What Is an RC Time Constant?

An RC circuit consists of a resistor and capacitor. The time constant describes how quickly the capacitor voltage changes when the circuit is charging or discharging.

        R
+V ───/\/\/\───┬────
               │
               C
               │
              GND

The time constant is represented by the Greek letter tau:

For a simple RC circuit:

τ = R × C

where:

• τ = time constant in seconds
• R = resistance in ohms
• C = capacitance in farads

RC Time Constant Calculator

Example — 10 kΩ and 100 µF

Suppose an RC circuit uses a 10 kΩ resistor and a 100 µF capacitor.

R = 10,000 Ω

C = 100 µF
  = 0.0001 F

The time constant is:

τ = R × C

τ = 10,000 × 0.0001

τ = 1 second

Therefore, the RC time constant is 1 second.

What Happens During Charging?

When an initially discharged capacitor is connected to a DC voltage through a resistor, its voltage rises exponentially toward the supply voltage.

Vc(t) = Vs × (1 - e^(-t/τ))

After one time constant, the capacitor reaches approximately 63.2% of its final voltage.

After approximately five time constants, the capacitor is considered practically fully charged for most applications.

Charging Time Calculator

Charging Formula

The capacitor voltage during charging is:

Vc = Vs × (1 - e^(-t/τ))

To calculate the time required to reach a particular percentage of the final voltage:

t = -τ × ln(1 - Vc/Vs)

For example, reaching 90% of the final voltage takes approximately:

t ≈ 2.303τ

What Happens During Discharging?

When a charged capacitor is discharged through a resistor, its voltage decreases exponentially.

Vc(t) = V0 × e^(-t/τ)

After one time constant, approximately 36.8% of the original voltage remains.

Discharge Time Calculator

RC Time Constant Examples

1 kΩ and 100 µF

τ = 1,000 × 100 µF

τ = 0.1 s

The time constant is 100 ms.

10 kΩ and 10 µF

τ = 10,000 × 10 µF

τ = 0.1 s

The time constant is again 100 ms.

100 kΩ and 100 µF

τ = 100,000 × 100 µF

τ = 10 s

The time constant is 10 seconds.

RC Cutoff Frequency

An RC network also has a characteristic cutoff frequency.

fc = 1 / (2πRC)

Because:

τ = RC

the cutoff frequency can also be written as:

fc = 1 / (2πτ)

RC Cutoff Frequency Calculator

Relationship Between Time Constant and Cutoff Frequency

The time constant and cutoff frequency are directly related:

fc = 1 / (2πτ)

A larger time constant produces a lower cutoff frequency, while a smaller time constant produces a higher cutoff frequency.

Capacitor Voltage After a Given Time

RC Circuits in Electronics

RC circuits are commonly used in electronics for timing, filtering, signal coupling, smoothing and delay functions.

• Low-pass filters
• High-pass filters
• Timing circuits
• Delay circuits
• Power-supply smoothing
• Audio coupling networks
• Reset circuits
• Pulse shaping
• Oscillator timing networks

RC Low-Pass Filter

             R
Vin ─────/\/\/\────┬──── Vout
                   │
                   C
                   │
                  GND

At low frequencies, the capacitor has a relatively high reactance and the output can follow the input.

At higher frequencies, the capacitor's reactance decreases and more of the signal is shunted toward ground.

RC High-Pass Filter

Vin ───── C ─────┬──── Vout
                 │
                 R
                 │
                GND

A high-pass RC network allows higher-frequency components to pass more readily while attenuating low-frequency components.

Time Constant and Resistor Value

For a fixed capacitor, increasing the resistance increases the time constant.

R ↑
↓
τ ↑
↓
Slower charging and discharging

For a fixed resistor, increasing the capacitance also increases the time constant.

C ↑
↓
τ ↑
↓
Slower charging and discharging

Time Constant and Capacitor Value

A larger capacitor stores more charge for a given voltage:

Q = CV

Therefore, a larger capacitance generally takes longer to charge or discharge through the same resistance.

Practical Considerations

• Resistor tolerance affects the actual time constant.
• Capacitor tolerance affects the actual time constant.
• Electrolytic capacitors can have significant tolerance.
• Capacitor leakage can affect long time constants.
• Parasitic resistance and capacitance can affect fast circuits.
• The source resistance may become part of the effective R.
• The load resistance can change the effective RC network.

Common Mistakes

• Using microfarads as if they were farads.
• Using kilohms as if they were ohms.
• Forgetting that τ is measured in seconds.
• Assuming five time constants means mathematically exactly 100% charge.
• Ignoring source and load resistance.
• Ignoring capacitor leakage in long-duration timing circuits.
• Confusing time constant with cutoff frequency.

Key Points

• The RC time constant is τ = RC.
• After one time constant, a charging capacitor reaches approximately 63.2%.
• After five time constants, it reaches approximately 99.3%.
• During discharge, approximately 36.8% remains after one time constant.
• The cutoff frequency is fc = 1/(2πRC).
• Larger R or C produces a longer time constant.
• Real circuits can have additional resistance and capacitance that affect the result.