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
CNU EE 11.1-2
§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 =
CNU EE 11.1-3
§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
CNU EE 11.1-4
§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
CNU EE 11.1-5
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
CNU EE 11.1-6
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
CNU EE 11.1-7
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
¥ ¥¥ ¥
CNU EE 11.1-8
- 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
CNU EE 11.1-9
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 +×××++
= --
¥
CNU EE 11.1-10
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
CNU EE 11.1-11
CNU EE 11.1-12
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
CNU EE 11.1-13
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, >
CNU EE 11.1-14
CNU EE 11.1-15
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±
CNU EE 11.1-16
CNU EE 11.1-17
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