A 4th-order 100 μA Diode-RC-based filter with 5 dBm-IIP₃ at 24 MHz cut-off frequency

Abstract

This paper presents a 4th-order low-pass continuous-time Diode-RC filter with 24 MHz cut-off frequency. The filter structure is based on the combination of positive (passive) and negative (using cross-coupled MOS transistors) cells to synthesize complex-conjugate-poles pairs. Moreover, the filter is designed and optimized to have high-linearity along all the filter pass-band. The filter exhibits 16 dBm-$\text {IIP}_3$ with 2–3 MHz input tones and 5 dBm-$\text {IIP}_3$ with 23–24 MHz, whereas the current consumption remains limited to $100 \, \upmu \hbox {A}$ from 1.8 V supply voltage.

Appendices

Appendix 1

The quality factor trimming detailed in Sect. 3.3 is based tuning of each capacitors in the circuit in Fig. 4, which would be adjusted in order to compensate to any mismatch between the impedances R and $R^*$, defined in Figs. 1 and 2. The transfer function is calculated starting from the single-ended model shown in Fig. 18. The Kirchhoff’s current law is applied at all nodes, obtaining the system (18).

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{v_\textit{in} - v_a}{R_1} -v_a \cdot s\cdot C_1 - (-v_a - v_b)\cdot g_\textit{m,n}=0 \\ - (-v_a - v_b)\cdot g_\textit{m,n} - v_b \cdot s \cdot C_2 - \dfrac{v_b - v_c}{R_3} = 0 \\ \dfrac{v_b - v_c}{R_3} -v_c \cdot s\cdot C_3 - (-v_\textit{out} - v_c)\cdot g_\textit{m,p} =0\\ (-v_\textit{out} - v_c)\cdot g_\textit{m,p} - s\cdot C_3 \cdot v_\textit{out} =0 \end{array}\right. } . \end{aligned}$$

(18)

Fig. 18

Single-ended filter model with all nodes marked

By solving the system, the filter transfer function H(s) is derived in (19).

$$\begin{aligned} H(s) = \frac{n_0}{d_4 s^4 + d_3 s^3 +d_2 s^2 + d_1s +n_0 }, \end{aligned}$$

(19)

with

$$\begin{aligned} n_0&=g_\textit{m,n} g_\textit{m,p}\\ d_4&=C_1 C_2 C_3 C_4 R_1 R_2\\ d_3&=C_1 C_2 C_4 R_1 (1- R_3 g_\textit{m,p} ) + C_1 C_3 C_4 R_1 (1+ R_3 g_\textit{m,n} ) \\&\quad +\,C_2 C_3 C_4 R_3 (1- R_3 g_\textit{m,n} ) + C_1 C_2 C_3 R_1 R_3 g_\textit{m,p} \\ d_2&=C_1 C_2 R_1 g_\textit{m,p} \\&\quad +\,C_2 C_4 (1- R_1 g_\textit{m,n} - R_3 g_\textit{m,p} + R_1 R_3 g_\textit{m,p} g_\textit{m,n} ) \\&\quad +\, C_3 C_4 (1- R_1 g_\textit{m,n} + R_3 g_\textit{m,n} ) \\&\quad +\, C_1 C_4 (R_1 g_\textit{m,n} - R_1 g_\textit{m,p} - R_1 R_3 g_\textit{m,p} g_\textit{m,n} ) \\&\quad +\,C_1 C_3 (R_1 g_\textit{m,p} + R_1 R_3 g_\textit{m,p} g_\textit{m,n} ) \\&\quad +\,C_2 C_3 (R_3 g_\textit{m,p} - R_1 R_3 g_\textit{m,p} g_\textit{m,n} )\\ d_1&=C_1 R_1 g_\textit{m,p} g_\textit{m,n} + C_2 g_\textit{m,p} (1- R_1 g_\textit{m,n} ) \\&\quad +\,C_3 (g_\textit{m,p} + ( R_1 - R_3 ) g_\textit{m,n} g_\textit{m,p} \\&\quad +\, C_4 (g_\textit{m,n} + ( R_1 - R_3 ) g_\textit{m,n} g_\textit{m,p} ). \end{aligned}$$

In presence of process variation, each of the terms could deviate from its nominal value, modifying therefore the denominator coefficients and, consequently, the poles frequency $f_o$ and quality factor Q. The Q-trimming approach used in this paper is similar to the one presented in [9], where all of the spreads are modeled to be caused by the resistances and the transconductances. As shown in Figs. 5 and 6 it is possible to adjust each of the capacitors value to correct this deviation. However, due to the complexity of these terms, a closed-form equation relating the Q’s and different denominator coefficients is not achievable and, therefore, the capacitances $C_{1{-}4}$ could not be directly expressed as a function of the other impedances process spread. This means that a numerical solution to for the trimming system is required.

Appendix 2

The main source of distortion of the proposed biquad cell (Fig. 7) comes from $\textit{MN}_1$ and $\textit{MN}_2$ transistors, which act like Source Follower and whose structure is reported in Fig. 19. According to the analysis reported in [10], the $\text {IIP}_3$ in a source follower is calculated as shown in (20).

$$\begin{aligned} \text {IIP}_3= \frac{V_\textit{in}}{\sqrt{\left| \text {IM}_3\right| }}, \end{aligned}$$

(20)

where $\text {IM}_3$ is the third order intermodulation product.

Fig. 19

Source follower

For the Source Follower the $\text {IM}_3$ is calculated as in (21) [10].

$$\begin{aligned} \text {IM}_3= -\frac{3}{8} \cdot \frac{T}{(1+T)^2} \cdot \left( \frac{2\cdot V_\textit{in,g}}{(1+T)\cdot V_\textit{ov}}\right) ^2, \end{aligned}$$

(21)

where $V_\textit{ov}$ is the overdrive voltage of transistor $M_1$ and T is the loop gain which is calculated in (22).

$$\begin{aligned} T=g_m\cdot r_\textit{DS} = 2 \frac{V_E}{V_\textit{ov}}, \end{aligned}$$

(22)

where $g_m$ is the transconductance of transistor $M_1$, $r_\textit{DS}$ and $V_E$ are respectively the output resistance and the Early voltage of transistor $M_2$.

In the proposed biquad cell (Fig. 7), the input voltage $V_\textit{in}$ of the Source-Follower in Fig. 19, corresponds to the differential voltage at the Gates $V_\textit{in,g}$ of $\textit{MN}_1$ and $\textit{MN}_2$, whereas the loop gain $T_\textit{bc}$ can be written as in (23).

$$\begin{aligned} T_\textit{bc} = g_\textit{m,n}\cdot Z_s = \frac{g_\textit{m,n}\cdot r_o}{1+i\cdot \omega \cdot C_2 \cdot r_o}. \end{aligned}$$

(23)

where $Z_s$ represents the parallel of the output resistance of $\textit{MN}_{b,1}$ ($r_o$) and $C_2$, whereas $g_\textit{m,n}$ is the transconductance of $\textit{MN}_1$.

$V_\textit{in,g}$ is derived in (24).

$$\begin{aligned} V_\textit{in,g}=(V_\textit{in}^+-V_\textit{in}^- )\cdot H(s)\cdot \frac{1+T_\textit{bc}}{T_\textit{bc}}. \end{aligned}$$

(24)

By replacing $V_\textit{in}$ with $V_\textit{in,g}$ and T with $T_\textit{bc}$ in (21) and by combining (21) with (22), the $\text {IIP}_3$ of the proposed biquadratic cell is derived in (25).

$$\begin{aligned} \begin{aligned} \text {IIP}_3&\approx \sqrt{\frac{8}{3}} \cdot \dfrac{V_\textit{ov}}{\left| H(s) \right| } \cdot \frac{\left| 1+i \cdot \frac{\omega \cdot C_2 \cdot r_o}{1+g_\textit{m,n} \cdot r_o} \right| }{\left| 1+i \cdot \omega \cdot C_2 \cdot r_o\right| ^\frac{3}{2}}\\&\quad \cdot \, \left( 1 + g_{m,n} \cdot r_o \right) \cdot \sqrt{g_{m,n} \cdot r_o } \approx \\&\approx \frac{8}{\sqrt{3}} \cdot \left( \dfrac{V_A}{V_\textit{ov}} \right) ^\frac{3}{2} \cdot \dfrac{1}{\left| H(s) \right| }\cdot \dfrac{\left| 1 + i \cdot \frac{\omega \cdot C_2}{g_{m,n}}\right| }{\left| 1+i \cdot \omega \cdot C_2 \cdot r_o\right| ^\frac{3}{2}}, \end{aligned} \end{aligned}$$

(25)

where $V_\textit{ov}$ is the overdrive of $\textit{MN}_1$ MOS transistor, and $V_A$ is the Eraly voltage of $\textit{MN}_{b,1}$ MOS transistor. The approximation in (25) is valid if $g_{m,n}\cdot r_o \gg 1$.