Rlccircuit Impedance Calculator

Impedance is the AC generalization of resistance, and unlike resistance it changes with frequency. Inductive reactance XL = 2 pi f L grows in proportion to frequency, while capacitive reactance XC = 1 / (2 pi f C) shrinks with it, so an inductor approaches a short circuit at DC and an open circuit at high frequency while a capacitor does the reverse. In a series branch the two reactances subtract because they are 180 degrees apart, giving Z = sqrt(R squared + (XL - XC) squared) with a phase angle of arctan((XL - XC) / R). A parallel network is handled through admittances instead: Y = 1/R + j(2 pi f C - 1 / (2 pi f L)), and Z is the reciprocal of its magnitude. Power factor is cos of the phase angle, equal to R/Z in the series case, and real power is V x I x cos phi while the reactive component circulates without being consumed. Working values make the frequency dependence concrete. At 1 kHz a 10 mH inductor presents 62.8 ohms while a 100 microfarad capacitor presents only 1.59 ohms, so the branch is strongly inductive and current lags voltage. Below the crossover at f0 = 1 / (2 pi sqrt(LC)) the capacitor dominates and current leads; above it the inductor dominates and current lags; at f0 the reactances cancel and the series impedance collapses to R. A consequence worth designing for is voltage magnification: near series resonance the voltage across L or across C equals Q times the source voltage and can far exceed it, so component voltage ratings must be chosen accordingly. The analysis assumes ideal parts and a single sinusoidal frequency in steady state, so winding resistance, capacitor equivalent series resistance, and self resonance are all excluded, and non-sinusoidal drive must be handled by superposing the harmonics individually.

Before the 1890s alternating current design was largely empirical, because analyzing circuits with inductance and capacitance meant solving differential equations for every case. Oliver Heaviside supplied much of the vocabulary and the operational methods, coining the term impedance in 1886 and later introducing reactance and admittance as companion quantities, while recasting circuit problems in a form engineers could manipulate algebraically. The decisive step was representing impedance as a complex number. Arthur Kennelly published a paper titled Impedance for the American Institute of Electrical Engineers in 1893 that set out the complex representation explicitly, and in the same year Charles Proteus Steinmetz presented his symbolic method at the International Electrical Congress in Chicago, developing it further in his 1897 treatise on alternating current phenomena. Their approach reduced steady state AC analysis to arithmetic with complex quantities, so that reactances could be added like resistances once phase was carried along. This arrived precisely when it was needed, during the contest between direct and alternating current distribution, and it allowed engineers at firms such as General Electric and Westinghouse to design transformers, transmission lines, and motors predictively. The convention of writing the imaginary unit as j rather than i comes from this electrical engineering lineage, adopted to avoid confusion with the symbol for instantaneous current, and phasor analysis in essentially the 1893 form remains the standard method taught in circuits courses today.