Impedance Calculator

Introduction:

Resistance alone does not describe how a resistor, inductor, and capacitor oppose alternating current. Inductors and capacitors add frequency-dependent reactance, so a series RLC circuit needs a complex impedance with both a real part and an imaginary part.

Resistance, R

The real part of impedance. It dissipates energy and lies on the horizontal phasor axis.

Inductive reactance, X_L

A positive imaginary contribution that grows with frequency and inductance.

Capacitive reactance, X_C

A positive magnitude that is subtracted from inductive reactance and falls as frequency or capacitance rises.

Impedance, Z

The combined opposition to sinusoidal current, expressed in ohms with magnitude and phase.

When inductive reactance is larger, the impedance angle is positive and current lags the source voltage. When capacitive reactance is larger, the angle is negative and current leads. Equal reactances cancel in the ideal series model, leaving a resistive balance at resonance.

Magnitude controls the current drawn from a specified root mean square (RMS) source voltage, while phase determines the split between real and reactive power. Power factor is the real resistance divided by impedance magnitude, so it approaches 1 as net reactance approaches zero.

At any one operating frequency, these impedance, current, and power relationships assume ideal components in steady sinusoidal operation. Real inductors and capacitors have equivalent series resistance, tolerance, parasitic elements, temperature drift, and frequency-dependent losses. Include known losses in resistance or use measured reactances when those effects matter.

How to Use This Tool:

Use known reactances when they already come from a measurement or specification; use component values when frequency should determine them.

1. Choose Known reactances or Component values and frequency, then enter the total series resistance in the desired unit.
2. For known reactances, enter positive magnitudes for Inductive reactance XL and Capacitive reactance XC. For components, enter frequency, inductance, and capacitance with their units.
3. Read the Impedance ledger for rectangular and polar impedance, phase direction, current lead or lag, and power factor. Power factor also appears in the summary. Use Circuit reading for the substituted calculation path.
4. Open Advanced to add an RMS source voltage for current and ideal power estimates. Changing display precision affects visible rounding only.

Technical Details:

All supported resistance, frequency, inductance, capacitance, and voltage units are converted to ohms, hertz, henries, farads, and volts before calculation. Both input paths then use the same signed reactance and complex-impedance equations.

Formula Core:

For component values, angular frequency is 2π times frequency. Inductive and capacitive reactances oppose each other on the imaginary axis.

Rectangular impedance preserves the sign of net reactance. Magnitude and phase convert the same point to polar form.

f is frequency in hertz, L inductance in henries, C capacitance in farads, R resistance in ohms, X net reactance in ohms, and φ impedance angle in degrees. Resonant frequency is reported for the component-value path.

Taking source voltage as the phase reference, current phase θ_I is the negative of impedance phase φ_Z, the φ defined above.

An entered RMS voltage V adds current and ideal power estimates without changing impedance.

Current is in amperes, real power P in watts, reactive power Q in volt-amperes reactive, and apparent power S in volt-amperes. Negative reactive power marks capacitive behavior.

Frequency response:

The primary impedance, current, and power results describe the operating frequency. Frequency response extrapolates impedance magnitude from 0.1× to 10× that frequency, holding resistance constant and treating inductance and capacitance as fixed ideal values. Inductive reactance scales with the frequency ratio, while capacitive reactance scales inversely. The chart marks the operating point and the ideal LC minimum when it falls within the plotted range.

In known-reactance mode, the frequency axis is relative. The sweep assumes both supplied reactances refer to one common operating frequency and follow the fixed ideal L/C model away from it. Measured reactances at one frequency alone do not establish actual broadband behavior.

Rule Core:

Near-zero reactance needs a tolerance so floating-point noise does not create a false phase label. The tolerance is the greater of 10⁻¹² Ω and 10⁻¹⁰ times impedance magnitude.

A zero resistance with exactly cancelling reactances gives zero impedance and is rejected because current would be unbounded for any nonzero ideal voltage. Negative component magnitudes are also invalid.

Worked Examples:

References:

• RLC Series Circuits with AC, OpenStax University Physics Volume 2.
• RLC Series AC Circuits, OpenStax College Physics 2e.