Dieser Leitfaden ist derzeit auf Englisch verfügbar.
Technology / Technical guide
Filter-Capacitor Reactive Power and Cyclic Stored Energy
A capacitor can carry alternating current while having zero average power in an ideal sinusoidal model. Energy enters and leaves its electric field within each cycle. Reactive power quantifies that phase-related exchange under a convention; it is neither stored energy itself nor continuous heat consumption.
Joules, watts and var describe different parts of the same cycle.
Use one declared branch, passive current direction and RMS voltage before moving between waveform and phasor descriptions.
01 / One-branch model
Define the quantities before assigning a phase.
Consider one ideal constant capacitor C with voltage v(t) = √2Vrms cos(ωt), where ω is a positive angular frequency in rad/s.
Consider one ideal constant capacitor C with voltage v(t) = √2Vrms cos(ωt), where ω is a positive angular frequency in rad/s. Define current as entering the terminal labelled positive voltage. This is a hypothetical sinusoidal branch, not an identified MARSTON filter or output interface.
Under that passive reference, i(t) = C dv/dt = −√2ωCVrms sin(ωt). Current leads voltage by a quarter cycle. OpenStax develops the capacitive phase relationship and RMS distinction; the voltage peak √2Vrms must not be silently substituted for Vrms in later power expressions.
02 / Current magnitude
Zero average current does not mean zero RMS current.
The displayed alternating current averages to zero over complete periods, but Irms = ωCVrms is nonzero.
The displayed alternating current averages to zero over complete periods, but Irms = ωCVrms is nonzero. RMS expresses a magnitude-related quantity rather than the signed arithmetic average. A component can therefore carry an appreciable alternating current without transporting net charge in one complete ideal cycle.
At unchanged C and Vrms in this model, doubling frequency doubles Irms. This proportionality does not establish a practical ripple-current limit or an accepted converter operating frequency. It isolates the ideal branch relation while retaining the same assumed voltage waveform and capacitance.
03 / Instantaneous power
Power alternates between entering and leaving the field.
Multiplying the declared voltage and current gives p(t) = −ωCVrms² sin(2ωt).
Multiplying the declared voltage and current gives p(t) = −ωCVrms² sin(2ωt). Its sign alternates and its average is zero over a whole number of steady cycles. Instantaneous positive power stores energy; negative power returns energy through the chosen electrical port.
OpenStax distinguishes this reversible exchange from resistor dissipation. The factor 2ω means the instantaneous-power pattern repeats twice per voltage cycle in this example. It does not mean the applied voltage or the capacitor current has acquired a doubled fundamental frequency.
04 / Stored energy
Maximum energy and mean energy are distinct.
The stored field energy is W(t) = ½Cv² = CVrms²cos²(ωt).
The stored field energy is W(t) = ½Cv² = CVrms²cos²(ωt). It has maximum Wmax = CVrms² and cycle mean Wmean = ½CVrms². Differentiating W reproduces the stated instantaneous power under the passive convention.
Although the mean stored energy is positive, its change over a complete cycle is zero. It is therefore not a continuously accumulating energy store in this steady state. The energy-accounting guide explains why a storage level differs from its rate of change.
05 / Reactive sign
A power convention determines the sign, not the energy unit.
Using RMS phasors and complex power S = V times the conjugate of I, with inductive lagging load Q positive, this capacitor has Qc = −ωCVrms² in var.
Using RMS phasors and complex power S = V times the conjugate of I, with inductive lagging load Q positive, this capacitor has Qc = −ωCVrms² in var. Its magnitude satisfies |Qc| = ωWmax in this specific ideal sinusoidal model.
The minus sign identifies capacitive phase behaviour under the declared convention. It does not mean stored energy is negative. Qc concerns a single branch, so a three-phase total or a different branch-voltage definition cannot be inserted without establishing the appropriate aggregation.
06 / Nonideal boundary
Real loss and distorted excitation require additional terms.
Equivalent resistance and dielectric loss introduce positive average dissipation.
Equivalent resistance and dielectric loss introduce positive average dissipation. The loss-tangent guide explains one sinusoidal equivalent representation. Non-sinusoidal excitation also requires its applicable waveform or harmonic account; one fundamental-frequency Q expression does not represent every loss or current contribution.
This page confirms no filter capacitor, capacitance, phase arrangement or delivered frequency for MARSTON equipment. It supplies no power-factor adjustment or component-selection procedure. Its purpose is to distinguish reversible cycle energy from average heat and convention-dependent reactive power.
Related technical guides: Dielectric Loss Tangent: Phase, Frequency and Equivalent Models · Capacitor Dielectrics: Polarisation, Construction and Energy · Inductor Air Gaps: Magnetic Reluctance and Stored Energy.
General background: OpenStax: Capacitive Phase and RMS Quantities · OpenStax: Instantaneous and Average AC Power. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Voltage | v = √2Vrms cos(ωt) | Single ideal sinusoid and passive current reference. |
| RMS current | Irms = ωCVrms | Constant C and positive ω. |
| Instantaneous power | p = −ωCVrms²sin(2ωt) | Average zero over complete steady cycles. |
| Stored energy | Wmax = CVrms² in J | Peak energy differs from cycle mean. |
| Reactive power | Qc = −ωCVrms² in var | S = VI* and inductive load Q positive. |
Engineering questions.
Does zero average power imply no alternating current?
No. The ideal capacitor has nonzero RMS current while returning its received cycle energy.
Is reactive power measured in joules?
No. Its conventional unit is var; stored energy is in joules.
Does negative Q mean negative stored energy?
No. It identifies capacitive behaviour under the chosen power sign convention.
Does power repeating at twice frequency mean current also does?
No. The factor comes from multiplying the fundamental voltage and current waveforms.
Can the branch formula be used as an unqualified three-phase total?
No. Branch voltages, arrangement and aggregation must be declared.
Does an actual capacitor always have zero average loss?
No. Real dissipative mechanisms need additional terms beyond the ideal model.
Technical enquiry
Discuss the technical question.
A capacitor can carry alternating current while having zero average power in an ideal sinusoidal model. Share the model and applicable operating or design information for an engineering discussion.