Transistor equation solving

[AdvaRes]

Hi all,
I was reading the following paper:
https://ece.uwaterloo.ca/~cdr/pubs/maymandi/3.pdf.

I dont understand the transistor behaviour given by the relation (1) in the paper.

This equation is not for transistor in linear or saturation region. Could somebody explain this new expression and when we use it ?

In a second step the autor try to calculate the voltage Vo using the equation (3).

And he say

I applyed the laplace transform and than I applied the inverse laplace transform for the relation to calculate Vo(t). I found

The result is different. I dont know what mistake I made in my calculation.
Could somebody help ?
Thanks in advance.

 

[erikl]

I dont understand the transistor behaviour given by the relation (1) in the paper.

This equation is not for transistor in linear or saturation region. Could somebody explain this new expression and when we use it ?

I think this equation presents an approximation to the \[(V_{GS} - V_T)^2\] dependency for the drain current in strong inversion at full velocity saturation (in saturation region), s. e.g. David M. Binkley "Tradeoffs and Optimization in Analog CMOS Design" Sect. 2.4.3 , Equs. (2.15) & (2.17), pp. 16 & 17 .

 

[Teddy]

erikl is right - in case you consider large gate length devices (1um and up) - that is what you find in books. But in sub -micron technologies those equations are barely close to reality.
Good example is calculation when device is in saturation compared to simulated vdsat.
To answer the question in detail I recommend to look at BSIM 3 and higher spice models and equations for the devices.

 

[dgnani]

Hi all,
I was reading the following paper:
https://ece.uwaterloo.ca/~cdr/pubs/maymandi/3.pdf.

I dont understand the transistor behaviour given by the relation (1) in the paper.

This equation is not for transistor in linear or saturation region. Could somebody explain this new expression and when we use it ?

In a second step the autor try to calculate the voltage Vo using the equation (3).

And he say

I applyed the laplace transform and than I applied the inverse laplace transform for the relation to calculate Vo(t). I found

The result is different. I dont know what mistake I made in my calculation.
Could somebody help ?
Thanks in advance.

It looks like someone already found what the first equation means, note that Kn is not the usual spice level 1 parameter because it has dimension of current over voltage (rather than current over voltage squared).

As of your differential equation, remember that the general solution (for any initial condition) is the sum of the solution of the homogeneous equation (i.e. without the first K1 on the right side) which for a first order equation has one undefined parameter and a particular solution (e.g. a constant solution) of the complete equation

 

[AdvaRes]

Hi,
Thank you so much Erikl, Teddy and dgnani.
I understand now.
Concerning the second question. The Laplace transform applied to Eq(1) gives the following transfer function:
Vo=-k/(l*k+C*s)
(Here k=K1, l=λ7 and C=CL1)
The result is given here
https://www.wolframalpha.com/input/?i=InverseLaplaceTransform[-k%2F%28k*l%2Bc*s%29%2Cs%2Ct]

When we apply the initial condition like used by the author the results is different.

 

[dgnani]

Hi,
Thank you so much Erikl, Teddy and dgnani.
I understand now.
Concerning the second question. The Laplace transform applied to Eq(1) gives the following transfer function:
Vo=-k/(l*k+C*s)
(Here k=K1, l=λ7 and C=CL1)
The result is given here
https://www.wolframalpha.com/input/?i=InverseLaplaceTransform[-k%2F%28k*l%2Bc*s%29%2Cs%2Ct]

When we apply the initial condition like used by the author the results is different.

Did not check on wolfram alpha but your Laplace transform is wrong:
C -> C/s
f'(t) -> sF(s)-f(0)

is all you need and you get exactly the paper result

 

[AdvaRes]

Yeah !
The mistake was the f(0).
But C is a constant so it remain as it is. If I change it by C/S the exponential part will not appear in the Vo expression.
The result obtained is Vo(t)=Vdd.exp(-t/τ).

 

[dgnani]

By C->C/s I mean any constant transforms accordingly, specifically in the case of your equation
K->K/s
constants transform as Heavyside (step) functions
Your equation
-C dV/dt = K + Kλ V
becomes, using F(s) as lapl transform of V(t):
-C[s F-V(t=0)]=K/s + Kλ F

Which gives you the correct result

 

[AdvaRes]

You're right. If I use the laplace transform that you provide I get correct result.
Thank you so much dgnani.
Thank you all for helping me.
Cheers,
Advares.

 

[elgiovo]

I assume kn = mun * Cox; if this is the case, the expression for the short channel saturation current is wrong, since it has not to be divided by 2L. (or it is not a current, dimensionally). Moreover, remember that it's just an approximation valid for L->0.