vault backup: 2026-09-14 10:25:08

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education/computer engineering/ECE2290/Ch 7, RL RC Circuits.md

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+
+Given an RL circuit, it starts with an initial condition. 
+
+KVL defines a first order homogeneous ordinary differential equation with constant coefficients:
+
+$$ 
+	L \frac{di(t)}{dt} + Ri(t) = 0
+	\to
+L \frac{di(t)}{dt} = -Ri(t)
+$$
+
+Steps for solving RC natural response circuits:
+1. Determine the initial inductor current, $I_0$, by analyzing the circuit for $t < 0$.
+
+ 2. Calculate the time constant, $\tau  RC$, where $R$ is the equivalent resistance connected to the capacitor for $t \ge 0$. 
+
+3. Write the equation for the capacitor voltage:
+$$ v(t) = V_o ^{e - \frac{t}{\tau}}, t \ge 0  $$
+4. Calculate other quantities of interest using the capacitor voltage
+
+- The *natural* response of a circuit is where there is an initial condition, but nothing else is driving the circuit.
+- The *step* response of a circuit describes the behavior of a circuit when a sudden change is introduced (switch toggled, et cetera)