Today Homework 1st Order Active Filters Cascaded Filters Desired ...

Today
2/1/11 Lecture 3
• Higher Order Active Filters
– Cascaded Filters and Higher Order Filter Response
– VCVS filter design
– Butterworth Active Filters
• Design
• Performance
• Homework
– See next slide
• Reading
– H&H Ed 2 268-276
• Lab this week
– Lab 2
– Do pre-lab of lab 2 BEFORE lab on Thursday
– Lab 1 and 2a due Friday at 10am
• Quiz
Homework
Due 2/8/11 HW3
1. Calculate the complex transfer functions for the two active filters
on the next slide. Write expression for the magnitudes of their
gains and phases as functions of frequency. What are the cut-off
(3dB) frequencies of the two filters? What are their gains at zero
frequency? What are the phase shifts of their outputs (relative to
the inputs) at very low and very high frequencies?
2. Design a 2- pole Butterworth low pass filter with cut-off
frequency ~30kHz.
Rest of HW delayed until next time
1. What is the formula for its gain as a function of frequency? What
is its attenuation (in dB) at f=3f c ?
2. Design a 4- pole Butterworth high pass filter with cut-off
frequency ~60kHz. What is the formula for its gain as a function
of frequency? What is its attenuation (in dB) at f=f c /2?
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 1
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 2
Vin
1 st Order Active Filters
INVERTING
C 2
R 1
R 2
0
-
+
V in
Vout
R
NON-INVERTING
C
+
_
R a
R b
V out
Cascaded Filters
Cascaded filters multiply to give total transfer function
T(f)=T 1 (f)T 2 (f)T 3 (f) . . .
Cascaded passive filters increase the order of the total filter.
Number of cascaded FIRST order filters = filter order
1 1
T
( f) for f f
( f / f )
total _ n
2
n
c
1 ( f / fc
) c
|Voltage Slope in stopband| = order of filter
Cascaded first order filters have weak knees.
n
1
T
total _ n( fc
)
2
Each cascaded passive filter’s loads previous section.
– Degrades response
n
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 3
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 4
Desired Low-Pass Filter Response
Order = | Slope in log-log plot |
Near Ideal Low Pass Filters
(with Gain =1)
Gain
1
1 st Order
0.1
0.01
0.001
3 rd Order
f c 10f c 100f c 1000f c
f
LPF with different roll-off rates
Desired High-Pass Filter Response
Order = | Slope in log-log plot |
Near Ideal High Pass Filters
Gain
1
1 st Order
0.1
0.01
0.001
3 rd Order
0.01f c 0.1f c f c f
HPF with different roll-off rates
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 5
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 6
1

Cascaded Active Filters
Cascaded passive filters increase the order of the total filter.
Same is true for active filters
Each cascaded passive filter’s loads previous section.
Not true with active filters
• OpAmp has low current input => High input impedance
• OpAmp provides lots of current => Low output impedance
Cascaded first order filters have weak knees.
True with active filters
But . . .
By crafting filter designs in cascaded active filters
• Sharper knees, better time response, OR flatter phase response
V in
Sallen-Key 2 nd Order Low-Pass Filter
A.K.A. Low Pass VCVS (voltage-controlled voltage-source)
R 1 R 2
+
_
C 2
C 1
V out
Passband Gain = K
Dual filtering at high f
R a =(K-1)R b
Different filter designs have
R b -Different K values
-Different RC values
•RC determines f c , BUT
•f c not always 1/(2RC)
The roll-off rate for a two-pole (2 nd order) filter is
20 decades/decade in Voltage OR -40 dB/decade in POWER.
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 7
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 8
Design of 2 nd Order Active Filters
All the 2 nd order active filter circuits have the same basic design
– Frequency selective RC circuit can be
• Band-pass (see H&H Figure 5.16)
• Low-pass
R
• High-pass
C
C
R
R
C
R
C
V in
Frequency
selective
RC circuit
Higher order (>2) active filters are cascaded 2 nd order circuits
– Built up by cascading basic filter circuits: V out_previous => V in_next
– Only one VCVS and one op-amp is needed per every two orders
+
_
R a
R b
V out
2 nd Order Butterworth Design
1-stage (2-pole) filter design:
Start with desire f c
Butterworth: RC=1/(2f c ) and R a =(K-1)R b
Typically R b =R in RC
R is typically 10-100K ohm.
(not hard rule)
Butterworth Bessel
Poles K f n K
2 1.59 1.27 1.27
V in
Frequency
selective
RC circuit
+
_
R a
R b
V out
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 9
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 10
Higher Order Butterworth Design
VCVS Low-Pass Filter Design:
R
C
R
f c is desired 3dB frequency
of total n-pole filter
C
Butterworth Bessel
Poles Stage(n) K n f n K n
2 1 1.59 1.27 1.27
4 1 1.15 1.43 1.08
2 2.24 1.61 1.76
6 1 1.07 1.61 1.04
2 1.59 1.69 1.36
3 2.48 1.91 2.02
Butterworth:
RC circuit is the same for all stages (Determined by desired f c )
Only the gain changes for each stage
RC=1/(2f c ) and R a =(K n -1)R b
Typically gains increase down the line to avoid dynamic range issues
Total Gain of multi-stage filter = product of the K n ’s
For high pass filter: Same design table except:
Use high pass VCVS
Use 1/f n to determine RC
2-stage (4-pole) Filter designs:
Butterworth:
R 1 C 1 = R 2 C 2 =1/(2f c )
R a1 = (K 1 -1)R b1 = (1.15-1)R b1
R a2 = (K 2 -1)R b1 = (2.24-1)R b2
V in
Frequency
selective
R 1 C 1 circuit
4 th Order Butterworth
Stage 1
Frequency
Stage 2
+ V out V in selective +
_
R 2 C 2 circuit _
R a1 =(K 1 -1)R b1
R b1
Butterworth Bessel
Poles Stage(n) K n f n K n
4 1 1.15 1.43 1.08
2 2.24 1.61 1.76
R 1 = R b1
R 2 = R b2
V out
R a2 =(K 2 -1)R b2
R b2
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 11
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 12
2

Power
Gain (dB)
T
( f)
Butterworth Response
1
2
(1 ( f / f ) n
c
T
( f)
1
2
Voltage
Gain: T(f)
1
0.1
0.01
0.001
0.0001
0.00001
0.000001
Butterworth High Pass Filter Response
Same design table except:
Use high pass VCVS
Use 1/f n to determine RC
n=1
2
3
4
5
6
Order n
T
( f)
T
( f)
1
2
1
f
2
(1 ( / ) n
c
f
Voltage
Gain: T(f)
1
0.1
0.01
0.001
0.0001
0.00001
0.000001
f/f c
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 13
Based with permission on lectures by John Getty
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 14
R1
10kohm
V1
1V
0.71V_rms
1000Hz
0Deg
Based with permission on lectures by John Getty
Review Butterworth Design
R2
10kohm
Swap the
locations of
the caps and
resistors to
change the
type of filter.
C1
0.01uF
C3
0.01uF
1
2
R3
U1
12.3kohm
1.5kom
R6
10kohm
Choose caps and
resistors to adjust
cut-off frequency. Use
resistor in RC that is
equal to or close to R b
3
R4
10kohm
Physics 262: Laboratory Electronics II Spring 2011 Lect 3 Page 15
R5
10kohm
Same caps and
resistors in each stage,
sets cut-off frequency.
Gains are set by
the Butterworth
polynomial and
should not be
changed
C2
0.01uF
C4
0.01uF
1
2
R8
10kohm
U2
Different gains in
each stage, as per
design table.
R7
3
1.5kohm
12.4kom
3