# Linearizing Transistors

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The primary objective of this exercise is to practice linearizing the model of a nonlinear device, in this case the bipolar junction transistor (BJT). The BJT is another type of transistor similar in many ways to the MOSFET (which is itself a type of transistor), but it operates using different physical phenomenon (no field effect) and its behavior is therefore governed by a different set of equations. All the linearizing and stuff we've been doing with the MOSFETs we can do with BJTs as well, though.

Shown below is:

• [left] the symbol for an NPN BJT. It has three terminals, Base, Emitter, and Collector.
• [center left] a simplified large-signal model for the NPN BJT operating in its forward-active region (FAR) defined by v_{BE} \gt 0 and v_{CB} \gt 0;
• [center right] a linearized small-signal model for operation in the FAR;
• [right] a more-common equivalent small-signal model. Note that the FAR of a BJT is much like the saturation region of a MOSFET in that the base-emitter voltage controls the collector and emitter currents in the former while the gate-source voltage controls the drain and source currents in the latter.

The current through a BJT in the FAR is the following:

i^{*} = I_S \left( e^{\frac{v_{BE}}{V_T}}-1\right)

(note this is just like the expression i=\frac{K}{2}\left(v_{GS}-VT\right)^2 in the case of a MOSFET)

For this exercise, first, determine r_e and g_m in terms of I_E (the bias current, similar to I_D in a MOSFET), I_S (a component parameter similar to K in a MOSFET), V_T (the thermal voltage, another component parameter) and \alpha_F (see diagrams and equations above) by directly linearizing the large-signal model around the bias current I_E. In doing so, let I_S and V_T be the saturation current and thermal voltage (Different than MOSFET threshold voltage!!), respectively, of the base-emitter diode in the large-signal model.

Use IE, IS, VT, and aF for I_E, I_S, V_T and \alpha_F, respectively.

r_e =

g_m =

Determine \beta and r_{\pi} so that the right-most small-signal model is equivalent to the center-right small-signal model.

\beta =

r_{\pi} =