I²t Calculator (Joule Integral)

Calculate Joule integral for capacitor discharge analysis and fuse protection design.

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Calculate I²tFind I²t from known V, C, and R_eqCalculate R_minFind minimum resistance for fuse protection

Calculate I²t

Calculate Joule integral (I²t) from voltage, capacitance, and equivalent resistance. Useful for analyzing capacitor discharge energy and fuse protection.

Voltage (V)

Capacitance (C)

Equivalent Resistance (Rₑq)

Results

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Calculate I²t (Joule Integral)

The I²t (pronounced "I-squared-t") is the Joule integral that quantifies the thermal energy delivered by a current pulse. It's fundamental for fuse protection design because every fuse has an I²t rating indicating its melting energy capacity. This calculation determines how much thermal stress a capacitor discharge creates, allowing you to verify if your fuse will survive or properly protect the circuit.

Step 1: Initial Energy Stored in Capacitor

When a capacitor is charged to voltage V, it stores energy E = ½CV². This is the total energy available for discharge through the circuit.

E = \frac{1}{2} C V^2

Step 2: Peak Discharge Current

At the moment of discharge (t = 0), the capacitor acts as a voltage source V with series resistance R_eq. The peak current follows Ohm's law.

I_{\text{peak}} = \frac{V}{R_{\text{eq}}}

Step 3: Exponential Current Decay

The discharge current decays exponentially with time constant τ = R_eqC. The current at any time t is given by the exponential decay equation.

i(t) = I_{\text{peak}} \cdot e^{-t/\tau}

Step 4: Joule Integral Calculation

The I²t integral is calculated by integrating i²(t) from t = 0 to ∞. For an exponential discharge, this integral has a closed-form solution. The result shows that exactly half of the capacitor's energy (½ × ½CV² = ¼CV²) contributes to the I²t value when divided by R_eq.

I^2t = \int_0^{\infty} i^2(t) \, dt = \frac{V^2 C}{2 R_{\text{eq}}}

ℹ️ This is the total thermal energy dissipated, expressed in ampere-squared-seconds (A²·s)

Step 5: Fuse Protection Verification

Compare the calculated I²t with the fuse's I²t rating from its datasheet. The fuse will blow if I²t_discharge ≥ I²t_fuse. For reliable protection, ensure I²t_fuse / I²t_discharge ≥ 1.5 (safety factor).

\text{Safety Factor} = \frac{I^2t_{\text{fuse}}}{I^2t_{\text{discharge}}} \geq 1.5

Where:

I^2t

= Joule integral (thermal energy)A²·s

V

= VoltageV

C

= CapacitanceF

R_{\text{eq}}

= Equivalent series resistanceΩ

I_{\text{peak}}

= Peak discharge currentA

E

= Initial capacitor energyJ

📘 Usage Notes

• I²t units are squared: 1 A²·s = 1,000,000 mA²·s. Always verify unit consistency when comparing with datasheet values.
• Fuse I²t rating: Found in fuse datasheets, typically specified at rated voltage. Some datasheets provide I²t vs time curves - use the value for your expected discharge duration.
• Equivalent resistance (R_eq): Includes all series resistances in the discharge path: ESR of capacitor + trace resistance + switch contact resistance + fuse resistance.
• Safety factor guidance: Use SF = 1.5-2.0 for normal conditions, SF = 2.0-3.0 for harsh environments or critical applications. Higher safety factors protect against component tolerances and aging.
• Mode A (Calculate I²t): Use this to verify if an existing circuit will blow a fuse during capacitor discharge, or to select an appropriate fuse rating.
• Mode B (Calculate R_min): Use this to design current-limiting resistance that prevents nuisance fuse blowing while maintaining protection during faults.
• Exponential discharge: This calculator assumes simple RC discharge. For circuits with inductance (LC or RLC), use specialized tools as the current waveform and I²t calculation differ.
• Repetitive pulses: If the capacitor discharges repeatedly, verify the fuse can handle the duty cycle. Most fuse I²t ratings assume single-pulse or infrequent operation.
• Cold vs hot resistance: Fuse resistance increases significantly when hot. Use cold resistance for I²t calculations, but verify voltage drop and power dissipation at operating current.
• Practical resistor selection: After calculating R_min, select the next higher standard value (E24/E96 series). Verify the resistor's pulse power rating can handle the peak power V²/R.