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