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RC Circuit · Capacitor Charging/Discharging Characteristics (Square Wave Excitation)

Square wave voltage \(v_{\text{in}}(t)\) applied to a series RC circuit | Observe the charging/discharging transient response of capacitor voltage \(v_C(t)\)

Signal Source & RC Parameters

1.00 V
0.2 V5 V
50.0 Hz
10 Hz500 Hz

50% duty cycle square wave; frequency affects charging/discharging time ratio

1.00 kΩ
0.01 kΩ (10Ω)1 kΩ (1000Ω)
10.0 μF
0.2 μF20 μF

Time constant \(\tau = R \cdot C\), determines charging/discharging speed

Time Constant & Charging/Discharging Rules

⏱️ Time Constant \(\tau\) \(\tau = R \cdot C\) Unit: seconds (s)
📐 RC Circuit Charging/Discharging Formulas
Charging phase ( \(v_{\text{in}} = V_m\) ): \(v_C(t) = V_m + (V_0 - V_m)e^{-t/\tau}\)
Discharging phase ( \(v_{\text{in}} = 0\) ): \(v_C(t) = 0 + (V_0 - 0)e^{-t/\tau}\)
\(\tau = RC\) determines transient speed: after \(1\tau\) rises/drops ~63.2%, after \(5\tau\) essentially reaches steady state.
The relationship between square-wave frequency and \(\tau\) determines the shape of the capacitor-voltage waveform.
💡 Key observations: When period \(T \ll \tau\), capacitor charging/discharging is incomplete, appearing as an approximate triangle wave; when \(T \gg \tau\), capacitor voltage can fully charge or discharge, waveform approaches a square wave.
With the default parameters, R = 1 kΩ and C = 10 μF, so τ = 0.01 s. The 0.02 s period leaves one time constant for each half-cycle, making the exponential response easy to observe.

RC Circuit Response · Input Square Wave \(v_{\text{in}}(t)\) and Capacitor Voltage \(v_C(t)\)

Time Constant \(\tau = R \cdot C\) | Exponential Charging/Discharging
Blue staircase waveform: input square wave voltage \(v_{\text{in}}\); Orange curve: capacitor voltage \(v_C(t)\), reflecting RC charging/discharging transient process.

🔍 Waveform Observation Points

• Square wave rising edge: capacitor begins charging, voltage rises exponentially toward \(V_m\).
• Square wave falling edge: capacitor discharges through resistor, voltage decays exponentially toward 0.
• A larger time constant \(\tau = RC\) produces slower charging and discharging, so the waveform changes more gradually.
• At low frequencies, the capacitor can charge and discharge fully; at high frequencies, its voltage approaches a triangular waveform.

⚙️ Parameter Influence Rules

\(v_C(t)\) charging rate is determined by \(\tau = R \cdot C\).
Increasing \(R\) or \(C\) → \(\tau\) increases → charging/discharging slows down, waveform distortion becomes more pronounced;
Increasing square wave frequency \(f\) → period decreases → charging/discharging time shortens, capacitor voltage amplitude decreases.
Amplitude \(V_m\) only changes voltage amplitude, not the time constant.