Complex Number Calculator

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phasor impedance calculator RLC

Complex impedance and phasors: RLC circuit calculator

On the Phasors tab, enter R, L, C, the frequency and the source voltage and pick series or parallel. The tool computes the impedances, the total impedance in rectangular and polar form, the current and the resonance.

Impedance as a complex number

In sinusoidal steady state, voltages and currents become phasors, complex numbers with magnitude and phase. A resistor has impedance ZR = R, an inductor ZL = jωL and a capacitor ZC = 1/(jωC) = −j/(ωC), with ω = 2πf. Electrical engineering writes j instead of i because i is already the symbol for current. Ohm’s law holds with complex numbers: I = V/Z.
In series the impedances add: Z = ZR + ZL + ZC. In parallel the reciprocals add, 1/Z = 1/ZR + 1/ZL + 1/ZC; for two impedances this gives Z₁Z₂/(Z₁ + Z₂). The sign of the imaginary part of Z tells the behavior: positive is inductive (current lags voltage), negative is capacitive (current leads) and zero is resistive. The power factor is cos φ, with φ the argument of Z.
At resonance in a series RLC circuit, ωL = 1/(ωC) and the reactances cancel: f₀ = 1/(2π√(LC)). With L = 10 mH and C = 100 µF, f₀ ≈ 159.15 Hz, and at that frequency Z = R. The tool shows f₀ whenever L and C are filled in. To add or combine impedances given as complex numbers, use the Operations tab with j in the options.

Worked circuits

Two impedances in series

Input: (4+3i) + (4-3i)
Expected output: 8

Z = 8 Ω: the reactances cancel.

The same two in parallel

Input: (4+3i)(4-3i)/8
Expected output: 25/8

Z₁Z₂/(Z₁ + Z₂) = 25/8 = 3.125 Ω.

Current from Ohm’s law

Input: 100/(3+4i)
Expected output: 12 − 16i

I = 12 − 16j A, with |I| = 20 A for V = 100∠0° V.

Impedance magnitude

Input: 3+4i
Expected output: |Z| = 5

Z = 3 + j4 Ω is inductive, with φ of about 53.13°.

Capacitive circuit

Input: 6-8i
Expected output: |Z| = 10

Negative imaginary part: the current leads the voltage.

Resonant frequency

Input: L = 10 mH, C = 100 µF
Expected output: f₀ ≈ 159 Hz

1/(2π·√(10⁻²·10⁻⁴)) = 1/(2π·10⁻³), about 159.15 Hz.

Full tool FAQ

How do you divide complex numbers?

Multiply numerator and denominator by the conjugate of the denominator, so the denominator becomes a real number. In (3 + 4i) ÷ (1 − 2i), the conjugate of 1 − 2i is 1 + 2i. The numerator is (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i, and the denominator is 1² + 2² = 5. The result is −1 + 2i.

How do you find the modulus and argument of a complex number?

How do you convert from polar to rectangular form?

What is De Moivre's formula and how do you compute powers?

How do you compute roots of complex numbers?

What is i to the power n, for large n?

How do you solve a quadratic equation with a negative discriminant?

What is the difference between i and j?

How do you compute the impedance of an RLC circuit?

Which input formats are accepted and how precise is it?

Frequently asked questions

Why does electrical engineering use j instead of i?

Because i is the traditional symbol for electric current. Mathematically j and i are the same number, with j² = −1. The tool accepts both letters as input, and the Phasors tab always shows j.

What does the phase of the impedance mean?

It is how far the voltage leads the current. With positive φ the current lags the voltage (inductive), with negative φ it leads (capacitive) and with zero φ they are in phase (resistive). The cosine of the phase is the power factor.

How does the tool treat an element that is not in the circuit?

Leave the field blank. In series, a missing element is a short (impedance 0); in parallel, it is an open circuit (no branch). Do not type 0 for L or C: inductance and capacitance must be positive.

Next steps

Impedance as a complex number

In sinusoidal steady state, voltages and currents become phasors, complex numbers with magnitude and phase. A resistor has impedance ZR = R, an inductor ZL = jωL and a capacitor ZC = 1/(jωC) = −j/(ωC), with ω = 2πf. Electrical engineering writes j instead of i because i is already the symbol for current. Ohm’s law holds with complex numbers: I = V/Z.

In series the impedances add: Z = ZR + ZL + ZC. In parallel the reciprocals add, 1/Z = 1/ZR + 1/ZL + 1/ZC; for two impedances this gives Z₁Z₂/(Z₁ + Z₂). The sign of the imaginary part of Z tells the behavior: positive is inductive (current lags voltage), negative is capacitive (current leads) and zero is resistive. The power factor is cos φ, with φ the argument of Z.

At resonance in a series RLC circuit, ωL = 1/(ωC) and the reactances cancel: f₀ = 1/(2π√(LC)). With L = 10 mH and C = 100 µF, f₀ ≈ 159.15 Hz, and at that frequency Z = R. The tool shows f₀ whenever L and C are filled in. To add or combine impedances given as complex numbers, use the Operations tab with j in the options.

Worked circuits

Two impedances in series

Input: (4+3i) + (4-3i)
Expected output: 8

Z = 8 Ω: the reactances cancel.

The same two in parallel

Input: (4+3i)(4-3i)/8
Expected output: 25/8

Z₁Z₂/(Z₁ + Z₂) = 25/8 = 3.125 Ω.

Current from Ohm’s law

Input: 100/(3+4i)
Expected output: 12 − 16i

I = 12 − 16j A, with |I| = 20 A for V = 100∠0° V.

Impedance magnitude

Input: 3+4i
Expected output: |Z| = 5

Z = 3 + j4 Ω is inductive, with φ of about 53.13°.

Capacitive circuit

Input: 6-8i
Expected output: |Z| = 10

Negative imaginary part: the current leads the voltage.

Resonant frequency

Input: L = 10 mH, C = 100 µF
Expected output: f₀ ≈ 159 Hz

1/(2π·√(10⁻²·10⁻⁴)) = 1/(2π·10⁻³), about 159.15 Hz.

Full tool FAQ

How do you divide complex numbers?

How do you find the modulus and argument of a complex number?

How do you convert from polar to rectangular form?

What is De Moivre's formula and how do you compute powers?

How do you compute roots of complex numbers?

What is i to the power n, for large n?

How do you solve a quadratic equation with a negative discriminant?

What is the difference between i and j?

How do you compute the impedance of an RLC circuit?

Which input formats are accepted and how precise is it?

Frequently asked questions

Why does electrical engineering use j instead of i?

Because i is the traditional symbol for electric current. Mathematically j and i are the same number, with j² = −1. The tool accepts both letters as input, and the Phasors tab always shows j.

What does the phase of the impedance mean?

How does the tool treat an element that is not in the circuit?

Leave the field blank. In series, a missing element is a short (impedance 0); in parallel, it is an open circuit (no branch). Do not type 0 for L or C: inductance and capacitance must be positive.