Foundations — Understand RL transient behavior

Before you can read the parent note RL Transient Behavior, you must be able to read every letter and squiggle it uses without pausing. This page builds each one from nothing, in an order where each piece rests on the one before it.

1. Current $i$ — the thing that flows

Picture a pipe with water flowing. The current is not the water — it is the rate of water passing a marked line each second. In this whole topic, $i$ is the star: it starts somewhere, changes, and settles.

Why the topic needs it: the entire RL story is the shape of the curve $i(t)$ — how the flow builds up or dies away after you flip a switch.

2. Voltage $V$ and $v$ — the push

Why the topic needs it: the source $V$ is the cause; the current $i$ is the effect. We track how the total push $V$ splits between the resistor and the inductor.

3. Sign convention — which way is "positive"?

Before we add voltages up, we must agree on which direction counts as positive. Otherwise a minus sign appears out of nowhere and the reader is lost.

Why the topic needs it: without a fixed sign convention, $V=iR+Ldi/dt$ is ambiguous. With it, the same equation covers rising, falling, and negative currents automatically.

4. Resistance $R$ and Ohm's law — the friction

Picture a narrow section of pipe: it resists flow, and the "pressure drop" across it grows if either more water flows ($i$) or the pipe is narrower (bigger $R$).

Why the topic needs it: the resistor is one of the two components in the loop, and $v_R=iR$ is one of the two voltage terms in the loop equation.

5. The inductor $L$ — current inertia

Why the topic needs it: this "inertia" is the entire reason the current cannot jump instantly. Without $L$, there would be no transient at all — just Ohm's law. See Inductor Fundamentals for where this comes from.

6. Rate of change $\frac{di}{dt}$ — how fast the flow changes

This is the symbol that scares people. Let us build it from zero.

Why this tool and not another? We need a quantity that captures change over time. Plain subtraction ($i_2-i_1$) needs two separate moments; the derivative gives the rate at a single instant, which is exactly what the inductor reacts to. That is why the derivative — and not, say, an average — is the right tool here.

7. The key inductor law $v_L=L\frac{di}{dt}$

Read it in plain words: the harder you try to change the current, the harder the inductor pushes back.

• Current steady ($di/dt=0$) → $v_L=0$ → the inductor is just plain wire (a "short").
• Try to change current instantly (jump in zero time) → $di/dt=\infty$ → $v_L=\infty$. Impossible! So current is forced to be continuous: $i(0^{+})=i(0^{-})$.
• Current decreasing ($di/dt<0$) → $v_L<0$ → the inductor's voltage flips sign and it feeds current back (the discharge case from §3).

Why the topic needs it: this single law, combined with the loop equation, produces the whole exponential curve.

8. Kirchhoff's Voltage Law — the loop rule

Why the topic needs it: this is the starting line of the whole derivation. See Kirchhoff's Voltage Law.

9. Two math buttons: $\ln$ and $\int$

The derivation in the parent note uses two symbols you should recognize, even if you never solve one by hand.

10. The exponential $e^{-t/\tau }$ — the shape of "easing"

Key facts about $e^{-t/\tau }$ you will use:

• At $t=0$: $e^0=1$ (full value).
• As $t\to \infty$: $e^{-\infty }=0$ (faded away).
• At $t=\tau$: $e^{-1}\approx 0.368$ (dropped to 36.8% of its start).

Why the topic needs it: the settling current is a pure exponential — that is the shape of every RL transient.

11. The initial current $I_0$ — where the story starts

Why the topic needs it: the discharge formula begins at $I_0$; without naming it we could not write the decay curve.

12. The time constant $\tau =\frac{L}{R}$ — the pace-setter

Why the topic needs it: $\tau$ is the one number that summarizes the whole curve's speed. After about $5\tau$ the transient is over (see Steady-State Analysis).

13. Putting it together — the two curves you are heading toward

Now every symbol below is defined. These are the two results the parent note derives; here they simply show you the destination.

Every letter here — $i$, $V$, $R$, $L$, $\tau$, $e$, $I_0$ — was built in the sections above. You are ready.

Equipment checklist

Self-test: can you say each of these out loud before reading the parent note?