Eecs 16b Lecture (Time-varying Complex Exponential Inputs)

Prev meeting: EECS 16B Lecture (16A review and PMOS) 202006231100

V_in(t) with a resistor and capacitor

Mathematically, we want to solve the diff eq

$x(t) + ax(t) = g(t)$

Homogenous part should be similar to last time.

Ways to solve this:

1. Piecewise approximation

   It’s kinda hard to do it (fa19 note 2 pg 8)

2. Integrating factor

   Choose $y(t)$ such that $y(t) = ay(t)$, we can use product rule in reverse to get something like this

   $(x(t)y(t)) = g(t)y(t)$

   $(x(t)(y(t)) = g(t)y(t)$

   We know that $y(t) = e^{at}$

   So ultimately we have that $x(t) = e^{-at}g() e^{a} d+ ke^{-at}$

   In the circuit, we have that

   $V_0(t) + V_0(t) =$

   So comparing the equations, we have that

   $V_0(t) = e{-t/RC}e{/RC}d{} + Ke^{-t/RC}$

   If you plug in values assuming $V_in(t) = VDD$, $V_0(0) = 0$, $g(t) = VDD / RC$ (constant values), you can see that a bunch of stuff cancels out and it works (matches yesterday)

Didn’t take the best notes on this, but it’s ($e^{st}$) world ($V = e^{st}$), similarly for $I$

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uid: 202006251100 tags: #meetings #ee16b

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