gain boosting in cascode

I am trying to analyze this circuit. the book says the gain is approximately equal to gm1(gm2ro2ro1)(gm3ro3). So I would like to how to analyze this circuit from a design point of view. In other words, how to tell the gain by just looking at the circuit?

Maybe neglect effect of length modulation?. Then will be only gm....

It's from Razavi's book.
gm3ro3 is the gain of the amplifier to drive M2.

Added after 18 seconds:

It's from Razavi's book.
gm3ro3 is the gain of the amplifier to drive the gate of M2.

Its from razavi book chapter 9 i suppose there this circuit and its fundamentals are discussed in detail... you better refer to that book....

thank you for replying. Yes, indeed, it is from chapter 9 Razavi book. i am having trouble understanding how to get the gain quation Av = gm1gm2ro2ro1gm3ro3. The book did not explain how to get that equation. And I dont really know how to figure it out. Do I have to draw a small signal model for this circuit? But then I want to learn how to fugure out the gain from a design point of view. I mean, I cant draw a small signal model for every circuit that i want to analyze. I will never be a good designer if I do it that way. I want to be able to figure out the approximate gain of a particular circuit by just looking at it. But how?

Added after 20 minutes:

I would also like to know how to figure out the min and max Vout...
The min Vout is Vod2+Vgs3. But how to get that equation?

The effective gain of the Circuit is gm1*Rout.
Rout is the effective output Resistance Looking into the Drain of M3. This effective Rout is boosted by M2 an is Given by (gm3*gm2*r02)*(r03)*r01. hence the output gain is gm1*Rout.

Iam a bit Confused with Over drive Voltages. Vout min is Such that the Output Voltage (DC) required to keep the Transistors edge of Saturation(Such that They Dont go into Triode).

Thank you for replying.
I dont understand why would the effective output impedance be looking into the drain of M3. The nodes that are connected to the drain of M1 are drain of M2 and gate of M3. So i would think that the effective output impedance of M1 is looking into at those two nodes (drain of M2 and gate of M3).

Since we are taking output at M3 and the current whic flows in M1 is gm1*vi, (vi is the voltage at the gate of M1), this in turn flows in drain to source of M3. so the Votage at Drain of M3 is equal to gm1*vi*Rout(of M3, this Rout with respect to Ground)

The o/p impedence....

o/p impedence of a source degenerated circuit when you r looking from M2's drain(without M3) is gm2*ro2*ro1....

Now this is boosted by the gain of transistor M3 i.e gm3*ro3
Now the o/p impedence with M3 is gm2*ro2*ro1*gm3*ro3

Swing..
Without M3 Minimum voltage at o/p would be Vod2+Vod1 (i.e only V0d1 required at node x)

With M3

at x you will need some voltage for M3 in saturation...... so at x you have an extra headgroom remember Vod can be in few hundreds milli volts but not Vgs3 which is Vod3+Vt
and Vt≈400mvto 600mv depending upon your technology

so minimum voltage at x should be Vgs3... which is a very healthy value for overdriving M1(to be in saturation).Hence minimum value at Vout is Vod2+Vgs3...
Hope this helps you to understand...

yeah it does help a lot. thank you.

its just the normal cascode Rout, gm2ro2ro1 multiplied by the gain of M3, gm3ro3 and that's the overall Rout and the driving transistor is M1 so gain is -gm1(Rout). I believe it is inverted.

The minimum output swing is just the Vgs of M3 plus the VDSsat of M2