Foundations — Understand RC charging - discharging time constants

Before you can read a single line of the parent note Understand RC charging - discharging time constants, you need to own the vocabulary. This page introduces every symbol and notation the topic uses, in the order that each one is needed to understand the next. No symbol appears before it is drawn — each section builds only on the ones above it.

1. Voltage $V$ — the "push"

Figure 1: two water tanks side by side, one tall and one short. A red double-headed arrow marks the height difference between the water levels — that height gap is the picture of voltage: the bigger the gap, the stronger the push driving water (charge) from the high tank to the low one.

Why the topic needs it: the whole story is about the capacitor's voltage $V_C$ climbing toward the source voltage $V_s$ and how long that climb takes. Without a clear idea of "push", nothing else makes sense.

• The source voltage $V_s$ is the fixed push from a battery or supply.
• The capacitor voltage $V_C$ is the push across the capacitor, which changes over time.
• The resistor voltage $V_R$ is the push across the resistor.

The little subscript (the small letter after $V$) just says which component's voltage we mean. It is not multiplication.

2. Current $I$ — the "flow"

Why the topic needs it: the resistor's job is to limit current, and the capacitor charges because current flows into it. Current is the link between "how much charge" and "how much time".

3. Charge $Q$ and the capacitor law $Q=C{V}_C$

The relationship between them:

Figure 2: a straight red line rising from the origin, plotting stored charge $Q$ (vertical axis) against capacitor voltage $V_C$ (horizontal axis). A marked point shows that doubling the voltage doubles the charge — the line is straight because $Q=C{V}_C$, and its steepness is the capacitance $C$.

See Capacitors and stored energy for what all that stored charge is good for.

4. Resistance $R$ and Ohm's law $V_R=IR$

Why the topic needs it: Ohm's law is the only rule connecting the resistor's voltage to the current. It's how the resistor enters the equations.

5. Time $t$ — the running clock

Why the topic needs it: everything in this topic is a story over time — the voltage isn't one number, it's a value that changes as $t$ ticks forward. Every formula that follows will have $t$ in it, so we pin it down now before it appears.

6. The derivative $\frac{d{V}_C}{dt}$ — "rate of change"

This is the one piece of notation that scares people. Let's build it from zero, now that we have both $V_C$ and $t$.

Figure 3: a black curve of capacitor voltage $V_C$ rising with time $t$. A red straight line touches the curve at one point — this tangent's tilt is the derivative $d{V}_C/dt$ at that instant. Where the curve is steep (early on), the red line is tilted sharply, meaning fast change and lots of current; where it flattens, the red line lies flat, meaning almost no current.

• If $\frac{d{V}_C}{dt}$ is large, voltage is climbing steeply → lots of current flowing in.
• If $\frac{d{V}_C}{dt}=0$, voltage is flat → no current, the bucket is full.
• A jump in voltage would mean an infinite slope → infinite current. Impossible with a real resistor. This is why capacitor voltage ramps smoothly and never steps.

For the full machinery of equations containing derivatives, see First-order differential equations.

7. Kirchhoff's Voltage Law — "the pushes add up"

Why the topic needs it: KVL is the single equation that ties the whole loop together. Substituting Ohm's law and the capacitor law into it produces the differential equation the parent note solves.

8. The time constant $\tau =RC$ — the "sip length"

Now we combine the straw and the bucket into a single number. This is the symbol the whole topic revolves around, and only now — with $R$, $C$, and $t$ all defined — is it safe to introduce.

Why the topic needs it: every formula in the parent note is written in units of $\tau$. Time always shows up as the ratio $t/\tau$ ("how many sips have passed"), which is what makes the "$63.2\%$-per-$\tau$" rule universal.

9. The exponential $e^{-t/\tau }$ — "natural decay"

With $e$, $t$, and $\tau$ all in hand, we can finally read the curve that governs everything.

Figure 4: a red decay curve $e^{-t/\tau }$ starting at height 1 and falling toward zero, with the horizontal axis measured in multiples of $\tau$. A dashed marker shows that after exactly one $\tau$ the curve has dropped to $0.368$ (about $37\%$), not to a half — the fall is steep at first and gentler later.

• $e^{-t/\tau }$ falls from 1 → used for discharging and for current.
• $1-e^{-t/\tau }$ rises from 0 → used for charging voltage.

Prerequisite map

The diagram below shows how these foundations feed into the topic. Read the abbreviations as: KVL = Kirchhoff's Voltage Law ($V_s=V_R+V_C$); ODE = the differential equation $RC\frac{d{V}_C}{dt}+V_C=V_s$; Solution = the final charging/discharging voltage formulas the parent note derives.

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, re-read that section before tackling the parent note.

Connections

• Capacitors and stored energy
• Kirchhoff's Voltage Law
• First-order differential equations
• RC low-pass and high-pass filters
• 555 timer circuits
• Digital signal rise time and integrity