AN121
Dataforth Corporation
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DID YOU KNOW The number "googol" is ten raised to the hundredth power or 1 followed by 100 zeros. Edward Kasner (18781955) a noted mathematician is best remembered for the "googol". Dr. Kasner asked his nephew, Milton Sirotta, what he would call a number with 100 zeros; nine-year-old Milton suggested "googol." and the word "googol" was born. Dr. Kasner topped Milton with a bigger number the "googolplex", which is googol raised to the googol power. Some estimate that writing the digits in a googolplex requires more space than in the known universe.
Low Pass Filter Rise Time vs Bandwidth
Preamble Scores of text books and hundreds of papers have been written about numerous filter topologies that have a vast spectrum of behavioral characteristics. These filters use a variety of circuit topologies made possible by today's integrated circuits. Modern microprocessors even provide a means whereby software can be used to develop unique digital filters without analog circuitry. Clearly, it is far beyond the scope of this application note to cover all these filter topics. The objective of this application note is to examine the low pass (LP) filter topology attributes that are common to both the leading edge rise time response to an input step voltage and the amplitude frequency response. The following bullet list represents the focus and strategy used in this Application Note. It is assumed that readers are familiar with the fundamental basics of circuit analysis. Basic circuit analysis fundamentals will be mentioned to stimulate the reader's memory. Equations will be given without detail derivation. The examination of the LP filter's time and frequency response will be predominately a MATLAB graphical approach (graphs are worth hundreds of words and equations) as opposed to the typical text book approach, which often analyzes filter behavior using the position of circuit poles in the left hand side of the s-plane. More on "poles" later. The LP filter topology of choice for analysis is the Sallen-Key Active 2-pole circuit with a passive RC section added to give an active-passive 3-pole LP filter. Filter analysis is limited to leading edge rise times and frequency response over the bandwidth. Phase analysis is not included. Only filters without numerator zeros will be analyzed. More on "zeros" later. A Few Little Reminders Filter circuit topologies contain resistors, capacitors, and inductors (modern LP filters seldom use inductors). In the early days (100+ years ago) solutions to filter circuits used differential equations since capacitor and inductor behaviors were (are) derivatives of time functions. Fortunately, numerous analytical giants have given us analytical tools for exploring circuit behavior. For example; (a) Charles Proteus Steinmetz introduced the "phasor" with complex numbers for circuit analysis. (b) Pierre Simon Laplace developed a mathematical transformation that when applied to functions of time introduces a new variable "s" that obeys simple rules of arithmetic. (c) Jean Baptiste Joseph Fourier showed that a typical time function can be expressed as a sum of individual sinusoidal terms each with their individual amplitude, frequency, and phase. (d) Leonhard Euler developed the famous exponential equation "Exp(j*x) = Cos(x) + j*Sin(x)". Of course, we must not forget Georg Simon Ohm and Robert Gustav Kirchhoff. Without the contributions of these giants, circuit analysis in both time and frequency domains would be most difficult. The following list illustrates some reminders of R, C, and L behavior.

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