Impedance and admittance (top) and
inductor (bottom). Since the
amplitude of the current and voltage
sinusoids are the same, the
absolute value of
impedance is 1 for both the capacitor and the inductor (in whatever units the graph is using). On the other hand, the
phase difference between current and voltage is −90° for the capacitor; therefore, the
complex phase of the
impedance of the capacitor is −90°. Similarly, the
phase difference between current and voltage is +90° for the inductor; therefore, the complex phase of the impedance of the inductor is +90°. 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: \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: • is time; • and are the voltage and current as a function of time, respectively; • and indicate the amplitude of the voltage and current, respectively; • \omega is the
angular frequency of the AC current; • \varphi is the displacement angle; • and are the complex-valued voltage and current, respectively; • and are the complex
impedance and
admittance, respectively; • \mathcal{R_e} indicates the
real part of a
complex number; and • j \equiv \sqrt{-1\ } is the
imaginary unit. The impedance and admittance may be expressed as complex numbers that can be broken into real and imaginary parts: \begin{align} Z &= R + jX \\ Y &= G + jB ~. \end{align} where is resistance, is conductance, is
reactance, and is
susceptance. These lead to the
complex number identities \begin{align} 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{align} which are true in all cases, whereas \ R = 1/G\ is only true in the special cases of either DC or reactance-free current. The
complex angle \ \theta = \arg(Z) = -\arg(Y)\ is the phase difference between the voltage and current passing through a component with impedance . For
capacitors and
inductors, this angle is exactly -90° or +90°, respectively, and and are nonzero. Ideal resistors have an angle of 0°, since is zero (and hence also), and and reduce to and 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. is the reciprocal of (\ Z = 1/Y\ ) for all circuits, just as R = 1/G for DC circuits containing only resistors, or AC circuits for which either the reactance or susceptance happens to be zero ( or , respectively) (if one is zero, then for realistic systems both must be zero).
Frequency dependence A key feature of AC circuits is that the resistance and conductance can be frequency-dependent, a phenomenon known as the
universal dielectric response. One reason, mentioned above is the
skin effect (and the related
proximity effect). Another reason is that the resistivity itself may depend on frequency (see
Drude model,
deep-level traps,
resonant frequency,
Kramers–Kronig relations, etc.) ==Energy dissipation and Joule heating==