# Maximum mid-band gain of s.s. model MOSFET amplifier?

Schematics

I need to find the maximum possible gain for this NMOS cascode circuit.

Here is the s.s. model:

Work

I used nodal-matrix analysis (basis is KCL) to find the gain ($$\A_v\$$):

$$\begin{bmatrix} \frac{1}{r_{o1}}+\frac{1}{r_{o2}} & -\frac{1}{r_{o2}} \\ -\frac{1}{r_{o2}} & \frac{1}{r_{o2}}+\frac{1}{(R_{L1}\,||\,50)} \\ \end{bmatrix} \cdot \begin{bmatrix} V_A \\ V_B \\ \end{bmatrix} = \begin{bmatrix} -gm1\cdot V_{in} - gm2\cdot V_A \\ gm2\cdot V_A \\ \end{bmatrix}$$

$$\Longrightarrow \begin{bmatrix} \frac{1}{r_{o1}}+\frac{1}{r_{o2}}+gm2 & -\frac{1}{r_{o2}} \\ -\frac{1}{r_{o2}}-gm2 & \frac{1}{r_{o2}}+\frac{1}{(R_{L1}\,||\,50)} \\ \end{bmatrix} \cdot \begin{bmatrix} V_A \\ V_B \\ \end{bmatrix} = \begin{bmatrix} -gm1\cdot V_{in} \\ 0 \\ \end{bmatrix}$$

$$\Longrightarrow V_A = D/C,\quad V_B = -\frac{v_{in}\cdot (r_{o1}\cdot gm1)\cdot (r_{o2}\cdot gm2 + 1)\cdot (R_{L1}\,||\,50)}{r_{o1}\cdot r_{o2}\cdot gm2+r_{o1}+r_{o2}+(R_{L1}\,||\,50)}$$

$$V_B = V_{out} = -\frac{v_{in}\cdot (r_{o1}\cdot gm1)\cdot (r_{o2}\cdot gm2 + 1)\cdot (R_{L1}\,||\,50)}{r_{o1}\cdot r_{o2}\cdot gm2+r_{o1}+r_{o2}+(R_{L1}\,||\,50)}$$

$$\Longrightarrow \frac{V_{out}}{V_{in}} = A_v = -\frac{(r_{o1}\cdot gm1)\cdot (r_{o2}\cdot gm2 + 1)\cdot (R_{L1}\,||\,50)}{r_{o1}\cdot r_{o2}\cdot gm2+r_{o1}+r_{o2}+(R_{L1}\,||\,50)}$$

This is the part where I am stuck; I was told that increasing $$\gm\$$ and $$\R_L\$$ will reveal the maximum gain but I don't really understand what that entails. Thus far, my interpretation of "increasing $$\gm\$$ and $$\R_L\$$" has been:

$$\lim\limits_{(gm1,gm2,R_{L1}) \to (\infty,\infty, \infty)} A_v$$

However, I've had trouble finding a reliable/straight-forward method for calculating these simultaneous/multi-variable limits, so I assumed that my interpretation was incorrect. For instance, when I apply this interpretation using Wolfram (the only calculator I could find to compute these types of limits), I receive the following output:

(limit does not exist, is path dependent, or cannot be determined)

If you can provide a better interpretation, please enlighten me.

Remarks

• I'd like to know the max-gain but I am more interested in being taught a correct/straight-forward method that is applicable to any similar problem.
• Although this is most likely a question of mathematical understanding, it belongs here b.c. many EE's might have already solved this problem or they might bring enough insight about the circuit (NMOS cascode is recognizable/typical) to derive a solution (or provide the final answer at-least).
• Basically it will be zero because Q2's gate is tied to ground. – Andy aka Oct 1 '18 at 9:10
• @Andyaka I don't see how you're correct since, for any reasonable insertion of values, my A_v eq returns a non-zero answer. It seems to tend towards -infinity, as R_L1 and gm's increase. – Landon Oct 1 '18 at 15:31
• @Andyaka And on a fundamental level, a MOSFET's gate being tied to ground doesn't imply that it's output is 0. If the gate and the source were tied to ground (V_GS = V_G - V_S = 0) then the s.s. dep. current source would be 0 & output would be 0. But that isn't the case here. – Landon Oct 1 '18 at 15:45
• I never said your output would be zero; I was answering your title question and saying the gain would be zero. But please sit and wait for that answer. – Andy aka Oct 1 '18 at 16:08
• @Andyaka I doubt it will receive an answer ),; – Landon Oct 2 '18 at 16:37