Application: low-pass filter Sallen–Key topology

1 application: low-pass filter

1.1 poles , zeros
1.2 design choices
1.3 example
1.4 input impedance

application: low-pass filter

figure 2: unity-gain low-pass filter implemented sallen–key topology

an example of unity-gain low-pass configuration shown in figure 2. operational amplifier used buffer here, although emitter follower effective. circuit equivalent generic case above with

z

1


=

r

1


,


z

2


=

r

2


,


z

3


=


1

s

c

1





,


z

4


=


1

s

c

2





.


{\displaystyle z_{1}=r_{1},\quad z_{2}=r_{2},\quad z_{3}={\frac {1}{sc_{1}}},\quad z_{4}={\frac {1}{sc_{2}}}.}

the transfer function second-order unity-gain low-pass filter is

h
(
s
)
=



ω

0


2




s

2


+
2
α
s
+

ω

0


2





,


{\displaystyle h(s)={\frac {\omega _{0}^{2}}{s^{2}+2\alpha s+\omega _{0}^{2}}},}

where undamped natural frequency




f

0




{\displaystyle f_{0}}

, attenuation



α


{\displaystyle \alpha }

, , q factor



q


{\displaystyle q}

(i.e., damping ratio



ζ


{\displaystyle \zeta }

) given by

ω

0


=
2
π

f

0


=


1


r

1



r

2



c

1



c

2







{\displaystyle \omega _{0}=2\pi f_{0}={\frac {1}{\sqrt {r_{1}r_{2}c_{1}c_{2}}}}}

and

2
α
=
2
ζ

ω

0


=



ω

0


q


=


1

c

1





(


1

r

1




+


1

r

2




)

=


1

c

1





(




r

1


+

r

2





r

1



r

2





)

.


{\displaystyle 2\alpha =2\zeta \omega _{0}={\frac {\omega _{0}}{q}}={\frac {1}{c_{1}}}\left({\frac {1}{r_{1}}}+{\frac {1}{r_{2}}}\right)={\frac {1}{c_{1}}}\left({\frac {r_{1}+r_{2}}{r_{1}r_{2}}}\right).}

so,

q
=



ω

0



2
α



=




r

1



r

2



c

1



c

2





c

2



(

r

1


+

r

2


)




.


{\displaystyle q={\frac {\omega _{0}}{2\alpha }}={\frac {\sqrt {r_{1}r_{2}c_{1}c_{2}}}{c_{2}\left(r_{1}+r_{2}\right)}}.}

the



q


{\displaystyle q}

factor determines height , width of peak of frequency response of filter. parameter increases, filter tend ring @ single resonant frequency near




f

0




{\displaystyle f_{0}}

(see lc filter related discussion).

poles , zeros

this transfer function has no (finite) zeros , 2 poles located in complex s-plane:

s
=
−
α
±



α

2


−

ω

0


2




.


{\displaystyle s=-\alpha \pm {\sqrt {\alpha ^{2}-\omega _{0}^{2}}}.}

there 2 zeros @ infinity (the transfer function goes 0 each of s terms in denominator).

design choices

a designer must choose



q


{\displaystyle q}

,




f

0




{\displaystyle f_{0}}

appropriate application.



q


{\displaystyle q}

value critical in determining eventual shape. example, second-order butterworth filter, has maximally flat passband frequency response, has



q


{\displaystyle q}

of



1

/



2




{\displaystyle 1/{\sqrt {2}}}

. comparison, value of



q
=
1

/

2


{\displaystyle q=1/2}

corresponds series cascade of 2 identical simple low-pass filters.

because there 2 parameters , 4 unknowns, design procedure typically fixes ratio between both resistors between capacitors. 1 possibility set ratio between




c

1




{\displaystyle c_{1}}

,




c

2




{\displaystyle c_{2}}





n


{\displaystyle n}

versus



1

/

n


{\displaystyle 1/n}

, ratio between




r

1




{\displaystyle r_{1}}

,




r

2




{\displaystyle r_{2}}





m


{\displaystyle m}

versus



1

/

m


{\displaystyle 1/m}

. so,

r

1





=
m
r
,





r

2





=
r

/

m
,





c

1





=
n
c
,





c

2





=
c

/

n
.






{\displaystyle {\begin{aligned}r_{1}&=mr,\\r_{2}&=r/m,\\c_{1}&=nc,\\c_{2}&=c/n.\end{aligned}}}

as result,




f

0




{\displaystyle f_{0}}

,



q


{\displaystyle q}

expressions reduced to

ω

0


=
2
π

f

0


=


1

r
c





{\displaystyle \omega _{0}=2\pi f_{0}={\frac {1}{rc}}}

and

q
=



m
n



m

2


+
1



.


{\displaystyle q={\frac {mn}{m^{2}+1}}.}




figure 3: low-pass filter, implemented sallen–key topology, fc = 15.9 khz , q = 0.5

starting more or less arbitrary choice e.g. c , n, appropriate values r , m can calculated in favor of desired




f

0




{\displaystyle f_{0}}

,



q


{\displaystyle q}

. in practice, choices of component values perform better others due non-idealities of real operational amplifiers. example, high resistor values increase circuit s noise production, whilst contributing dc offset voltage on output of opamps equipped bipolar input transistors.

example

for example, circuit in figure 3 has




f

0


=
15.9
 

khz



{\displaystyle f_{0}=15.9~{\text{khz}}}

,



q
=
0.5


{\displaystyle q=0.5}

. transfer function given by

h
(
s
)
=


1

1
+





c

2


(

r

1


+

r

2


)

⏟






2
ζ


ω

0




=


1


ω

0


q





s
+





c

1



c

2



r

1



r

2



⏟




1

ω

0


2






s

2





,


{\displaystyle h(s)={\frac {1}{1+\underbrace {c_{2}(r_{1}+r_{2})} _{{\frac {2\zeta }{\omega _{0}}}={\frac {1}{\omega _{0}q}}}s+\underbrace {c_{1}c_{2}r_{1}r_{2}} _{\frac {1}{\omega _{0}^{2}}}s^{2}}},}

and, after substitution, expression equal to

h
(
s
)
=


1

1
+





r
c
(
m
+
1

/

m
)

n

⏟






2
ζ


ω

0




=


1


ω

0


q





s
+





r

2



c

2



⏟




1



ω

0




2






s

2





,


{\displaystyle h(s)={\frac {1}{1+\underbrace {\frac {rc(m+1/m)}{n}} _{{\frac {2\zeta }{\omega _{0}}}={\frac {1}{\omega _{0}q}}}s+\underbrace {r^{2}c^{2}} _{\frac {1}{{\omega _{0}}^{2}}}s^{2}}},}

which shows how every



(
r
,
c
)


{\displaystyle (r,c)}

combination comes



(
m
,
n
)


{\displaystyle (m,n)}

combination provide same




f

0




{\displaystyle f_{0}}

,



q


{\displaystyle q}

low-pass filter. similar design approach used other filters below.

input impedance

the input impedance of second-order unity-gain sallen–key low-pass filter of interest designers. given eq. (3) in cartwright , kaminsky as

z
(
s
)
=

r

1






s

′

2



+

s
′


/

q
+
1



s

′

2



+

s
′

k

/

q



,


{\displaystyle z(s)=r_{1}{\frac {s ^{2}+s /q+1}{s ^{2}+s k/q}},}

where




s
′

=


s

ω

0






{\displaystyle s ={\frac {s}{\omega _{0}}}}

,



k
=



r

1




r

1


+

r

2





=


m

m
+
1

/

m





{\displaystyle k={\frac {r_{1}}{r_{1}+r_{2}}}={\frac {m}{m+1/m}}}

.

furthermore,



q
>




1
−

k

2



2





{\displaystyle q>{\sqrt {\frac {1-k^{2}}{2}}}}

, there minimal value of magnitude of impedance, given eq. (16) of cartwright , kaminsky, states that

|

z
(
s
)


|


min


=

r

1




1
−



(
2

q

2


+

k

2


−
1

)

2




2

q

4


+

k

2


(
2

q

2


+

k

2


−
1

)

2


+
2

q

2





q

4


+

k

2


(
2

q

2


+

k

2


−
1
)







.


{\displaystyle |z(s)|_{\text{min}}=r_{1}{\sqrt {1-{\frac {(2q^{2}+k^{2}-1)^{2}}{2q^{4}+k^{2}(2q^{2}+k^{2}-1)^{2}+2q^{2}{\sqrt {q^{4}+k^{2}(2q^{2}+k^{2}-1)}}}}}}.}

fortunately, equation well-approximated by

|

z
(
s
)


|


min


≈

r

1





1


q

2


+

k

2


+
0.34






{\displaystyle |z(s)|_{\text{min}}\approx r_{1}{\sqrt {\frac {1}{q^{2}+k^{2}+0.34}}}}

for



0.25
≤
k
≤
0.75


{\displaystyle 0.25\leq k\leq 0.75}

.



k


{\displaystyle k}

values outside of range, 0.34 constant has modified minimal error.

also, frequency @ minimal impedance magnitude occurs given eq. (15) of cartwright , kaminsky, i.e.,

ω

min


=

ω

0







q

2


+



q

4


+

k

2


(
2

q

2


+

k

2


−
1
)




2

q

2


+

k

2


−
1




.


{\displaystyle \omega _{\text{min}}=\omega _{0}{\sqrt {\frac {q^{2}+{\sqrt {q^{4}+k^{2}(2q^{2}+k^{2}-1)}}}{2q^{2}+k^{2}-1}}}.}

this equation can approximated using eq. (20) of cartwright , kaminsky, states that

ω

min


≈

ω

0






2

q

2




2

q

2


+

k

2


−
1




.


{\displaystyle \omega _{\text{min}}\approx \omega _{0}{\sqrt {\frac {2q^{2}}{2q^{2}+k^{2}-1}}}.}






^ stop-band limitations of sallen–key low-pass filter.
^ cartwright, k. v.; e. j. kaminsky (2013). finding minimum input impedance of second-order unity-gain sallen-key low-pass filter without calculus (pdf). lat. am. j. phys. educ. 7 (4): 525–535.