**4. Second-Order** *Gm***-***C* **Bandpass Filter**

The second-order *Gm*-*C* BPF illustrated in Figure 10 has been implemented by using the linearized transconductor described in the previous section and depicted in Figure 7, which is based in turn on the bootstrapped bulk-driven voltage buffer shown in Figure 2b. The filter structure incorporates four transconductors in order to be able to set independently the center frequency, *ω*0, the gain at the center frequency, |*H*(*ω*0)|, and the quality factor, *Q*. In our application, only *ω*<sup>0</sup> is intended to be swept, whereas |*H*(*ω*0)| and *Q* will have fixed values. Nevertheless, the configuration selected allows for keeping constant a given quality factor while the center frequency is swept. In addition, there is an additional degree of freedom in the structure that allows for maximizing the dynamic range of the BPF. Indeed, the other node in the filter, *vOUT*,*LP* in Figure 10, provides a lowpass response. The lowpass response presents an overdamping at the frequency of the poles that is a function of the quality factor selected for the BPF. As a consequence, a noticeable peak appears at that node at *ω*0, thus limiting the dynamic response of the overall biquad. This fact can be avoided with the structure illustrated in Figure 10, as the value of *Q* can be set through the ratios of the active (transconductance) or the passive (capacitor) elements, which allows for decreasing the overall gain of the lowpass response, thus decreasing the maximum signal amplitude achieved at *vOUT*,*LP* at the center frequency of the BPF.

**Figure 10.** Second-order *Gm*-*C* bandpass filter.

The transfer function of the selected BPF can be written as:

$$H(s)\_{BP} = \frac{\frac{G\_{m1}}{C\_2}s}{s^2 + \frac{G\_{m4}}{C\_2}s + \frac{G\_{m2}G\_{m3}}{C\_1C\_2}}\tag{14}$$

where *Gmi*, with *i* = 1 to 4, represents the effective transconductance of the *i*-th transconductor and *C*<sup>1</sup> and *C*<sup>2</sup> are integrated capacitors. The gain at the center frequency, |*H*(*ω*0)|, the center frequency, *ω*0, and the quality factor, *Q*, can be obtained from (14) in a straightforward manner and expressed as:

$$|H(\omega\_0)| = \frac{G\_{\text{m1}}}{G\_{\text{m4}}} \tag{15a}$$

$$
\omega\_0 = \sqrt{\frac{G\_{m2}G\_{m3}}{C\_1C\_2}}\tag{15b}
$$

$$Q = \sqrt{\frac{C\_2}{C\_1} \cdot \frac{G\_{m2}G\_{m3}}{G\_{m4}^2}}\tag{15c}$$

The intended application of the BPF is the separation of signals with different frequencies in a multi-frequency bioimpedance measurement system. Thus, the selectivity of the filter must be relatively high, which requires a moderately high value of the quality factor. A hand-analysis of the response at node *vOUT*,*LP* of the filter reveals that an optimal choice in order not to limit the dynamic range of the BPF response is obtained when *C*<sup>1</sup> = *C*<sup>2</sup> = *C*. Thus, the following equality has been established for the transconductances *Gm*<sup>2</sup> = *Gm*<sup>3</sup> = *k* · *Gm*<sup>4</sup> = *k* · *Gm* so that the factor *Q* is equal to parameter *k*. In addition, transconductors *Gm*<sup>1</sup> and *Gm*<sup>4</sup> have been sized to be equal, *Gm*<sup>1</sup> = *Gm*<sup>4</sup> = *Gm*, in order to have a gain at the center frequency equal to unity. Therefore, the expressions in (15a–15c) can be rewritten as:

$$|H(\omega\_0)| = 1\tag{16a}$$

$$
\omega\_0 = k \cdot \frac{G\_m}{\mathbb{C}} \tag{16b}
$$

$$Q = k \tag{16c}$$

The factor *k* has been achieved by properly sizing the pseudo-resistor in each transconductor, whereas the rest of the *V*-to-*I* converter has been kept equal. The response of the BPF, in particular the center frequency, can be programmed by fixing voltage *VBULK* to an appropriate value and by tuning the value of the control voltage *VTUN* around it. For *VTUN* = *VDD*, the transconductors achieve their minimum transconductance value, thus leading to the lowest value of *ω*0. Conversely, when *VTUN* reaches the minimum reliable value, the *Gm* is maximized and also is the value of the center frequency.
