# Small Signal Model

Continuing on from my last post, let’s talk about how linearizatzion actually helps us with transistors. Going back to those transistor current-voltage curves, they’re fairly nonlinear. Reference picture below (note: reference was quickly made by me in numpy, units not to scale):

While this graph may not be perfect unit or curve wise, it gets the point across. We have a linear region, then a saturated region. This nice piecewise graph is great on paper, but we also need a circuit schematic that shows this behavior too. What we really want is the schematic for the saturated region, as that is the simplest part. If $V_{gs}$ goes up, so does the current $I_{ds}$, but the drain voltage $V_{ds}$ doesn’t move. Let’s learn how to model that!

##### MOSFET

The MOSFET is slightly easier to model, so let’s start with that. There’s really only two components, a current source and a resistor.

In saturated mode, this holds pretty well for us! The gate node is just an open node, which aligns with the idea of the gate drawing nearly no current. An open circuit draws no power at all, so check that off. The drain-source is a current source in parallel with a resistor, but the it’s not a normal current source. The current is instead proportional to some constant $g_m$ and the input gate voltage. Again, makes sense with the plot above. Some scaling exists between the gate voltage and the drain current, but it’s a linear scale.

Now, the most important part is the first three words of the last paragraph! This model is only valid in saturated mode, not in linear mode. All bets are off then!

#### BJT

The BJT model is just a little bit more difficult. It’s effectively the same as before, but now we just add one small resistor.

Instead of having no current into the gate node of a MOSFET, a BJT has some leakage current draining into it. Since we called them “current controlled-currents sources” before, now we’re just going to defend that explanation. We have a current source that depends on the input (base-emitter) voltage, but that input voltage can also be thought of as just the input current, since it only sees a resistor!

The naming scheme here is pretty confusing, as this model only works in forward active mode, not saturated. While this is the same “state” as before with the MOSFET, the naming scheme is swapped around here.