Differential Amplifier with Source Degeneration

differential source degeneration

How to derive the linear region range of operation for the diffamp below? I know the linear range increases by Ib*2*R compared to regular diffamp but how to derive this relation?

Thanks

differential amplifier source degeneration

Gate-Source and Resistor Voltages
VG1=VDSAT*sqrt(I1/IB)+VTH
VG2=VDSAT*sqrt(I2/IB)+VTH
VR=2*R(I1-I2) (I would use R instead of 2R but I follow the picture)

I1: Drain Current M1
I2: Drain Current M2
VDSAT: Saturation Voltage at Bias Point VIN=0


Define Voltage Loop Equation
VIN=VG1-VG2+VR

Substitute
VIN=VDSAT*sqrt(I1/IB)+VTH-VDSAT*sqrt(I2/IB)-VTH+2*R*(I1-I2)

Simplify
VIN=VDSAT*(sqrt(I1/IB)-sqrt(I2/IB))+2*R(I1-I2)

Redefine Variables
x=VIN/VDSAT
y=(I1-I2)/(2*IB)

then
I1/IB=(I1+I2)/(2*IB)+(I1-I2)/(2*IB)=1+y
I2/IB=(I1+I2)/(2*IB)-(I1-I2)/(2*IB)=1-y
(I1-I2)=2*IB*y

Substitute Variables
x*VDSAT=VDSAT*(sqrt(1+y)-sqrt(1-y))+4*R*IB*y
x=(sqrt(1+y)-sqrt(1-y))+y*a

with
a=4*R*IB/VDSAT

if a=0 then solve equation for a simple diffpair
x=(sqrt(1+y)-sqrt(1-y))
and
y=1/2*x*sqrt(4-x^2)

The last equation is the known transfer function of a MOS diffpair
If a>0 the implicite equation could not be solved analytical
Try using Matlab/Mathcad or others to solve numerical and plot

The factor "a" define the linearisation of the diffpair!

differential amplifier degeneration

VG1=VDSAT*sqrt(I1/IB)+VTH
VG2=VDSAT*sqrt(I2/IB)+VTH

Click to expand...

Sorry for my ignorance but how did u get the above expression.

Thanks

source degeneration differential

Razavi;s book got a chapter on linearization..