RC Timing

Time Constant

resistance times capacitance

An RC circuit has an exact time scale tau equal to resistance times capacitance. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

An RC pair sets a time scale

The resistor and capacitor together set a time constant. This book uses the exact time scale, not a numeric exponential midpoint.

τ=RC\tau = R C
RC time constantThe resistor and capacitor set the timing scale.6 V3 ohm4 F

Multiply resistance by capacitance

The resistance is 3 ohm and the capacitance is 4 farads.

τ=RC=3 ohm⋅4 F\tau = R C = 3\ \text{ohm}\cdot 4\ \text{F}
RC time constantThe resistor and capacitor set the timing scale.6 V3 ohm4 F

The time constant is in seconds

The exact time constant is 12 seconds.

τ=12 s\tau = 12\ \text{s}
RC time constantThe resistor and capacitor set the timing scale.6 V3 ohm4 F

Several RC pairs make exact time scales

Each row multiplies resistance by capacitance. The diagram shows the middle row; no row claims an exponential midpoint voltage.

RCVτQfUf2 ohm3 F6 V6 s18 C54 J3 ohm4 F6 V12 s24 C72 J4 ohm5 F6 V20 s30 C90 J\begin{array}{c|c|c|c|c|c}R&C&V&\tau&Q_f&U_f\\2\ \text{ohm}&3\ \text{F}&6\ \text{V}&6\ \text{s}&18\ \text{C}&54\ \text{J}\\3\ \text{ohm}&4\ \text{F}&6\ \text{V}&12\ \text{s}&24\ \text{C}&72\ \text{J}\\4\ \text{ohm}&5\ \text{F}&6\ \text{V}&20\ \text{s}&30\ \text{C}&90\ \text{J}\\\end{array}
RC time constantThe middle table row is the checked diagram.6 V3 ohm4 F