Transcript EE 311: Electrical Engineering Lab Active Filter Design (Sallen

EE 311: Junior EE Lab
Sallen-Key Filter Design
J. Carroll
9/17/02
Background Theory
• Filter applications include:
– power supplies to attenuate undesirable ripple
– audio circuits for bass and treble control
– band limiting a signal before it is sampled
• Four basic filter types:
–
–
–
–
high-pass
low-pass
band-pass
band-reject or notch
Background Theory
• Filters fall in one of two categories:
– Passive: consist of only passive elements
• i.e., resistors, inductors and capacitors
– Active: consist of passive and active devices
• such as transistors or op-amps
• can’t amplify output of passive filter to produce active filter
• op-amps typically chosen over transistors
– All things equal, active filters have responses equal to
or better than conventional passive filters
•
•
•
•
reduced insertion loss
can amplify desired frequencies
simple design and ease of tuning
does not require the use of inductors
Band-pass Filter Performance
• Center (Resonant) Frequency fo
– frequency for maximum filter gain, the geometric
mean of the two half-power frequencies fo  fl  fh
• Lower and Upper Cutoff Frequencies fl and fh
– half-power frequencies are 3dB less than the gain at
the center frequency
• Maximum Gain,
Ho
– ratio of Vo to Vi at the filter's center or resonant
frequency, often expressed in dB
• Bandwidth   fh  fl
– difference between the upper and lower filter cutoff
frequencies, closely related to the passband
Band-pass Filter Performance
• Quality Factor Q  fo / 
– dimensionless figure of merit used to measure the
selectivity of a filter expressed as ratio of center
frequency to bandwidth
Ho
HO
Sallen-Key Band-pass Filter
Ks
Vo
R1C1
T (s)  
1
1
1 K 
R1  R 2
Vi
 1
2
s 



s 
 R1C1 R3C1 R3C 2 R 2C1  R1R 2 R3C1C 2
Vo
Rf
K   1
Vx
Ri
(1)
(2)
Equal Component Sallen-Key Filter
R3
R1  R 2 
 R and C1  C 2  C
2
Ks
RC
T ( s) 
3 K   1 
s 
s  

 RC   RC 
2
(3)
(4)
2
3 K 
3 K 
 1 

 
  4

RC 
RC 
RC 



s 
2
2
1, 2
2
(9)
In order to ensure stability of the filter, we must ensure that the poles
of the transfer function lie in the left-half of the complex s-plane, or
s   0. Thus, we must ensure that the gain of the op-amp is less
than 3, i.e., K  3 .
1, 2
Standard Second Order Filter
 
H  s
Q

T (s) 
 
s   s  
Q
o
o
2

(5)
2
o
1
1
K
 
, Q
, Ho 
RC
3 K
3 K
Note: For your design, let
R3
R1  R 2 
 Ri  2 k
2
Frequency Scaling
•Frequency scaling is a method of changing a filter’s
frequency of operation
•This method is extremely useful once one has designed a
filter with a satisfactory response (i.e.,  , Q, and Ho )
and then merely wants to change, for example, the center
frequency
•To increase the center frequency of a filter without
affecting any of its other characteristics (i.e., Q and Ho),
we can simply divide all frequency determining
capacitors or divide all frequency determining resistors
by the desired scaling factor
•As an example, to triple the center frequency, divide all
capacitor values by 3 or divide all resistor values by 3
Sallen-Key Low-pass Filters
Sallen-Key Low-pass Filters
•Pass all frequencies from zero up to the corner frequency, and
blocks all frequencies above this value
•In actual filters, there is a transition region between the
passband and the stopband
•The frequency response of the low-pass filter is not, however,
as straightforward to analyze as that of the band-pass filter
•For quality factors less than 0.5, the poles of the transfer
function are real
•For quality factors greater than 0.5, the poles are complex
•For quality factors >0.707, the frequency response peaks
above H just beyond the corner frequency
•This peak can be quite large for large quality factors
•Quality factors =0.707 produces a maximally flat response,
i.e., the sharpest fall-off near the corner without any peaks
larger than H
o
o
Closing Remarks
• Let’s quickly examine the Pre-lab questions
• Make sure you work all of the problems,
especially the PSpice problems
• See the class website for various resources
related to this lab, including PDF documents,
M-files, filter design programs, etc.