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Second Order Filters Overview
• What’s different about second order filters
• Resonance
• Standard forms
• Frequency response and Bode plots
• Sallen-Key filters
• General transfer function synthesis
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 1
Prerequisites and New Knowledge
Prerequisite knowledge
• Ability to perform Laplace transform circuit analysis
• Ability to solve for the transfer function of a circuit
• Ability to generate and interpret Bode plots
New knowledge
• Ability to design second-order filters
• Knowledge of the advantages of transfer functions with complexpoles and zeros
• Understanding of resonance
• Familiarity with second-order filter properties
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 2
Motivation
Y (s) =
(
∑Mk=0 bksk
∑N
k=0 aksk
)
X(s) = H(s)X(s)
• Second order filters are filters with a denominator polynomial thatis of second order
• Usually the roots (poles) are complex
p1,2 = −ζωn ± jωn
√
1 − ζ2
• In many applications, second order filters may be sufficient
• Are also the building blocks for higher-order filters
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 3
Standard Form
H(s) =N(s)
s2
ω2n
+ sQωn
+ 1
• Second order filters can be designed as lowpass, highpass,bandpass, or notch filters
• All four types can be expressed in standard form shown above
• N(s) is a polynomial in s of degree m ≤ 2
– If N(s) = k, the system is a lowpass filter with a DC gain of k
– If N(s) = k s2
ω2n
, the system is a highpass filter with a high
frequency gain of k
– If N(s) = k sQωn
, the system is a bandpass filter with a
maximum gain of k
– If N(s) = k(
1 − s2
ω2n
)
, the system is a notch filter with a gain
of k
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 4
Unity Gain Lowpass
A unity-gain lowpass second-order transfer function is of the form
H(s) =ω2
n
s2 + 2ζωns + ω2n
=1
1 + 2ζ sωn
+(
sωn
)2
• ωn is called the undamped natural frequency
• ζ (zeta) is called the damping ratio
• The poles are p1,2 = (−ζ ±√
ζ2 − 1) ωn
• If ζ ≥ 1, the poles are real
• If 0 < ζ < 1, the poles are complex
• If ζ = 0, the poles are imaginary: p1,2 = ±jωn
• If ζ < 0, the poles are in the right half plane (Rep > 0) and thesystem is unstable
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 5
Unity Gain Lowpass Continued
The transfer function H(s) can also be expressed in the following form
H(s) =1
1 + 2ζ sωn
+(
sωn
)2=
1
1 + sQωn
+(
sωn
)2
where
Q ,1
2ζ
The meaning of Q, the Quality factor, will become clear in thefollowing slides.
H(jω) =1
1 −(
ωωn
)2
+ jωQωn
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 6
Unity Gain Lowpass Magnitude Response
20 log10 |H(jω)| = 20 log10
∣
∣
∣
∣
∣
1 −(
ω
ωn
)2
+jω
Qωn
∣
∣
∣
∣
∣
−1
= −20 log10
√
(
1 − ω2
ω2n
)2
+
(
ω
Qωn
)2
For ω ≪ ωn,
20 log10 |H(jω)| ≈ −20 log10 |1| = 0 dB
For ω ≫ ωn,
20 log10 |H(jω)| ≈ −20 log10
ω2
ω2n
= −40 log10
ω
ωn
dB
At these extremes, the behavior is identical to two real poles.
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 7
Lowpass Magnitude Continued
20 log10 |H(jω)| = −20 log10
√
(
1 − ω2
ω2n
)2
+
(
ω
Qωn
)2
For ω = ωn,
20 log10 |H(jωn)| = −20 log10
1
Q= 20 log10 Q = QdB
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 8
Lowpass Magnitude
10−2
10−1
100
101
102
−60
−40
−20
0
20
40
Frequency (rad/sec)
Mag
(dB
)
Complex Poles
Q=0.100Q=0.500Q=0.707Q=1.000Q=2.000Q=10.000Q=100.000
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 9
Matlab Code
function [] = BodeComplexPoles();
N = 1000;
wn = 1;
Q = [0.1 0.5 sqrt (1/2) 1 2 10 100];
w = logspace(-2,2,N);
nQ = length(Q);
mag = [];
phs = [];
legendLabels = cell(nQ ,1);
for c1 = 1:nQ
sys = tf([wn^2] ,[1 wn/Q(c1) wn ^2]);
[m, p] = bode(sys ,w);
mag = [mag reshape(m,N ,1)];
phs = [phs reshape(p,N ,1)];
legendLabels c1 = sprintf(’Q=%5.3f’,Q(c1));
end
figure(1);
FigureSet(1,’Slides’);
h = semilogx(w,20* log10(mag));
set(h,’LineWidth ’,1.5);
grid on;
xlabel(’Frequency (rad/sec)’);
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 10
ylabel(’Mag (dB)’);
title(’Complex Poles’);
xlim ([w(1) w(end)]);
ylim ([-60 45]);
box off;
AxisSet (8);
legend(legendLabels );
print(’BodeMagComplexPoles’,’-depsc’);
figure(2);
FigureSet(2,’Slides’);
h = semilogx(w,phs);
set(h,’LineWidth ’,1.5);
grid on;
xlabel(’Frequency (rad/sec)’);
ylabel(’Phase (deg)’);
title(’Complex Poles’);
xlim ([w(1) w(end)]);
ylim ([ -190 10]);
box off;
AxisSet (8);
legend(legendLabels );
print(’BodePhaseComplexPoles’,’-depsc’);
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 11
Lowpass Phase
∠H(jω) = ∠1
1 −(
ωωn
)2
+ jωQωn
For ω ≪ ωn,∠H(jω) ≈ ∠1 = 0
For ω ≫ ωn,
∠H(jω) ≈ ∠1
− ωQωn
= ∠ − Qωn
ω= ∠ − 1 = −180
For ω = ωn,
∠H(jω) ≈ ∠1
Qj= ∠
1
j= ∠ − j = −90
At the extremes, the behavior is identical to two real poles. At othervalues of ω near ωn, the behavior is more complicated.
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 12
Unity Gain Lowpass Phase
10−2
10−1
100
101
102
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Frequency (rad/sec)
Pha
se (
deg)
Complex Poles
Q=0.100Q=0.500Q=0.707Q=1.000Q=2.000Q=10.000Q=100.000
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 13
Lowpass Comments
H(jω) =1
1 +(
jωQωn
)
+(
jωωn
)2
• The second-order response attenuates twice as fast as a first-orderresponse (40 dB/decade)
• Generally better than the cascade of two first-order filters
• Offers additional degree of freedom (Q), which is the gain nearω = ωn
• Q may range from 0.5 to 100
• Is usually near Q = 1
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 14
Lowpass Maximum Magnitude
What is the frequency at which |H(jω)| is maximized?
H(jω) =1
1 +(
jωQωn
)
+(
jωωn
)2|H(jω)| =
1√
(
1 − ω2
ω2n
)2
+(
ωQωn
)2
• For high values of Q, the maximum of |H(jω)| > 1
• This is called peaking
• The largest Q before the onset of peaking is Q = 1√2≈ 0.707
– Said to be maximally flat
– Also called a Butterworth response
– In this case, |H(jωn)| = −3 dB and ωn is the cutoff frequency
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 15
Lowpass Maximum Continued
If Q > 0.707, the maximum magnitude and frequency are as follows:
ωr = ωn
√
1 − 1
2Q2|H(jωr)| =
Q
1 − 14Q2
• ωr is called the resonant frequency or the damped natural
frequency
• As Q → ∞, ωr → ωn
• For sufficiently large Q (say Q > 5)
– ωr ≈ ωn
– |H(jωr)| ≈ Q
• Peaked responses are useful in the synthesis of high-order filters
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 16
Example 1: Passive Lowpass
v s ( t ) v o ( t )
-
+
50 mH R
200 nF
Generate the bode plot for the circuit shown above.
H(s) =1
LC
s2 + RL
s + 1LC
=ω2
n
s2 + 2ζωns + ω2n
=ω2
n
s2 + ωn
Qs + ω2
n
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 17
Example 1:Passive Lowpass Continued
ωn =
√
1
LC= 10 k rad/s
ζ =R
2L
√LC =
R
2
√CL = R × 0.001
Q =1√LC
L
R=
√
L
C
1
R=
500
R
R = 5 Ω ζ = 0.005 Q = 100 Very Light Damping
R = 50 Ω ζ = 0.05 Q = 10 Light Damping
R = 707 Ω ζ = 1.41 Q = 0.707 Strong Damping
R = 1 kΩ ζ = 1 Q = 0.5 Critical Damping
R = 5 kΩ ζ = 5 Q = 0.1 Over Damping
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 18
Example 1: Lowpass Magnitude Response
102
103
104
105
106
−120
−100
−80
−60
−40
−20
0
20
40
Frequency (rad/sec)
Mag
(dB
)
Resonance Example
R = 5 ΩR = 50 ΩR = 707 ΩR = 1 kΩR = 5 kΩR = 50 kΩ
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 19
Example 1: Lowpass Phase Response
102
103
104
105
106
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Frequency (rad/sec)
Pha
se (
deg)
Resonance Example
R = 5 ΩR = 50 ΩR = 707 ΩR = 1 kΩR = 5 kΩR = 50 kΩ
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 20
Practical Second-Order Filter Tradeoffs
• There are many standard designs for second-order filters
• A library of first and second-order filters is all you need tosynthesize any transfer function
• The ideal transfer functions are identical
• Engineering design tradeoffs are practical
– Cost (number of op amps)
– Complexity
– Sensitivity to component value variation
– Ease of tuning
– Ability to vary cutoff frequencies and/or gains
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 21
Practical Second-Order Filters
• Popular practical second-order filters include
– KRC/Sallen-key filters
– Multiple-feedback filters: more than one feedback path
– State-variable filters
– Biquad/Tow-Thomas filters filters
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 22
Example 2: Lowpass Sallen-Key Filter
v o ( t )
-
+ v s ( t )
R 1 R 2
C 1
C 2
R b
R a
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 23
Example 2: Questions
• Is the input impedance finite?
• What is the output impedance?
• How many capacitors are in the circuit?
• What is the DC gain?
• What is the “high frequency” gain?
• What is the probable order of the transfer function?
• Solve for the transfer function
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 24
Example 2: Work Space
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 25
Example 2: Work Space
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 26
Example 2: Tuning
|H(0)| =Ra + Rb
Ra
= k
ωn =1√
R1C1R2C2
Q =1
√
R2C2
R1C1
+√
R1C2
R2C1
+ (1 − k)√
R1C1
R2C2
• 6 circuit parameters (four resistors, two capacitors)
• 3 constraints
– DC gain, k
– Natural frequency
– Q
• Prudent to pick K = 1, since gain is often added at in the firstand/or last stages
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 27
Example 2: Tuning Tips
|H(0)| =RA + RB
RA
ωn =1√
R1C1R2C2
Q =1
√
R2C2
R1C1
+√
R1C2
R2C1
+ (1 − K)√
R1C1
R2C2
1. If you select R1 = R2 = R and C1 = C2 = C, the equationssimplify considerably
• Watch out for coarse tolerances on components
2. Helps guard against op amp limitations if you select k = 1
• Gain is often used at end or beginning of filter stages
3. Best to start of with two capacitances in a ratio C1
C2
≥ 4Q2
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 28
Example 3: Design
Design a Sallen-Key lowpass filter with a Q = 1/√
2, corner
frequency of 1000 Hz, and DC gain of 1.
• Redraw the circuit with a DC gain of 1
• Arbitrarily pick C2 = 1 nF, a common capacitor value that ischeaply and readily available
Hint: Recall that the solution to ax2 + bx + c = 0 is x = −b±√
b2−4ac2a
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 29
Example 3: DC Gain Solution
|H(0)| =RA + RB
RA
= 1 +RB
RA
Pick RB = 0 and RA = ∞
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 30
Example 3: Solution
Define R1 = mR2 and C1 = nC2
ωn =1√
R1C1R2C2
=1
R2C2
√nm
Q =1
√
R2C2
R1C1
+√
R1C2
R2C1
+ (1 − K)√
R1C1
R2C2
=1
√
1mn
+√
mn
√mn√mn
=
√mn
1 + m
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 31
Example 3: Solution
Q2 =mn
1 + 2m + m2
m2 + 2m + 1 =mn
Q2
1
2m2 + m +
1
2=
mn
2Q2
1
2m2 +
(
1 − mn
2Q2
)
m +1
2= 0
β ,n
2Q2− 1
1
2m2 − βm +
1
2= 0
am2 + bm + c = 0
m =−b ±
√b2 − 4ac
2a= β +
√
β2 − 1
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 32
Example 3: Solution
1. Arbitrarily pick C2 = 10 nF, a common capacitor value that ischeaply and readily available
2. Select n such that C1
C2
≥ 4Q2, to ensure solution to quadratic
equation for m = R1
R2
is real valued
• Pick n = 8
• Then C1 = 8Q2C2 = 80 nF
3. Solve for m = β +√
β2 − 1, given β = n2Q2 − 1
• m = 1
4. Solve for R2 = 1C2ωn
√nm
• R2 = 2.33 kΩ
5. Solve for R1 = mR2, given m and R2
• R1 = 13.6 kΩ
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 33
Example 3: Magnitude Response
102
103
104
105
−60
−50
−40
−30
−20
−10
0
10
20
Frequency (rad/sec)
Mag
(dB
)
Complex Poles
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 34
Example 3: Phase Response
102
103
104
105
−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Frequency (rad/sec)
Pha
se (
deg)
Complex Poles
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 35
Example 3: MATLAB Code
function [] = SallenKeyLowpass();
RA = inf;
RB = 0;
C2 = 10e-9;
Q = 1;
fn = 1000;
%==========================================================
% Preprocessing
%==========================================================
dCGain = 1 + RB/RA;
wn = 2*pi*fn;
n = 8*Q^2;
b = n/(2*Q^2) -1;
m = b + sqrt(b^2 -1);
C1 = n*C2;
R2 = 1/(wn*C2*sqrt(n*m));
R1 = m*R2;
fprintf(’C1: %5.3f nF\n’,C1*1e9);
fprintf(’C2: %5.3f nF\n’,C2*1e9);
fprintf(’R1: %5.3f kOhms\n’,R1*1e -3);
fprintf(’R2: %5.3f kOhms\n’,R2*1e -3);
N = 1000;
functionName = sprintf(’%s’,mfilename ); % Get the function name
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 36
w = logspace(2,5,N);
mag = [];
phs = [];
fileIdentifier = fopen([ functionName ’.tex’],’w’);
sys = tf([ dCGain] ,[1/(wn^2) 1/(wn*Q) 1]);
[m, p] = bode(sys ,w);
bodeMagnitude = reshape(m,N,1);
bodePhase = reshape(p,N,1);
figure(1);
FigureSet(1,’Slides’);
h = semilogx(w,20* log10(bodeMagnitude ),’r’);
set(h,’LineWidth ’,1.5);
grid on;
xlabel(’Frequency (rad/sec)’);
ylabel(’Mag (dB)’);
title(’Complex Poles’);
xlim ([w(1) w(end)]);
ylim ([-60 20]);
box off;
AxisSet (8);
fileName = sprintf(’%s-Magnitude ’,functionName );
print(fileName ,’-depsc’);
fprintf(fileIdentifier ,’%%==============================================================================\n’
fprintf(fileIdentifier ,’\\ newslide\n’);
fprintf(fileIdentifier ,’\\ slideheading Example \\arabicexc: Magnitude Response\n’);
fprintf(fileIdentifier ,’%%==============================================================================\n’
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 37
fprintf(fileIdentifier ,’\\ includegraphics[scale=1] Matlab/%s\n’,fileName);
figure(2);
FigureSet(2,’Slides’);
h = semilogx(w,bodePhase ,’r’);
set(h,’LineWidth ’,1.5);
grid on;
xlabel(’Frequency (rad/sec)’);
ylabel(’Phase (deg)’);
title(’Complex Poles’);
xlim ([w(1) w(end)]);
ylim ([ -190 10]);
box off;
AxisSet (8);
fileName = sprintf(’%s-Phase’,functionName );
print(fileName ,’-depsc’);
fprintf(fileIdentifier ,’%%==============================================================================\n’
fprintf(fileIdentifier ,’\\ newslide\n’);
fprintf(fileIdentifier ,’\\ slideheading Example \\arabicexc: Phase Response\n’);
fprintf(fileIdentifier ,’%%==============================================================================\n’
fprintf(fileIdentifier ,’\\ includegraphics[scale=1] Matlab/%s\n’,fileName);
fprintf(fileIdentifier ,’%%==============================================================================\n’
fprintf(fileIdentifier ,’\\ newslide \n’);
fprintf(fileIdentifier ,’\\ slideheading Example \\arabicexc: MATLAB Code \n’);
fprintf(fileIdentifier ,’%%==============================================================================\n’
fprintf(fileIdentifier ,’\t \\ matlabcode Matlab/%s.m\n’,functionName );
fclose(fileIdentifier);
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 38
Example 4: Highpass Sallen-Key Filter
v o ( t )
-
+ v s ( t ) R b
R a
R 1
C 1 C 2
R 2
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 39
Example 4: Questions
• Is the input impedance finite?
• What is the output impedance?
• How many capacitors are in the circuit?
• What is the DC gain?
• What is the “high frequency” gain?
• What is the probable order of the transfer function?
• Solve for the transfer function
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 40
Example 4: Work Space
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 41
Example 4: Work Space
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 42
Example 4: Solution
H(s) = K
(
sωn
)2
1 + sQωn
+(
sωn
)2
K = 1 +RB
RA
ωn =1√
R1C1R2C2
Q =1
√
R1C1
R2C2
+√
R1C2
R2C1
+ (1 − K)√
R2C2
R1C1
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 43
Example 5: Bandpass Sallen-Key Filter
v o ( t )
-
+
v s ( t ) R b
R a
R 3
C 2
R 2
R 1
C 1
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 44
Example 5: Questions
• Is the input impedance finite?
• What is the output impedance?
• How many capacitors are in the circuit?
• What is the DC gain?
• What is the “high frequency” gain?
• What is the probable order of the transfer function?
• Solve for the transfer function
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 45
Example 5: Work Space
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 46
Example 5: Work Space
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 47
Transfer Function Synthesis
You can build almost any transfer function with integrators, adders, &subtractors
Y (s) = H(s)X(s) =N(s)
D(s)X(s) =
X(s)
D(s)N(s) = U(s)N(s)
where
U(s) ,1
D(s)X(s)
D(s) , bnsn + bn−1sn−1 + · · · + b1s + b0
X(s) = bnsnU(s) + bn−1sn−1U(s) + · · · + b1sU(s) + b0U(s)
snU(s) =1
bn
(
X(s) − bn−1sn−1U(s) − · · · − b1sU(s) − b0U(s)
)
Y (s) = U(s)N(s)
= ansnU(s) + an−1sn−1U(s) + · · · + a1sU(s) + a0U(s)
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 48
Transfer Function Synthesis Diagram
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 49
Transfer Function Synthesis Clean Diagram
-
+
y ( t )
x ( t )
ΣΣΣΣ
Σ
Σ
Σ
Σ1
bn
1
s
1
s
1
s
bn−1
b2
b1
b0
anan−1 a2 a1 a0
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 50
Example 6: Transfer Function Synthesis
Draw the block diagram for the following transfer function:
H(s) =s3 + 5s2 + 2
s4 + 17s3 + 82s2 + 130s + 100
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 51
Analog Filters Summary
• There are many types of filters
• Second-order filters can implement the four basic types
– Second-order
– Lowpass & Highpass
– Bandpass & Bandstop
– Notch
• For a given H(s), there are many implementations & tradeoffs
• Second order filters are fundamentally different than first-orderfilters because they can have complex poles
– Causes a resonance
J. McNames Portland State University ECE 222 Second Order Filters Ver. 1.04 52