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sallen-key 滤波器设计资料1.pdf
Analysis of the
SallenĆKey Architecture
Application
Report
July 1999 – Revised September 2002
Mixed Signal Products
SLOA024B
IMPORTANT NOTICE
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Copyright 2002, Texas Instruments Incorporated
1 Introduction
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1
Contents
2 Generalized Circuit Analysis
2.1 Gain Block Diagram
2.2 Ideal Transfer Function
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3 Low-Pass Circuit
3.1 Simplification 1: Set Filter Components as Ratios
3.2 Simplification 2: Set Filter Components as Ratios and Gain = 1
3.3 Simplification 3: Set Resistors as Ratios and Capacitors Equal
3.4 Simplification 4: Set Filter Components Equal
3.5 Nonideal Circuit Operation
3.6 Simulation and Lab Data
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4 High-Pass Circuit
4.1 Simplification 1: Set Filter Components as Ratios
4.2 Simplification 2: Set Filter Components as Ratios and Gain=1
4.3 Simplification 3: Set Resistors as Ratios and Capacitors Equal
4.4 Simplification 4: Set Filter Components as Equal
4.5 Nonideal Circuit Operation
4.6 Lab Data
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5 Summary and Comments About Component Selection
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13
Analysis of the Sallen-Key Architecture
iii
Figures
List of Figures
1 Basic Second Order Low-Pass Filter
2 Unity Gain Sallen-Key Low-Pass Filter
3 Generalized Sallen-Key Circuit
4 Gain-Block Diagram of the Generalized Sallen-Key Filter
5 Low-Pass Sallen-Key Circuit
6 Nonideal Effect of Amplifier Output Impedance and Transfer Function
7 Test Circuits
8 Effect of Output Impedance
9 High-Pass Sallen-Key Circuit
10 Model of High-Pass Sallen-Key Filter Above fc
11 High-Pass Sallen-Key Filter Using THS3001
12 Frequency Response of High-Pass Sallen-Key Filter Using THS3001
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iv
SLOA024B
Analysis of the Sallen-Key Architecture
James Karki
ABSTRACT
This application report discusses the Sallen-Key architecture. The report gives a general
overview and derivation of the transfer function, followed by detailed discussions of
low-pass and high-pass filters, including design information, and ideal and non-ideal
operation. To illustrate the limitations of real circuits, data on low-pass and high-pass
filters using the Texas Instruments THS3001 is included. Finally, component selection is
discussed.
1 Introduction
Figure 1 shows a two-stage RC network that forms a second order low-pass filter.
This filter is limited because its Q is always less than 1/2. With R1=R2 and C1=C2,
Q=1/3. Q approaches the maximum value of 1/2 when the impedance of the
second RC stage is much larger than the first. Most filters require Qs larger than
1/2.
VI
R1
R2
C2
VO
C1
=
Vo
Vi
)
1C2R2C1Rs
(
2
+
1
+
(
1C1R1C2R2C1Rs
+
) 1
+
Figure 1. Basic Second Order Low-Pass Filter
Larger Qs are attainable by using a positive feedback amplifier. If the positive
feedback is controlled—localized to the cut-off frequency of the filter—almost any
Q can be realized, limited mainly by the physical constraints of the power supply
and component tolerances. Figure 2 shows a unity gain amplifier used in this
manner. Capacitor C2, no longer connected to ground, provides a positive
feedback path. In 1955, R. P. Sallen and E. L. Key described these filter circuits,
and hence they are generally known as Sallen-Key filters.
R1
VI
R2
C2
C1
+
–
VO
Figure 2. Unity Gain Sallen-Key Low-Pass Filter
The operation can be described qualitatively:
• At low frequencies, where C1 and C2 appear as open circuits, the signal is
simply buffered to the output.
• At high frequencies, where C1 and C2 appear as short circuits, the signal is
shunted to ground at the amplifier’s input, the amplifier amplifies this input to
its output, and the signal does not appear at Vo.
• Near the cut-off frequency, where the impedance of C1 and C2 is on the same
order as R1 and R2, positive feedback through C2 provides Q enhancement
of the signal.
1
Generalized Circuit Analysis
2 Generalized Circuit Analysis
The circuit shown in Figure 3 is a generalized form of the Sallen-Key circuit,
where generalized impedance terms, Z, are used for the passive filter
components, and R3 and R4 set the pass–band gain.
Vf
VI
Z1
Z2
Z4
Vp
Vn
+
–
3
Z
VO
R4
R3
Figure 3. Generalized Sallen-Key Circuit
To find the circuit solution for this generalized circuit, find the mathematical relationships
between Vi, Vo, Vp, and Vn, and construct a block diagram.
KCL at Vf:
Vfǒ 1
KCL at Vp:
Vpǒ 1
Z1 )
1
Z2 )
Z4Ǔ + Viǒ 1
1
Z1Ǔ ) Vpǒ 1
Z2Ǔ ) Voǒ 1
Z4Ǔ
Z2 )
Z3Ǔ + Vfǒ 1
1
Z2Ǔ å Vf + Vpǒ1 )
Z2
Z3Ǔ
Substitute Equation (2) into Equation (1) and solve for Vp:
Vp + Viǒ
Z2Z3Z4
Z2Z3Z4 ) Z1Z2Z4 ) Z1Z2Z3 ) Z2Z2Z4 ) Z2Z2Z1Ǔ )
Z2Z3Z4 ) Z1Z2Z4 ) Z1Z2Z3 ) Z2Z2Z4 ) Z2Z2Z1Ǔ
Z1Z2Z3
Voǒ
KCL at Vn:
Vnǒ 1
R3 )
R4Ǔ + Voǒ 1
1
R4Ǔ å Vn + Voǒ R3
R3 ) R4Ǔ
2.1 Gain Block Diagram
By letting: a(f) = the open-loop gain of the amplifier, b + ǒ R3
c +
Z2Z3Z4 ) Z1Z2Z4 ) Z1Z2Z3 ) Z2Z2Z4 ) Z2Z2Z1
Z2Z3Z4
R3 ) R4Ǔ,
,
d +
Z2Z3Z4 ) Z1Z2Z4 ) Z1Z2Z3 ) Z2Z2Z4 ) Z2Z2Z1
Z1Z2Z3
,
and Ve = Vp – Vn, the generalized Sallen-Key filter circuit is represented in
gain-block form as shown in Figure 4.
2
SLOA024B
(1)
(2)
(3)
(4)
Generalized Circuit Analysis
VI
c
Ve
+
+
–
d
b
a(f)
VO
Figure 4. Gain-Block Diagram of the Generalized Sallen-Key Filter
From the gain-block diagram the transfer function can be solved easily by
observing, Vo = a(f)Ve and Ve = cVi + dVo – bVo. Solving for the generalized
transfer function from gain block analysis gives:
Vo
ȡ
bǓȧȧ
Vi + ǒc
1 )
Ȣ
1
1
aǒfǓb *
ȣ
ȧȧ
d
Ȥ
b
2.2 Ideal Transfer Function
Assuming a(f)b is very large over the frequency of operation, 1
transfer function from gain block analysis becomes:
a(f)b [ 0, the ideal
Vo
ȣ
ȡ
bǓȧ
Vi + ǒc
1
ȧ
d
1 *
Ȥ
Ȣ
b
b + K, c +
N2
D
By letting 1
, where N1, N2, and D are the
numerators and denominators shown above, the ideal equation can be rewritten
as:
, and d +
N1
D
K
ȡ
ȣ
Vo
Vi +ȧ
ȧ
. Plugging in the generalized impedance terms gives the
D
N1 * K@N2
Ȣ
Ȥ
N1
ideal transfer function with impedance terms:
Vo
Vi +
Z1Z2
Z3Z4 )
Z1
Z3 )
K
Z2
Z3 )
Z1ǒ1*KǓ
Z4 ) 1
(5)
(6)
(7)
Analysis of the Sallen-Key Architecture
3
Low-Pass Circuit
3 Low-Pass Circuit
The standard frequency domain equation for a second order low-pass filter is:
HLP +
2
K
)
* ǒ f
fcǓ
jf
Qfc ) 1
(8)
Where fc is the corner frequency (note that fc is the breakpoint between the pass
band and stop band, and is not necessarily the –3 dB point) and Q is the quality
factor. When f<>fc, Equation (8) reduces to * Kǒfc
f Ǔ
, and
signals are attenuated by the square of the frequency ratio. With attenuation at
higher frequencies increasing by a power of 2, the formula describes a second
order low-pass filter.
Figure 5 shows the Sallen-Key circuit configured for low-pass:
Z1 + R1, Z2 + R2, Z3 +
1
,
sC1
Z4 +
1
sC2
, and K + 1 )
R4
R3
.
R1
R2
VI
C1
C2
+
–
R4
R3
VO
Figure 5. Low-Pass Sallen-Key Circuit
From Equation (7), the ideal low-pass Sallen-Key transfer function is:
Vo
Vi
(Ip) +
By letting
s2(R1R2C1C2) ) s(R1C1 ) R2C1 ) R1C2(1 * K)) ) 1
K
(9)
1
Ǹ
fc +
2p R1R2C1C2
s + j2pf,
equation (9) follows the same form as Equation (8). With some simplifications,
these equations can be dealt with efficiently; the following paragraphs discuss
commonly used simplification methods.
R1C1 ) R2C1 ) R1C2(1 * K)
, and Q +
Ǹ
R1R2C1C2
,
4
SLOA024B
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