Butterworth Design

# Butterworth Design - ELCT 301 Lab Report#5 Butterworth...

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ELCT 301 Lab Report #5 Butterworth Filter Design Due: 4/25/2008 ___________________ Jared Tucker Department of Electrical Engineering University of South Carolina Columbia, SC 29208

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Laboratory Grade: Pre-lab computation ____ of 10 Technical Content ____ of 60 Format/Presentation Clarity ____ of 20 Other ____ of 10 Late Deductions ____ ____ of 100 Student Comments: Grader Comments:
Butterworth Filter Design I. Introduction The topic of this lab revolved around the design and construction of a Sallen Key, 4 th order Butterworth filter. The concept of higher order filters may sound complex however, it is my goal here to demonstrate the simplicity of higher order design once the original transfer function is derived. II. Background First off what is a filter anyway? Perhaps the simplest analogy is that of a window screen. If bugs didn’t exist, there would be no need of screens. However, because we don’t want mosquitoes sucking our blood at night, we put up screens to keep out undesired intruders. This is precisely what a filter does. It removes any undesired signal(s) from the signal(s) of interest. So the mechanical filter is pretty simple to visualize. The electrical filter is a little more involved. Let us consider the simple circuit of Figure (1). (a) Output Resistor (b) Output Capacitor Figure 1. RC Filter Circuit We know that the capacitive impedance is defined as C j Z c = ϖ 1 (1) where f π 2 = . This is what makes the capacitor so very useful. The impedance of the capacitor depends on the frequency. Thus, if one were to increase the frequency, the resistance of the capacitor would decrease. While the converse holds for a decreasing frequency increasing the resistance. Taking the limit as C j Z f c = 1 (2) then Z c would approach 0 or short circuit. Conversely

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C j Z f c = ϖ 1 0 (3) Z c would approach or open circuit. If we rewrite (3) in the frequency form we achieve: C S Z c = 1 (4) where S is merely j , nothing fancy. It’s just a simpler way of representing (1). For every circuit there is a transfer function G(s). The transfer function tells us information about the circuit or system if you will. The transfer function is nothing more than a ratio of two polynomials. The poles (eigenvalues) of the transfer function which are the roots of the denominator well always remain the same as long as the system is not changed. The zeroes of the transfer function G(s) (numerator) will change depending on where the output of the circuit is coming from. Let us again consider the circuit of Figure (1). If we were to take Vo across R as our output, the transfer function would become RC s s s G 1 ) ( + = (5) taking the limit of (5) as s would yield 1. Then taking the limit of (5) as 0 s would yield 0. This brings me too perhaps the most intuitive way of analyzing any transfer function. Consider Figure (2) Figure (2) Limit Graph for High pass filter I call this the 30 second filter analysis. Not knowing a thing about circuit theory and by
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Butterworth Design - ELCT 301 Lab Report#5 Butterworth...

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