What impedance is and why it is not just resistance
Impedance is the AC generalisation of resistance: the ratio of voltage to current in a circuit driven by a sinusoid. What makes it different from resistance is that voltage and current need not peak at the same instant. A resistor's voltage and current are exactly in step. An inductor's voltage leads its current by 90°, because voltage across an inductor depends on the rate of change of current. A capacitor's voltage lags its current by 90°, for the mirror-image reason. Impedance carries both the size of the ratio and the timing offset.
That timing is why the three contributions do not simply add. The resistive voltage and the reactive voltage are a quarter-cycle apart, so they combine like the two legs of a right triangle: the total is √(R² + X²), not R + X. The inductive and capacitive voltages, being a half-cycle apart from one another, do subtract directly, which is why the net reactance is X_L − X_C rather than a sum. A 10 Ω resistor in series with 10 Ω of inductive reactance gives 14.14 Ω, not 20 Ω; put 10 Ω of capacitive reactance in with them and you are back to 10 Ω.
The angle matters as much as the magnitude. Its cosine is the power factor, the fraction of the volt-amperes the source supplies that turn into real work. A branch with 0.6 power factor draws 67% more current than a resistive load doing the same job, and that extra current heats conductors, loads transformers and, on a commercial tariff, costs money. Reactive elements return their energy to the source each half-cycle rather than consuming it, so they contribute nothing to the watts but everything to the amps.
All of this assumes one steady sinusoidal frequency. If the excitation contains harmonics, work each frequency separately — the reactances differ at each one — and combine the results. For the individual reactance terms alone, the inductive reactance calculator and capacitive reactance calculator are quicker.
Reading the formula as a triangle
Draw the impedance triangle and every result on this page becomes obvious. Put R along the horizontal axis. Put the net reactance X = X_L − X_C along the vertical axis, upward when inductive reactance wins and downward when capacitive wins. The hypotenuse is |Z| and the angle it makes with the horizontal is φ. Then:
|Z| = √(R² + X²)is Pythagoras.φ = arctan(X/R)is the angle of that hypotenuse, positive for a net inductive branch and negative for a net capacitive one.PF = cos φ = R/|Z|is the adjacent side over the hypotenuse.
The same triangle scales into the power triangle. Multiply every side by the current squared and the horizontal leg becomes real power in watts, the vertical leg becomes reactive power in VAR and the hypotenuse becomes apparent power in volt-amperes. That is why power factor and phase angle carry the same information.
Both reactances depend on frequency, and in opposite directions. X_L = 2πfL rises without limit as frequency rises; X_C = 1/(2πfC) falls towards zero. Somewhere they are equal, and at that frequency the vertical leg of the triangle vanishes: the branch looks purely resistive, the impedance falls to R alone and the current peaks. That is series resonance, at f₀ = 1/(2π√(LC)), and the resonant frequency calculator covers the Q and bandwidth that go with it.
Sign conventions cause more confusion than the algebra. A positive phase angle here means the voltage leads the current, which is the same as saying the current lags — the normal state for motors and transformers. A negative angle means the current leads, which is what a capacitor-dominated branch does. Power factor itself is always reported as a positive number between 0 and 1, with the words “lagging” or “leading” carrying the sign.
Worked example: 10 Ω, 50 mH and 100 µF on 120 V at 60 Hz
A series branch of 10 Ω, 50 mH and 100 µF is connected across 120 V, 60 Hz. Find the impedance, the phase and the current.
- Angular frequency. ω = 2πf = 6.283185 × 60 = 376.9911 rad/s.
- Inductive reactance. X_L = ωL = 376.9911 × 0.050 = 18.8496 Ω.
- Capacitive reactance. X_C = 1/(ωC) = 1/(376.9911 × 1.0000 × 10⁻⁴) = 1/0.0376991 = 26.5258 Ω.
- Net reactance. X = 18.8496 − 26.5258 = −7.6763 Ω. Negative, so this branch is capacitive at 60 Hz.
- Impedance. |Z| = √(10² + (−7.6763)²) = √(100 + 58.9251) = √158.9251 = 12.6066 Ω.
- Phase. φ = arctan(−7.6763 / 10) = arctan(−0.76763) = −37.511°. The current leads the voltage.
- Power factor. PF = R/|Z| = 10 / 12.6066 = 0.7932, leading.
- Current. I = V/|Z| = 120 / 12.6066 = 9.5189 A.
- Power. P = I²R = 9.5189² × 10 = 90.609 × 10 = 906.1 W. Apparent power S = V × I = 120 × 9.5189 = 1,142.3 VA. Check: P/S = 906.1/1,142.3 = 0.7932, the power factor again.
Where does resonance sit for these parts? f₀ = 1/(2π√(0.050 × 1.0000 × 10⁻⁴)) = 1/(2π√(5.0000 × 10⁻⁶)) = 1/(6.283185 × 2.23607 × 10⁻³) = 1/0.0140496 = 71.176 Hz. That is above 60 Hz, which confirms the sign we found: below resonance the capacitive term is the larger one. Drive the same branch at 71.176 Hz and the impedance drops to 10.000 Ω and the current rises to 12.000 A, its maximum.
How to read the phase angle and power factor
Start with the sign of the phase angle, because it tells you what kind of load you have. A positive angle — current lagging — is the signature of motors, transformers, ballasts and solenoids, anything built around a winding. A negative angle — current leading — comes from capacitance: long lightly loaded cables, over-sized correction capacitor banks, and some switch-mode front ends. Both cost you the same in extra current, and utilities penalise both.
Then read the power factor against the standard. Most industrial tariffs set a threshold around 0.90 to 0.95 lagging and bill demand on kVA or apply a penalty below the threshold; the exact figure is set by your utility's tariff document, so read yours rather than assuming a number. A power factor of 0.79 means the source supplies 1/0.79 = 1.27 times the current a unity-power-factor load would need for the same watts, and conductor heating goes as current squared, so 1.27² = 1.60 times the I²R loss in the feeder. That is the real cost of a poor power factor even where there is no tariff penalty. Correcting it is what the power factor correction calculator sizes.
Compare the two reactances directly. Whichever is larger sets the character of the branch, and the difference between them is what actually appears in the impedance. Two components of 100 Ω and 90 Ω reactance contribute only 10 Ω of net reactance, and the individual voltage across each of them is still large — which is the trap in series-resonant circuits, where the voltage across L and across C can each far exceed the supply while the branch as a whole looks benign.
Finally, check the current against the components' ratings, not just against the source. A 9.52 A current through a 100 µF capacitor puts 9.52 × 26.53 = 252.5 V across it — more than the 120 V supply. Capacitor voltage ratings and inductor current ratings are separate constraints that the impedance magnitude alone does not reveal.
Reactance at common frequencies
| Component | 50 Hz | 60 Hz | 400 Hz | 1 kHz |
|---|---|---|---|---|
| 1 mH inductor | 0.3142 Ω | 0.3770 Ω | 2.513 Ω | 6.283 Ω |
| 50 mH inductor | 15.708 Ω | 18.850 Ω | 125.66 Ω | 314.16 Ω |
| 1 H inductor | 314.16 Ω | 376.99 Ω | 2,513.3 Ω | 6,283.2 Ω |
| 1 µF capacitor | 3,183.1 Ω | 2,652.6 Ω | 397.89 Ω | 159.15 Ω |
| 10 µF capacitor | 318.31 Ω | 265.26 Ω | 39.789 Ω | 15.915 Ω |
| 100 µF capacitor | 31.831 Ω | 26.526 Ω | 3.9789 Ω | 1.5915 Ω |
Every inductive row scales linearly with frequency and every capacitive row scales inversely, so you can read any other value off by proportion — a 25 mH coil is half the 50 mH row, and 400 Hz aircraft power gives an inductor about 6.67 times the reactance it has at 60 Hz.
Assumptions and common errors
- Adding R and X arithmetically. They are 90° apart, so 10 Ω of resistance with 10 Ω of reactance gives 14.14 Ω, not 20 Ω. Only X_L and X_C subtract directly, because they are 180° apart from each other.
- Treating an inductor as pure inductance. Real coils have winding resistance and, at higher frequencies, core loss and self-capacitance. Measure the branch resistance rather than assuming it is negligible; it is what sets the power factor and the current at resonance.
- Using this on a distorted waveform. The whole method assumes one sinusoid. With harmonics present, each order sees a different reactance, and the true power factor includes a distortion term that no impedance triangle captures.
- Ignoring the voltage across individual reactive elements. Near series resonance those voltages are magnified by the circuit Q and can be many times the supply, which is invisible if you only look at |Z|.
- Mixing peak and RMS values. The impedance is the same either way, but the current and power results only mean what you expect when the voltage you entered is RMS.
- Applying the series formula to a parallel circuit. Parallel branches combine by admittance, not by adding impedances, and a parallel tank behaves in the opposite way at resonance — maximum impedance rather than minimum.
Key terms
- Reactance (X)
- The opposition to alternating current from energy storage rather than dissipation, in ohms. Inductive reactance is positive and rises with frequency; capacitive reactance is negative and falls with frequency.
- Impedance (Z)
- The complex ratio of voltage to current, combining resistance and reactance. Its magnitude sets the current and its angle sets the phase relationship.
- Power factor
- The cosine of the phase angle, equal to real power divided by apparent power. Described as lagging when the current lags the voltage and leading when it leads.
- Series resonance
- The frequency at which inductive and capacitive reactances are equal, so they cancel and the branch impedance falls to its resistance alone.
Where this fits in AC analysis
The series RLC branch is the smallest circuit that contains everything AC analysis has to handle: dissipation, two kinds of energy storage, a frequency-dependent response and a phase relationship. Every larger analysis is built from it. Motor equivalent circuits are an RL series branch with a second branch for the magnetising current; transmission line models are series RL with shunt capacitance; a loudspeaker's electrical impedance is a series RLC around its mechanical resonance.
In power work the numbers here feed directly into two decisions. First, conductor sizing: it is the current, not the power, that heats a conductor, and a poor power factor raises the current for the same watts — the wire size and ampacity calculator takes that current and returns the conductor. Second, correction: knowing the reactive power lets you size the capacitor bank that cancels it. For three-phase systems the same triangle applies per phase, with the √3 factors that the three-phase power calculator handles.
In electronics the same arithmetic describes filters and matching networks. A series RC branch is a first-order high-pass or low-pass depending on where you take the output, which is what the RC filter calculator works out in decibels rather than ohms. Adding the inductor makes the response second-order and introduces the resonant peak.
When you need more than a single-frequency answer — a full frequency response, a transient, or a network with several loops — the tools are circuit simulation and Laplace-domain analysis. The impedance you compute here is exactly what a simulator evaluates at each frequency point, so the two agree by construction, and hand-checking one frequency against a simulation is the fastest way to catch a wrong component value.