RC Circuit Impedance Calculator and Formula

An RC circuit impedance calculator finds the total complex impedance of a resistor and capacitor at a chosen frequency. Enter resistance in ohms, capacitance in farads, and frequency in hertz to get the impedance’s real and imaginary parts, its magnitude, and its phase angle.

Formula

For a series RC circuit, the total impedance is the resistor impedance plus the capacitor impedance:

Z = R + 1/(jωC) = R − j/(2πfC)

• Z is complex impedance, measured in ohms (Ω).
• R is resistance in ohms (Ω).
• j is the imaginary unit, with j² = −1. In electrical engineering it is conventionally written j rather than i.
• ω is angular frequency in radians per second, equal to 2πf.
• f is frequency in hertz (Hz), or cycles per second.
• C is capacitance in farads (F). The calculator accepts microfarads (µF) and converts them to farads by multiplying by 0.000001.

The real part is R, and the imaginary part is −1/(2πfC). The magnitude is |Z| = √(R² + X_C²), where the capacitive reactance magnitude is X_C = 1/(2πfC). The phase angle is θ = atan2(−X_C, R); because R is positive here, this is equivalent to arctan(−X_C/R). The calculator reports the phase in degrees.

Worked Example

Using the calculator’s default values, R = 1,000 Ω, C = 1 µF = 0.000001 F, and f = 1,000 Hz:

1. Calculate capacitive reactance: X_C = 1/(2π × 1,000 × 0.000001) ≈ 159.15 Ω.
2. Find the real part: Re(Z) = R = 1,000 Ω.
3. Find the imaginary part: Im(Z) = −159.15 Ω, so Z ≈ 1,000 − j159.15 Ω.
4. Calculate magnitude: |Z| = √(1,000² + 159.15²) ≈ 1,012.58 Ω. Rounded to two decimal places, the calculator shows 1,012.58 Ω.
5. Calculate phase: θ = arctan(−159.15/1,000) ≈ −9.04°.

The negative phase indicates that the impedance is capacitive. The calculator’s separate real and imaginary outputs are rounded for display; calculations of magnitude and phase use the formula values before display rounding.

How RC Impedance Works

A resistor opposes current without storing energy in an electric field, so its ideal impedance is simply R. A capacitor stores and releases energy as the voltage changes. Its opposition to alternating current is called capacitive reactance, and it depends on frequency: at higher frequencies, a capacitor offers less reactance.

Impedance combines resistance and reactance as a complex quantity rather than adding their magnitudes directly. The real and imaginary components are perpendicular in the complex plane, which is why the magnitude uses the Pythagorean formula. In a series circuit, current is the same through both components, and their impedances add directly.

This result applies to a series resistor and capacitor driven by a sinusoidal signal at one frequency, using ideal component models. It describes the ratio of voltage to current as a phasor: Z = V/I. It is not, by itself, a DC resistance reading or a complete prediction of a circuit containing other components.

When to Use the Calculator

Use it to estimate how a series RC pair responds at a particular AC frequency, compare the resistor’s effect with the capacitor’s reactance, or check a hand calculation. For example, if reactance is much larger than resistance, the magnitude is dominated by the capacitor. If reactance is much smaller than resistance, the total impedance is close to R.

To use the result, read the magnitude as the overall voltage-to-current ratio in ohms. The real and imaginary parts show the components that determine phase. The phase angle is the angle of impedance, not necessarily the phase of current relative to voltage: since I = V/Z, current phase relative to voltage has the opposite sign. For a capacitive series RC circuit, current leads voltage, while the impedance phase is negative.

Units and Common Mistakes

• Convert capacitance correctly. One microfarad is 10⁻⁶ F; one nanofarad is 10⁻⁹ F and one picofarad is 10⁻¹² F. Enter the equivalent value in microfarads. For instance, 100 nF is 0.1 µF.
• Use frequency in hertz. If a source is specified in kilohertz, multiply by 1,000 before entering it. Do not enter angular frequency ω in the frequency field; the formula already converts f using 2π.
• Do not add R and X_C as ordinary positive numbers. They occupy different directions in the complex plane. Add the real and imaginary components, then calculate magnitude.
• Keep the sign of the imaginary part. Capacitive reactance contributes a negative imaginary component under the usual engineering convention. Dropping that sign can give the wrong phase interpretation.
• Distinguish series from parallel. These formulas are for a resistor and capacitor in series. A parallel RC circuit has a different total impedance and cannot be calculated by treating its components as a series pair.

Resistance and impedance are commonly expressed in ohms in both metric and US customary electrical work; there is no separate US customary unit conversion for them. Frequency is in hertz, and capacitance is in farads or commonly smaller SI multiples such as µF, nF, and pF.

Limits and Real Components

Real capacitors are not ideal. Their equivalent series resistance, leakage, lead inductance, and variation with frequency or temperature can affect measured behavior. A resistor also has tolerances and may have parasitic capacitance or inductance at sufficiently high frequencies. The simple formula omits these effects, as well as source impedance and wiring.

The calculator accepts nonnegative frequency and capacitance. At zero frequency or zero capacitance, the ideal capacitive reactance tends toward infinity, so the calculator’s division cannot produce a finite result; use a positive frequency and capacitance for a numerical answer. At DC, an ideal capacitor behaves as an open circuit after transients settle, rather than as a finite impedance described by this AC calculation.

Frequently Asked Questions

What is the impedance of a series RC circuit?
Z = R − j/(2πfC). Its real part is resistance, its imaginary part is negative capacitive reactance, and its magnitude is √(R² + X_C²).

Why does capacitor impedance decrease when frequency increases?
Because X_C = 1/(2πfC), reactance is inversely proportional to frequency when capacitance is fixed. Doubling frequency halves the ideal capacitive reactance.

Is the impedance angle positive or negative for an RC circuit?
For a series resistor and capacitor, the impedance angle is negative because its imaginary component is negative. This corresponds to current leading voltage in the ideal circuit.

Can I use this for a parallel RC circuit?
No. This calculator uses the series-impedance formula. For a parallel circuit, add admittances (the reciprocals of impedances) and then take the reciprocal to find total impedance.

What happens at zero frequency?
The ideal capacitor’s reactance tends to infinity as frequency approaches zero. At steady-state DC, an ideal capacitor acts as an open circuit; real components may still have leakage or other nonideal behavior.

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in r | Resistance | ohm | 1000 | 0
in c | Capacitance | uF | 1 | 0
in f | Frequency | Hz | 1000 | 0
out mag | Impedance magnitude | ohm | sqrt(r^2 + (1 / (2 * pi * f * c * 0.000001))^2) | 2
out re | Real part | ohm | r | 2
out im | Imaginary part | ohm | -1 / (2 * pi * f * c * 0.000001) | 2
out phase | Phase angle | degrees | atand(im / re) | 2