Formula used
Complex impedance: Z = R + jX for series-type models, where R is resistance and X is reactance.
- |Z| = √(R² + X²)
- ω = 2πf (angular frequency)
- Inductive reactance: XL = ωL
- Capacitive reactance: XC = 1/(ωC) (series uses -XC)
- Series RLC reactance: X = ωL − 1/(ωC)
Parallel models: compute admittance Y = G + jB with G = 1/R.
- Parallel RC: B = ωC
- Parallel RL: B = −1/(ωL)
- Parallel RLC: B = ωC − 1/(ωL)
- |Y| = √(G² + B²), then |Z| = 1/|Y|
How to use this calculator
- Select a mode that matches your circuit connection.
- Enter resistance, frequency, and component values as needed.
- Choose units for each input using the dropdowns.
- Press Calculate to view results above the form.
- Use Download CSV or Download PDF to export.
Example data table
| Mode | Inputs | Output |Z| (Ω) | Notes |
|---|---|---|---|
| Direct (R & X) | R = 100 Ω, X = -50 Ω | 111.8034 | Capacitive reactance gives negative imaginary part. |
| Series RL | R = 10 Ω, L = 10 mH, f = 1 kHz | 62.8320 | X = ωL ≈ 62.8319 Ω dominates magnitude. |
| Series RC | R = 1 kΩ, C = 100 nF, f = 1 kHz | 1879.0880 | X = −1/(ωC) ≈ −1591.55 Ω. |
| Parallel RC | R = 1 kΩ, C = 100 nF, f = 1 kHz | 846.7330 | Parallel branch reduces |Z| versus R alone. |
Professional guide
1) Why impedance magnitude matters
In AC analysis, impedance describes how a circuit resists current when voltage alternates. The magnitude |Z| is the single value most often used for sizing sources, estimating current, and comparing designs at a chosen frequency. For a sinusoidal source, RMS current is approximately I = V/|Z| when the network is linear and steady state is reached.
2) What |Z| represents mathematically
Impedance is complex: Z = Re(Z) + jIm(Z). The calculator reports the rectangular form and then computes |Z| = √(Re(Z)² + Im(Z)²). This magnitude is always non‑negative, even when the imaginary part is negative (capacitive). The phase angle shows whether current leads or lags voltage.
3) Direct entry using resistance and reactance
If you already know reactance X from measurements or datasheets, use the Direct mode. Enter R and X in ohms. Positive X indicates inductive behavior, while negative X indicates capacitive behavior. This path is helpful when you have an equivalent circuit model from an instrument such as an LCR meter.
4) Series RL, RC, and RLC behavior
For series circuits, reactances add algebraically. Inductive reactance grows linearly with frequency: XL = ωL. Capacitive reactance decreases with frequency: XC = 1/(ωC). In series RLC, X = ωL − 1/(ωC). At resonance, the two reactive terms cancel and |Z| approaches R.
5) Parallel models and admittance
Parallel networks are easiest through admittance Y = G + jB, where G = 1/R and B is the net susceptance. The tool computes |Y| = √(G² + B²) and returns |Z| = 1/|Y|. This method reflects how parallel branches share the same voltage while currents add.
6) Frequency response and design checks
Because ω = 2πf, changing frequency can dramatically change |Z|, especially with small capacitors or inductors. Use this calculator to check worst‑case current at high frequency for capacitors, and at low frequency for inductors. Compare multiple frequencies by recalculating and exporting results for documentation.
7) Interpreting phase angle
The phase angle is derived from atan2(Im, Re). A positive angle typically indicates inductive dominance (current lags), while a negative angle indicates capacitive dominance (current leads). Near 0°, the load is close to resistive and real power transfer is more efficient for a given RMS voltage.
8) Practical tips for reliable inputs
Use consistent units and realistic component values. Remember that real components include parasitics: inductors have series resistance, capacitors have ESR, and wiring adds inductance at high frequency. If you have measured impedance at a test frequency, Direct mode can capture that behavior quickly and accurately.
FAQs
1) Can |Z| be less than resistance R?
Yes. In parallel networks, additional reactive branches can increase admittance, reducing |Z| below R. In series networks, |Z| is always at least R because |Z| = √(R² + X²).
2) What does a negative imaginary part mean?
A negative Im(Z) indicates capacitive behavior. It means current leads voltage for a sinusoidal excitation. The magnitude |Z| remains positive because it is computed from squared components.
3) Why does inductive reactance increase with frequency?
Inductive reactance is XL = ωL. As frequency increases, ω increases, so the inductor opposes changes in current more strongly, raising the reactive component of impedance.
4) Why does capacitive reactance decrease with frequency?
Capacitive reactance follows XC = 1/(ωC). Higher frequency means larger ω, which reduces XC. The capacitor then allows more AC current for the same voltage.
5) What happens at resonance in a series RLC circuit?
At resonance, ωL equals 1/(ωC), so the net reactance is near zero. The impedance becomes mostly resistive, and |Z| approaches R, maximizing current for a fixed RMS voltage.
6) What is the difference between series and parallel selection?
Series selection assumes the same current flows through R and reactive elements. Parallel selection assumes the same voltage across branches and uses admittance to combine effects. Choose the option matching your wiring and schematic.
7) Is the phase angle enough to compute power factor?
For a linear sinusoidal load, power factor magnitude is approximately cos(phase angle). The sign indicates leading or lagging. Real systems may include harmonics, so measured power factor can differ from this ideal estimate.