Calculate Impedance: Formula, Examples, and AC Circuits

Calculate impedance from resistance and reactance using the AC-circuit formula Z = √(R² + X²) for a simple series circuit. Enter resistance and reactance in ohms; the calculator returns impedance in ohms, along with the phase angle that describes how voltage and current are shifted.

Formula

For a series AC circuit represented by resistance and a single net reactance, impedance magnitude is:

Z = √(R² + X²)

• Z is impedance magnitude, in ohms (Ω).
• R is resistance, in ohms (Ω).
• X is net reactance, in ohms (Ω). Inductive reactance is positive and capacitive reactance is negative under the usual sign convention.

The phase angle is θ = arctan(X/R), expressed here in degrees. The calculator uses the signed reactance, so a negative input produces a negative phase angle. Its magnitude calculation squares X, so positive and negative reactance of equal size give the same impedance magnitude.

This two-input formula treats resistance and reactance as perpendicular components of a complex quantity. More generally, impedance is Z = R + jX, where j is the imaginary unit used in electrical engineering. Its magnitude is √(R² + X²); the calculator’s Z is that magnitude, not the complex value itself.

Worked Example

Using the calculator’s default values, resistance is 30 Ω and net reactance is 40 Ω.

1. Square each component: R² = 30² = 900 Ω² and X² = 40² = 1,600 Ω².
2. Add the squares: 900 + 1,600 = 2,500 Ω².
3. Take the square root: Z = √2,500 = 50.00 Ω.
4. Find the phase angle: θ = arctan(40/30) = 53.13°.

The result is a 50.00 Ω impedance magnitude with a positive, inductive phase angle of 53.13°. The result is not 70 Ω: resistance and reactance are not added as ordinary numbers.

What Impedance Means in an AC Circuit

Impedance is the total opposition a circuit presents to alternating current. It combines resistance, which dissipates electrical energy as heat, with reactance, which arises from energy storage and release in capacitors and inductors. Like resistance, impedance is measured in ohms, but it can also carry a phase relationship.

For a sinusoidal signal in a linear circuit, Ohm’s law can be written with complex quantities as V = IZ. Here voltage V, current I, and impedance Z are phasors: they encode both magnitude and phase. If only magnitudes are needed, the impedance magnitude relates voltage and current amplitudes by |Z| = |V|/|I|, provided the same RMS or peak convention is used for both.

Finding Reactance from Components

When the circuit details are known, reactance can be calculated from frequency and component values. For an ideal inductor, X_L = 2πfL; for an ideal capacitor, X_C = −1/(2πfC) using the signed convention in this article. Frequency f is in hertz, inductance L in henries, and capacitance C in farads; each reactance is in ohms.

In a series circuit containing one ideal inductor and one ideal capacitor, net reactance is X = X_L + X_C. Their effects oppose each other. At the resonant frequency, their reactances cancel in this ideal series model, leaving Z = R. Real components also have losses and parasitic effects, so measured behavior can differ from the ideal calculation.

How to Use and Interpret the Result

• Enter R and the net X at the signal frequency of interest. Use the same circuit configuration and operating frequency for both values.
• Use ohms for both entries. If a value is given in kilo-ohms, multiply it by 1,000 before entering it; for milli-ohms, divide by 1,000.
• Use a negative X for net capacitive reactance and a positive X for net inductive reactance. If you know only the magnitude of reactance, the impedance magnitude can still be found, but the phase-angle sign cannot.
• Read Z as the magnitude of impedance, not as a resistance that is necessarily dissipating power. Read θ as the phase of voltage relative to current under the stated sign convention.

For example, if R = 30 Ω and X = −40 Ω, the magnitude remains 50.00 Ω, but the angle becomes −53.13°, indicating capacitive rather than inductive behavior. An angle of zero means the net reactance is zero; it does not mean the circuit has no resistance.

Common Mistakes and Limits

A frequent error is adding R and X directly. The correct magnitude formula uses the square root of the sum of squares. Another is treating inductive and capacitive reactance as both positive when combining them in a series circuit; their signs matter because they oppose. Keep units consistent and avoid rounding intermediate reactance values too early.

This calculator applies only when the circuit can be represented by one resistance and one net reactance in series, or when those values already describe the total input impedance at a particular frequency. A parallel circuit cannot generally be reduced by simply inserting one component’s R and X into this formula. Convert each branch to complex impedance or use admittance (the reciprocal of impedance) and combine the branches appropriately.

For a non-sinusoidal signal, impedance can vary across frequency components, and a single number may not describe the full response. Components can also be nonlinear or frequency-dependent. In those cases, use the circuit’s frequency response or a suitable measured/modelled complex impedance rather than assuming one fixed value.

Frequently Asked Questions

What is the basic formula for impedance?
For resistance R and net reactance X in a simple series model, the magnitude is Z = √(R² + X²). The complex impedance is R + jX.

Can impedance be calculated from voltage and current?
Yes. For sinusoidal steady-state measurements, impedance magnitude is voltage magnitude divided by current magnitude, in ohms. To determine complex impedance and phase, you also need the phase difference between voltage and current.

Why is impedance not just resistance plus reactance?
Resistance and reactance are components at right angles in the complex plane. Their combined magnitude is therefore found with the Pythagorean formula, not ordinary addition.

Does frequency affect impedance?
Usually. Inductive reactance increases with frequency, while the magnitude of capacitive reactance decreases with frequency. Resistance may also vary with frequency in real components and circuits.

What does a negative phase angle mean?
With the convention used here, a negative angle means net capacitive reactance. A positive angle means net inductive reactance; zero means the net reactance is zero.

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in r | Resistance | ohm | 30 | 0
in x | Reactance | ohm | 40 | 0
out z | Impedance | ohm | sqrt(r^2 + x^2) | 2
out theta | Phase angle | degrees | atand(x / r) | 2