Impedance Calculation Formula: AC Circuits Explained

The impedance calculation formula for a sinusoidal AC circuit is Z = V / I, where voltage and current are expressed as phasors; for a resistor, inductor, and capacitor in series, it is Z = R + j(ωL − 1/(ωC)). Enter resistance, inductance, capacitance, and frequency to calculate the series circuit’s impedance magnitude, in ohms, along with its phase angle.

Formula

For a series circuit containing resistance, inductance, and capacitance, the complex impedance is:

Z = R + j(X_L − X_C) = R + j(ωL − 1/(ωC))

The calculator reports the magnitude and phase, rather than the complex number itself:

• |Z| = √(R² + (X_L − X_C)²), in ohms (Ω).
• θ = tan⁻¹((X_L − X_C)/R), in degrees.

Here, R is resistance in ohms, X_L is inductive reactance in ohms, X_C is capacitive reactance in ohms, L is inductance in henries (H), C is capacitance in farads (F), f is frequency in hertz (Hz), and ω = 2πf is angular frequency in radians per second. The symbol j means √−1; electrical engineering uses j because i commonly denotes current.

The reactance formulas are X_L = 2πfL and X_C = 1/(2πfC). The calculator treats the supplied values as ideal component values and uses the series RLC model.

Worked Example

Using the defaults in the calculator: R = 100 Ω, L = 0.1 H, C = 0.00001 F (10 μF), and f = 60 Hz.

1. Angular frequency: ω = 2π × 60 ≈ 376.991 rad/s.
2. Inductive reactance: X_L = 2π × 60 × 0.1 ≈ 37.70 Ω.
3. Capacitive reactance: X_C = 1/(2π × 60 × 0.00001) ≈ 265.26 Ω.
4. Net reactance: X = X_L − X_C ≈ −227.56 Ω.
5. Magnitude: |Z| = √(100² + (−227.56)²) ≈ 248.57 Ω.
6. Phase: θ = tan⁻¹(−227.56/100) ≈ −66.29°.

The negative net reactance and negative phase indicate a capacitive result under the stated sign convention. The magnitude is the first calculator output and is rounded to two decimal places.

What Impedance Means

Impedance describes how much a circuit opposes alternating current and how voltage and current are shifted in time relative to one another. It extends the idea of resistance to AC: resistance dissipates energy, while ideal inductors and capacitors store and return energy, creating reactance and phase shift. Impedance is measured in ohms, like resistance, but it can be represented by a complex value.

In general, for a sinusoidal steady-state circuit, Z = V/I, where V and I are complex voltage and current phasors. This is not always a simple division of two ordinary meter readings: their phase relationship matters. For a purely resistive circuit, voltage and current are in phase and impedance equals resistance. For a reactive circuit, impedance has both magnitude and angle.

Reading the Result and Choosing a Formula

The magnitude |Z| describes the voltage-to-current ratio for the specified frequency and circuit model. With RMS phasor voltage, the corresponding current magnitude is |I| = |V|/|Z|. For example, a 10 V RMS source across an impedance magnitude of 248.57 Ω gives about 0.0402 A RMS, provided the circuit is represented by this series impedance and the source voltage is across it.

• Resistor only: Z = R.
• Inductor only: Z = j2πfL.
• Capacitor only: Z = 1/(j2πfC) = −j/(2πfC).
• Series RLC: add the complex impedances, so the reactances subtract as in the calculator.

For components in parallel, do not add their impedances directly. Use reciprocal addition, 1/Z_total = 1/Z₁ + 1/Z₂ + …, treating each Z as complex, or combine admittances first. Circuit topology determines which expression applies; the calculator here is specifically for a series RLC circuit.

Frequency, Resonance, and Units

Impedance can change with frequency even if the component values stay fixed. Inductive reactance rises in proportion to frequency, while capacitive reactance falls as frequency rises. In an ideal series RLC circuit, the reactances cancel at the resonant frequency f₀ = 1/(2π√(LC)). At resonance, net reactance is zero and the ideal series impedance magnitude equals R.

Use consistent SI units in the formula: hertz, henries, and farads. Common conversions include 1 mH = 0.001 H, 1 μF = 0.000001 F, and 1 kHz = 1,000 Hz. A frequent error is entering microfarads as if they were farads; 10 μF is 0.00001 F, not 10 F. Resistance and impedance are in ohms; US customary units do not provide a different standard unit for these electrical quantities.

Assumptions and Common Mistakes

The result is an idealized calculation for sinusoidal steady-state operation with constant component values. Real components can have tolerances, parasitic resistance, inductance, or capacitance, and their behavior may vary with frequency, temperature, signal level, and construction. A more detailed model may therefore differ from this basic result.

• Do not use the series formula for a parallel circuit without converting the network appropriately.
• Do not confuse impedance magnitude with resistance. The magnitude combines resistance and net reactance; it is not generally just R.
• Keep track of the sign of net reactance. Positive means inductive under this convention; negative means capacitive.
• Use RMS values consistently for ordinary AC magnitude calculations, and preserve phase when working with phasors.
• Rounding intermediate reactances can slightly change the final result. Keep full calculator precision until the final display.

For nonsinusoidal waveforms, a single-frequency impedance may not describe the whole signal: analyze the frequency components separately when the circuit is linear. This formula also does not establish whether a component or circuit is suitable for a particular application; practical design requires appropriate component data and circuit-specific checks.

Frequently Asked Questions

What is the basic impedance formula?
For AC phasors, Z = V/I. For a series RLC circuit, Z = R + j(2πfL − 1/(2πfC)); its magnitude is the square root of resistance squared plus net reactance squared.

Is impedance the same as resistance?
No. Resistance is the real, energy-dissipating part of impedance. Impedance also includes reactance, which accounts for the phase effects of inductors and capacitors and depends on frequency.

Why does impedance depend on frequency?
Inductive reactance is 2πfL and increases with frequency, while capacitive reactance is 1/(2πfC) and decreases. Their combined effect changes the circuit’s impedance and phase.

How do you calculate impedance in a parallel circuit?
For branches with complex impedances, add their reciprocals: 1/Z_total = Σ(1/Z_branch). Then take the reciprocal of the resulting complex admittance. Do not use direct series addition for parallel branches.

What does a negative phase angle mean?
With the convention used here, a negative angle means the circuit is net capacitive: current leads voltage. A positive angle means it is net inductive, with current lagging voltage.

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in r | Resistance | ohm | 100 | 0
in l | Inductance | H | 0.1 | 0
in c | Capacitance | F | 0.00001 | 0
in f | Frequency | Hz | 60 | 0
out z | Impedance magnitude | ohm | sqrt(r^2 + (2 * pi * f * l - 1 / (2 * pi * f * c))^2) | 2
out x | Net reactance | ohm | 2 * pi * f * l - 1 / (2 * pi * f * c) | 2
out a | Phase angle | degrees | atand(x / r) | 2