BJT EM1 model

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Author

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BJT EM1 model

Jonathan Kirwan

05-05-06

Jonathan Kirwan

Jim Thompson

Jonathan Kirwan

Posted by Jonathan Kirwan on May 5, 2006, 5:18 pm

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I decided to spend a little hobbyist time and see what I could make of
the DC parts of the EM model for the BJT. This basically means the
EM1 plus the Early voltage of the EM3 model, as I gather it.

I started out with the very simple case of a 2N3906 BJT and attempted
to make predictions about a clearly saturated case to start. The
2N3906 is modeled with the following key values:

Bf = 300
Br = 4
Is = 1E-14
Vaf = 100

In the saturated case I tried, I presumed a current source arrangement
like this:

--> FIGURE 1

: 1mA
:
: |
: |
: |/c
: .1mA --|

: |>e

: |
: ---
: ///

A basic beta=10 situation.

I decided to use the hybrid-PI form of EM1. After some algebraic
manipulations, I arrived at the following equations:

Ie + Ic * (1 + 1/Bf)
Iec = ---------------------------
1 - (1 + 1/Bf) * (1 + 1/Br)

Ic + Ie * (1 + 1/Br)
Icc = ---------------------------
1 - (1 + 1/Bf) * (1 + 1/Br)

Ib = Icc/Bf + Iec/Br

Vbc = (kT/q) * ln( 1 + Iec/Is )

Vbe = (kT/q) * ln( 1 + Icc/Is )

where Ie is the actual emitter current (all currents measured as
positive-in, negative out), Ic is the actual collector current, Ib is
the actual base current, Iec and Icc are model currents, and Vbc and
Vbe are probably obvious and I can compute Vce = Vbe - Vbc.

From the inputs from my above example of Ic = 1mA, Ie = 1.1mA, I get:

Iec = 3.8033e-4
Icc = 1.4754e-3
Ib = 1e-4 (as expected)
Vbc = 630.115mV
Vbe = 665.179mV
Vce = 35.064mV

This turns out to match the spice modeling of Vce and Vbe pretty well.
So that part is okay.

When I turned to examine the normal region, though, as opposed to the
saturated region, I run immediately into a model problem. Let me
illustrate:

--> FIGURE 2

: 1mA
:
: |
: |
: |/c
: 3uA --|

: |>e

: |
: ---
: ///

In this case, I already know that Bf is about 300 and so this is not
too far from achievable. But my task is to now predict the voltage at
the collector.

However, the problem is that in the normal region I expect that the
current contributions from the reverse-biased Vbc to be negligible and
the base current should be independent of Vce and only depending on
Ic. All this suggests to me that the above equations are going to run
into trouble. And they do. Let's see:

From the inputs of Ic = 1mA, Ie = 1.003mA, I get:

Iec = -1.3115e-6
Icc = 9.9836e-4
Ib = 3e-6 (as expected)
Vbc = 483.634mV + j*81.257mV
Vbe = 655.076mV

Ah. A complex number for Vbc. Not so good. (Note that the model Iec
value is negative and substantially larger than Is.)

This makes some sense. The expectation is that Ib = Ic/Bf and this is
independent of Vce, in the simple case without Vaf figured in. And so
there is no particular Vce that then suggests itself. Any would work.
And in the above case, Ib < Ic/Bf by some small factor. Not equal.

Which brings me to fold in EM3's Vaf parameter. Since now a once-flat
curve will have a tilt to it, I can simply follow out along the Ib
line to find the appropriate Vce needed to hit the Ic I have.

But I needed to figure out the key elements of the triangle formed
with Vaf. For this, I decided to look at this case:

--> FIGURE 3

: Vx
:
: |
: |
: |/c
: Vx ---|

: |>e

: |
: ---
: ///

Here, I'm basically taking Vce = Vbe, or Vbc=0. In this case, the
value of Ic can be computed from Vbe directly, and Ib set to 1/Bf of
that figure. This provides me with an Ic(Vbc=0) benchmark for the
triangle. I can then extrapolate:

(Vaf + Vbc) Icc
-------------------- = -------
(Vaf + Vbe_computed) Ib * Bf

What I'm going is ratioing the modeled Icc value against the Ib value
multiplied by the expected forward beta, Bf. That ratio, if larger
than one, should be reflected by pushing Vbc higher than Vbe, to
compensate. I chose the Vbe here to be the one I computed using the
modeled Icc, arbitrarily setting Vbc=0, and assuming that then the
case would be that Icc=Ib*Bf, so that if Icc is actually higher, then

Vbc must be >0.

From this, I solved for Vbc as:

Icc * ( Vaf - Vbe )
Vbc = ------------------- - Vaf
Ib * Bf

Here, I computed that Vbc = 11.656V, which is close to the modeled
value in spice and using the fuller spice model for that 2N3906.

So I think I'm on the right track here. But I'm interested in
specific corrections to my concepts in the basic EM DC model of the
BJT.

Jon

Posted by Jonathan Kirwan on May 5, 2006, 5:21 pm

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On Fri, 05 May 2006 21:18:51 GMT, Jonathan Kirwan

>From the inputs from my above example of Ic = 1mA, Ie = 1.1mA, I get:

Actually, I used: Ic = 1mA, Ie = -1.1mA. Note that Ie is negative.

Jon

Posted by Jim Thompson on May 5, 2006, 5:30 pm

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On Fri, 05 May 2006 21:18:51 GMT, Jonathan Kirwan

>I decided to spend a little hobbyist time and see what I could make of
>the DC parts of the EM model for the BJT. This basically means the
>EM1 plus the Early voltage of the EM3 model, as I gather it.
>
>I started out with the very simple case of a 2N3906 BJT

[snip]

>
>Here, I computed that Vbc = 11.656V, which is close to the modeled
>value in spice and using the fuller spice model for that 2N3906.
>
>So I think I'm on the right track here. But I'm interested in
>specific corrections to my concepts in the basic EM DC model of the
>BJT.
>
>Jon

Posted by Jonathan Kirwan on May 5, 2006, 5:53 pm

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On Fri, 05 May 2006 14:30:44 -0700, Jim Thompson

>On Fri, 05 May 2006 21:18:51 GMT, Jonathan Kirwan
>

>>I decided to spend a little hobbyist time and see what I could make of
>>the DC parts of the EM model for the BJT. This basically means the
>>EM1 plus the Early voltage of the EM3 model, as I gather it.
>>
>>I started out with the very simple case of a 2N3906 BJT

>[snip]

>>
>>Here, I computed that Vbc = 11.656V, which is close to the modeled
>>value in spice and using the fuller spice model for that 2N3906.
>>
>>So I think I'm on the right track here. But I'm interested in
>>specific corrections to my concepts in the basic EM DC model of the
>>BJT.
>>
>>Jon

>
>NE, IKF, RB, RBM and ISE are critical parameters you haven't
>considered.

First off, I really just wanted to do calculations on the basis of EM1
and keep only to a very simple set of cases -- not delving into
corrections needed at rather high or very low current domains. Those
parameters, those I recognize such as NE and RB, are for EM2 and
beyond. Do you know if these would materially impact the magnitude of
my calculations for the cases mentioned?

NE defaults to 1.5 in the spice I'm using (the 2N3904 model I'm using
doesn't specify it, so the default is used.) And I think that one
applies at very low currents. RB is 20 ohms. I don't see that as
posing a significant change, at this point.

Anyway, I'm first trying to make sure I can manage to get into the
ballpark with the EM1 model. Since I ran into a problem in projecting
the Vce for the normal case (not saturated) right away, I was then
_forced_ to reach into the EM3 model for the Vaf figure. No choice.
But I want to pick up only what is important to begin, in following
the DC behavior, before adding in 2nd order, 3rd order, etc.
corrections.

Jon

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