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.

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To compute the input-referred noise voltage, we short the input to ground, obtaining

[latex] \overline{V_{n 1, ext { out }}^2}=4 k T \gamma\left(g_{m 1}+g_{m 3} ight)\left(r_{O 1} \| r_{O 3} ight)^2 [/latex] (7.96)

Thus, the input-referred noise voltage, [latex] V_{n, i n} [/latex], must satisfy this relationship:

[latex] \overline{V_{n, i n}^2}\left(g_{m 1}+g_{m b 1} ight)^2\left(r_{O 1} \| r_{O 3} ight)^2=4 k T \gamma\left(g_{m 1}+g_{m 3} ight)\left(r_{O 1} \| r_{O 3} ight)^2 [/latex] (7.97)

where the voltage gain from [latex] V_{ ext {in }} ext { to } V_{ ext {out }} [/latex] is approximated by [latex] \left(g_{m 1}+g_{m b 1} ight)\left(r_{O 1} \| r_{O 3} ight) [/latex]. It follows that

[latex] \overline{V_{n, i n}^2}=4 k T \gamma \frac{\left(g_{m 1}+g_{m 3} ight)}{\left(g_{m 1}+g_{m b 1} ight)^2} [/latex] (7.98)

As expected, the noise is proportional to [latex] g_{m 3} [/latex].

To calculate the input-referred noise current, we open the input and note that the output noise voltage due to [latex]M_3[/latex] is simply given by [latex] \overline{I_{n 3}^2} R_{ ext {out }}^2 , ext { where } R_{ ext {out }}=r_{O 3} \|\left[r_{O 2}+\left(g_{m 1}+g_{m b 1} ight) r_{O 1} r_{O 2}+r_{O 1} ight] [/latex] denotes the output impedance when the input is open. The reader can prove that, in response to an input current [latex]I_{in}[/latex], the circuit generates an output voltage given by

[latex] V_{o u t}=\frac{\left(g_{m 1}+g_{m b 1} ight) r_{O 1}+1}{r_{O 1}+\left(g_{m 1}+g_{m b 1} ight) r_{O 1} r_{O 2}+r_{O 2}+r_{O 3}} r_{O 3} r_{O 2} I_{i n} [/latex] (7.99)

Dividing [latex] I_{n 3} R_{ ext {out }} [/latex] Rout by this gain to refer the noise of [latex]M_3[/latex] to the input, we have

[latex] \left.I_{n, i n} ight|_{M 3}=\frac{r_{O 2}+\left(g_{m 1}+g_{m b 1} ight) r_{O 1} r_{O 2}+r_{O 1}}{r_{O 2}\left[\left(g_{m 1}+g_{m b 1} ight) r_{O 1}+1 ight]} I_{n 3} [/latex] (7.100)

which reduces to

[latex] \begin{aligned} \left.I_{n, i n} ight|_{M 3} & \approx I_{n 3} & (7.101)\ & \approx 4 k T \gamma g_{m 3} &(7.102) \end{aligned} [/latex]

if any [latex]g_mr_O[/latex] product is much greater than unity. Since the noise current of [latex]M_2[/latex] directly adds to the input, we have

[latex] \overline{I_{n, i n}^2}=4 k T \gamma\left(g_{m 2}+g_{m 3} ight) [/latex] (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 [latex]I_{n1}[/latex] when the input is left open even though the source of [latex]M_1[/latex] sees a finite degeneration [latex](r_{O2})[/latex]. In Problem 7.31, we refer this noise to the input and prove that it is still negligible.

Question 7.20: Calculate the input-referred thermal noise voltage and curre...

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