![]() ![]() The remaining two design steps are somewhat intuitive, but the rules are simple. The equation produced in Step 3 is now the defining equation for a network of integrator blocks that will perform the required filtering function. Rearrange to Obtain an Expression for V OUT Divide by the Highest Power of S to Obtain S² × V OUT ω 0/q × S × V OUT ω 0² × V OUT = ω 0² × V IN Step 2. The expression obtained at each stage is given below. Three basic mathematical steps are required. ![]() This is because the final circuit will be composed of integrators, i.e., functions of 1/S. Note that all frequency-dependent terms (occurrences of S) in the resulting expression must appear in the denominator terms. Transfer function of a second-order filter with lowpass response.Ī sequence of mathematical steps is then applied to the transfer function to obtain an expression of the form: The technique can be employed for any filter type and is easily extended to higher order systems.įigure 3. The equation in Figure 3, which represents a second-order lowpass-filter response, will be used for illustration. The design process starts with the required filter transfer function. Transfer function of the integrator circuit block in Figure 1. ![]() The response (output) of this circuit to the input voltage is gain diminishing with frequency at a rate of 6dB per octave with unity gain occurring at a frequency in hertz of 1/2 πCR.įigure 2. The transfer functions of the integrator in Figure 1 and its symbolic representation are shown in the expression in Figure 2. Op-amp-based (linear) integrator circuit block and symbolic representation. The method can be applied to both continuous-time and switching-filter (e.g., switched capacitor filter) designs.įigure 1. The system uses simple op-amp integrator blocks, an example of which is shown in Figure 1. However, there is an alternative to this approach, a method that is precise, easy to apply, and uses integrator blocks and some simple mathematical manipulation to produce filter responses of any order. When a nonstandard filter response is required, it is often left to the circuit designer to produce a solution using his or her own "standard" set of filter networks. Much literature and software has been published on the design and implementation of standard filter responses. The method is precise, easy to apply, and an alternative to a \"standard\" set of filter networks when a nonstandard filter response is needed. This application note explains a method that uses integrator blocks and some simple mathematical manipulation to produce filter responses of any order. ![]()
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