Teaching guide for the module

RC Transients: Capacitor charge and discharge: Teacher's Guide

Teaching guide for the Capacitor Charge & Discharge simulator: explain in class the exponential transient of an RC circuit, the time constant $τ = R C$ , the practical $5 τ$ rule and the energy stored in a capacitor. Designed for electronics and electrical engineering teachers.

Module: Capacitor Charge & Discharge · Virtual oscilloscope of the RC circuit

Physical phenomenon

When a capacitor of capacitance $C$ is connected in series with a resistor $R$ and a DC source $V_{0}$ , the voltage across the capacitor does not change instantaneously: it grows exponentially from its initial value up to $V_{0}$ . The same happens during discharge: opening the circuit onto a resistor, the voltage decays exponentially towards zero.

The describing equations are:

$v_{C} (t) = V_{0} \cdot (1 - e^{- t / τ}) (charge)$

$v_{C} (t) = V_{0} \cdot e^{- t / τ} (discharge)$

where $τ = R \cdot C$ is the time constant, the only parameter governing the speed of the transient. After $5 τ$ the capacitor is considered charged (or discharged) at 99.3%: the technical convention used to declare the transient "complete".

The energy stored at steady state in the capacitor is $W = \frac{1}{2} C V_{0}^{2}$ .

Key concepts

Time constant $τ = R C$ : measured in seconds, depends only on the components, not on the applied voltage.
Exponential behaviour: non-linear. The growth rate is highest at the beginning and slows down progressively.
Practical $5 τ$ rule: operational convention for "transient complete".
Initial current $I_{0} = V_{0} / R$ : at the first instant a discharged capacitor behaves like a short circuit.
Steady-state current: zero in DC (the capacitor "blocks" direct current once the transient ends).
Stored energy: the capacitor is an electrostatic energy storage device, not a dissipator like a resistor.
Voltage continuity: $v_{C} (t)$ cannot change in steps: this is the fundamental property that justifies the exponential behaviour.

How to use it in the classroom

Opening: observing the transient. Press CHARGE with default values and observe the exponential curve growing on the oscilloscope. Have students verbalise the difference from a linear ramp: the slope is not constant, it rises rapidly at first and then slows down. Visually point to the $τ$ marker and the $5 τ$ marker on the trace.

Development: exploring $τ$ . Keep $V_{0}$ fixed and vary first only $R$ , then only $C$ , then both. Guiding question: "What changes if I double R? And if I double C? What if I double both?" Have students compute $τ$ mentally before reading it from the KPIs. This is the moment when students grasp that $τ$ is a circuit property, independent of the applied voltage.

Deeper exploration: discharge. Press DISCHARGE after charging the capacitor. Observe that the discharge curve is the "mirrored negative" of the charge curve, same $τ$ , same exponential behaviour. Highlight that the capacitor returns the stored energy: it is not dissipated at the same instant it was supplied, but is given back to the circuit when the source is excluded.

Final exercise. Assign numerical values (e.g. $R = 10 k Ω$ , $C = 100 μ F$ , $V_{0} = 12 V$ ) and ask students to compute $τ$ , $5 τ$ , $I_{0}$ and $W$ before verifying them in the simulator.

Real-world examples

Camera flash. A high-capacitance capacitor charges slowly through a resistance from the battery, then discharges in a few milliseconds onto the xenon lamp: the stored energy is released instantly in an intense flash.
Cardiac defibrillator. Same principle as the camera flash, different scale: the capacitor stores high-voltage energy and releases it in a controlled pulse on the patient's chest.
Delayed switching of lights and motors. RC circuits are used to generate predictable delays, the transient is the timer.
Power supply filters. The smoothing capacitor after a rectifier "fills" the dips of the rectified voltage by exploiting exponential discharge between consecutive peaks.
Pushbutton debouncing. A small RC across a pushbutton eliminates mechanical contact bouncing, providing a clean edge to digital circuits.

Classroom discussion questions

If I double R and halve C, what happens to $τ$ ? And to the time $5 τ$ ?
At the initial instant of charging, why is the current at its maximum and the voltage on the capacitor zero?
A "discharged" capacitor and a "wire" behave the same at instant zero. Why?
Can I instantaneously discharge a capacitor through zero resistance? What would happen in the real world?
Two equal capacitors in parallel form double the capacitance: what changes for $τ$ ? And for the stored energy at the same $V_{0}$ ?

Related modules

Ohm's Law & Power Management: Ohm's Law is the foundation of transient analysis: at steady state, the capacitor is an "open circuit" and all the current is the ohmic one.
AC Behaviour (R, L, C): the same RC circuit seen in sinusoidal regime instead of transient: the time constant $τ$ translates into the phase shift $φ$ and the impedance $Z_{C}$ .
Filters: the RC low-pass filter is the frequency-domain application of the same circuit: the cutoff frequency $f_{c} = 1/ (2 π R C)$ is tightly linked to $τ$ .