Impedance Calculator — Free Online Calculator
Calculate AC circuit impedance from resistance and reactance. Series and parallel combinations.
How to Use This Calculator
Enter resistance (R) and reactance (X) to calculate total impedance.
The Formula Explained
Impedance Z = √(R² + X²). Phase angle θ = arctan(X/R). Power factor = cos(θ) = R/Z.
AC Impedance: Beyond Simple Resistance
AC circuit analysis requires considering impedance, not just resistance. Impedance is the vector sum of resistance and reactance, representing the total opposition to AC current flow. For a simple resistor, impedance equals resistance and the current is in phase with voltage. Add an inductor or capacitor, and the current becomes partially out of phase, with reactance adding a frequency-dependent component to the total impedance. Understanding impedance is essential for analyzing motor circuits, transformer behavior, filter design, and transmission line characteristics.
The key equations: inductive reactance XL = 2πfL, capacitive reactance XC = 1/(2πfC), total impedance magnitude Z = sqrt(R² + (XL - XC)²), and phase angle φ = arctan((XL - XC) / R). Power factor in an AC circuit equals cos(φ), so a circuit with high reactance has a large phase angle and low power factor. Industrial power factor correction adds capacitance to cancel the inductance from motors, bringing power factor closer to unity and reducing total current for the same real power delivery.
Worked Example: Motor Starting Current
A 10 HP motor at 460V three-phase has winding resistance 0.5 ohm per phase and starting leakage reactance 1.2 ohms per phase. Impedance at locked rotor: Z = sqrt(0.5² + 1.2²) = 1.3 ohm per phase.
Starting current per phase: V_phase / Z = 460/sqrt(3) / 1.3 = 265 / 1.3 = 204 amps. Compare to full-load current for 10 HP at 460V: 14 amps (NEC Table 430.250). Starting current ratio: 204 / 14 = 14.6x.
Wait, that is too high. Real-world motor locked-rotor current is typically 6-7x FLA, not 14x. Let me reconsider — the full motor model includes magnetizing reactance, which is much larger than leakage reactance. At startup, the motor is approximately: source impedance + winding impedance, but the effective starting impedance is dominated by leakage plus rotor reflected resistance. Real-world: locked rotor current for a 10 HP motor is about 90A (6.4x × 14A FLA).
Key takeaway: motor starting current is a complex function of winding impedance, and simplified models underestimate it. Use NEMA code letters on the motor nameplate (A through V) to determine starting kVA per HP for accurate sizing of protection devices and upstream feeders.
Worked Example: Power Factor Correction Capacitor
An industrial facility has 500 kW of real power with 0.75 power factor. Apparent power: 500 / 0.75 = 667 kVA. Reactive power: sqrt(667² - 500²) = 441 kVAR (lagging, inductive).
Target power factor: 0.95. New apparent power: 500 / 0.95 = 526 kVA. New reactive power: sqrt(526² - 500²) = 164 kVAR.
Capacitor bank size needed: 441 - 164 = 277 kVAR. At 480V three-phase, this is a 280 kVAR capacitor bank (standard size). Capacitance per phase: C = kVAR / (2πf × V² × 3) = 277,000 / (2π × 60 × 480² × 3) = 266 µF. Most capacitor banks are sold in pre-packaged units at standard kVAR ratings, with automatic step control to match load variation.
Impedance Calculation Mistakes
1. Adding R and X arithmetically. They add as a vector sum (Pythagorean), not simple sum. Z = sqrt(R² + X²), not R + X.
2. Forgetting frequency dependence. XL and XC both depend on frequency. Calculations at 60 Hz do not apply to variable frequency drive outputs or high-frequency switching circuits.
3. Using wrong sign for capacitive vs inductive reactance. Inductive reactance is positive; capacitive is negative. In the formula Z = sqrt(R² + (XL - XC)²), they subtract, so at resonance they cancel.
4. Assuming impedance is constant. Motor impedance varies with speed (locked rotor vs running vs no-load). Transformer impedance varies slightly with load due to saturation. Line impedance varies with temperature. For precision work, account for operating conditions.
5. Ignoring mutual inductance. Parallel conductors have mutual inductance that affects circuit impedance. For high-accuracy calculations in long runs or tightly packed conductors, include mutual coupling.
Common Impedance Values
14 AWG copper wire: about 2.5 ohm per 1,000 ft resistance, 0.05 ohm per 1,000 ft reactance at 60 Hz in conduit.
500 kVA 480V transformer: about 0.02 ohm equivalent impedance at 5% impedance rating.
Small motor winding (1 HP): 5-15 ohms at full speed; 1-3 ohms at locked rotor.
Large motor winding (100 HP): 0.05-0.3 ohms at full speed; 0.01-0.05 ohms at locked rotor.
Power factor capacitor at 480V, 50 kVAR: about 4.6 ohms capacitive reactance per phase.
Audio speaker: 4, 6, or 8 ohms nominal (varies with frequency).
Standards for Impedance Measurement
IEEE 112 — Standard test procedure for polyphase induction motors and generators including impedance measurements. ANSI C57.12 — Transformer testing standards including impedance calculation. IEEE 519 — Recommended practices for harmonic control including impedance of power systems at harmonic frequencies.
Impedance: AC opposition that combines resistance and reactance
Impedance is the AC analog of resistance. It is the total opposition a circuit presents to alternating current, combining pure resistance (independent of frequency) with reactance (frequency-dependent inductive and capacitive components). The result is a complex number with magnitude Z and phase angle theta.
The formula and what it does
R is resistance. X_L is inductive reactance = 2 x pi x f x L (L in henries). X_C is capacitive reactance = 1 / (2 x pi x f x C) (C in farads). Net X = X_L - X_C, positive for inductive loads, negative for capacitive. Z is the magnitude; phase angle is theta = arctan(X/R).
Worked example
Scenario: Motor with 5 ohm resistance and 8 ohm inductive reactance at 60 Hz. Find impedance and PF.
Z = sqrt(25 + 64) = sqrt(89) = 9.43 ohm. Theta = arctan(8/5) = 58 degrees. PF = cos(58) = 0.53. Low PF, typical for an unloaded motor. At full load mechanical resistance dominates, R goes up, and PF rises to 0.85+.
Common mistakes to avoid
undefinedFrequently asked questions
When does impedance equal resistance?
At DC (frequency = 0), or for purely resistive loads, or at resonance (where X_L = X_C and net X = 0). In those cases Z = R.
Why does impedance change with frequency?
Reactance depends on frequency. Inductive reactance rises with frequency; capacitive falls. At one specific frequency (resonance) they cancel, leaving only resistance. Below or above that frequency, impedance grows.
How does this affect motor sizing?
Motor starting impedance is much lower than running impedance (locked-rotor is mostly the inductance of the stator alone). Locked-rotor current is 4-7x running current, which is why time-delay protection is required.
What is the impedance of a transformer?
Typically 4-7 percent on its rated base, meaning short-circuit current is 14-25x rated current at the secondary. This impedance limits short-circuit faults and protects downstream equipment.
How do I measure impedance?
LCR meter is the standard tool. Cheap multimeters cannot measure reactance directly. For a working motor, indirect calculation from V, I, and PF measurements gives effective Z.
Why does NEC Chapter 9 Table 9 list separate R, X, and Z for conductors?
On large feeders at high current, the inductive reactance of the conductor adds non-trivially to the resistive drop. Table 9 lets you account for that on commercial designs.