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Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. The concept of electrical impedance generalises Ohm's law to AC circuit analysis. Unlike electrical resistance, the impedance of an electric circuit can be a complex number, but the same unit, the ohm, is used for both quantities. Oliver Heaviside coined the term "impedance" in July of 1886.
In general, the solutions for the voltages and currents in a circuit containing resistors, capacitors and inductors (in short, all linearly behaving components) are solutions to a linear ordinary differential equation. It can be shown that if the voltage and current sources in the circuit are sinusoidal and of constant frequency, the solutions take a form referred to as AC steady state. Thus, all of the voltages and currents in the circuit are sinusoidal and have constant amplitude, frequency and phase.
In AC steady state, v(t) is a sinusoidal function of time with constant amplitude Vp, constant frequency f, and constant phase :
where represents the imaginary unit () means the real part of the complex number z.
The phasor representation of v(t) is the constant complex number V:
For a circuit in AC steady state, all of the voltages and currents in the circuit have phasor representations as long as all the sources are of the same frequency. That is, each voltage and current can be represented as a constant complex number. For DC circuit analysis, each voltage and current is represented by a constant real number. Thus, it is reasonable to suppose that the rules developed for DC circuit analysis can be used for AC circuit analysis by using complex numbers instead of real numbers.
The impedance of a circuit element is defined as the ratio of the phasor voltage across the element to the phasor current through the element:
It should be noted that although Z is the ratio of two phasors, Z is not itself a phasor. That is, Z is not associated with some sinusoidal function of time.
For DC circuits, the resistance is defined by Ohm's law to be the ratio of the DC voltage across the resistor to the DC current through the resistor:
where and above are DC (constant real) values.
Just as Ohm's law is generalized to AC circuits through the use of phasors, other results from DC circuit analysis such as voltage division, current division, Thevenin's theorem, and Norton's theorem generalize to AC circuits.
For a resistor:
For a capacitor:
For an inductor:
For derivations, see Impedance of different devices (derivations).
The term reactance refers to the imaginary part of the impedance. Some examples:
A resistors impedance is R (its resistance) and its reactance is