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Why, by shorting the drains in Fig \$14.15\$ in this manner, will it allow the \$G_m\$ plots to add together? What will happen to the \$G_m\$ plot if we swap the shorting transistors, i.e., shorting transistor \$M_1\$ and \$M_4\$ drains, and \$M_2\$ and \$M_3\$ drains?.
Reference of Both pictures: Razavi, Design of Analog CMOS Integrated Circuits

The circuit fig 4.19 after linearization technique For Example4.6, figure 4.19 enter image description here

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Why, by shorting the drains in Fig 14.15 in this manner, will it allow the \$G_m\$ plots to add together?

Because you're paralleling conductances. Say, one transistor has 1mS conductance at a given G-S voltage, another has 2mS conductance. Parallel them, and you have 3mS conductance. You do this for every G-S voltage, so the individual \$G_m(V_{gs})\$ functions simply add together: $$ G_{m\text{(parallel)}}(V_{gs}) \approx \sum_i {G_m}_i(V_{gs}).$$

Same as when you're paralleling resistors: their conductance (reciprocal of resistance) adds together.

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In the example, there's effectively a differential pair with 3W on each side. If you do what you suggest, you'll end up with 2W on one side and 4W on the other, won't be balanced and you'll have introduced an offset.

Gm transfers can add like that because it's a transconductance, i.e. the output is a current and both are driven by the same voltage

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    \$\begingroup\$ Not "effectively a differential pair with 3W on each side", since the two pairs have separate source connections. \$\endgroup\$
    – John Doty
    Feb 18 at 22:34
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    \$\begingroup\$ @JohnDoty if W does gm, then 3W does 3*gm. That's what I was referring to. \$\endgroup\$
    – Designalog
    Feb 18 at 22:36
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    \$\begingroup\$ It doesn't really do that here. Yes, they add, but gm is a function of differential input voltage, and the curves are offset. And that's the linearization trick. \$\endgroup\$
    – John Doty
    Feb 18 at 22:40
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    \$\begingroup\$ Look at the gm curves in Figure 14.15 in the question. \$\endgroup\$
    – John Doty
    Feb 18 at 22:58
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    \$\begingroup\$ @JohnDoty My answer was to the question of why don't we connect the 2W/L with the 2W/L and the other ones; giving a plausible answer as to why we don't do that. I'm not explaining the linearization trick. Finally, gm can be a function of anything you decide to abstract away. Each transistor has its own gm, and given that they are driven by the same voltage, and have similar size, so I don't see why 3*gm is not a good approximation. Cheers. \$\endgroup\$
    – Designalog
    Feb 19 at 12:22

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