**Overview**
Electrolytic capacitors are essential components in electronic devices, especially in switching power supplies. Their lifespan and operating conditions directly impact the reliability and longevity of the power supply systems they are part of. In many cases, when a capacitor fails—particularly when it bulges or leaks electrolyte—it often leads to disputes between manufacturers and suppliers. The power supply manufacturer may blame the capacitor's quality, while the capacitor producer might argue that the design of the power supply is at fault. This back-and-forth highlights the importance of understanding how electrolytic capacitors perform under different conditions.
This article provides an in-depth analysis of the life and safety of electrolytic capacitors, offering engineers practical insights for better design and selection.
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**1 Arrhenius**
**1.1 Arrhenius Equation**
The Arrhenius equation is a widely used empirical formula that describes how the rate of a chemical reaction changes with temperature. Inside an electrolytic capacitor, there are metal plates (typically aluminum) and an electrolyte, both of which undergo chemical processes that affect the capacitor’s lifespan. The Arrhenius equation helps quantify how temperature influences these reactions.
The Arrhenius equation can be expressed as:
**k = A * e^(-Ea/RT)**
Or in logarithmic form:
**ln(k) = ln(A) - Ea/(R*T)**
Where:
- **k** is the reaction rate
- **A** is the pre-exponential factor
- **Ea** is the activation energy
- **R** is the gas constant
- **T** is the absolute temperature (in Kelvin)
**1.2 Arrhenius Conclusion**
According to the Arrhenius equation, as temperature increases, the chemical reaction rate also increases. Typically, for every 10°C rise in temperature, the reaction rate (and thus the rate of capacitor degradation) can increase by 2 to 10 times. Similarly, for each 10°C drop in temperature, the life of the capacitor can double. This means that ambient temperature is a critical factor in determining the lifespan of electrolytic capacitors.
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**2 Electrolytic Capacitor Life Analysis**
**1) Formula:**
Based on the Arrhenius equation, the formula for calculating the service life of an electrolytic capacitor is:
**L = L₀ × 2^[(T₀ - T)/10]**
Where:
- **L** is the expected life at ambient temperature **T** (in hours)
- **Lâ‚€** is the rated life at maximum operating temperature **Tâ‚€** (in hours)
- **T₀** is the rated maximum operating temperature (in °C)
- **T** is the actual ambient temperature (in °C)
**2) Analysis:**
When the capacitor operates at its maximum rated temperature (**T = Tâ‚€**), the calculated life is equal to the rated life (**L = Lâ‚€**). For example, if the rated life is 8000 hours, this corresponds to about 0.9 years.
If the operating temperature is 10°C lower than the maximum, the life doubles:
**L = L₀ × 2^(10/10) = 2 × L₀**
So, 8000 hours becomes 16,000 hours, or approximately 1.8 years.
This clearly shows that the Arrhenius-based calculation aligns with real-world performance trends.
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**3 Electrolytic Capacitor Life Calculation**
In practice, the lifespan of an electrolytic capacitor is influenced by two main factors: **ambient temperature** and **ripple current**.
Ripple current causes internal heating due to the resistance of the capacitor’s equivalent series resistance (ESR). The higher the ripple current, the more heat is generated, which accelerates the aging process and reduces the capacitor’s lifespan.
To account for this, engineers must calculate the **power loss**, **temperature rise**, and **synthetic ripple current** when designing circuits.
**3.1 Ripple Current Calculation**
Ripple current is determined by the charging and discharging cycles of the capacitor. It involves several steps:
- Calculating capacitance
- Determining charging and discharging times
- Computing charging and discharging currents
- Summing up the total ripple current (Irms)
**3.2 Power Loss Calculation**
Power loss in the capacitor is calculated using:
**P = I² × R**
Where **I** is the RMS ripple current and **R** is the ESR.
**3.3 Heating Formula**
The temperature rise caused by ripple current is given by:
**ΔT = I² × R / (β × S)**
Where:
- **ΔT** is the temperature rise (°C)
- **I** is the RMS ripple current (A)
- **R** is the ESR (Ω)
- **β** is the heat dissipation coefficient (W/°C/cm²)
- **S** is the surface area of the capacitor (cm²)
**3.4 Synthetic Ripple Current**
Since ripple current can consist of multiple frequency components, the effective ripple current is calculated using root mean square (RMS) values across all frequencies.
**3.5 Rated Working Temperature**
Capacitors have a rated temperature (e.g., 105°C) and a maximum allowable temperature rise (e.g., 5°C) under rated ripple current. If the actual ripple current exceeds this, the temperature rise increases, reducing the capacitor’s lifespan.
**3.6 Final Life Calculation**
Taking into account both temperature and ripple current, the final life formula is:
**L = L₀ × 2^[(T₀ - T - ΔT)/10]**
Where:
- **ΔT** is the temperature rise due to ripple current
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**4 Example**
Consider a capacitor rated at **33 μF / 200 V / 105°C**, with a rated life of **8000 hours**, and a maximum allowed ripple current of **195 mA at 120 Hz**. It is used in a **110 V / 60 Hz** circuit with an ambient temperature of **55°C**.
After analyzing various waveforms (triangle, sine, and combined), calculating the ripple current, and determining the resulting temperature rise, the estimated life of the capacitor is significantly reduced when ripple current is considered. Without accounting for ripple current, the life might be around **8000 hours**, but with proper consideration, it could be as low as **4000 hours** or less.
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**5 Conclusion**
This example demonstrates how ripple current has a significant impact on the lifespan of electrolytic capacitors. Engineers must consider both **ambient temperature** and **ripple current** when designing power supplies. Choosing high-quality capacitors with good sealing and proper thermal management can help extend their life and prevent failures such as bulging, leakage, or even explosions.
By ensuring a safe working environment and a well-designed circuit, the risk of electrolytic capacitor failure can be minimized, leading to more reliable and longer-lasting electronic systems.
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