Question 7.20: Calculate the input-referred thermal noise voltage and curre...
Calculate the input-referred thermal noise voltage and current of the circuit shown in Fig. 7.49 assuming that all of the transistors are in saturation.
To compute the input-referred noise voltage, we short the input to ground, obtaining
\overline{V_{n 1, \text { out }}^2}=4 k T \gamma\left(g_{m 1}+g_{m 3}\right)\left(r_{O 1} \| r_{O 3}\right)^2 (7.96)
Thus, the input-referred noise voltage, V_{n, i n} , must satisfy this relationship:
\overline{V_{n, i n}^2}\left(g_{m 1}+g_{m b 1}\right)^2\left(r_{O 1} \| r_{O 3}\right)^2=4 k T \gamma\left(g_{m 1}+g_{m 3}\right)\left(r_{O 1} \| r_{O 3}\right)^2 (7.97)
where the voltage gain from V_{\text {in }} \text { to } V_{\text {out }} is approximated by \left(g_{m 1}+g_{m b 1}\right)\left(r_{O 1} \| r_{O 3}\right) . It follows that
\overline{V_{n, i n}^2}=4 k T \gamma \frac{\left(g_{m 1}+g_{m 3}\right)}{\left(g_{m 1}+g_{m b 1}\right)^2} (7.98)
As expected, the noise is proportional to g_{m 3} .
To calculate the input-referred noise current, we open the input and note that the output noise voltage due to M_3 is simply given by \overline{I_{n 3}^2} R_{\text {out }}^2 , \text { where } R_{\text {out }}=r_{O 3} \|\left[r_{O 2}+\left(g_{m 1}+g_{m b 1}\right) r_{O 1} r_{O 2}+r_{O 1}\right] denotes the output impedance when the input is open. The reader can prove that, in response to an input current I_{in}, the circuit generates an output voltage given by
V_{o u t}=\frac{\left(g_{m 1}+g_{m b 1}\right) r_{O 1}+1}{r_{O 1}+\left(g_{m 1}+g_{m b 1}\right) r_{O 1} r_{O 2}+r_{O 2}+r_{O 3}} r_{O 3} r_{O 2} I_{i n} (7.99)
Dividing I_{n 3} R_{\text {out }} Rout by this gain to refer the noise of M_3 to the input, we have
\left.I_{n, i n}\right|_{M 3}=\frac{r_{O 2}+\left(g_{m 1}+g_{m b 1}\right) r_{O 1} r_{O 2}+r_{O 1}}{r_{O 2}\left[\left(g_{m 1}+g_{m b 1}\right) r_{O 1}+1\right]} I_{n 3} (7.100)
which reduces to
\begin{aligned} \left.I_{n, i n}\right|_{M 3} & \approx I_{n 3} & (7.101)\\ & \approx 4 k T \gamma g_{m 3} &(7.102) \end{aligned}if any g_mr_O product is much greater than unity. Since the noise current of M_2 directly adds to the input, we have
\overline{I_{n, i n}^2}=4 k T \gamma\left(g_{m 2}+g_{m 3}\right) (7.103)
Again, the noise is proportional to the transconductance of the two current sources. In the above calculations, we have neglected the effect of I_{n1} when the input is left open even though the source of M_1 sees a finite degeneration (r_{O2}). In Problem 7.31, we refer this noise to the input and prove that it is still negligible.