Banerjee - Automated Electronic Filter Design

Basic loop equations are derived from Kirchhoffs current/voltage laws (KCL/KVL). These loop equations are differential equations, sometimes nonlinear. Typically, these differential equations are converted to more tractable algebraic equations, using Laplace transforms, effectively going from the time to frequency domain.

In the filter transfer function H ( s ), the Laplace transform of the unit impulse response of the filter is obtained by evaluating H ( s ) at s = jw (in general, s = a + jw and s = jw represent a pure sinusoidal input). For calculation purposes, it is the ratio of the output to the input voltage in the frequency domain. The transfer function is almost always in a denominator-numerator polynomial form. The goal is to determine the roots of these two polynomials in the complex s -plane to obtain the poles and zeros. Poles and zeros are, respectively, the roots of the denominator and numerator polynomials. Only poles in the left half of the complex s -plane guarantee filter stability and need to be complex conjugates of each other to ensure real-valued coefficients in the differential equations representing the filter. Most importantly, for the generic case, the pole (and zero) values are analytical expressions involving the capacitors, inductors, and resistors to be used in the filter. Both the denominator and numerator polynomials need to be factorized to extract the zeros and poles of the transfer function. Sometimes, instead of using the transfer function, expressions for the filter insertion loss or loss magnitude versus frequency are used [], but that scheme involves first evaluating the loss expression in terms of the filter component values and then using a judicious combination of heuristics, ladder network, and precalculated table values.

The most difficult step is to use the values of the poles and zeros evaluated previously and to determine values of each of the resistive and reactive components (capacitors, inductors) in the filter circuit. In the generic case, using the analytical expressions for the values of poles and zeros in combination with the predefined numerical values of design specifications (cutoff frequency, pass/stop band ripple, etc.) is an extremely complicated, manual, and thus error-prone/time-consuming/trial-and-error process. The designer can also utilize the causality and stability conditions and set resistor values to 1 to aid/simplify this calculation. Clearly, this step is feasible for low-order filters only.