Parallel Impedance Calculator and Formula

Calculate the equivalent impedance of parallel branches by entering each branch’s resistance and reactance, expressed in ohms. The calculator combines their complex admittances and returns the equivalent resistance, reactance, magnitude, and phase angle.

Formula

For two parallel impedances, the direct complex formula is Z_eq = Z₁Z₂ / (Z₁ + Z₂). The calculator uses the equivalent admittance form, which avoids treating complex quantities as ordinary real-number fractions.

• Z is impedance, measured in ohms (Ω).
• R is resistance, the real part of impedance, in ohms.
• X is reactance, the imaginary part, in ohms. Positive X is inductive under the usual engineering sign convention; negative X is capacitive.
• Y = 1/Z = G + jB is admittance, in siemens (S). G is conductance and B is susceptance.
• j denotes the imaginary unit, with j² = −1.

For each branch, Z_k = R_k + jX_k. Its admittance is G_k + jB_k = (R_k − jX_k)/(R_k² + X_k²). Parallel admittances add: G_eq = G₁ + G₂ and B_eq = B₁ + B₂. Inverting gives R_eq = G_eq/(G_eq² + B_eq²) and X_eq = −B_eq/(G_eq² + B_eq²). Magnitude is |Z_eq| = √(R_eq² + X_eq²), and phase is atan2(X_eq, R_eq).

Worked Example

With the default inputs, the branches are Z₁ = 10 + j0 Ω and Z₂ = 20 + j0 Ω. Their admittances are 1/10 = 0.1 S and 1/20 = 0.05 S, so the combined admittance is 0.15 S with zero susceptance. Inverting gives Z_eq = 1/0.15 = 6.666… Ω. The calculator therefore displays equivalent resistance 6.67 Ω, reactance 0.00 Ω, magnitude 6.67 Ω, and phase 0.00°.

How Parallel Impedance Works

Every parallel branch has the same voltage across it, while current divides among the branches. Since admittance is current per unit voltage, branch admittances add directly. This is why impedance is not generally added in parallel: impedance is the reciprocal of admittance, and complex reciprocals involve both resistance and reactance.

For purely resistive branches, the familiar shortcut applies: 1/R_eq = 1/R₁ + 1/R₂. For two resistors, that can also be written R_eq = R₁R₂/(R₁ + R₂). The result is below the resistance of either branch when both resistances are positive. That rule does not apply to arbitrary complex impedances.

Using the Calculator

• Enter each branch’s resistance and reactance in ohms. Use the same frequency and sign convention for both branches.
• For an ideal resistor, enter its resistance and set reactance to zero.
• For an ideal inductor or capacitor, enter its reactance as X. Inductive reactance is positive; capacitive reactance is negative.
• Read the first output as equivalent resistance. The next outputs give reactance, magnitude, and phase angle.

The inputs describe the branch impedance at a particular operating frequency; the calculator does not infer frequency or component values. If you have impedance given as magnitude and phase, convert it first: R = |Z| cos(θ), X = |Z| sin(θ), with θ in degrees or radians as appropriate.

Interpreting Results and Common Mistakes

Resistance and reactance are signed rectangular components. Magnitude is always nonnegative, while the phase angle indicates whether the combined impedance is inductive or capacitive. A negative equivalent reactance is not a negative magnitude; it means the net behavior is capacitive under the stated convention.

• Adding impedances directly: that is the series rule, not the parallel rule. For parallel branches, add admittances or use the two-impedance formula.
• Ignoring signs: capacitive reactance must be negative in the R + jX representation. Replacing it with a positive number changes the result.
• Mixing units or frequencies: all values must be in ohms and correspond to the same frequency. Reactance changes with frequency.
• Rounding too early: retain precision during intermediate calculations, then round the final display. Near cancellation, small rounding differences can noticeably affect the answer.
• Reading phase from a simple ratio: the calculator’s displayed phase uses the equivalent components. In unusual cases, an atan2 calculation is needed to retain the correct quadrant.

Limits and Related Cases

This calculator handles two branches with finite real-valued resistance and reactance. It is a steady-state AC calculation for linear impedances, not a transient analysis. Real components can vary with frequency, temperature, signal level, and other conditions, so use impedance data appropriate to the circuit’s operating conditions.

A zero-ohm branch is an ideal short circuit; it makes the parallel equivalent zero, but the displayed admittance method divides by R² + X² and cannot evaluate that exact input. An ideal open circuit has infinite impedance and is likewise outside the calculator’s finite-input range; omit an open branch when calculating the remaining network. For three or more branches, add all branch admittances and invert their sum. A complex-number-capable circuit tool may be more convenient for large networks.

Frequently Asked Questions

What is the formula for two impedances in parallel?
Z_eq = Z₁Z₂/(Z₁ + Z₂). For several branches, use 1/Z_eq = Σ(1/Z_k), treating each impedance as a complex number.

Can I use the resistor parallel formula for AC impedance?
Only when the branches are purely resistive. With reactance, use complex arithmetic or convert each impedance to admittance first.

Why do parallel admittances add?
All parallel branches have the same voltage, and their currents sum. Dividing total current by that shared voltage makes the total admittance the sum of the individual admittances.

Does frequency affect parallel impedance?
Often, yes. Inductive reactance is X_L = 2πfL, while capacitive reactance is X_C = −1/(2πfC). Changing frequency can therefore change both the equivalent impedance and its phase.

What does a negative equivalent reactance mean?
It means the net reactance is capacitive using the R + jX sign convention. It does not mean the impedance magnitude is negative.

Related

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in r1 | Branch 1 resistance | ohm | 10 | 0
in x1 | Branch 1 reactance | ohm | 0 | -
in r2 | Branch 2 resistance | ohm | 20 | 0
in x2 | Branch 2 reactance | ohm | 0 | -
out req | Equivalent resistance | ohm | (r1/(r1^2+x1^2)+r2/(r2^2+x2^2))/((r1/(r1^2+x1^2)+r2/(r2^2+x2^2))^2+(x1/(r1^2+x1^2)+x2/(r2^2+x2^2))^2) | 2
out xeq | Equivalent reactance | ohm | -(x1/(r1^2+x1^2)+x2/(r2^2+x2^2))/((r1/(r1^2+x1^2)+r2/(r2^2+x2^2))^2+(x1/(r1^2+x1^2)+x2/(r2^2+x2^2))^2) | 2
out mag | Equivalent impedance magnitude | ohm | sqrt(req^2+xeq^2) | 2
out phase | Equivalent phase angle | degrees | atand(xeq/req) | 2