Impedance polar to rectangular converter

Enter magnitude and phase angle to obtain resistance and reactance, or enter resistance and reactance to recover magnitude and phase. Results are expressed in ohms, the angle is reported in degrees, and the sign of reactance is identified as inductive, capacitive, or purely resistive. The same deterministic calculation is useful for hand checks, phasor work, network combinations, and automated engineering worksheets.

Choose the representation you already know

Start by selecting the form of the impedance you have, not the form you want. For polar input, supply the non-negative magnitude in ohms and the phase angle in degrees. A phase angle may be positive, negative, or outside the usual one-turn interval because coterminal angles describe the same phasor. For rectangular input, supply resistance R and reactance X in ohms. Reactance is signed: positive X represents an inductive impedance and negative X represents a capacitive impedance. Both resistance and reactance may be zero, and negative resistance is accepted because active and small-signal network models can legitimately contain it. The converter always returns both forms, so there is no separate output-direction switch to misconfigure. It also reports a principal phase angle from minus 180 through 180 degrees and labels the reactance type. Keep all values in ohms; if your source values are in kilohms or milliohms, scale them consistently before entering them and scale the returned components back afterward. This avoids silently mixing prefixes within one impedance.

Understand the trigonometry behind the conversion

An impedance in polar form is written as Z equals magnitude at angle theta. Its horizontal component is resistance, calculated as magnitude times cosine theta, while its vertical component is reactance, calculated as magnitude times sine theta. In the reverse direction, magnitude is the hypotenuse formed by R and X, so the converter uses the numerically stable hypotenuse operation. The phase comes from a two-argument arctangent, atan2 of X and R, rather than an ordinary arctangent of X divided by R. That distinction preserves the correct quadrant when resistance is negative and remains defined when resistance is zero. A zero impedance has no unique mathematical phase, so this capability reports zero degrees as a practical convention. Values extremely close to zero after trigonometric calculation are normalized to zero, removing distracting negative-zero output and floating-point residue. Other results are rounded to stable significant precision, which is ample for normal network calculations while ensuring repeatable API and browser output. You can therefore compare results directly without chasing insignificant binary arithmetic noise.

Use the result in AC network calculations

Rectangular form is usually the convenient choice when adding impedances in series because resistances add directly and reactances add directly. Polar form is usually more convenient for multiplication, division, gain ratios, and phasor interpretation because magnitudes multiply or divide while angles add or subtract. This converter bridges those steps without changing the electrical meaning of the impedance. For example, convert each series element to rectangular form, add all R terms and all X terms, then convert the total back to polar form for magnitude and phase. The returned reactance label is also a quick sign check: an inductive result should have positive reactance and a positive phase for positive resistance, while a capacitive result should have negative reactance and a negative phase. Be careful when using the label alone with negative resistance, since the principal phase can lie in a different quadrant even though the X sign still identifies reactance. The calculation does not incorporate frequency or derive impedance from component values; it converts an impedance already evaluated at the frequency of interest. Preserve that frequency context in your surrounding worksheet or application.