a. Redraw the above circuit using thermal noise current source.

b. Write the expression for total output thermal noise voltage.

c. Write the expression for output thermal noise voltage referred to the gate of MI.

d. Why should gm, be maximized and gm 2 be minimized in the above circuit?

Solution:

a) circuit using thermal noise current:

Representing the thermal noise of $M_1$ and $M_2$ by current sources and noting that they are uncorrelated we write.

$ V_{n, 0 u t}^2=4 \mathrm{KT}\left(r g m_1+g m_2\right)\left(r_0 111 r_2\right)^2 $

Since the voltage gain is equal to $g m_1\left(\mathrm{rO}_1 11\left(\mathrm{YO}_2\right)\right.$, the total noise voltage referred to the gate of $M_1$ is

$ v_{n_1,{ }^2 \text { n }}=4 k T\left(r g m_1+r g m_2\right) \frac{1}{g m_1^2}....(1) $

$$ V_{{\text {sin }}}^2=4 k T n\left(\frac{1}{g m_1}+\frac{g m_2}{g m_1^2}\right)-\text { (3.) } $$

To compute the total output noise we integrate (1) across the bond,

$ V_{n_1^2 \text { out, } t_{0 t}}=\int_0^{\infty} \frac{4 k T r\left(g m_1+g m_2\right)\left(r_0 \| r_{02}\right)^2 d f}{1+\left(r 0.11 r_{02}\right)^2 c_2^2(2 \pi f)^2}\\ $

Why gm, should it be maximized and gm should it be minimized?

depends upon $g m_1 8 g m_2$, confirming that $g m_2$ must be minimized because $m_2$ serves as a current source rather than a trans conductor.

$\rightarrow$ The noise currents of both transistors flow through re, $11 \mathrm{ro}_2$. However $M_1$ \& $M_2$ both exhibit different noise effects.

$\Rightarrow$ This because as $\mathrm{gm}$, increases, the output noise voltage rises in proportion to $\sqrt{g m}$, whereas the voltage gain of the stage increases in proportion to $\mathrm{gm}$.

$\rightarrow$ As a result, the input-referred noise voltage decreases.

$\rightarrow$ Such a trend doesnat apply to $\mathrm{M}_2$ $\rightarrow$ Hence $g m$, must be maximised and $g m_2$ must be minimised in the given circuit.