structural:reformatting
Impedance and admittance

When an alternating current flows through a circuit, the relation between current and voltage across a circuit element is characterized not only by the ratio of their magnitudes, but also the difference in their phases. For example, in an ideal resistor, the moment when the voltage reaches its maximum, the current also reaches its maximum (current and voltage are oscillating in phase). But for a capacitor or inductor, the maximum current flow occurs as the voltage passes through zero and vice versa (current and voltage are oscillating 90° out of phase, see image below). Complex numbers are used to keep track of both the phase and magnitude of current and voltage:

u (t) = R e (U 0 ⋅ e j ω t) i (t) = R e (I 0 ⋅ e j (ω t + φ)) Z = U I Y = 1 Z = I U {\displaystyle {\begin{array}{cl}u(t)&=\operatorname {\mathcal {R_{e}}} \left(U_{0}\cdot e^{j\omega t}\right)\\i(t)&=\operatorname {\mathcal {R_{e}}} \left(I_{0}\cdot e^{j(\omega t+\varphi)}\right)\\Z&={\frac {U}{\ I\ }}\\Y&={\frac {\ 1\ }{Z}}={\frac {\ I\ }{U}}\end{array}}}

where:

The impedance and admittance may be expressed as complex numbers that can be broken into real and imaginary parts: Z = R + j X Y = G + j B. {\displaystyle {\begin{aligned}Z&=R+jX\\Y&=G+jB~.\end{aligned}}}

where R is resistance, G is conductance, X is reactance, and B is susceptance. These lead to the complex number identities R = G G 2 + B 2 , X = − B G 2 + B 2 , G = R R 2 + X 2 , B = − X R 2 + X 2 , {\displaystyle {\begin{aligned}R&={\frac {G}{\ G^{2}+B^{2}\ }}\,\qquad &X={\frac {-B~}{\ G^{2}+B^{2}\ }}\,\\G&={\frac {R}{\ R^{2}+X^{2}\ }}\,\qquad &B={\frac {-X~}{\ R^{2}+X^{2}\ }}\,\end{aligned}}} which are true in all cases, whereas R = 1 / G {\displaystyle \ R=1/G\ } is only true in the special cases of either DC or reactance-free current.

The complex angle θ = arg (Z) = − arg (Y) {\displaystyle \ \theta =\arg(Z)=-\arg(Y)\ } is the phase difference between the voltage and current passing through a component with impedance Z. For capacitors and inductors, this angle is exactly -90° or +90°, respectively, and X and B are nonzero. Ideal resistors have an angle of 0°, since X is zero (and hence B also), and Z and Y reduce to R and G respectively. In general, AC systems are designed to keep the phase angle close to 0° as much as possible, since it reduces the reactive power, which does no useful work at a load. In a simple case with an inductive load (causing the phase to increase), a capacitor may be added for compensation at one frequency, since the capacitor's phase shift is negative, bringing the total impedance phase closer to 0° again.

Y is the reciprocal of Z (Z = 1 / Y {\displaystyle \ Z=1/Y\ }) for all circuits, just as R = 1 / G {\displaystyle R=1/G} for DC circuits containing only resistors, or AC circuits for which either the reactance or susceptance happens to be zero (X or B = 0, respectively) (if one is zero, then for realistic systems both must be zero).