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RL Circuit · Inductor Charging/Discharging Characteristics (Square Wave Excitation)

Square wave voltage \(v_{\text{in}}(t)\) applied to a series RL circuit | Observe the charging/discharging transient response of inductor current \(i_L(t)\)
Inductor current unit: milliamps (mA)

Signal Source & RL 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

500 Ω
20 Ω2000 Ω
500 mH
10 mH1000 mH

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

Time Constant & Charging/Discharging Rules

⏱️ Time Constant \(\tau\) \(\tau = L/R\) Unit: seconds (s)
📐 RL Circuit Charging/Discharging Formulas (Inductor Current)
Charging phase ( \(v_{\text{in}} = V_m\) ): \(i_L(t) = \frac{V_m}{R} + \left(I_0 - \frac{V_m}{R}\right)e^{-t/\tau}\)
Discharging phase ( \(v_{\text{in}} = 0\) ): \(i_L(t) = 0 + (I_0 - 0)e^{-t/\tau}\)
\(\tau = L/R\) determines transient speed: after \(1\tau\) current rises/drops ~63.2%, after \(5\tau\) essentially reaches steady state.
The ratio of square wave frequency to \(\tau\) affects the inductor current waveform shape (exponential charging/discharging).
Inductor current unit: milliamps (mA), displayed values in the chart are converted to mA.
💡 Key observations: When period \(T \ll \tau\), current charging/discharging is incomplete, appearing as an approximate triangle wave; when \(T \gg \tau\), current can fully charge or discharge, waveform approaches a square wave.
With the default parameters, R = 500 Ω and L = 500 mH, so τ = 0.001 s. Each 0.01 s half-cycle lasts 10τ, allowing the current to settle to approximately 99.995% of its final value. The steady-state current is \(I_{max}=V_m/R = 2\,\text{mA}\).

RL Circuit Response · Input Square Wave \(v_{\text{in}}(t)\) and Inductor Current \(i_L(t)\) (mA)

Time Constant \(\tau = L/R\) | Exponential Charging/Discharging
Blue staircase waveform: input square wave voltage \(v_{\text{in}}\); Orange curve: inductor current \(i_L(t)\) (mA), reflecting RL charging/discharging transient process.

🔍 Waveform Observation Points

• Square wave rising edge: inductor current begins exponential rise toward \(V_m/R\) (mA).
• Square wave falling edge: inductor discharges through resistor, current decays exponentially toward 0.
• A larger time constant \(\tau = L/R\) produces slower current rise and decay, so the waveform changes more gradually.
• At low frequencies, the current can rise and decay fully; at high frequencies, it approaches a triangular waveform.

⚙️ Parameter Influence Rules

\(i_L(t)\) charging rate is determined by \(\tau = L/R\).
Increasing \(L\) or decreasing \(R\) → \(\tau\) increases → charging/discharging slows down, waveform distortion becomes more pronounced;
Increasing square wave frequency \(f\) → period decreases → charging/discharging time shortens, current amplitude decreases.
Amplitude \(V_m\) changes steady-state current value \(V_m/R\) (mA), but not the time constant.