘)(1 + 𝑘)𝑠 + 𝑘𝑐 2 (1 + 𝛼𝑘 + 𝛼)𝑠 2 1

=𝑘

1

𝑘(𝑠2 + 2 )× 𝑘(1+𝛼𝑘+𝛼) 𝑐

1 + (2−𝜇+𝛼𝑘)(1+𝑘)𝑠+𝑘(1+𝛼𝑘+𝛼)𝑠2 𝑐2 𝑐

Data sheet for THS4011CD THS4011 Ro(out resitance)

12

Io (output current)

75 -110mA

Vo(output volatge)

±3.7 - ±13.5

Vn(voltge noise)

7.5nv/Hz

unity gain bandwidth

270-290Mhz

vcc (Supply volatge)

±5-±15V

SR (slow rate)

310v/us

Setting time

37ns

Ri(input resitance)

2M

Ci(input capictor)

1.2pF

CMMR

110dB

input offset volatge

1mv

40

Glossary  -Aliasing – refers to an effect that causes different signals to become indistinguishable when sampled. -the distortion or artifact that results when the signal reconstructed from samples is different than the original continuous signal.  -Attenuates –reduces amplitude of the signal  Data acquisition – is the process of sampling of real world physical conditions and conversion of the resulting samples into digital numeric values that can be manipulated by a computer.  Stop band- is a band of frequencies between specified limits  Corner frequency are the frequencies where the stop-band and the transition bands meet in a filter specification.  Acoustic instruments:- electronic musical instruments  FCC- federal communication commission office of engineering and technology policy and rules division   LTI :- linear time invariant system  Component :-A device with two or more terminals into which, or out of which, charge may flow.  Transfer function:- The relationship of the currents and/or voltages between two ports. Most often, an input port and an output port are discussed and the transfer function is described as gain or attenuation.

41

Reference 1) V.K Aater. Network Theory and Filter Design, 1985 2) F.F Kuo, Network Analysis and synthesis , 3) Gobind Daryanani Principles of active network synthesis and design,1976 4) www.national.com 5) James D. McCabe, network analysis, Architecture, and Design,2007 6) Steven T. Karris, Signals and Systems, with MATLAB® Applications,2003 7) Wai-Kai Chen, Passive, active and digital filters, 8) Wikipedia,the free encyclopedia 9) en.wikipedia.org/wiki/Notch_signaling_pathway 10) www.thefreedictionary.com/notch 11) www-k.ext.com 12) www.myspace.com/notchonline 13) www.satsignal.eu 14) www.discovercircuit.com 15) www.me.cmu.edu

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