What is the formula for capacitors in parallel?

For capacitors in parallel, the equivalent capacitance is simply the sum of all individual capacitances: C_eq = C1 + C2 + C3. Each capacitor has the same voltage across it, and the total charge equals the sum of individual charges. The equivalent capacitance is always larger than any single capacitor.

How do you calculate energy stored in a capacitor?

The energy stored in a capacitor is calculated using E = 1/2 × C × V², where C is the capacitance in farads and V is the voltage across the capacitor in volts. This can also be expressed as E = Q²/(2C) or E = 1/2 × Q × V, where Q is the charge in coulombs. The energy is measured in joules.

Capacitor Bank Simulator

Series • Parallel • Mixed — Equivalent Capacitance • Charge • Energy — Simulate • Explore • Practice • Quiz

Mode

RC Circuit

RLC Circuit 📖 User Guide

Configuration

Capacitors

V_supply (V) V

Show Charges Animate Charging

Presets

C_eq

0 μF

Q_total

0 μC

E_total

0 mJ

V_supply

12.0 V

📖 Learning panels

Σ Live equations — values substituted from current state

💡 What-if coach — insights from current values

User Guide — Capacitor Bank Simulator

1 Overview

The Capacitor Bank Simulator lets you build series, parallel, and mixed capacitor configurations and observe how charge, voltage, and energy distribute across each component. You can calculate equivalent capacitance (C_eq), total charge (Q), total energy stored (E = ½CV²), individual capacitor voltages, and individual charges in real time. The canvas displays animated charging behaviour and colour-coded charge distribution.

This tool is designed for electrical engineering students studying circuit analysis, power systems engineers working with power factor correction capacitor banks, electronics technicians designing filter circuits, and physics learners exploring electrostatic energy storage and reactive power compensation.

2 Building the Circuit

Capacitor Bank simulator interface preview

The simulator opens in Simulate mode with a series configuration: C1 = 100 μF, C2 = 220 μF, C3 = 330 μF, and V_supply = 12 V. To begin:

Choose the number of capacitors (1–6, default 3) with the Capacitors selector — a slider is generated for each.
Select a Configuration: Series (capacitors in a chain), Parallel (capacitors across the same two nodes), or Mixed.
In Mixed, choose the number of Series groups: the capacitors split as evenly as possible into that many series-connected groups, with the capacitors inside each group wired in parallel. 1 group = all parallel, N groups = all series, and anything in between gives a series-parallel array (e.g. 3 series × 2 parallel).
Adjust each C1…Cn slider (1–1000 μF each) to set individual capacitance values, or type an exact value in its box.
Change the V_supply slider (1–24 V) to modify the charging voltage.
Toggle Show Charges and Animate Charging for visual feedback.
Try Presets: Equal Series, Equal Parallel, Filter Bank, or Power Factor correction.

3 Energising the Circuit

Simulate mode provides the interactive capacitor bank workspace. The canvas shows the circuit schematic with animated charge distribution. Key controls:

Configuration pills: Series gives 1/C_eq = 1/C1 + 1/C2 + 1/C3 (C_eq always less than the smallest capacitor). Parallel gives C_eq = C1 + C2 + C3 (C_eq always greater than any single capacitor). Mixed combines both topologies.
C1, C2, C3 sliders: Set individual capacitances. In series, all capacitors store the same charge Q but voltage divides inversely with capacitance. In parallel, all capacitors see the same voltage but charge distributes proportionally to capacitance.
V_supply slider: Changes the applied voltage. Energy stored scales as V², so doubling voltage quadruples energy.

Readout cards display: C_eq, Q_total, E_total, V_supply, plus individual voltages (V_C1, V_C2, V_C3), charges (Q_C1, Q_C2, Q_C3), and energies (E_C1, E_C2, E_C3) for each capacitor.

4 Circuit Theory

Explore mode organises capacitor theory into two categories:

Basics: Covers capacitance definition (C = Q/V), parallel plate capacitor formula (C = εA/d), dielectric materials, energy storage (E = ½CV²), and the relationship between charge, voltage, and capacitance.
Combinations: Explains series capacitor banks (same charge, voltage divides), parallel banks (same voltage, charge distributes), mixed configurations, and practical applications including power factor correction, pulse power systems, and capacitor voltage dividers.

Select any concept card for detailed explanations with formulas and interactive canvas illustrations.

5 Try a Problem

Practice mode generates problems such as: “Three capacitors of 100, 220, and 330 μF are connected in series. Find C_eq”, “A 470 μF capacitor is charged to 15 V. Calculate the energy stored in millijoules”, or “In a series bank, if Q = 500 μC and C1 = 100 μF, find V_C1.” Enter your answer and receive step-by-step feedback.

Quiz mode tests your knowledge with 15 multiple-choice questions covering series and parallel capacitance, energy storage, charge distribution, voltage division, and practical capacitor bank applications. Review your score and detailed explanations at the end.

6 Tools, Export & Keyboard Shortcuts

Editable inputs: Type an exact value into any C1/C2/C3/V_supply number box, or drag the slider — both stay in sync.
Live equations panel: The collapsible Learning panels below the readouts render the active configuration’s formulas (C_eq, Q, E, voltage/charge division) in classical notation with your current values substituted, plus a What-if coach with insights.
Show Calculations: Click the calculator button on the canvas to open a step-by-step derivation of equivalent capacitance, total charge, energy, and per-capacitor voltage/charge in SI units.
Export: Use the action bar to export results as CSV (all values) or the circuit diagram as a PNG image for study sheets and reports.
Right-click the circuit canvas for a context menu: Copy C_eq, Export PNG/CSV, toggle charges, and reset charging.
Keyboard shortcuts: Ctrl+Z undo, Ctrl+Shift+Z redo, R reset. Undo/Redo also have buttons in the action bar.
Sound: Practice and Quiz modes play a short tone on correct/incorrect answers (generated in-browser, no files).

7 Field Tips

In series, the smallest capacitor determines C_eq (it is always less than the smallest value). In parallel, capacitances simply add up.
Series capacitors share the same charge Q but the smallest capacitor gets the highest voltage — this is important for voltage rating considerations.
Parallel capacitors share the same voltage, so the largest capacitor stores the most charge and energy.
Try the Power Factor preset to see how capacitor banks are used in industrial power systems to compensate reactive power and improve power factor.
Energy scales with V²: doubling the supply voltage from 6 V to 12 V quadruples the total stored energy.
For the mixed configuration, identify which capacitors are in series and which are in parallel, then simplify step by step — just like solving a resistor network.
Use the individual readout cards (V_C1, Q_C1, E_C1) to verify your manual calculations against the simulator’s results.

Understanding Capacitor Banks — Free Interactive Series & Parallel Simulator

Capacitor bank simulator showing three capacitors in series and parallel configurations with voltage source and live readouts for equivalent capacitance, voltage distribution across each capacitor, and total energy stored, plus a series/parallel/mixed toggle — Toggle between series, parallel, and mixed configurations to see how equivalent capacitance and voltage distribution change instantly.

A capacitor bank is a group of capacitors connected together in series, parallel, or mixed configurations to achieve a desired equivalent capacitance, voltage rating, or energy storage capacity. Capacitor banks are used extensively in power systems for power factor correction, in electronics for energy storage and filtering, and in pulse power applications like defibrillators and laser systems. This interactive simulator lets you build series, parallel, and mixed capacitor circuits with one to six capacitors and observe how charge, voltage, and energy distribute across each component in real time.

Series Capacitor Banks — Voltage Division

When capacitors are connected in series, the reciprocal of the equivalent capacitance equals the sum of the reciprocals: 1/C_eq = 1/C₁ + 1/C₂ + 1/C₃. The equivalent capacitance is always less than the smallest individual capacitor. All capacitors in series store the same charge Q, but the supply voltage divides among them inversely proportional to their capacitance — the smallest capacitor gets the largest voltage. Series connections are used when a higher voltage rating is needed than any single capacitor can handle.

Parallel Capacitor Banks — Charge Storage

When capacitors are connected in parallel, the equivalent capacitance is the sum: C_eq = C₁ + C₂ + C₃. Each capacitor sees the same voltage (the supply voltage), and the total charge equals the sum of individual charges. The charge on each capacitor is proportional to its capacitance: Q = CV. Parallel banks are the most common configuration for increasing total capacitance in power factor correction and energy storage applications.

Energy Storage in Capacitors

The energy stored in a capacitor is given by E = ½CV², which can also be expressed as E = Q²/(2C) or E = ½QV. In a series bank, although each capacitor stores the same charge, the energy distribution depends on both capacitance and voltage. In a parallel bank, each capacitor stores energy proportional to its capacitance since all share the same voltage. The total energy stored equals the sum of individual energies, which also equals ½C_eqV_supply².

Power Factor Correction — The Real Industrial Application

Industrial loads (induction motors, transformers, welders) draw inductive current that lags voltage. This means the apparent power S (VA) exceeds the actual power P (W) by a factor 1/cosφ. Utility companies charge for apparent power because their generators and transmission lines have to carry the full VA. Industrial customers pay penalty rates if power factor drops below about 0.85.

Solution: capacitor banks in parallel with the inductive load. Capacitors draw leading current that cancels the inductive lag, raising power factor toward unity. A 100 kW factory with PF = 0.7 lagging draws 143 kVA from the grid. Add 71 kVAr of capacitor bank: power factor jumps to 0.96, apparent power drops to 104 kVA, monthly electricity bill drops significantly.

Sizing a Power Factor Correction Bank — A Worked Example

A factory draws 100 kW at PF = 0.7 lagging on a 415 V three-phase supply. How much capacitor bank is needed to raise PF to 0.95?

Step	Working	Result
Initial reactive power	Q₁ = P·tan(arccos 0.7)	102 kVAr (inductive)
Target reactive power	Q₂ = P·tan(arccos 0.95)	33 kVAr (inductive)
Capacitor bank needed	Q_cap = Q₁ − Q₂	69 kVAr
Capacitance per phase (star-connected)	C = Q_cap/(3·V_ph²·ω) where V_ph = 240 V, ω = 314	~1270 µF per phase
Monthly demand-charge savings (typical)	Roughly proportional to (kVA reduction) at industrial rates	15−30% on demand portion of bill

Capacitor banks pay back the investment cost in 1−3 years for medium-to-large industrial customers. Modern installations use automatic step controllers that switch capacitor banks in and out as load varies, keeping PF tightly regulated.

Why Capacitor Banks Need Reactors

Modern factories also have non-linear loads (variable-frequency drives, switching power supplies) that inject harmonics into the supply. Capacitor banks present low impedance at higher frequencies, so harmonics flow into them preferentially. The result: overheated capacitors, blown fuses, premature failure. Solution: add a detuning reactor (inductor) in series with each capacitor bank. The reactor + capacitor combination resonates below the 5th harmonic (the most common offender at 250 Hz on a 50 Hz system), preventing harmonic amplification. Modern PFC installations always include reactors.

References

IEEE Std 18-2012 — Standard for Shunt Power Capacitors.
IEC 60831-1 — Shunt power capacitors of the self-healing type for AC systems.
Bisquerra-Gil, V. et al. — Power Quality in Electrical Systems. McGraw-Hill.

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