Question 10.10: This example illustrates the advantages of cascoding by comp...
This example illustrates the advantages of cascoding by comparing the performance of a cascode amplifier with that of a common-source amplifier in two cases:
(a) The resistance of the signal source is significant, Rsig = 10 kΩ.
(b) Rsig is negligibly small.
Assume all MOSFETs have gm = 1.25 mA/V, ro = 20 kΩ, Cgs = 20 fF, Cgd = 5 fF, Cdb = 5 fF, and CL (excluding Cdb) = 10 fF. For case (a), let RL = ro = 20 kΩ for both amplifiers. For case (b), let RL = ro = 20 kΩ for the CS amplifier and RL = Ro for the cascode amplifier. For all cases, determine Av, fH, and ft .
(a) For the CS amplifier:
A0 = gmro = 1.25 × 20 = 25 V/V
Av = −gm (RL || ro) = −gm (ro || ro)
= −\frac{1}{2} A_{0} = −12.5 V/V
To obtain τH we use Eq. (10.87) and note that Rsig = R_{sig}^{′} and that here CL does not include Cdb, thus
τ_{H} = b_{1} = C_{gs}R_{gs} + C_{gd}R_{gd} + C_{L}R_{C_{L}} (10.87)
τ_{H} = C_{gs}R_{sig} + C_{gd} [(1 + g_{m}R_{L}^{′})R_{sig} + R_{L}^{′}] + (C_{L} + C_{db}) R_{L}^{′}
where
R_{L}^{′} = r_{o} || R_{L}= r_{o} || r_{o} = 10 kΩ
τH = 20 × 10 + 5 [(1 + 12.5) 10 + 10] + (10 + 5)10
= 200 + 725 + 150 = 1075 ps
Thus,
f_{H} = \frac{1}{2π × 1075 × 10^{−12}} = 148 MHz
ft = |Av| fH = 12.5 × 148 = 1.85 GHz
For the cascode amplifier:
Ro = 2ro + (gmro) ro = (2 × 20) + (25 × 20) = 540 kΩ
Av = −gm (Ro || RL)
= −1.25 (540 || 20) = −24.1 V/V
R_{in2} = \frac{r_{o} + R_{L}}{g_{m}r_{o}} = \frac{r_{o} + r_{o}}{g_{m}r_{o}} = \frac{2}{g_{m}} = \frac{2}{1.25} = 1.6 kΩ
Rd1 = ro || Rin2 = 20 || 1.6 = 1.48 kΩ
Using Eq. (10.109),
τH = Rsig [Cgs1 + Cgd1 (1 + gm1Rd1)] + Rd1 (Cgd1 + Cdb1 + Cgs2) + (RL || Ro) (CL + Cgd2) (10.109)
τH = Rsig [Cgs1 + Cgd1 (1 + gm1Rd1)]
+ Rd1 (Cgd1 + Cdb1 + Cgs2)
+ (RL || Ro) (CL + Cdb2 + Cgd2)
= 10 [20 + 5 (1 + 1.25 × 1.48)]
+1.48 (5 + 5 + 20)
+ (20 || 540) (10 + 5 + 5)
= 342.5 + 44.4 + 385.7
= 772.6 ps
f_{H} = \frac{1}{2π × 772.6 × 10^{−12}} = 206 MHz
ft = 24.1 × 206 = 4.96 GHz
Thus cascoding has increased both the dc gain and the 3-dB frequency, with the combined effect being an increase of ft by a factor of 2.7.
(b) For the CS amplifier:
Av = −12.5 V/V
τ_{H} = (C_{gd} + C_{L} + C_{db}) R_{L}^{′}
= (5 + 10 + 5) 10 = 200 ps
f_{H} = \frac{1}{2π × 200 × 10^{−12}} = 796 MHz
ft = 12.5 × 796 = 9.95 GHz
For the cascode amplifier:
RL = Ro = 540 kΩ
Av = −gm (Ro || RL)
= −1.25 (540 || 540) = −337.5 V/V
R_{in2} = \frac{r_{o} + R_{L}}{g_{m}r_{o}} = \frac{20 + 540}{1.25 × 20} = 22.4 kΩ
Rd1 = ro1 || Rin2 = 20 || 22.4 = 10.6 kΩ
τH = Rd1 (Cgd1 + Cdb1 + Cgs2) + (RL || Ro) (CL + Cgd2 + Cdb2)
= 10.6 (5 + 5 + 20) + (540 || 540) (10 + 5 + 5)
= 318 + 5400 = 5718 ps
f_{H} = \frac{1}{2π × 5718 × 10^{−12}} = 27.8 MHz
ft = 337.5 × 27.8 = 9.39 GHz
Thus cascoding increases the dc gain from 12.5 V/V to 337.5 V/V. This increase has been obtained at the cost of a decrease in fH by approximately the same factor, resulting in the unity-gain frequency (which, in this case, is equal to the gain–bandwidth product) remaining nearly constant.