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Inductor and Capacitor Instantaneous Power Curves

Sinusoidal AC current reference \(i(t)=I_m\sin(2\pi f t)\) | Dashed line: current waveform · Solid line: instantaneous power waveform
Voltage amplitude \(V_m\) affects power amplitude | Average power of pure energy storage components is zero

Signal Source & Component Parameters

4.00 V
0.2 V5 V

Increasing voltage makes power peak significantly higher than current amplitude for easier observation

50.0 Hz
10 Hz500 Hz

Frequency changes inductive/capacitive reactance → affects current amplitude and power peak

4.0 mH
0.2 mH20 mH

Inductor current lags voltage by 90°, power \(p_L(t)=\frac{V_m^2}{2\omega L}\sin(2\omega t)\)

3000.0 μF
500 μF5000 μF

Capacitor current leads voltage by 90°, power \(p_C(t)= -\frac{V_m^2 \omega C}{2}\sin(2\omega t)\)

Reactive Power & Power Factor

Inductor Reactive \(Q_L\)
— var
Capacitor Reactive \(Q_C\)
— var
\(\cos\varphi_L\) (Inductor)
0.00
\(\cos\varphi_C\) (Capacitor)
0.00
Pure inductor/capacitor voltage and current have a phase difference of ±90°, power factor is 0, average power is zero, and reactive power equals the power peak.

Inductor · Current \(i_L(t)\) and Power \(p_L(t)\)

Phase Diff \(+90^\circ\) (Current Lags Voltage)
Dashed: Current \(i_L(t)\) (A)   |   Solid: Instantaneous Power \(p_L(t)\) (W) — Bidirectional energy exchange, average power is zero

Capacitor · Current \(i_C(t)\) and Power \(p_C(t)\)

Phase Diff \(-90^\circ\) (Current Leads Voltage)
Dashed: Current \(i_C(t)\) (A)   |   Solid: Instantaneous Power \(p_C(t)\) (W) — Negative sine power waveform, opposite phase to inductor

Theoretical Basis (Current Reference)

Let \(i(t)=I_m\sin(\omega t)\), then inductor voltage \(v_L = L\frac{di}{dt}= \omega L I_m \cos(\omega t)\),
Instantaneous power \(p_L = \frac{\omega L I_m^2}{2}\sin(2\omega t)\).
Capacitor voltage \(v_C = \frac{1}{C}\int i\,dt = -\frac{I_m}{\omega C}\cos(\omega t)\),
Instantaneous power \(p_C = -\frac{I_m^2}{2\omega C}\sin(2\omega t)\).
Where \(I_m = V_m/(\omega L)\) (inductor) or \(I_m = V_m \omega C\) (capacitor).

Waveform Observation Key Points

• Current reference is a sine wave (red dashed), power waveform is a double-frequency sine.
• Inductor power \(p_L\) is always a positive sine half-cycle (absorb → release), capacitor power \(p_C\) is a negative sine.
• Increasing voltage \(V_m\) or decreasing frequency/reactance makes power peak clearly exceed current peak.
• Pure energy storage component average power = 0, reactive power \(Q = \pm \frac{V_m^2}{2X}\).