In general, audio equalizers need second-order bandpass filters. Such cells require an easy and independent tuning of their parameters: the natural or central frequency, ω₀; the quality factor, Q; and the maximum bandpass gain, k. The use of cells with independent adjustments could require state-variable topologies. Unfortunately, this sort of structure usually needs at least three operational amplifiers. The basis for an alternative uses SAB (single-amplifier-biquadratic) filters. These cells allow obtaining second-order bandpass filters, but they have two main drawbacks: The quality factors that you can obtain with these cells have a practical maximum limit, and you cannot independently tune the three characteristic parameters.

This Design Idea instead uses the TQE (transimpedance-Q-enhancement) structure in an audio-equalizer (Figure 1). This cell has two advantages when you use it in equalizer circuits: You can adjust the three characteristic parameters in an independent way, but it uses only two operational amplifiers per cell. Reference 1 presents the generic TQE topology. Figure 1 shows the configuration that implements a bandpass filter based on the generic structure. This structure, which processes current-input signals, shows low-impedance input without the resistor R_IN. Considering that R₁ and R₃ are equal in value and that all the capacitors are equal to C, the transimpedance, Z(s), is:

However, by adding R_IN, the input has high impedance, allowing the processing of voltage input signals because R_IN provides the required voltage-to-current conversion. In this way, the input-to-output transfer function, H(s), is:

Thus, the circuit implements a second-order bandpass-transfer function; the following equations yield the central frequency, ω₀, and the quality factor, Q:

and the value of the gain, k, is:

Thus, you can make the adjustments of ω₀, Q, and k with R₁, R₂, and R_IN, respectively.

You can use the bandpass cell in Figure 1 in an audio equalizer. Figure 2 shows a possible implementation of a graphic equalizer. The basis for the circuit is a bank of bandpass TQE cells. Note that the cells are TQE with low-impedance input. Thus, the input network’s R_IN converts V_IN(t) and V_OUT(t) to the corresponding input current, I_IN(t). Adjusting potentiometer R_IN with its wiper to the far left (X_I→0) accentuates the frequency band that the corresponding cell covers in the overall circuit output. On the other hand, positioning the wiper of R_IN to the far right (X_I→1) causes a large amount of negative feedback to occur at this same frequency, thus causing attenuation in the forward-signal path. In each case, the remaining filters’ TQE_I receives percentages of both the input signal, V_IN(t), and the output signal, V_OUT(t), in ratios that their respective potentiometer settings determine.

You can derive the overall transfer function from Figure 2. The output voltage, V_OUT(s), of the equalizer is:

where Z_I(s) are the cells’ transimpedances, and

If you define

then the transfer function of the equalizer becomes

Now, you can investigate the effect of various settings of the potentiometers. For example, in the case with all of the controls centered, X_I equals 0.5 for each band. Then, V_OUT(s)/V_IN(s)=–1, as you would expect in a typical equalizer’s response. Setting band I=1 to a value of X_I and all other bands flat—that is, X_I=0.5 for I=2, 3, ... n), you obtain:

which represents a bandpass filter with unity gain, or 0 dB, in the stopband and a gain of A_O at resonance, and M is a constant representing the average value of the complete summation. M is approximately 1.3, or approximately 2.3 dB (Reference 2). Note that this gain can be higher—that is, boost—or lower than one. Considering as typical values M=1.3, A=1, and k=1, you can simplify the equation for passband gain A_O, which is equal to the ratio of the s term coefficients, as:

As an example, consider the case of an octave-band equalizer with 10 bands. In this case, the value of the quality factor for each band is about 1.42 (Reference 3), and the typical central frequencies of the 10 sections are 32 Hz to 16 kHz. Adjusting R_IN in the input of the cell TQE with its wiper to the left boosts the frequency band that the corresponding cell covers in the overall circuit output. For instance, if X₁ is 0.1, then A_O is approximately 13 dB. On the other hand, positioning the wiper of R_IN to the right causes attenuation in the forward-signal path. So, if X₁ is 0.9, then A_O is approximately –13 dB. You must have a minimum input impedance in each cell for the input voltage, V_IN(t), and the feedback voltage, V_OUT(t). Thus, the inclusion of two resistors in series with each potentiometer R_IN in Figure 2 guarantees this resistance.