# Biasing + Small Signals

Transistors are incredibly useful devices with incredibly complex explanations. I touched on some basic ideas of how to use them previously, but that doesn’t do them enough justice. To truly do them justice, I would write several posts about device physics, quantum mechanics, and other low level theory that I really just don’t care about. While I absolutely should care, I realize that it’s not critical information for most of the circuits I’ll be designing.

Instead, I want to talk about a way of modeling these transistors in a manner that gives us useful parameters and is more intuitive than balls of variables and equations. While many models exist, the one I’m most comfortable with is the hybrid pi model. But before that, let’s break it down why we need it.

##### Small Signals

The name for this modeling term is really just a form of analog linearization. Imagine putting a sine wave into the equations we got for transistors - you’d get a whole mess of output values as you change between all the different states. Defining the output of a circuit as a piecewise function is totally useless! If we put two or three transistors back to back, we’d get like 10 different outputs with all these small, unique cases.

Now instead of defining our input as some sine wave, let’s define it a bit more formally. We’ll say our input is $V_{B} + \delta_V$, with the $\delta_V >> V$. In this case, we can our bias voltage ($V_B$) is constant, and the sum is also effectively constant. We can then plug and chug into our equations from last time to determine a mode of operation. Once we get that far, we can linearize our system for very small inputs, also known as small signals.

##### Linearization

I keep dropping this word, so let’s quickly define linearization. All it means is that for some weird erratic signal, we can eventually zoom in on one part of it enough that it will look like a linear signal. While this isn’t true for every signal in the world (thanks, pure mathematics), it’s pretty true for everything we’ll see in electronics.

Above we see two graphs, one showing a normal sine wave for a few cycles, and the other one at the same frequency but far more zoomed in. The second one isn’t a perfect line, but it’s pretty close. This is exactly what we’re going to do in our I-V curves for transistors! We’ll set that bias voltage, and move very small voltages around it such that we don’t change the state.