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Connecting capacitors in series raises their permissible DC voltage but lowers the net capacitance:
\[ \frac{1}{C{\text{eq}}}= \sum{k=1}^{n}\frac{1}{Ck}\qquad\Longrightarrow\qquad C{\text{eq}}<\min(C_k) \]
Every capacitor in the string carries exactly the same charge \(Q\); the individual voltages divide in inverse proportion to the capacitances.
Real-world spreads in capacitance and leakage current lead to unequal DC‐voltage sharing. Designers therefore add balancing networks (passive resistors or active circuits) and derate the components.
Key points
• Same charge, inverse-C voltage division
• Net voltage rating ≈ sum of individual ratings (subject to balancing)
• Net capacitance goes down, ESR and ESL add up
• Balancing resistors (or active balancers) are mandatory for electrolytics and high-voltage DC strings
Ideal series behaviour
• Current through the string is identical: \(i(t)\) is common.
• Charge on each device: \(Q_i = \int i(t)\,dt = Q\).
• Voltage on each device: \(V_i = \dfrac{Q}{Ci}\).
• Energy stored: \(E = \tfrac12 C{\text{eq}}V{\text{tot}}^{2}\). Because \(C{\text{eq}}\) is small, energy density per microfarad drops.
Influence of parasitics
• Equivalent series resistance (ESR) and inductance (ESL) add linearly:
‑ ESR\({\text{eq}} = \sum R{\text{ESR},k}\)
‑ ESL\({\text{eq}} = \sum L{\text{ESL},k}\)
• Result: higher losses and lower self-resonant frequency vs. a single capacitor of equal standalone voltage rating.
Voltage imbalance mechanisms
• Capacitance tolerance (±5 … ±20 % typical)
• DC-leakage mismatch, temperature dependence, ageing (especially electrolytics, supercaps)
• Dielectric absorption and charge redistribution after disconnection
Mitigation strategies
a) Passive balancing resistors
Derating guidelines (commonly referenced in IEC 60384, UL 810, MIL-PRF-39018)
• Electrolytics: operate at ≤ 70 % of combined rating, ≤ 0.8·\(V_{\text{cap}}\) if no balancing.
• Film capacitors: derate 10 – 20 % for long-term reliability.
• High-energy pulse strings (Marx generators): further derating or triggered grading gaps.
Practical design example
Desired: 800 V DC bus, ≥ 220 µF.
Components on hand: 4 × 470 µF/250 V electrolytics.
• Wide-bandgap (SiC, GaN) inverters run at 800–1000 V: designers increasingly use stacked film capacitors with integrated active balancing.
• Super-capacitor modules (48 V systems, UPS) ship with built-in MOSFET equalizers; passive resistor strings waste too much energy.
• Standards such as AEC-Q200 Rev D now specify test methods for stacked capacitor reliability in traction EV and renewable-energy converters.
• Research into thin-film polymer dielectrics (PPN, PEEK) promises higher per-unit voltage, reducing the need for large series stacks.
Analogy: Water tanks in series. Equal volume of water (charge) flows through each, but a narrower tank (smaller C) builds a higher water level (voltage) to hold that same volume.
Series energy penalty: If you need both high \(V\) and high \(C\), series connection is inefficient—the stored energy scales with \(C_{\text{eq}}\). Prefer single high-voltage parts or parallel low-ESR film blocks when space permits.
• Safety: Exceeding a capacitor’s rating can lead to violent venting or fire. Compliance with UL 810, IEC 61010, and CE Low-Voltage Directive is mandatory in commercial equipment.
• Environmental: Older capacitors may contain PCBs or electrolytes requiring special disposal; WEEE and RoHS restrict hazardous substances.
• Reliability: In mission-critical and medical devices, uneven voltage distribution can be a latent hazard—balancing networks and derating are considered part of a “reasonably foreseeable misuse” analysis per ISO 14971.
Select balancing resistors:
\[ R \le \frac{V{\text{cap}}}{10\,I{\text{leak,max}}} \]
Use 0.5–2 MΩ/ W metal-film, ≥ 1 / 4 W for ≤ 600 V segments.
• Formulas assume capacitors have negligible cross-coupling and identical temperature coefficients; real parts deviate.
• AC behaviour: With sinusoidal excitation, impedance—not static capacitance—governs division; DC balancing resistors don’t affect AC significantly.
• Ceramic capacitors (class-2 dielectrics) exhibit strong voltage-coefficient; effective C drops at high bias, further aggravating imbalance.
• Active balancing ICs for >100-series-cell super-cap modules (e.g., TI LM4030, LTC3350).
• High-K ferroelectric thin films to achieve ≥ 200 V rating per 0402 package, reducing need for stacking.
• Ageing models for leakage current drift in organic electrolytics—adaptive balancing algorithms.
• Partial-discharge monitoring as an early-warning method for series film-capacitor banks (CIGRÉ WG A3-32).
A series string of capacitors shares current and charge, splits voltage inversely to capacitance, and yields an equivalent capacitance lower than any single unit while allowing a higher composite voltage rating. Because manufacturing tolerances and leakage currents skew the voltage distribution, engineers must provide balancing networks, adequate derating, and validation testing—especially in high-reliability or high-energy applications. Understanding parasitics, standards, and modern balancing techniques is essential for safe and efficient implementation.
