Consider the cascode stage shown in Fig. 6.35(a), where the load resistor is replaced by an ideal current source. Neglecting the capacitances associated with M1, representing Vin and M1 by a Norton equivalent as in Fig. 6.35(b), and assuming γ = 0, compute the transfer function.

Question

Consider the cascode stage shown in Fig. 6.35(a), where the load resistor is replaced by an ideal current source. Neglecting the capacitances associated with [latex]M_1[/latex], representing [latex]V_{in}[/latex] and [latex]M_1[/latex] by a Norton equivalent as in Fig. 6.35(b), and assuming γ = 0, compute the transfer function.

Accepted Answer

Since the current through [latex]C_X[/latex] is equal to [latex]−V_{out}C_Y s − I_{in}[/latex], we have [latex]V_X = −(V_{out}C_Y s + I_{in})/(C_X s)[/latex], and the smallsignal drain current of [latex]M_2[/latex] is [latex]−g_{m2}(−V_{out}C_Y s − I_{in})/(C_X s)[/latex]. The current through [latex]r_{O2}[/latex] is then equal to [latex]−V_{out}C_Y s − g_{m2}(V_{out}C_Y s + I_{in})/(C_X s)[/latex]. Noting that [latex]V_X[/latex] plus the voltage drop across [latex]r_{O2}[/latex] is equal to [latex]V_{out}[/latex], we write

[latex]−r_{O2} \left[(V_{out}C_Y s + I_{in})\frac{g_{m2}}{C_X s} + V_{out}C_Y s\right] − (V_{out}C_Y s + I_{in}) \frac{1}{C_X s} = V_{out}[/latex] (6.82)

That is

[latex]\frac{V_{out}}{I_{in}}= \frac{−g_{m2}r_{O2} + 1}{C_X s} · \frac{1}{1 + (1 + g_{m2}r_{O2})\frac{C_Y}{C_X}+ C_Y r_{O2}s}[/latex] (6.83)

which, for [latex]g_{m2}r_{O2} \gg 1 and g_{m2}r_{O2}C_Y /C_X \gg 1 (i.e., C_Y > C_X )[/latex], reduces to

[latex]\frac{V_{out}}{I_{in}}≈ \frac{− g_{m2}}{C_X s} \frac{1}{\frac{C_Y}{C_X}g_{m2} + C_Y s}[/latex] (6.84)

and hence

[latex]\frac{V_{out}}{V_{in}}=\frac{−g_{m1}g_{m2}}{C_YC_X s}\frac{1}{g_{m2}/C_X + s}[/latex] (6.85)

The magnitude of the pole at node X is still given by [latex]g_{m2}/C_X[/latex] . This is because at high frequencies (as we approach this pole), [latex]C_Y[/latex] shunts the output node, dropping the gain and suppressing the Miller effect of [latex]r_{O2}[/latex].

Question 6.16: Consider the cascode stage shown in Fig. 6.35(a), where the ...

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