RC Circuit Calculator: Time Constant & Energy

Introduction

This tool allows you to calculate the Time Constant ($\tau$) and the Stored Energy ($E$) of a capacitor in an RC circuit.

• Time Constant ($\tau$): Calculated using Capacitance ($C$) and Load Resistance ($R$).
• Capacitor Energy ($E$): Calculated using Voltage ($V$), Capacitance ($C$), or Charge ($Q$).

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1. Understanding the RC Time Constant ($\tau$)

What is the Time Constant? The Time Constant (represented by the Greek letter Tau, $\tau$) is a measure of how quickly a capacitor charges or discharges through a resistor. A capacitor cannot change its voltage instantly; the series resistor limits the current, creating a time delay.

The Formula: The transient response time is measured in seconds:

$\tau = R \times C$

Where:

• $\tau$: Time Constant (Seconds)
• $R$: Resistance (Ohms, $\Omega$)
• $C$: Capacitance (Farads, F)

Key Rule: It takes exactly 5 Time Constants ($5\tau$) for a capacitor to be considered "fully charged" (reaching ~99.3% of source voltage) or "fully discharged" (dropping to ~0.7%).

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2. Charging vs. Discharging Tables

The behavior of Voltage and Current differs depending on whether the capacitor is charging or discharging.

RC Charging Table (Voltage Rises, Current Falls)

When connected to a DC supply, the capacitor voltage increases while the charging current decreases.

Time Constant	Calculation	Capacitor Voltage ($V_c$)	Circuit Current ($I$)
0.5 $\tau$	$0.5 \times RC$	39.3%	60.7%
0.7 $\tau$	$0.7 \times RC$	50.3%	49.7%
1.0 $\tau$	$1.0 \times RC$	63.2%	36.8%
2.0 $\tau$	$2.0 \times RC$	86.5%	13.5%
3.0 $\tau$	$3.0 \times RC$	95.0%	5.0%
4.0 $\tau$	$4.0 \times RC$	98.2%	1.8%
5.0 $\tau$	$5.0 \times RC$	99.3% (Full)	0.7% (Zero)

RC Discharging Table (Both Fall)

When the source is removed and the capacitor discharges through a resistor, both voltage and current decrease over time.

Time Constant	Calculation	Capacitor Voltage ($V_c$)	Discharge Current ($I$)
0.5 $\tau$	$0.5 \times RC$	60.7%	60.7%
0.7 $\tau$	$0.7 \times RC$	49.7%	49.7%
1.0 $\tau$	$1.0 \times RC$	36.8%	36.8%
2.0 $\tau$	$2.0 \times RC$	13.5%	13.5%
3.0 $\tau$	$3.0 \times RC$	5.0%	5.0%
4.0 $\tau$	$4.0 \times RC$	1.8%	1.8%
5.0 $\tau$	$5.0 \times RC$	0.7% (Empty)	0.7% (Zero)

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3. Capacitor Energy Calculations

What is Capacitor Energy? A capacitor stores potential energy in the electric field created between its plates. Unlike a battery, which stores energy chemically, a capacitor stores energy electrostatically.

The Energy Formula: The energy stored is half the product of the charge and the voltage. The standard formula is:

$E = \frac{1}{2}CV^2$

Where:

• $E$: Energy (Joules, J)
• $C$: Capacitance (Farads, F)
• $V$: Voltage (Volts, V)

Alternative Formulas: Using the relationship $Q = C \times V$, we can express energy in three ways:

1. Using C and V: $E = \frac{1}{2}CV^2$
2. Using Q and V: $E = \frac{1}{2}QV$
3. Using Q and C: $E = \frac{Q^2}{2C}$

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4. General Electrical Energy (Mains Power)

While capacitors store Joules, large-scale electrical consumption (like household appliances) is often calculated in Kilowatt-hours (kWh).

Formula:

$E = P \times t$

• $E$: Energy (kWh)
• $P$: Power (Kilowatts, kW)
• $t$: Time (Hours, h)

Note: To convert Watts to Kilowatts, divide by 1,000. To convert Seconds to Hours, divide by 3,600.