Impedance Calculator
Calculate total electrical impedance magnitude (|Z|), complex impedance form (R ± jX), and phase angle (θ) for RL, RC, and RLC AC circuits instantly using verified engineering formulas.
Impedance Calculator
How to Use the Impedance Calculator
Determining total electrical impedance in alternating current (AC) circuits is critical for power distribution sizing, transformer selection, and filter tuning. Follow these simple steps to operate the calculator:
- 1Select Calculation Mode: Choose between Resistance and Reactance, Resistance and Inductance with Frequency (RL), or Resistance and Capacitance with Frequency (RC).
- 2Enter Resistance (R): Input the circuit resistance in Ohms (Ω) or Kilo-ohms (kΩ).
- 3Enter Reactance or Component Values: Input reactance (X) in Ω, or enter inductance (H, mH) / capacitance (μF, F) and frequency (Hz, kHz).
- 4Click Calculate: Click the Calculate Impedance button to compute the AC circuit impedance parameters.
- 5Read Results: Review total impedance magnitude (|Z|), complex vector notation (R ± jX), and phase angle (θ).
How to Calculate Electrical Impedance (Sizing Guide)
Electrical impedance (Z) generalizes Ohm's law to alternating current (AC) circuits. While pure resistance opposes AC current independently of frequency, inductive and capacitive reactances introduce frequency-dependent phase shifts between voltage and current. Total impedance is calculated as a vector hypotenuse using the Pythagorean theorem: Z = √(R^2 + X^2).
Real-Life Impedance Sizing Scenarios
Scenario 1: Industrial AC Motor Feeder Branch Sizing
An AC induction motor operates on a 480V 60Hz supply. The motor winding exhibits a resistance R = 10 Ω and an inductive reactance XL = 15 Ω. Sizing total feeder impedance and phase angle:
|Z| = √(10^2 + 15^2) = √(100 + 225) = √325 = 18.03 Ω
θ = atan(15 ÷ 10) = atan(1.5) = 56.31° (Current lags voltage by 56.31°).
Scenario 2: High-Frequency RF Transmission Line Matching
A high-frequency RF transmission line operates with a series resistance R = 50 Ω and a capacitive reactance XC = 50 Ω. Sizing total line impedance:
|Z| = √(50^2 + (-50)^2) = √(2500 + 2500) = √5000 = 70.71 Ω
θ = atan(-50 ÷ 50) = atan(-1) = -45.0° (Current leads voltage by 45.0°).
Step-by-Step Sizing Guide & Formulas
|Z| = √(R^2 + X^2)
θ = atan2(X, R)
X = XL - XC = (2 × π × f × L) - (1 ÷ (2 × π × f × C))
Step 1: Determine resistance R and net reactance X. Assume R = 10 Ω and XL = 15 Ω (X = +15 Ω).
Step 2: Square resistance and reactance values:
10^2 = 100; 15^2 = 225
Step 3: Add squared terms and take the square root:
|Z| = √(100 + 225) = √325 = 18.03 Ω
Step 4: Compute phase angle θ:
θ = atan(15 ÷ 10) = 56.31°
Final Answer: Total impedance magnitude is 18.03 Ω with a 56.31° lagging phase angle.
AC Circuit Impedance Reference Table
The lookup table below shows calculated impedance magnitude (|Z|) and phase angle (θ) for standard combinations of resistance (R) and reactance (X) in series AC circuits:
| Resistance (R) | Reactance (X) | Calculated Impedance (|Z|) | Phase Angle (θ) | Circuit Behavior |
|---|---|---|---|---|
| 5 Ω | 5 Ω | 7.07 Ω | 45.0° | Inductive (+45.0°) |
| 10 Ω | 5 Ω | 11.18 Ω | 26.6° | Inductive (+26.6°) |
| 10 Ω | 10 Ω | 14.14 Ω | 45.0° | Inductive (+45.0°) |
| 10 Ω | 15 Ω | 18.03 Ω | 56.3° | Inductive (+56.3°) |
| 20 Ω | 10 Ω | 22.36 Ω | 26.6° | Inductive (+26.6°) |
| 25 Ω | -25 Ω | 35.36 Ω | -45.0° | Capacitive (-45.0°) |
| 50 Ω | -20 Ω | 53.85 Ω | -21.8° | Capacitive (-21.8°) |
| 100 Ω | 50 Ω | 111.80 Ω | 26.6° | Inductive (+26.6°) |
transformer impedance calculation
Transformer impedance defines internal opposition to AC current flow, determining short-circuit fault levels and voltage drops under full-load operation:
- Formula in Ohms:
Z_secondary (Ω) = (V_secondary^2 ÷ (kVA × 1,000)) × (%Z ÷ 100) - Percentage Impedance:
%Z = (V_short_circuit ÷ V_rated_primary) × 100 - Fault Current Impact: Lower %Z yields smaller internal voltage drops under load but results in significantly higher prospective short-circuit fault currents.
earth loop impedance calculation
Earth fault loop impedance (Zs) measures the total impedance of the fault path traversed by current during an insulation breakdown between a phase conductor and earthed metalwork:
- Formula:
Zs = Ze + (R1 + R2) - Ze: External earth loop impedance measured at the supply intake terminals.
- R1 + R2: Sum of line conductor resistance (R1) and circuit protective earth conductor resistance (R2).
- Purpose: Ensures fault current is high enough to trip circuit breakers or fuses within prescribed safety disconnection times (e.g. 0.4s).
transformer impedance calculation formula
The standard formulas to calculate transformer impedance in percentage (%Z) and physical Ohms (Ω) are:
- Percentage Impedance Formula:
%Z = (I_full_load × Z_ohms ÷ V_rated) × 100 - Physical Impedance Formula:
Z_ohms = R_winding + jX_leakage = √(R^2 + X^2) - Primary-Side Ohms:
Z_primary (Ω) = (V_primary^2 ÷ (kVA × 1,000)) × (%Z ÷ 100)
3 winding transformer impedance calculation
Three-winding transformers (Primary P, Secondary S, Tertiary T) feature three short-circuit test impedances (Zps, Zpt, Zst). Star-equivalent branch impedances (Zp, Zs, Zt) are calculated as:
- Primary Branch:
Zp = 0.5 × (Zps + Zpt - Zst) - Secondary Branch:
Zs = 0.5 × (Zps + Zst - Zpt) - Tertiary Branch:
Zt = 0.5 × (Zpt + Zst - Zps)
fault loop impedance calculation as3000
Under AS/NZS 3000 (Australia/New Zealand Wiring Rules), maximum allowable fault loop impedance (Zs) guarantees automatic disconnection of protective devices within 0.4 seconds for 230V final sub-circuits:
- Formula:
Zs_max = (U0 × 0.8) ÷ Ia - U0: Nominal AC phase voltage to earth (230 Volts).
- 0.8 Factor: Accounts for conductor temperature rise during fault conditions.
- Ia: Disconnection trip current for circuit breakers (e.g. 5 × In for Type C breakers).
percentage impedance calculation
Percentage impedance (%Z) expresses the internal impedance voltage drop of a transformer as a percentage of rated voltage when full-load current flows:
- Formula:
%Z = √(%R^2 + %X^2) - Short-Circuit Fault Current Sizing:
I_sc_max = I_full_load ÷ (%Z ÷ 100) - Example: A transformer with 5% impedance delivers a maximum short-circuit fault current equal to 20 times (100 / 5) its full-load current.
Frequently Asked Questions (FAQs)
Calculate total impedance magnitude (|Z|) by combining resistance (R) and net reactance (X) using the Pythagorean theorem: |Z| = √(R^2 + X^2).
In complex vector notation, Z impedance is written as Z = R + jX, where R is resistance in Ohms, X is reactance in Ohms, and j is the imaginary unit (√-1). The magnitude is |Z| = √(R^2 + X^2) and phase angle is θ = atan2(X, R).
The general series AC circuit impedance formula is Z = √(R^2 + (XL - XC)^2), where XL is inductive reactance (2πfL) and XC is capacitive reactance (1 / 2πfC).
Measure impedance using an LCR meter, impedance analyzer, or by applying a known AC voltage signal and measuring the resulting AC current: |Z| = V_rms ÷ I_rms.
An example is an 8 Ohm audio speaker, or an AC motor winding having 10 Ohms resistance and 15 Ohms inductive reactance, resulting in a total impedance of √(10^2 + 15^2) = 18.03 Ohms.
The three fundamental components contributing to AC impedance are pure Resistance (R), Inductive Reactance (XL), and Capacitive Reactance (XC).
Other terms for impedance include AC resistance, total alternating current opposition, or complex electrical load.