LP filter specifications.
The filter mask is shown in the above figure and the corresponding specifications are given below
Exercise 11
Problem 1
Christian Enz
Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland
In this problem, we want to design a $G_m$-$C$ filter using the cascade approach.
LP filter specifications.
The filter mask is shown in the above figure and the corresponding specifications are given below
The fillter mask is shown below.
We now will find the minimum filter order $N$ for matching the above mask with a Butterworth approximation.
It can be shown that for a Butterworth approximation, the minimum order required to satify the specification (the filter mask) is
The corresponding transfer magnitude is shown below.
It can be shown that the transfer function is then given by \begin{equation*} T(s) = \frac{\omega_0}{s+\omega_0} \cdot \frac{\omega_0^2}{s^2 + \tfrac{\omega_0}{Q}\,s + \omega_0^2}, \end{equation*} where $\omega_0$ is the resonance frequency and $Q$ the quality factor of the 2nd -order section which are given by
There are several solutions for implementing the desired transfer function with a $G_m$-$C$ filter. We can choose the simplest 1st-order section shown below
First-order section.
which transfer function is given by \begin{equation*} T(s) = \frac{\omega_c}{s + \omega_c} \end{equation*} with $\omega_c = G_m/C_1$. The 2nd-order section can be implemented by a Tow-Thomas 2nd-order section as shown below.
Second-order section.
which has the following transfer function \begin{equation*} T(s) = \frac{\omega_0^2}{s^2 + \frac{\omega_0}{Q}\,s + \omega_0^2} \end{equation*} with \begin{align*} \omega_0 &= \sqrt{\frac{G_{m1}}{C_1} \cdot \frac{G_{m2}}{C_2}},\\ Q &= \frac{G_{m1} \cdot G_{m2}}{G_{m3}} \cdot \frac{C_2}{C_1}. \end{align*}
In this design we can choose all the $G_m$ to be equal and because $Q=1$ we can also choose all the capacitances to be equal. This leads to the final filter shown below.
Complete 3rd-order low-pass Gm-C filter.
The transconductance $G_m$ or capacitance $C$ still need to set for the given specifications. We can use an additional constraint on the integrated thermal noise at the output
If we assume that all the OTAs contribute equally to the output noise, then \begin{equation*} V_{nout}^2 \cong 3 \cdot \frac{k\,T}{C}. \end{equation*} from which we can estimate the smallest value of the capacitance \begin{equation*} C_{min} = 3 \cdot \frac{k\,T}{V_{nout}^2} \end{equation*} leading to
We finally choose
The transconductance follows as
We can now check the transfer function.
The above design can be verified by simulations using ngspice using the follwoing schematic.
Schematic used for the ngspice simulation.
Starting NGSpice simulation... Simulation executed successfully. ****** ** ngspice-43 : Circuit level simulation program ** Compiled with KLU Direct Linear Solver ** The U. C. Berkeley CAD Group ** Copyright 1985-1994, Regents of the University of California. ** Copyright 2001-2024, The ngspice team. ** Please get your ngspice manual from https://ngspice.sourceforge.io/docs.html ** Please file your bug-reports at https://ngspice.sourceforge.io/bugrep.html ** Creation Date: Jul 13 2024 10:19:33 ****** Batch mode Comments and warnings go to log-file: ./Simulations/ngspice/GmC.ac.log Contents of the log file: ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ Using SPARSE 1.3 as Direct Linear Solver Note: vin: has no value, DC 0 assumed Using SPARSE 1.3 as Direct Linear Solver Note: vin: has no value, DC 0 assumed Note: No compatibility mode selected! Circuit: * 3rd-order butterworth low-pass filter Doing analysis at TEMP = 27.000000 and TNOM = 27.000000 No. of Data Rows : 1 ASCII raw file "./Simulations/ngspice/Simple_OTA.ac.op" Doing analysis at TEMP = 27.000000 and TNOM = 27.000000 No. of Data Rows : 405 hdc = -2.207331e-05 at= 1.000000e+02 fc = 1.999516e+04 Note: Simulation executed from .control section
Tdc = -0.000 dB fc = 2.000e+04 Hz
