Question 3.3: Let us calculate the 3 dB frequency of the differential OTA ...
Let us calculate the 3 dB frequency of the differential OTA (Fig. 2.20), designed in Chapter 2.
The design goals were: transconductance G_m = 3 mS, B_o = 1 and the DC current of the input transistors I_{D1} = 1 mA (consequently the total current consumption, 4 mA).
The calculated dimensions (in micrometers) for 0.35 micron AMS CMOS technology were
M1, M11: 24/0.35,
M2, M12: 50/0.35,
M4, M14: 48/0.35,
M5, M15: 111/1,
M3: 230/1.
From (3.65), the 3 dB frequency (frequency corresponding to the pole of the gain
function) can be written as
s_0=+\frac{g_{m3}}{C_{dg3}} and s_p=-\frac{g_{m2}}{(C_{gs2}+C_{gs3})+C_{xp}} (3.65)
f_{3dB}=\frac{1}{2\pi}\frac{g_{m2}}{(C_{gs2}+C_{gs3})+C_{xp}}
g_{m2}=\sqrt{2(KP)_p (W/L)_2 \left|I_{D2}\right|},(KP)_P
=\mu_p C_{ox}=137\times 4.56\times 10^{-7}=0.624\times 10^{-4}\left[A/V^2\right]
g_{m2}=\sqrt{2\times (0.624\times 10^{-4})\times (50/0.35)\times 1}=4.22\times 10^{-3}S=4.22 mS,
(C_{gs2}+C_{gs4})=\frac{2}{3}(W_2+W_4)L C_{ox}+(W_2+W_4)CDGO
Inserting dimensions in micrometers, specific capacitances in fF/μm² and fF/μm:
(C_{gs2}+C_{gs4})=\frac{2}{3}(50+48)\times 0.35\times 4.56+(50+48)\times 0.12\cong 110 fF
C_{xp}=(W_1+W_2)XC_j+(W_1+W_2+2X)C_{jsw}+(W_1+W_2+W_4)CDGO
C_{xp}=(24+50)\times 0.85\times 0.94+2\times (24+50+1.7)\times 0.25+(24+50+48)\times 0.12\cong 111.6 fF
f_{3dB}=\frac{1}{2\pi}\frac{4.3\times 10^{-3}}{(110+111.6)\times 10^{-15}}=3.09 GHz
The frequency characteristic obtained with PSpice simulation is shown in Fig. 3.33.
The hand calculation and simulation results for the low-frequency value of the transconductance are in perfect agreement (3 mS and 3.02 mS, respectively).
The disagreement of the 3 dB frequency (3.09 GHz vs. 2.36 GHz) is mainly owing to the approximations in the analytical expressions and the neglected parasitics.
