**DESIGN OF a 5 ^{th} ORDER BUTTERWORTH LOW-PASS FILTER
USING SALLEN & KEY CIRCUIT**

**Background Theory:**

Filters are classified according to the functions
that they are to perform, in terms of ranges of frequencies. We will be dealing
with the *low-pass* *filter*, which has the property that
low-frequency excitation signal components down to and including direct
current, are transmitted, while high-frequency components, up to and including
infinite ones are blocked. The range of low frequencies, which are passed, is called
the *pass band* or the *bandwidth* of the filter. It extends from ω=0 to ω= ω_{c}
rad/sec (f_{c} in Hz). The highest frequency to be transmitted is ω_{c},_{ }which
is also called the *cutoff frequency*. Frequencies above cutoff are
prevented from passing through the filter and they constitute the filter *stopband*.

The ideal response of a *low-pass filter* is
shown above. However, a physical circuit cannot realize this response. The
actual response will be in general as shown below.

It can be seen that a small error is allowable in
the *pass band, *while the transition from the *pass band *to the *stopband
*is not abrupt.

The sharpness of the transition from stop band to
pass band can be controlled to some degree during the design of a *low-pass*
*filter*.

The ideal low-pass filter response can be approximated by a rational function approximation scheme such as the Butterworth response.

**The
Butterworth Response**

_{}

Normalizing *H*_{0}=1 and _{}

Then _{}

_{}

finding the roots of *D(s)*

_{}

Example:

For n=5

All the poles are:

-1.0000

-0.8090 + 0.5878i

-0.8090 - 0.5878i

-0.3090 + 0.9511i

-0.3090 - 0.9511i

0.3090 + 0.9511i

0.3090 - 0.9511i

1.0000

0.8090 + 0.5878i

0.8090-0.5878

POLE LOCATIONS

The poles are distributed over the circle of
radius 1 (_{}). Never a pole in the imaginary axis.

Finding *H(s)* from *H(s) H(-s)*:

*H(s)* is assigned all RHS poles *H(-s) *is assigned all LHS poles

Following this procedure, the Butterworth LPF *H(s)*
(*H _{0}=1, w_{c}=1rad/sec*) can be found for various
filters of order n.

_{}

We can use MATLAB to get this denominator polynomial (Butterworth polynomial)

__In
MATLAB (code):__

all_poles=roots([(-1)^n, zeros(1,2*n-1),1])

poles=all_poles(find(real(all_poles)<0))

Den=poly(poles)

**Circuit Design:**

We want to design of a
fifth order Butterworth *low-pass* *filter* with a cutoff frequency
of 10KHz.

_{}

During the design we make use of magnitude and
frequency scaling and also of the uniform choice of _{}as a characterizing frequency will appear in all design
steps, except for the last where the de-normalized (actual) values will be
found.

**Circuit Implementation:**

Implementation of the circuit is done using the Sallen & Key Topology.

_{}

this is of the general form

_{}

If k=1,

_{}

taking _{}

_{}

To realize a 5^{th} order
BLPF one Sallen & Key stage with a single op-amp is required for every
complex-conjugate pole pair. Since n=5 (odd), an additional negative pole is
required and we use an RC/voltage follower. Also we made the choice of K=1,
which requires that the inverting op-amp circuit be replaced by a voltage
follower as shown below.

_{}

_{}

To find actual values:

Make all resistors=_{}

Frequency scaling=_{}

Multiplying each capacitor by _{}

_{}

**Performance Measures:**

Cutoff frequency=10KHz

Frequency (KHz) |
Vin (mV) |
Vout
(mV) |

1.502 |
500 |
498.5 |

2.009 |
500 |
493.5 |

5.782 |
500 |
493.7 |

9.001 |
500 |
487 |

9.6 |
500 |
387.5 |

10.04 |
500 |
245.7 |

11.01 |
500 |
225.7 |

12.04 |
500 |
187.5 |

13.02 |
500 |
115.6 |

14.02 |
500 |
90.62 |

15.06 |
500 |
75.00 |

16.01 |
500 |
62.50 |

17.01 |
500 |
47.50 |

Ideal response:

Actual response: From the recorded values after measurements.

Measured dB gain values vs. log frequency
values

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**Circuit Diagram:**

**Final Circuit**

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**Parts:**

Part |
Quantity |

LM324N Low Power Quad Operational Amplifier |
1 |

1 K Ω resistor |
5 |

0.016 μF capacitor |
1 |

0.019 μF capacitor |
1 |

0.013 μF capacitor |
1 |

0.051 μF capacitor |
1 |

0.0049 μF capacitor |
1 |

9 V battery |
2 |

**References:**

** **

Deliyannis T., Yichuang Sun, and J. K. Fidler
1999, ** Continuous-time Active Filter Design**, CRC Press, New York.

Van Valkenburg M. E., ** Analog Filter Design**,
1982, Oxford University Press, New York.

Chen Wai-Kai, ** Passive & Active Filter
Design**, Chapter 2, pp. 50-92

Huelsman L. P. and P. E. Allen, *Introduction
to the Theory and Design of Active Filters*

__RELATED WEB SITES:__

__ __

http://www.eecs.uic.edu/~jmorisak/blpf.htm

http://www.planetee.com/planetee/servlet/DisplayDocument?ArticleID=3634

http://ourworld.compuserve.com/homepages/f_ostrander/afilterw.htm

http://www.microchip.com/10/tools/analog/flab/index.htm

http://members.aol.com/johnpomann/speakers/theory.htm

http://www.dsp.rice.edu/software/RU-FILTER/butter/gui.html

http://members.aol.com/maxfro/private/butter.html

http://members.tripod.com/michaelgellis/filter.html

**Questions:**

Email
to jmorisak@eecs.uic.edu