Step Response of Rc Circuit
The step response of an RC circuit with resistance R and capacitance C is v_C(t) = V₀(1 − e^{−t/RC}) for charging and v_C(t) = V₀·e^{−t/RC} for discharging, where V₀ is the source voltage and τ = RC is the time constant. For R = 10kΩ and C = 100μF, the time constant is τ = 1 second, meaning the capacitor reaches 63.2% of V₀ in 1 second and is effectively fully charged after 5 seconds. The RC circuit is the fundamental first-order system in electrical engineering.
Deriving the RC Step Response from First Principles
Consider a series RC circuit connected to a voltage source V₀ at t = 0. Kirchhoff's voltage law gives V₀ = Ri(t) + v_C(t). Since i(t) = C·dv_C/dt, substituting yields the first-order ODE: RC·dv_C/dt + v_C = V₀. Taking the Laplace transform with zero initial conditions: RC·sV_C(s) + V_C(s) = V₀/s, so V_C(s) = V₀/[s(RCs + 1)]. Partial fractions and inverse transform give v_C(t) = V₀(1 − e^{−t/RC}). The current is i(t) = (V₀/R)e^{−t/RC} — maximum at t = 0 and decaying exponentially as the capacitor charges.
Key Formulas
Time Constant τ = RC: Physical Interpretation
The time constant τ = RC (in seconds when R is in ohms and C in farads) determines the speed of charging. Larger R slows charging because it limits current flow. Larger C slows charging because more charge must be stored. At t = τ: voltage reaches 63.2%, current drops to 36.8%. At t = 2τ: 86.5%. At t = 3τ: 95.0%. At t = 5τ: 99.3%. For circuit design, the '5τ rule' is standard: the circuit is considered settled after 5 time constants. To charge a capacitor faster, reduce either R or C. To create a longer delay, increase RC.
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Open CalculatorEnergy Analysis During Charging
During charging to voltage V₀, the energy delivered by the source is E_source = CV₀². The energy stored in the capacitor is E_cap = ½CV₀². The remaining ½CV₀² is dissipated as heat in the resistor — regardless of the resistance value. This remarkable 50% energy efficiency is a fundamental property of RC charging: whether R is 1Ω or 1MΩ, exactly half the source energy goes to the capacitor and half to heat. This inefficiency is why switched-capacitor power converters and resonant charging circuits (which use inductors) are preferred for high-efficiency energy transfer.
RC Circuit as a Low-Pass Filter
The same RC circuit that produces a step response also functions as a low-pass filter. The transfer function H(jω) = 1/(1 + jωRC) has magnitude 1/√(1 + (ωRC)²) and phase −arctan(ωRC). The −3 dB cutoff frequency is f_c = 1/(2πRC). For R = 10kΩ and C = 10nF, f_c = 1/(2π × 10⁴ × 10⁻⁸) ≈ 1.59 kHz. Signals below this frequency pass through nearly unchanged; signals above are increasingly attenuated. This dual interpretation — time-domain step response and frequency-domain filtering — are two views of the same underlying physics, connected by the Laplace transform.
Practical RC Circuit Considerations
Real capacitors have equivalent series resistance (ESR) that adds to R, slightly reducing the effective time constant. Electrolytic capacitors have significant ESR (0.1–10Ω) while ceramic capacitors have very low ESR (<0.01Ω). Capacitor leakage current acts like a parallel resistance, causing the charged capacitor to slowly discharge over time. For precision timing circuits, use polypropylene or polystyrene capacitors with minimal leakage. In digital circuits, RC time constants at logic gates determine switching speed — faster transitions require lower RC products, which is why modern CMOS processes minimize both on-resistance and parasitic capacitance.
Related Topics in step response analysis
Understanding step response of rc circuit connects to several related concepts: rc step response. Each builds on the mathematical foundations covered in this guide.
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