Welcome to EDAboard.com

Welcome to our site! EDAboard.com is an international Electronic Discussion Forum focused on EDA software, circuits, schematics, books, theory, papers, asic, pld, 8051, DSP, Network, RF, Analog Design, PCB, Service Manuals... and a whole lot more! To participate you need to register. Registration is free. Click here to register now.

Register Log in

Transistor equation solving

Status
Not open for further replies.

AdvaRes

Advanced Member level 4
Joined
Feb 14, 2008
Messages
1,166
Helped
113
Reputation
220
Reaction score
50
Trophy points
1,328
Location
At home
Activity points
7,444
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.
 

Attachments


erikl

Super Moderator
Staff member
Joined
Sep 9, 2008
Messages
8,112
Helped
2,689
Reputation
5,358
Reaction score
2,291
Trophy points
1,393
Location
Germany
Activity points
44,153
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 .
 
  • Like
Reactions: AdvaRes

    AdvaRes

    points: 2
    Helpful Answer Positive Rating

Teddy

Advanced Member level 1
Joined
Sep 15, 2004
Messages
461
Helped
103
Reputation
206
Reaction score
62
Trophy points
1,308
Activity points
3,861
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.
 
  • Like
Reactions: AdvaRes

    AdvaRes

    points: 2
    Helpful Answer Positive Rating

dgnani

Advanced Member level 1
Joined
Jul 25, 2009
Messages
425
Helped
160
Reputation
322
Reaction score
152
Trophy points
1,323
Location
USA
Activity points
4,007
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
 
  • Like
Reactions: AdvaRes

    AdvaRes

    points: 2
    Helpful Answer Positive Rating

AdvaRes

Advanced Member level 4
Joined
Feb 14, 2008
Messages
1,166
Helped
113
Reputation
220
Reaction score
50
Trophy points
1,328
Location
At home
Activity points
7,444

dgnani

Advanced Member level 1
Joined
Jul 25, 2009
Messages
425
Helped
160
Reputation
322
Reaction score
152
Trophy points
1,323
Location
USA
Activity points
4,007
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
http://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
 
  • Like
Reactions: AdvaRes

    AdvaRes

    points: 2
    Helpful Answer Positive Rating

AdvaRes

Advanced Member level 4
Joined
Feb 14, 2008
Messages
1,166
Helped
113
Reputation
220
Reaction score
50
Trophy points
1,328
Location
At home
Activity points
7,444
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).
 
Last edited:

dgnani

Advanced Member level 1
Joined
Jul 25, 2009
Messages
425
Helped
160
Reputation
322
Reaction score
152
Trophy points
1,323
Location
USA
Activity points
4,007
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

Advanced Member level 4
Joined
Feb 14, 2008
Messages
1,166
Helped
113
Reputation
220
Reaction score
50
Trophy points
1,328
Location
At home
Activity points
7,444
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

Junior Member level 1
Joined
Mar 15, 2009
Messages
16
Helped
2
Reputation
4
Reaction score
0
Trophy points
1,281
Location
Italy
Activity points
1,354
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.
 

Status
Not open for further replies.

Part and Inventory Search

Welcome to EDABoard.com

Sponsor

Top