Question about the single stage ＯＴＡ in Sansen's book

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Patrick Yang

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Hi,
I'm reading Sansen's Book, and occurs several questions. it's about the second non-dominant pole in current mirror in M3, I wonder how to calculate the pole, any convenient method to determine the value? and why the f_zero = 2*f_pole?
The slide is below. Thanks in advance!
 

Hi,
The pole value is simply gm3/Cn2 radians per second. The zero frequency can (roughly) be computed by understanding that at the zero, no current flows out of the output node. Thus,
the transconductance current of M2 entering the output node + the transconductance current of M1 mirrored by M3-M4 = 0.
This means,
gm1,2*Vin/2 + gm1,2*(Vin/2)*(1/(gm3+s*Cn))*gm3 = 0
Hence,
gm3=-(gm3+s*Cn)
So,
s=-2*gm3/Cn
This value of s is the zero frequency, which clearly is twice the second non-dominant pole.
 

The output current is contributed by 2 paths; 1st path: M1; 2nd path:M2->M3->M4; each path having a small signal current of 'gmvd/2' & opposite in magnitude. Since both of them are opposite they sum up @ the output node & hence you get output current equal to 'gmvd'. Now since there is pole in one path (M2->M3->M4) the current from this path will start to reduce after the pole frequency & eventually goes to zero current from gmvd/2. So now the effective output current is only through M1 -->which is gmvd/2 -->that defines the Zero frequency.
 

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