# Common Emitter Stage BJT Amplifier with Early Effect Inclusion

How can I find Rin, Rout and Gm parameters of a BJT amplifier in a common-emitter stage without neglecting the Early Effect?

We have this as homework and I couldn't find the exact same example in Behzad Razavi's Fundamentals of Microelectronics nor Adel Sedra's Microelectronic Circuits. Moreover, I couldn't derive the hybrid-pi model parameters (Rin, Rout, and Gm) by myself because things get really complex when I need to deal with this many resistances.

Some guidance only for homeworks:

Early effect (=the effective width of the base shrinks as reverse Vcb increases) can be seen as current gain increase as Vcb (and Vce) increases. That can be seen as AC conductance between C and E. If you have the curves Ic vs Vce at different base currents you see how Ic grows as Vce increases. As a simplest linearization you can insert in gain calculations to the AC model a leakakge resistor Va/Ic between C and E, where Va is the Early voltage and Ic is the operating point collector current.

I guess you are expected to use a large signal model and derive by yourself the linearized equations because you have the needed parameters such as VT and Is. Hopefully you can calculate partial derivatives to derive the linearized quantities such as Gm.

Here's a small screenshot taken from Early effect article in Wikipedia:

Gm is the partial derivative of Ic when the variable is Vbe. You must calculate the numerical value in the operating point.

• The Ebers-Moll (in its third edition, where it finally includes basewidth modulation) and Gummel-Poon models use $V_\text{BC}$ and not $V_\text{CE}$ in calculating the Early Effect. There is a separate effect that is included in the Gummel-Poon model (but not in the Ebers-Moll 3rd edition) called the Late Effect, which addresses basewidth modulation due to $V_\text{BE}$. Using $V_\text{CE}$ would appear to be to be an attempt to combine both the Early and Late Effects, but probably incorrectly.
– jonk
Feb 25, 2019 at 23:54
• See J. M. Early, "Effects of Space-Charge Layer Widening in Junction Transistors," Proc. IRE, Vol. 40, pp. 1401-1406. Both $\text{EM}_3$ and Gummel-Poon use a 1st order analysis of these effects, because they are 2nd order and it is sufficient to apply a 1st order analysis. (Also, you missed including the -1 term with the exponential. But that's a minor detail.) Because Early/Late effects are 2nd order and because of BJT variations anyway, my comments here are more just an academic issue.
– jonk
Feb 25, 2019 at 23:56
• @jonk maybe some leakage developed in the insulation between the wires => some crosstalk between different formulas. But Ic should grow as Vbe increases.
– user136077
Feb 26, 2019 at 0:20
• $I_\text{C}$ grows with increasing $V_\text{BE}$ simply because of the Shockley equation alone. This is what obscured the Late Effect as a separate entity for a while. (You don't see higher order effects until you understand the lower orders better and can measure them accurately.) However, there is a separate Late Effect which is due to basewidth modulation and not due to Shockley's well-known equation. I think the use of $V_\text{CE}$ for the Early Effect is at sloppy, if from an academic source. But I'm sure many engineers use it without a 2nd thought about the rest.
– jonk
Feb 26, 2019 at 0:27
• My point is simply that if someone is learning (academically) about the Early Effect (per the paper by Early) then they should stick to the facts and avoid confusion. It's about $V_\text{BC}$.
– jonk
Feb 26, 2019 at 0:29