# Thoughts on the T Model of an NPN Bipolar Transistor

#### MrAl

Joined Jun 17, 2014
11,681
Hello there,

I meant to post this sooner but some problems came up at home with the car that i have to attend to. It's been a problem because it is an electrical item and i dont have a schematic so i have to make one myself. Quite a challenge with many parts and many are SMD parts hard to identify.

Anyway, here are the drawings and the solutions to the DC bias point and AC gain. Some are with 're' and some are without.
I hope to do more with this as time goes on.

First the equations, then the drawings will be attached. See what you think and if you can reconcile 're' or 'gm' with the solutions.
Note R5 represents the notorious 're'. In the top diagram 're' is zero, in the bottom 're' is non zero and calculated from VT and the emitter current 'ie'.
Also note E3 in the solutions is Vbe in the drawings.

** Vdc out with R5=0:
Vc=(B*E2*R3*R6+E2*R3*R6+B*E2*R2*R6+E2*R2*R6+B*E3*R3*R4-B*E2*R3*R4+B*E3*R2*R4+E2*R2*R3)/(B*R3*R6+R3*R6+B*R2*R6+R2*R6+R2*R3)

** Vdc out with R5=non zero (Vbe as shown in the drawing):
Vc=(B*E2*R3*R6+E2*R3*R6+B*E2*R2*R6+E2*R2*R6+B*E2*R3*R5+E2*R3*R5+B*E2*R2*R5+E2*R2*R5
+B*E3*R3*R4-B*E2*R3*R4+B*E3*R2*R4+E2*R2*R3)/(B*R3*R6+R3*R6+B*R2*R6+R2*R6+B*R3*R5+R3*R5+B*
R2*R5+R2*R5+R2*R3)

** for AC only, R5=0:
(AC gain)=-(s^2*B*C1*C2*R2*R3*R4*R7)/((s*B*C1*R2*R3*R6+s*C1*R2*R3*R6+s*B*C1*R1*R3*R6+s*C1*R1*
R3*R6+B*R3*R6+R3*R6+s*B*C1*R1*R2*R6+s*C1*R1*R2*R6+B*R2*R6+R2*R6+s*C1*R1*R2*R3+R2*R3)*
(s*C2*R7+s*C2*R4+1))

** for AC only, R5 included:
(AC gain)=-(s^2*B*C1*C2*R2*R3*R4*R7)/((s*B*C1*R2*R3*R6+s*C1*R2*R3*R6+s*B*C1*R1*R3*R6+s*C1*R1*
R3*R6+B*R3*R6+R3*R6+s*B*C1*R1*R2*R6+s*C1*R1*R2*R6+B*R2*R6+R2*R6+s*B*C1*R2*R3*R5+s*C1*R2*R3*R5
+s*B*C1*R1*R3*R5+s*C1*R1*R3*R5+B*R3*R5+R3*R5+s*B*C1*R1*R2*R5+s*C1*R1*R2*R5+B*R2*R5+R2*R5+s*C1*
R1*R2*R3+R2*R3)*(s*C2*R7+s*C2*R4+1))

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#### WBahn

Joined Mar 31, 2012
30,286
If these are "thoughts on the T-model", why are you using the hybrid pi model?

Are you really claiming that your results are good to sixteen sig figs?

#### LvW

Joined Jun 13, 2013
1,772
Without going into details - a small-signal model (Pi-model) for DC analyses?

#### WBahn

Joined Mar 31, 2012
30,286
You have been claiming (in another thread) that you have developed this new analysis technique that dispenses with the small-signal model and takes into account the variability of the parameters. So you need to show what this new technique is and how to use it, not just show a circuit and what is supposedly the result of this new analysis technique. Unless, of course, you plan to put together an encyclopedia of every conceivable circuit and the corresponding results so that no one has to actually perform this new analysis technique.

#### MrAl

Joined Jun 17, 2014
11,681
If these are "thoughts on the T-model", why are you using the hybrid pi model?

Are you really claiming that your results are good to sixteen sig figs?
Hello again and thanks for the reply.

Well the info i found on the web says it is a T model and i do not see any Rpi but if you think it is a Hybrid Pi model then maybe you can explain why and maybe a link and i'll check it out. Everything i saw so far says it is a T model and i got that from the web because i dont remember this from 40 years ago and not all my books are with me right now.

No i am not claiming sixteen sig figs i am claiming 100 billion sig figs, really to any degree of accuracy you wish to employ in the calculation. However, that is for the circuit exactly as is no matter what you want to call it. That means that every component shown is exactly as indicated.
In fact, the choice of Rc (a ratio of two integers) i think is exact to an infinite number of digits in order to get Vdc out equal to exactly 5.000000000000000000000000000000000000000 etc., etc., volts DC.
If you see a problem with that though let me know.

This is part of the total solution and i will get more into this soon.

#### MrAl

Joined Jun 17, 2014
11,681
Without going into details - a small-signal model (Pi-model) for DC analyses?
You happen to have a link for that?

#### WBahn

Joined Mar 31, 2012
30,286
Without regard to which model you are trying to use (since you are trying to mix large and small signal models, it's hard to comment), you have claimed over and over that a cornerstone of your approach is that you don't assume that re is fixed. Yet in your solutions you use a fixed value or re (a.k.a, R5), namely 14.01304983388704 Ω. So how is that allowing for the variability of re with signal, which is what you claimed was the huge shortcoming of using the linearized small-signal model?

In your bottom circuit, you are claiming that all of the current, both the DC bias current and the AC signal current, flows through re. But the very definition of re requires that ONLY the small-signal current flows through it, because it is defined so as to represent the CHANGE in Vbe that results from a small CHANGE in Ie.

If you REALLY want to claim that you are letting re vary as Ie varies, that you are defining re to be V_T/i_e, and that the total emitter current flows through re, then simply replace re with a constant voltage source equal to V_T. After all, using that approach, the voltage across re is simply i_e*re, which is V_T.

#### LvW

Joined Jun 13, 2013
1,772
LvW said:
Without going into details - a small-signal model (Pi-model) for DC analyses?

You happen to have a link for that?
I dont understand.
Do you really need a link which explains to you why it is not very logical to use small-signal model for large signal (DC) analyses?

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#### LvW

Joined Jun 13, 2013
1,772
Well the info i found on the web says it is a T model and i do not see any Rpi but if you think it is a Hybrid Pi model then maybe you can explain why and maybe a link and i'll check it out. Everything i saw so far says it is a T model and i got that from the web because i dont remember this from 40 years ago and not all my books are with me right now.
In fact, the shown circuit contains neither the well-known T-model of a BJT nor the classical Pi-model.
It looks partly as a arbritary mixture of both, with a DC voltage source Vbe (DC !!) - in addition to the base voltage divider.
Such an unknown circuit deserves some explanations.

#### MrAl

Joined Jun 17, 2014
11,681
LvW said:
Without going into details - a small-signal model (Pi-model) for DC analyses?

I dont understand.
Do you really need a link which explains to you why it is not very logical to use small-signal model for large signal (DC) analyses?
No link to a Pi model which you said it was right?

#### MrAl

Joined Jun 17, 2014
11,681
In fact, the shown circuit contains neither the well-known T-model of a BJT nor the classical Pi-model.
It looks partly as a arbritary mixture of both, with a DC voltage source Vbe (DC !!) - in addition to the base voltage divider.
Such an unknown circuit deserves some explanations.
Ok then link to the well known T model of the BJT.

What is wrong with the Vbe ? Did you see the circuit in the previous thread or do you want me to post that again (the T model i was referring to) ?

#### MrAl

Joined Jun 17, 2014
11,681
Without regard to which model you are trying to use (since you are trying to mix large and small signal models, it's hard to comment), you have claimed over and over that a cornerstone of your approach is that you don't assume that re is fixed. Yet in your solutions you use a fixed value or re (a.k.a, R5), namely 14.01304983388704 Ω. So how is that allowing for the variability of re with signal, which is what you claimed was the huge shortcoming of using the linearized small-signal model?

In your bottom circuit, you are claiming that all of the current, both the DC bias current and the AC signal current, flows through re. But the very definition of re requires that ONLY the small-signal current flows through it, because it is defined so as to represent the CHANGE in Vbe that results from a small CHANGE in Ie.

If you REALLY want to claim that you are letting re vary as Ie varies, that you are defining re to be V_T/i_e, and that the total emitter current flows through re, then simply replace re with a constant voltage source equal to V_T. After all, using that approach, the voltage across re is simply i_e*re, which is V_T.
Hello,

"The definition of 're' requires that only the small signal current flows though it", and that's ok, but this is not that. If it was a mix, then it must be different than the usual. This will be different than the model you've seen in the past, but it starts with the usual model and goes from there. If you still dont think i am starting with the "T" model proper, then let me know why i'll surely look into it also. Keep in mind thought that i am STARTING with the T model as far as i know it to be, and so there will be variations on that model that deviate from the true model.

I will get to the part where we let 're' vary continuously, but you can not make 're' a constant voltage source because then the AC gain does not come out right. Note that if we forgot about 're' for a minute and then just adjusted RE (external) by increasing it, that would make the AC gain change.

Aside from all that, it would be interesting to see someone use the formulas presented to show how they can calculate that from 'gm' for example. We could then compare results. That might be even more interesting.

#### LvW

Joined Jun 13, 2013
1,772
Ok then link to the well known T model of the BJT.

What is wrong with the Vbe ? Did you see the circuit in the previous thread or do you want me to post that again (the T model i was referring to) ?
When you ask google for the "BJT T-model" you will get a lot of diagrams.
Regarding Vbe: In your diagrams, I see a base voltage divider (for creating the base bias) and - in addition - a voltage source Vbe=0.65 volts.
I did not say that something would be "wrong with Vbe" - I have ask for some explanation because this is a rather unusual configuration.
In particular, you should mention if your diagram is a small-signal equivalent diagram or if it is a real circuit diagram which allows DcC and large-signal calculations.

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#### MrAl

Joined Jun 17, 2014
11,681
When you ask google for the "BJT T-model" you will get a lot of diagrams.
Regarding Vbe: In your diagrams, I see a base voltage divider (for creating the base bias) and - in addition - a voltage source Vbe=0.65 volts.
Hi,

Oh you're away early this morning too huh

Yes Vbe is the base emitter voltage 'diode' which becomes a voltage source in the DC model when it is properly biased, and because of the other component values we can assume it is properly biased for all time.

I'll do another search thanks.

#### MrAl

Joined Jun 17, 2014
11,681
Ok here is what came up for the T model.
Note i am combining Fig 2 and Fig 3.

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#### WBahn

Joined Mar 31, 2012
30,286
Hello,

"The definition of 're' requires that only the small signal current flows though it", and that's ok, but this is not that. If it was a mix, then it must be different than the usual. This will be different than the model you've seen in the past, but it starts with the usual model and goes from there. If you still dont think i am starting with the "T" model proper, then let me know why i'll surely look into it also. Keep in mind thought that i am STARTING with the T model as far as i know it to be, and so there will be variations on that model that deviate from the true model.

I will get to the part where we let 're' vary continuously, but you can not make 're' a constant voltage source because then the AC gain does not come out right. Note that if we forgot about 're' for a minute and then just adjusted RE (external) by increasing it, that would make the AC gain change.

Aside from all that, it would be interesting to see someone use the formulas presented to show how they can calculate that from 'gm' for example. We could then compare results. That might be even more interesting.
You appear to be using something that started life as the current-driven T-model and then got bastardized with DC bias sources.

Why should anyone try to do anything with a bunch of formulas that appear out of nowhere.

Which beta do you use? The DC beta or the small-signal beta? Aside from being highly variable, beta is also quite nonlinear, so what value of beta are you using in your formulas?

#### LvW

Joined Jun 13, 2013
1,772
Ok here is what came up for the T model.
Note i am combining Fig 2 and Fig 3.
Yes - and that cannot work.
You must not combine a DC model with a model that contains dynamic (differential) quantities like re=1/gm.
In your diagram (post#1), you have assumed that re=1/gm=14.01 ohms would be a fixed static resistor (you have named it R5).
And this is simply wrong.
In order to reveal the discreapency - please, give us the information about the base current Ib in your drawings (post#1):
Is it the DC base current or the small-signal base current ?

Added: I have analyzed the expression as given by you for the AC gain (R5=0).
I am not sure if I have made an error (since it is a pretty large expression) - therefore my question:
What is the result for this gain for very high frequencies?
You have assumed that the capacitors do not present a short circuit and, therefore, the math requires to let the frequency approach infinity for such an analysis.

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#### MrAl

Joined Jun 17, 2014
11,681
Yes - and that cannot work.
You must not combine a DC model with a model that contains dynamic (differential) quantities like re=1/gm.
In your diagram (post#1), you have assumed that re=1/gm=14.01 ohms would be a fixed static resistor (you have named it R5).
And this is simply wrong.
In order to reveal the discreapency - please, give us the information about the base current Ib in your drawings (post#1):
Is it the DC base current or the small-signal base current ?

Added: I have analyzed the expression as given by you for the AC gain (R5=0).
I am not sure if I have made an error (since it is a pretty large expression) - therefore my question:
What is the result for this gain for very high frequencies?
You have assumed that the capacitors do not present a short circuit and, therefore, the math requires to let the frequency approach infinity for such an analysis.
Hi,

First off, it doesnt matter to me if it is considered somehow "wrong" this is something new as i have said.

Of course i did re static for now because that's the way the model works. You calculate the static value of re and then insert it into the circuit as shown as R5. Did you look at the diagram in post #15? That explains how it is done. That was my beef, sort of.

The gain for very high frequencies is zero. I am not considering high frequencies this would be for audio mostly, and mostly for theory anyway. However, my test frequency was 1kHz as usual for audio. We can look at other frequencies i guess but i wasnt really interested in that it's just a distraction at this point.

I did not assume the capacitors were not a short circuit or were a short circuit, either way i did not.
I assumed the capacitors were capacitors and have impedance 1/(s*C) as usual. With that you can effectively compute the gain or whatever at any frequency as long as you ignore the other capacitances not shown in the schematic.
If for some reason you want to investigate the response at infinite frequency you can let C go to infinity as you said or you can calculate the DC bias points where the caps connect and replace them with static DC constant voltage sources and then perturb the input by some small delta v.

If you like i can compute the AC gain at 1kHz or some other reasonable frequency, and if you like i can do the infinite frequency but it will be only in theory as the circuit is shown exactly with no extra components like parasitic stuff. It's still interesting but i just like to point that out.

#### MrAl

Joined Jun 17, 2014
11,681
You appear to be using something that started life as the current-driven T-model and then got bastardized with DC bias sources.

Why should anyone try to do anything with a bunch of formulas that appear out of nowhere.

Which beta do you use? The DC beta or the small-signal beta? Aside from being highly variable, beta is also quite nonlinear, so what value of beta are you using in your formulas?
Hi,

Ha ha, i like the wording. I dont think that is right though because as i posted in post #15 the circuit first has to be DC biased so that the 're' resistor can be calculated. That's the standard procedure with this model. After that 're' is taken to be constant and i think the reasoning behind that it that it doesnt change too much in a real circuit although it does change somewhat.

The formulas drop out of a nodal analysis anyone can do it.
The Beta is 50 in the calculations i did mostly, but in the formulas even THAT is not grounded in rock, you can set "B" to anything you like, and that also means that later when we want to check the design we can sweep the Beta to see how it changes things. We can compare the different Betas to the data sheet and that should give us a good idea how it would respond.
There are some limits of course at least for now. For one, only the parts shown in the diagram are considered and nothing else. That of course means that for now it would not be valid for very high frequencies, so i am sticking to audio for now.

#### WBahn

Joined Mar 31, 2012
30,286
The Beta is 50 in the calculations i did mostly, but in the formulas even THAT is not grounded in rock, you can set "B" to anything you like, and that also means that later when we want to check the design we can sweep the Beta to see how it changes things. We can compare the different Betas to the data sheet and that should give us a good idea how it would respond.
There are some limits of course at least for now. For one, only the parts shown in the diagram are considered and nothing else. That of course means that for now it would not be valid for very high frequencies, so i am sticking to audio for now.
You still don't get it.

You combined two circuits into one. Each of those circuits has a dependent current source that has a beta parameter in it. They are NOT the same beta! Make one of them beta1 and the other beta2. The base currents they are multiplying are NOT the same base currents! Make one of the Ib1 and the other Ib2.