Microelectronic Circuits II Ch11 : Filters and Tuned...

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CNU EE 11.1-1 Microelectronic Circuits II Ch 11 : Filters and Tuned Amplifiers 11.1 Filter Transmission, Types, and Specification 11.2 Filter Transfer Function 11.4 First-order and Second-order Filter Functions

Transcript of Microelectronic Circuits II Ch11 : Filters and Tuned...

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CNU EE 11.1-1

Microelectronic Circuits II

Ch 11 : Filters and Tuned Amplifiers

11.1 Filter Transmission, Types, and Specification11.2 Filter Transfer Function11.4 First-order and Second-order Filter Functions

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§Important building block of communications, instrumentation systems, electronic filter- Passive LC filters : use of inductors & capacitors; work well at high-frequencies

Inductor : large, physically bulky, nonideal characteristic, No monolithic form- Inductorless filters à active-RC filters : use of op amps, resistors & capacitors;

switched-capacitor filters : fully integrated monolithic filters- Tuned amplifier : radio & TV receiver à bandpass filter§Filter Transmission- Filter : linear circuit, general two-port network

- Filter transfer function T(s) :

- Filter transmission by s=jwà magnitude & phase :Magnitude of transmission in decibel à gain function : Attenuation function :

- Filter output Vo(jw) :

Filter Transmission, Types & Specification

( ) ( )( )sVsVsT

i

o=

( ) ( ) ( )wfww jejTjT =

( ) ( ) dBjTG ,log20 ww º( ) ( ) dBjTA ,log20 ww -º

( ) ( ) ( )www jVjTjV io =

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§Filter Types- frequency selection function : passing signals whose frequency spectrum lies within a specified range,

and stopping whose frequency spectrum falls outside this range- Filter passband : a frequency band (or bands) over which the magnitude of transmission is unity- Filter stopband : a frequency band (or bands) over which the magnitude of transmission is zero- Major filter types : (a) low-pass (LP), (b) high-pass (HP), (c) bandpass (BP) &

(d) bandstop (BS) or band-reject :: ideal vertical edge characteristics à brick-wall responses

Filter Transmission, Types & Specification

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§Filter Specification- Realistic specifications for the transmission characteristics of a low-pass filter - Upper deviation bound of the passband transmission, Amax (dB) : 0.05 ~ 3 dB - Stopband signals to be attenuated by at least Amin (dB) relative to the passband signals : 20 ~ 100 dB- Transition band from the passband edge wp to the stopband edge ws- Selectivity factor ws /wp : a measure of the sharpness of the low-pass filter response

Filter Transmission, Types & Specification

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Filter Transmission, Types & Specification§Low-pass Filter Specification- Passband edge wp- Maximum allowed variation in passband transmission Amax- Stopband edge ws- Minimum required stopband attenuation Amin- Ideal filter spec. : Lower Amax , higher Amin ,

selectivity ratio ws /wp closer to unity à higher order & more complex & expensive

§Transfer function whose specification meets the specification - Since the peak ripple is equal to Amax, passband ripple Amax & ripple bandwidth wp- Minimum stopband attenuation is equal to Amin, with the ripple peaks all equal

à equiripple in both the passband & the stopband- Filter approximation : The process of obtaining a transfer function that meets given specification

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Filter Transmission, Types & Specification§Transmission specification for a bandpass filter- approximation function does not ripple in the passband- The transmission decreases monotonically on both sides of the center frequency- The transmission attains the maximum allowable deviation at the two edges of the passband

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Filter Transfer Function- Filter transfer function T(s)

degree of denominator, N : filter degree ; stable filter circuit : numerator coefficients, a0, a1,…., aM & denominator coefficients, b0, b1,…., bN-1 : real number

- Factored polynomials form T(s)

numerator roots, z1, z2,…, zM : transfer function zeros, or transmission zerosdenominator roots, p1, p2,…, pN : transfer function poles, or natural modes

- Since in the filter stop band the transmission is required to be zero or small, the filter transmission zerosare usually placed on the jw axis at stop band frequencies

- low-pass filter has infinite attenuation (zero transmission) at two stopband frequencies : wl1 & wl2 àtransmission zeros at s = +jwl1 & s = +jwl2 à the other transmission zeros at s = -jwl1 & s = -jwl2 sincecomplex zeros occur in conjugate pairs ànumerator polynomial factors (s2+wl1

2)(s2+wl2 2)

- In the low pass filter, the transmission decreases toward – as w approaches à one or moretransmission zeros at s = à the number of transmission zeros at s = is N – M

- For a filter circuit to be stable, all its poles must lie in the left half of the s plane, and thus p1, p2,…, pNmust all have negative real parts

( )0

11

01

1

bsbsasasasT N

NN

MM

MM

+×××+++×××++

= --

--

NM £

( ) ( )( ) ( )( )( ) ( )N

MM

pspspszszszsa

sT-×××---×××--

=21

21

¥ ¥¥ ¥

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- sixth order (N=6) bandpass filter T(s)transmission zeros at s = +jwl1 & s = +jwl2,

one or more zeros at s = 0 &

Filter Transfer Function- Typical pole & zero locations for fifth order (N=5)

low-pass filter T(s)five poles : two pairs of complex-conjugate poles +

one real-axis pole All poles lies in the vicinity of the passband à high

transmission at passband frequencies five transmission zeros at

( ) ( )( )0

11

22

33

44

5

22

221

24

bsbsbsbsbsssa

sT ll

+++++++

=ww

¥

¥=±=±= sjsjs ll ,, 21 ww

( ) ( )( )0

55

6

22

221

25

bsbssssasT ll

+×××++++

=ww

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Filter Transfer Function- a fifth order (N=5) low-pass filter having all transmission zeros at infinity T(s)

No finite values of w at which the attenuation is infinite (zero transmission) à all zeros at s = à all-pole filter

- General filters : Transmission zeros are on the jw axis, in the stopband(s), w = 0 & w = - To obtain high selectivity, all the natural modes will be complex conjugate (except for the case of odd-

order filters, where one natural mode must be on the real axis) - The more selective the filter response is, the higher its order must be, and the closer its natural modes

are to the jw axis

¥

( )0

11

0

bsbsasT N

NN +×××++

= --

¥

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First-order & Second-order Filter Function§First-Order filters- The general first-order transfer function (bilinear transfer function)

a natural modes at s = -w0 , a transmission zero at s = -a0/a1, & a high-frequency gain that approaches a1The numerator coefficients, a0 and a1, determines the type of filter (i.e., LP, HP, etc.)

- Passive (RC) and active (op amp - RC) realizations- The Output impedance of the active circuits is very low (ideally zero) à cascading does not change the

transfer functions of the individual blocks

( )0

01

w++

=s

asasT

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First-order & Second-order Filter Function- all-pass filter : the transmission zero and the natural modes are symmetrically located relative to the jw

axis (mirror-image symmetry with respect to the jw axis)à the transmission of the all-pass filter is (ideally) constant at all frequenciesà its phase shows frequency selectivityà phase shifters

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First-order & Second-order Filter Function§Second-Order (biquadratic) filter functions- The general second-order transfer function :

a natural modes (poles) by w0 & Q : Q > 0.5 : complex-conjugate natural modes

- Location of the pair of complex-conjugate poles in the s planepole frequency w0 : radial distance of the natural modes (from the origin)pole quality factor (or pole) Q : distance of the poles from the jw axisThe higher the value of Q, the closer the poles are to the jw axis, and the more selective the filter

response becomes à infinite value of Q = poles on the jw axis = sustained oscillationà negative value of Q = poles in the right half of the s plane = oscillations

- The numerator coefficients, a0 , a1 and a2, determines the type of filter (i.e., LP, HP, etc.)- low-pass (LP) case : two transmission zeros at s = ; peak occurs only for - high-pass (HP) case : both transmission zeros at s = 0; peak occurs only for - bandpass (BP) case : one transmission zero at s = 0 (dc);& the other at s = ; magnitude response peaks

at w = w0 , center frequency ; selectivity of the filter by 3-dB bandwidth, w2 - w1 at which the magnitude response is 3dB below its maximum value (at w0)

à as Q increases, the bandwidth decreases & the bandpass filter become more selective

( ) ( ) 200

201

22

ww ++++

=sQs

asasasT

( )20

021 411

2, Qj

Qpp -±-= ww

¥ 21>Q21>Q

¥

( ) QBWQ

Q wwwwwww =-º±+= 1202

021 2411, >

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First-order & Second-order Filter Function§Second-Order (biquadratic) filter functions- notch filter, bandstop (BS) case :

If the transmission zeros are on jw axis, at the complex-conjugate locations ,then the magnituderesponse exhibits zero transmission at w = wn à notch occurs notch frequency wn

three cases : regular notch when wn = w0, low-pass notch when wn > w0, high-pass notch when wn < w0No transmission zeros at either s = 0 or s = à transmission at dc & at s = is finite¥ ¥

njw±

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First-order & Second-order Filter Function§Second-Order (biquadratic) filter functions- All-pass (AP) filter case :

two transmission zeros are in the right half of the s plane, at the mirror-image locations of the polesflat gain : the magnitude response is constant over all frequenciesfrequency selectivity is in its phase response