RC and RL Transient Circuits — Charging & Discharging¶
Aim¶
To study the exponential charging and discharging behaviour of RC and RL circuits, measure the time constant in each case, and verify its dependence on the circuit parameters.
Theoretical Background¶
In a series RC circuit charged through a resistor R from a battery of EMF ε₀, the capacitor voltage grows as V_C(t) = ε₀(1 − e^{−t/τ}) with time constant τ = RC, and discharges as V_C(t) = V₀ e^{−t/τ} once the battery is removed and the capacitor is shorted through R. Analogously, in a series RL circuit the current grows as I(t) = (ε₀/R)(1 − e^{−t/τ}) with time constant τ = L/R, and decays as I(t) = I₀ e^{−t/τ} when the source is removed. In both cases τ is the time for the quantity to reach (1 − 1/e) ≈ 63.2% of its final value while charging, or to fall to 1/e ≈ 36.8% of its initial value while discharging, and after about 5τ the transient is essentially complete.
Governing Formula¶
| Symbol | Meaning |
|---|---|
| τ | time constant of the circuit (s) |
| R | resistance (Ω) |
| C | capacitance (F) |
| L | inductance (H) |
| ε₀ | source EMF (V) |
Interactive Controls¶
Launch the simulator and use the following controls on the RC and RL Transient Circuits — Charging & Discharging panel:
| Control | Symbol | Range | Default |
|---|---|---|---|
| Resistance | R | 1–100 kΩ | 10 kΩ |
| Capacitance | C | 1–100 μF | 10 μF |
| Inductance (RL mode) | L | 0.1–10 H | 1 H |
| Source EMF | ε₀ | 1–20 V | 10 V |
| Circuit (0=RC, 1=RL) | mode | 0–1 | 0 |
Procedure¶
- For the RC circuit, close the switch and record V_C(t) on the oscilloscope; read off the time to reach 63.2% of ε₀ and compare with RC.
- Vary R (keeping C fixed) and verify τ scales proportionally; repeat varying C at fixed R.
- Discharge the capacitor through R and verify the decay time constant matches the charging time constant.
- Switch to the RL circuit and repeat: measure the growth time constant and verify τ = L/R.
- Vary L and R independently for the RL circuit and verify the corresponding scaling of τ.
- Compare the general shapes of the RC-charging and RL-growth curves and note they are functionally identical apart from which quantity (voltage vs current) is plotted.
Observation Table¶
| S. No. | Circuit type | R (kΩ) | C (μF) or L (H) | Measured τ (ms) | Calculated τ = RC or L/R (ms) | |---|---|---|---|---|---|| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Graph¶
Capacitor Charging Curve — 63.2% at t=τ, ~99% at t=5τ
Calculations¶
Use the governing formula above with your tabulated observations to compute the required result for each row. Show at least one complete sample calculation with units.
Result¶
| Circuit | τ measured | τ calculated | % deviation | |---|---|---|---|| | | | | |
Precautions¶
- Allow at least 5τ for the transient to settle before considering the circuit to be at steady state.
- Discharge the capacitor completely (or de-energize the inductor) before starting a fresh charging run.
- Vary only R or only C (or only L) between successive runs to isolate each dependence.
- Remember an ideal inductor's current cannot change instantaneously, and an ideal capacitor's voltage cannot change instantaneously — use this as a sanity check on the simulated traces.
Maximum Permissible Error¶
τ = RC is a simple product, so fractional errors in R and C add directly.
Viva-Voce Questions¶
- Derive the charging equation of an RC circuit from Kirchhoff's voltage law.
- What is the physical significance of the time constant τ, and how is it read off an exponential curve graphically?
- Why do the RC-charging and RL-growth curves have the same mathematical form?
- How much energy is ultimately stored in the capacitor after full charging, and how does this compare with the energy dissipated in R during charging?
- What happens to the time constant if R is doubled while C is halved?
- Explain the duality between RC and RL transient circuits in terms of the roles of voltage and current.