Transient RL/RC cheat sheet

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\begin{minipage}{0.75\linewidth}

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  \bfseries\color{white} Transient Response cheat-sheet for First-Order RC and RL circuits
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\begin{multicols}{3}


% RL Time Constant Box
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  \bfseries TIME CONSTANT FOR RL-CIRCUIT\\[0.5em]
  {\setlength{\abovedisplayskip}{-4pt}%
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   \[
     \tau = \dfrac{L}{R}
   \]
  }
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% RL Step Response Box
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  title=\textbf{\color{myNavy}\textit{RL}--Circuit}
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\[
i(t) = I_f + (I_0 - I_f)e^{-t/\tau}.
\]




\textbf{Step 1:} Determine the initial current, \( V_0 \), in the inductor. This usually involves analyzing the circuit for \( t < 0 \).

\textbf{Step 2:} Calculate the time constant, \( \tau \). To do this, you need to find the equivalent resistance attached to the inductor for \( t \ge 0 \) after zeroing any independent sources.

\textbf{Step 3:} Calculate the final value of the inductor current I$_f$, by analyzing the circuit as \( t \to \infty \).

\textbf{Step 4:} Write the equation for the inductor
current for \( t \ge 0 \) by substituting the values for
the initial current and the time constant into the
equation above. If there are no sources when \( t \ge 0 \), then I$_f$ = 0.
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% RC Time Constant Box
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  \bfseries TIME CONSTANT FOR RC-CIRCUIT\\[0.5em]
  {\setlength{\abovedisplayskip}{-2pt}%
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   \[
     \tau = RC
   \]
  }
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% RC Step Response Box
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\[
v(t) = V_f + (V_0 - V_f) e^{-t/\tau}.
\]


\textbf{Step 1:} Determine the initial voltage, \( V_0 \), in the capacitor. This usually involves analyzing the circuit for \( t < 0 \).

\textbf{Step 2:} Calculate the time constant, \( \tau \). To do this, you need to find the equivalent resistance attached to the capacitor for \( t \ge 0 \) after zeroing any independent sources.

\textbf{Step 3:} Calculate the final value of the capacitor voltage V$_f$, by analyzing the circuit as \( t \to \infty \).

\textbf{Step 4:} Write the equation for the capacitor
voltage for \( t \ge 0 \) by substituting the values for
the initial voltage and the time constant into the
equation above. If there are no sources when \( t \ge 0 \), then V$_f$ = 0.

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\columnbreak








% RL Step Response Box
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  title=\textbf{\color{myNavy}\textit{Key Reminders for First-Order RC/RL Circuits}}
]

\begin{enumerate}

\item \textbf{Continuity at $t=0$}
    \[
    i_L(0^+) = i_L(0^-), \qquad
    v_C(0^+) = v_C(0^-)
    \]

\item \textbf{DC Steady-State Behavior}
    \[
    \text{Inductor } \rightarrow \text{ short circuit}
    \]
    \[
    \text{Capacitor } \rightarrow \text{ open circuit}
    \]

\item \textbf{Finding the Time Constant $\tau$}
    \begin{itemize}
        \item Zero all independent sources:
        \[
        \text{Voltage source } \rightarrow \text{ short},
        \]
                \[
        \text{Current source } \rightarrow \text{ open}
                \]
        \item Find $R_{\text{eq}}$ seen by the inductor or capacitor in the $t \ge 0$ circuit, and input into formula for $\tau$.
    \end{itemize}

\end{enumerate}

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