# Hybrid Model - Why differential of voltage equals AC component voltage?

In the derivation Hybrid Model (Equivalent Circuit) of transistor, there is a step that replace differential $$\dV_1\$$ with AC component $$\v_1\$$. However, to the best of my knowledge, voltage equals sum of DC component and AC component, that is $$\V_1(t)=V_{DC}+v_1(t)\$$. If take derivatives on both side respect to time, as $$\V_{DC}\$$ is a constant, we get $$\\frac{dV_1}{dt}=0+\frac{dv_1}{dt}\$$ which simplifies to $$\dV_1 = dv_1\$$. So, I feel quite confused about this replacement.

Also, I think $$\dV_1\$$ means change (approximately) in voltage from time $$\t\$$ to $$\t+\Delta{t}\$$ while $$\v_1\$$ means the value of AC component voltage at time $$\t\$$, that seems to be quite different. Thus why shall we replace $$\dV_1\$$ by $$\v_1\$$ but not $$\dv_1\$$? Could anyone give me some hint? Thanks.

• You can find the non-linear hybrid-$\pi$ model here, along with the transport and injection models. These are large-scale models, though. The linearized hybrid-$\pi$ is based upon the non-linear hybrid-$\pi$ model exhibited above and is developed exactly as Tim Wescott discusses in his answer below.
– jonk
Dec 22, 2018 at 3:34
• Also, you should start to "see" things like $\text{d} V$ as nothing more than any other algebraic variable. You can cancel them out just like any other variable, move them around like any other variable, etc. The only difference between $\text{d} V$ and $V$ is that $\text{d} V$ can only hold infinitesimal values and $V$ can only hold finite values. They are both "just variables," otherwise. Note: the ratio of two infinitesimal variables can be a finite value. So $\frac{\text{d} V}{\text{d} t}=5$ just means that $\text{d} V=5\cdot\text{d} t$. Nothing more or less. Simple.
– jonk
Dec 22, 2018 at 3:42

The hybrid pi model is the result of doing something called linearization (which you ought to be able to search on). The notion is that you take the derivative around some input (in the video it appears to be $$\I_1\$$), build a linear model based on those derivatives, and then declare that the circuit behaves as the Taylor series approximation out to the first term.
• Finally I caught the idea that h - model is used when AC component is relatively smaller than DC component and low frequency so AC component voltage = $dU$. xD Jan 7, 2019 at 13:53