A biquad transfer function can be realized straightforwardly with a pair of integrators in series, each introducing a factor 1/s, with their outputs added to the input signal in a summer, as illustrated in the circuit at the left. This circuit looks complex at first, but if you carry out a circuit analysis its operation will become clear. Assume that the signal at the output of the summer, the leftmost operational amplifier, is x. Then find the signals that are added at the summer in terms of x, and finally solve for x in terms of the input signal v1. You will find that the result is in the form of a high pass transfer function. The outputs of the following two amplifiers will be a bandpass function and a lowpass function. This amplifier realizes the three basic amplifier types, which can then be combined as desired.

The effect of an integrator stage is illustrated at the right, to help in the analysis. The parameters f0, Q and the bandpass gain at the centre frequency are given in terms of the circuit parameters in the figure. These expressions are sufficient for the design of a filter. In any case, resistances should be 5k or greater to avoid overloading the operatonal amplifiers. The frequency-determining values of RF and C must be carefully matched if a high Q is to be realized.

A circuit I tested used the following values: C = 0.01μF, RF = 16k, RG = R = 10k, RQ = 1k (not an op-amp load). The amplifiers were LM833 dual op-amps, which are intended for audio applications. This circuit should have f0 = 1kHz, Q = 5.5 and ABP = 1. I applied a 2V p-p input signal whose frequency was measured by a counter. The input and ouput were displayed on an oscilloscope. The HP, LP and BP outputs should all be examined as the frequency is varied to either side of 1 kHz. Note the peaks on the HP and LP responses, and the phase relation between input and output at the peaks. Estimate the Q from the BP response by finding the 1/&root;2 points.

Some references show the input applied to the noninverting input of the summer. Of course, this changes the sign of the gains, but otherwise seems to work in about the same way. An accurate analysis is more difficult for this configuration. It does not seem to have any advantage over input to the inverting input.

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