As far as the BJT is operated in its forward region and it can be considered unidirectional (vce voltage variations does not have an impact on ib or vbe) a 1:1 correspondence between ib and vbe can be established so that a constant vbe bias corresponds to a specific constant ib bias and vice versa. Assuming this is the case, either ib or vbe can be equivalently used to specify point in the bias in the definition of the early voltage. Moreover, in BJTs, the Early voltage is normally almost independent of the bias (the ic-vce curves in forward active region converge to the same point in the negative vce axis, which corresponds to the Early voltage). This is due to the physics of the Early effect, which is related to the change of the depleted region in the ce junction induced by the reverse voltage.

Thank you for being the first to answer my question.

But I must confess that I am not too "happy" with your answer because it assumes that "vce voltage variations does not have an impact on ib or vbe". That means: The Ic=f(Vce) curves have a slope of zero. 

However, it is exactly the non-zero slope which defines the Early voltage. Moreover, what do you mean with "1:1 correspondence between ib and vbe"? Do you neglect the exponential relationship? It is obvious that all curves with Ib as a parameter have a DIFFERENT slope if compared with curves having Vbe as a fixed parameter .

And exactly this is the reason of my question.

I think that the point is that vce variations may have an impact on Ic, which is a quantity defined at the output port - and it is actually what gives rise to the slope of the ic-vce characteristics, but they may not have (almost) any impact on the quantities defined at the input port. i.e. if, in general we take ib and vce as independent port variables, i.e.: vbe = vbe(ib, vce) and ic = ic(ib, vce), in practice, it is vbe=vbe(ib) independently of vce with a very good approximation, whereas ic=ic(ib, vce) and this dependence on vce is actually related to the Early effect (change in the effective base lenght). The vbe=vbe(ib) hypothesis is typically maintained, to the best of my knowledge, in Early effect modelling in the literature (e.g. in [1]). Is the dependance of vbe on vce what you want to highlight? This is typically negligible in most cases since vce variations affect mostly the reverse biased c-b junction and not the forward biased b-e junction.  And this is - in my opinion - the reason why conditions (a) and (b) mentioned in your original question are substantially equivalent and often not specified.

By the way, if it is not the case, to be rigorous, I believe that in the vbe = vbe(ib, vce) and ic = ic(ib, vce) description (current amplifier description, inputs ib and vce), the Early voltage should be defined for constant base current bias and in an ib = ib(vbe, vce) and ic = ic(vbe, vce) description (transconductance description, inputs vbe and vce), it should be defined for a constant vbe bias.

The relation vbe=vbe(ib) is non linear (logarithmic, the inverse being exponential) nonetheless is a 1:1 map under the above hypotheses, i.e. for a value of ib there is one and only one value of vbe, independently of vce, if the BJT is in the active region. So you can label the ic-vce curves in terms of ib or equivalently in terms of the values of vbe that correspond to each value of ib, the curves are the same and show the same slope in the ic-vce axes. Obviously it is no longer the case if vbe=vbe(ib,vce). Probably this is the case you are considering in your question.

I hope this can clarify what I have previously stated and I hope it could be of help.

[1] Hart, B.L., "Modeling the Early Effect in Bipolar Transistors," Solid-State Circuits, IEEE Journal of , vol.18, no.1, pp.139,140, Feb 1983

doi: 10.1109/JSSC.1983.1051910

Again - thank you for explaining your opinion. However, I cannot agree to everything. Let me give some comments:

"Is the dependance of vbe on vce what you want to highlight? This is typically negligible in most cases since vce variations affect mostly the reverse biased c-b junction and not the forward biased b-e junction. "

I think, we cannot neglect this dependence because this is the CAUSE of the effect we are discussing.

* Following a Vce increase, the current Ic also increases (due to the shrinking of the base B-E junction). If Vbe is held constant, the base current Ib decreases (Ie=const because Vbe=const). This gives us the output characteristics with a certain positive slope for Vbe=const.

*However, if we want o keep keep Ib=const, we must slightly increase Vbe (to compensate the above mentioned Ib decrease). This gives us another set of curves Ic=f(Vce) for Ib=const.  

"...for a value of ib there is one and only one value of vbe, independently of vce, if the BJT is in the active region. So you can label the ic-vce curves in terms of ib or equivalently in terms of the values of vbe that correspond to each value of ib, the curves are the same and show the same slope in the ic-vce axes."

* As I have tried to explain - the exponential relationship between Vbe and Ib is NOT independent on Vce. This can be seen in some detailed datasheets which show different input curves Ib=f(Vbe) for different Vce values.

* To my understanding, my question has yet not been answered.

Thank you for your further comments. Now it seems to me the question is more clear. I agree that taking into account of vbe(ib,vce) could be needed for a sound physical description of the Early effect. I have just one more comment about:

"I think, we cannot neglect this dependence because this is the CAUSE of the effect we are discussing.

Following a Vce increase, the current Ic also increases (due to the shrinking of the base B-E junction). If Vbe is held constant, the base current Ib decreases (Ie=const because Vbe=const)"

Not only the collector current but also the emitter current increases for the same reason (both are related to the slope of the minority carrier distribution injected in the base region) even if the vbe voltage is constant. This is not immediate from the Ebers-Moll description, but it can be regarded as a change in the reverse saturation current of the b-e junction due to the Early effect, which affects at the same time ic and ie. What actually happens to the base current is more difficult to be discussed since it is mainly related to recombination in the base region, which is reduced (leading to a larger alpha_F factor) since the base region is shrunk and is also slightly affected by the minority carriers distribution changes. In most models used in simulations (e.g. the Gummel-Poon model), however, it is assumed the base current remains almost constant. And this leads to vbe=vbe(ib). Since the Gummel-Poon model or similar used in simulators are based on this assumption, I think that this can explain the results which you have obtained in simulations.

Paolo - thank you for your last reply. I hope this contribution can help to clarify the problem. At first, I can agree to the first sentence of your first reply: "As far as the BJT is operated in its forward region and it can be considered unidirectional".

But the question is: Is it realistic to assume unidirectional operation?

I think an answer can be derived as follows:

The slope of the Ic=f(Vce) curves at a certain fixed operational point (Ic,o, Vce,o) within the active region can be expressed using small-signal parameters:

* hybrid parameters: For Ib=const the slope is dI/dV=h22

* y-parameters: For Vbe=const the slope is dI/dV=y22.

The relationship between both parameters can be found elsewhere:

y22=h22 - h12*h21/h11=h22 - h12*gm=h22 - h12*Ic,o/Vt

Hence, the slopes of both curves (Ib=const resp. Vbe=const) are not identical, unless the last term can be neglected. Is it allowed to neglect it?

Example (typical values at Ic,o=1.3mA)):

h22=25µS, h12=260E-6

y22=25µS-260E-6*1.3mA/26mV=25µS -13µS

This example shows that - from the beginning - both slopes cannot be considered as equal (or nearly equal). Hence, the Early voltages derived from both slopes cannot be identical.

Therefore, the question still is: Which set of curves must be used for finding the „correct“ Early voltage?

Any comments or corrections are welcome.

Dear Lutz, your numerical example (numerical values seem very reasonable) shows that the effect of h12 can have a very relevant impact on the output characteristics of the BJT.

While discussing about the "correct" Early voltage, however, I believe that the point is that the Early voltage (intended as a constant large-signal model parameter), is a parameter of a model,  like Gummel-Poon's, which is not so accurate to take into account of the impact of vce on the input port (i.e. h12) and of other effects like the dependance of y22 on the Vce bias.

If the problem is just "how to extract the Early Voltage (Va) parameter" appearing in the Gummel-Poon model, considering that the independent variables of this model formulation are v_be and v_ce (y-formulation), to be consistent, I believe it should be extracted considering constant vbe bias and fitting experimental curves. The same approach could be extended to different models, depending on how their equations are formulated (y-type, or hybrid-type). Anyway a model like GP's will never give the correct picture of the situation under different base impedances, as you have observed in simulations. So it could be questioned whether information about the input port bias (voltage or current) in the definition of the Early voltage, which is a parameter defined in the framework of such models, provides any added value. The consideration that arises from your numerical example is that the degree of accuracy in describing the output characteristics of a BJT, which is implicit in the concept of "Early voltage", is rather poor, probably not acceptable in the specific case.

If the problem is to provide an accurate experimental description of a BJT, I think that it can be sufficient to specify the bias conditions of the test and to provide sufficient information (measured data) to highlight all relevant effects (e.g. h22 and y22 parameters).

Finally, if a correct description of the Early effect is needed, more accurate model formuations are required where the Early voltage itself is not constant but a bias-dependent (vbe and vce- dependent) parameter. An effort in this sense can be found e.g. in [1]. I hope it could be of interest.

[1] Jiann-Shiun Yuan; Liou, J.J., "An improved Early voltage model for advanced bipolar transistors," Electron Devices, IEEE Transactions on , vol.38, no.1, pp.179,182, Jan 1991

Really good question Lutz,

I will try to derive a relationship between the Early voltage VA for constant IB and constant VBE based on the basic relations in the bipolar transistor.

Assume that the transistor operates in the active mode such that theEJ i forward biased and he CJ is reverse biased.

According to the charge control model of the npn transistor the collector current is given by equation 1

IC= A Dn q delta n / Wbn with the symbol have their usual meaning and the reverse saturation current of the collector current is neglected.

The quasineutral base region width Wbn is that parameter what decreases by increasing VCE causing the collector current to increase. This effect is the Early effect..

Let us express Wbn by its value at VCB=0, Wbn0 and its incremental decrease dW because of increasing VCB and reverse biasing the collector junction, that is 

Wbn= Wbn0- dW, equation 2 , where dW is because of the collector pace charge widening in the base region.

Now let us expres IB in terms of the excess charges in the base region and emitter region

IB = A q delta n Wbn/ 2 Taun + A Dp q deltap / We ,wih Taun the minority carrier lifetime in the base region and We is the neutral region of the p-emitter.

The base current is composed of two components the recombination in the base region which is the first term, and the back infected current in the emitter which is the second term.

This IB relation is the most important one to explain the different observations concerning the Early voltage.

It is so, that The base current may be dominated by the recombination in the base region or may be dominated by back injected current. Integrated circuits very miniature transistors the base current is dominated by back injected base current.

In power transistors with the base width relatively larger, the base current may be dominated by the recombination in the base. With the improvement of technology, the base current is observed to be dominated by the back injection in the emitter.

There is remaining one fundamental equation to complete the required equations for the proof.

IT is the junction law of the EJ

delta n= nb0 exp VBE/ Vt with VBE>Vt,

delta p= pe0 exp VBE/ Vt with VBE>Vt,

The case of VBE is constant is straight forward,

Then delta n and delta p will be constants and the collector  current can be put in the form by combining equations 1 and 2

IC= ICSAT( I+ dW/ Wb0), With ICSAT, the collector current at VCB=0,

Now we come to the case of IB= constant and dominated by the back in injection in the emitter, IB= A q Dp delta p /we, From this equation if IB is costant, delta pwill be constant and from the junction law, VBE will be constant and consequently the collector current dependence on the neutral base width.

The last case, where the base current is dominated  by the recombination in the base region IB =A q delta n Wbn/ 2 Taun,

Combining with collector current and rearranging we get

IC= ICSAT ( 1- dw/ Wb0)^-2 = approximately ICSAT( 1+2 dw/ Wb0)

which means that the Early effect will increase in case of constant base current which is dominated by the recombination in the base.

In this way an explanation is introduced to Lutz observations concerning the Early effect in biploar transistors.

Dear Paolo, thank you again.

As far as my basic question is concerned, I think two of your statements are of importance: 

* I believe it (the early voltage) should be extracted considering constant vbe bias

* the Early voltage itself is not constant but a bias-dependent (vbe and vce- dependent) parameter. An effort in this sense can be found e.g. in [1]. I hope it could be of interest.

From this I derive - and I fully agree with it - that the present definition of the Early voltage requires constant Vbe bias (rather than constant Ib bias). I think, this view is supported also by the work referenced by you in [1]. Thank you for mentioning this document.

@Abdelhalim Zekry:

Thank you very much for your detailed and very helpful contribution (although the various formulas are not easy to read).

To me, the most important parts of your answer are:

The base current is composed of two components the recombination in the base region which is the first term, and the back infected current in the emitter which is the second term.

The base current may be dominated by the recombination in the base region or may be dominated by back injected current.

To me, this can explain the V(Early) difference between the both cases (Ib vs. Vbe const.). More than that, your conclusion (the Early effect will increase in case of constant base current which is dominated by the recombination in the base) fits perfectly to the known equation y22=h22 - h12*h21/h11 which indicates that h22(Ib=const)>y22(Vbe=const) at same DC operational conditions.

Resulting from the discussion - the situation now is much more clearer (for me) than before. Thank you all.

Sorry, but I must come back to the subject of discussion.

My present understanding is as folllows: We can define two DIFFERENT Early voltages VEA - depending on the used set of output characteristics (IB=const or VBE=const.). Both must be different because the conductance parameters h22 and y22, respectively, are different (as shown in one of my former comments).

In this context, it is woth mentioning that papers from Berkeley Univ. are defining  VEA for IB=const. and papers from Stanford Univ.  are defining VEA for VBE=const. Surprisingly, both are using the same approximation for the transistors dynamic output resistance ro~VEA/IC. (Although they have different definitions for VEA).

Lecture notes from the Columbia Univ.  define VEA also for VBE=const.

Even more confusing is a contribution from the Georgia Inst. of Tech.. Here, the Early voltage VEA is definde for IB=const. . However, in the same paper this parameter is used in the equation y22=IC/(VEA+VCE). As we know - y22 is the output conductance for VBE=const.

Can somebody explain these contradictions?

EDIT: Discovered just now: In another lecture note from Berkeley Univ. VBE is held constant. Now we have two different definitions from the same source.

Dear Lutz, could you reference that works? I believe that in all the cases the Early voltage is introduced as a parameter in models where the reverse feedback mechanism leading to nonzero h12 is not described. As a consequence, as far as such model formulations are concerned, the IB=const and the VBE=const definitions are equivalent, even though in the real world this is not the case. Since the "Early voltage" is included in those models to describe the BJT for any bias condition, any definition will not be satisfactory when compared with reality. Maybe one may choose an average of the VA values extracted from measurements for different base impedances (from 0ohm, i.e. constant VBE, to infinite, i.e. constant IB) so that to minimize the error in model predictions over different biasing conditions.

I believe this is the very same situation we have to address if we want to define the current gain beta (or equivalently alfa_F = beta/(1+beta) ) appearing the Ebers-Moll model. The Ebers-Moll model does not take into account of the Early effect at all, so that the IC-VCE curves in saturation are flat according with this model. When we define the current gain beta=IC/IB, we are giving a sound definition if we just consider the model in itself, nonetheless, if we consider that in practice IC depends on VCE because of the Early effect, we know that if VCE is not specified, the above definition gives different values of beta for different VCE values. What is the correct one? A definite answer cannot be given, in my opinion and a VCE value cannot be included in the definition of beta, but should be rather specified while describing the test conditions. The best could be averaging the results obtained from the definition considering different VCE values in a range of interest. The same can be extended to all quantities which are not explicitly considered in a model, even though they have an effect on the quantities involved in it, as it is the case considered in this discussion.

Paolo - thank you.  Presently, I am short in time and I will answer soon.

Therefore, only the links:

* http://www-inst.eecs.berkeley.edu/~ee105/sp04/handouts/lectures/Lecture22.pdf

* Georgia IT: http://users.ece.gatech.edu/~alan/ECE3040/Lectures/Lecture20-BJT%20Small%20Signal%20Model.pdf

* http://www.ee.columbia.edu/~bbathula/courses/SSDT/lect07.pdf

* http://www.eecs.berkeley.edu/~hu/Chenming-Hu_ch8.pdf

* Stanford: https://www.coursehero.com/file/7827746/ho3BJT/

Hi Paolo, I could agree with you if in all the contributions I was referring to the mentioning (definition) of the Early voltage would be in conjunction with model descriptions. But this is not the case. The Early effect and the corresponding Early voltage is explained in a chapter that deals with the basic functional properties of a BJT (transistor physics). Of course, I agree that this parameter has no fundamental meaning in practice. However, different definitions may cause confusion for students (beginners) and - I think - an unambigious definition should be possible.

Hi Lutz, sorry for my late reply. I have read your references and actually I have noticed the discrepancies you have pointed out in the previous comment. For what concerns the topic of this discussion, it is also interesting to observe in the second slide of your reference from Georgia IT., how the dependance on Vce of Ic and beta are properly devised so that ib is independent of Vce... i.e. to completely mask the effect we are discussing here... I found also interesting the anecdote about how James Early introduced the Early voltage in the book chapter...

More in general, while I agree that the Early effect (Ic increase for increasing Vce) is a physical phenomenon (it can be observed), I believe the Early voltage is not a physical quantity (as a matter of fact, it cannot be "measured" properly, and the same fact that the BJT Ic-Ib or Ic-Vbe curves converge to the same point is only an approximation) and it has a meaning only as a parameter of a model. Unfortunately, in different models it has a more or less slightly different definition (in one case it is an empirical correction to the Ebers-Moll model, in another case (e.g. Gummel-Poon model) is derived considering a charge-controlled model..., in one case a constant vbe is assumed, in one case a constant ib and almost always these two bias conditions are equivalent...), even if the same name is used, and this can be surely a cause of confusion. I believe this is the same concept you have expressed saying that it has "no fundamental meaning". On the other hand, when describing and teaching physical matters, more or less consciously we are always dealing with models, and it is so natural to do so that we are induced to assume that the model is the reality (I admit I inclined to this in my first answers, having in mind a specific description of the BJT). This is surely questionable in itself, but it provides a good self-consistent framework for teaching in which everithing is relatively easy to be explained and understood and fully makes sense. Following this approach, the operation of the BJT can be described with reference to any specific model (e.g. the Gummel-Poon model), as done in your references and the Early voltage should be defined as it appears in that model (for const. vbe, but considering const. ib would be the same), no matter if it does not provide a very good description of what happens in the real world (e.g. that the slope of ic-vce curves @vbe=const and @ib=const are different). Going to the lab, students could observe that in practice the model has its own limits and cannot be always trusted. This could be disappointing, but perhaps also stimulating.

Hi Paolo - thank you again for your wise contribution. Yes - that is the situation - and I agree with you:  We should treat the Early voltage only "as a parameter of a model".

And, moreover, we have to accept that different transistor models may have different values for this parameter.  Nevertheless, it is surprising that there is no contribution (as to my knowledge) which mentions that two alternative definitions exist.  Instead, without any comment or justification, one of the two alternatives (Ib or VBE const.) is used for defining VEA. That was - and still is - the background for my scepticism.

Thank you again, Lutz.

(PS: To me, it was surprising to note that only two forum members - you and A. Zekry - did respond to my inquiry. Am I the only one facing this problem?)

I have tried to find some physical sense in the quite artificial definition of the so-called "Early voltage" by asking a question, "Can the Early voltage be physically explained?":

https://www.researchgate.net/post/Can_the_Early_voltage_be_physically_explained

Cyril, you have mentioned the “quite artificial definition“ for the Early voltage.

May I ask you WHICH Definition you are talking about?

Because THIS was my initial question - we have TWO possible definitions: Using the slopes for Ib=const or Vbe=const.

The physical background is clear, however we have the choice to keep one of two parameters fixed. Only THIS is not clear as far as I know (see all the references I have given at the beginning of this thread)

Luciano - thank you for the information you have given.

Nevertheless, may I ask you: Do you have an answer to the basic question: IB=const or VBE=cinstant ?

Hi Luciano - molto grazie and thank you again for your contribution.

However, I am no too happy with it.

When you say that "it does not matter which one is fixed " (VBE or IB), you belive that in both cases the resulting Early voltage would be the same.

But, surely, this is NOT the case! Therefore, this columne (my question) !!

May I direct your attention to my answer No.6 of this columne? Here, I have tried to explain and to show why we get two different values for the Early voltage.

And this fact is the background of my doubts.

Grazie

Lutz