Editing Electrical impedance (section)

== Resistance vs reactance ==

Resistance and reactance together determine the magnitude and phase of the impedance through the following relations:

:<math>\begin{align}
     |Z| &= \sqrt{Z Z^*} = \sqrt{R^2 + X^2} \\
  \theta &= \arctan{\left(\frac{X}{R}\right)}
\end{align}</math>

In many applications, the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant.

=== Resistance ===
<!--[[File:Resistors.jpg|thumb|right|200px|A pack of resistors. [[Media:Resistors.jpg|Actual size]]]]-->
{{Main|Electrical resistance}}

Resistance <math>R</math> is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.

:<math>\ R = |Z| \cos{\theta} \quad</math>

=== Reactance ===
{{Main|Electrical reactance}}

Reactance <math> X</math> is the imaginary part of the impedance; a component with a finite reactance induces a phase shift <math> \theta</math> between the voltage across it and the current through it.

:<math>\ X = |Z| \sin{\theta} \quad</math>

A purely reactive component is distinguished by the sinusoidal voltage across the component being in quadrature with the sinusoidal current through the component. This implies that the component alternately absorbs energy from the circuit and then returns energy to the circuit. A pure reactance does not dissipate any power.

==== Capacitive reactance ====
<!--[[File:Photo-SMDcapacitors.jpg|thumb|right|200px|Capacitors: [[Surface-mount technology|SMD]] ceramic at top left; SMD tantalum at bottom left; [[through-hole]] tantalum at top right; through-hole electrolytic at bottom right. Major scale divisions are cm. [[Media:Resistors.jpg|Actual size]]]]-->
{{Main|Capacitance}}

A capacitor has a purely reactive impedance that is [[Inversely proportional#Inverse proportionality|inversely proportional]] to the signal [[frequency]]. A capacitor consists of two [[Electrical conduction|conductor]]s separated by an [[Electrical insulation|insulator]], also known as a [[dielectric]].

:<math>X_\mathsf{C} = \frac{-1\ ~}{\ \omega\ C\ } = \frac{-1\ ~}{\ 2\pi\ f\ C\ } ~.</math>

The minus sign indicates that the imaginary part of the impedance is negative.

At low frequencies, a capacitor approaches an open circuit so no current flows through it.

A DC voltage applied across a capacitor causes [[Electrical charge|charge]] to accumulate on one side; the [[electric field]] due to the accumulated charge is the source of the opposition to the current. When the [[potential]] associated with the charge exactly balances the applied voltage, the current goes to zero.

Driven by an AC supply, a capacitor accumulates only a limited charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge accumulates and the smaller the opposition to the current.

==== Inductive reactance ====
{{Main|Inductance}}

Inductive reactance <math>X_L</math> is [[Proportionality (mathematics)|proportional]] to the signal [[frequency]] <math>f</math> and the [[inductance]] <math>L</math>.

:<math>X_L = \omega L = 2\pi f L\quad</math>

An inductor consists of a coiled conductor. [[Faraday's law of induction|Faraday's law]] of electromagnetic induction gives the back [[Electromotive force|emf]] <math>\mathcal{E}</math> (voltage opposing current) due to a rate-of-change of [[magnetic flux density]] <math>B</math> through a current loop.

:<math>\mathcal{E} = -{{d\Phi_B} \over dt}\quad</math>

For an inductor consisting of a coil with <math>N</math> loops this gives:

:<math>\mathcal{E} = -N{d\Phi_B \over dt}\quad</math>

The back-emf is the source of the opposition to current flow. A constant [[direct current]] has a zero rate-of-change, and sees an inductor as a [[short-circuit]] (it is typically made from a material with a low [[resistivity]]). An [[alternating current]] has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

==== Total reactance ====

The total reactance is given by

:<math>{X = X_L + X_C}</math> (<math>X_C</math> is negative)

so that the total impedance is

:<math>\ Z = R + jX</math>

== Combining impedances ==
{{Main|Series and parallel circuits}}

The total impedance of many simple networks of components can be calculated using the rules for combining impedances in series and parallel. The rules are identical to those for combining resistances, except that the numbers in general are [[complex number]]s. The general case, however, requires [[equivalent impedance transforms]] in addition to series and parallel.

=== Series combination ===

For components connected in series, the current through each circuit element is the same; the total impedance is the sum of the component impedances.

[[File:Impedances in series.svg]]

:<math>\ Z_{\text{eq}} = Z_1 + Z_2 + \cdots + Z_n \quad</math>

Or explicitly in real and imaginary terms:
:<math>\ Z_{\text{eq}} = R + jX = (R_1 + R_2 + \cdots + R_n) + j(X_1 + X_2 + \cdots + X_n) \quad</math>

=== Parallel combination ===

For components connected in parallel, the voltage across each circuit element is the same; the ratio of currents through any two elements is the inverse ratio of their impedances.

:[[File:Impedances in parallel.svg]]

Hence the inverse total impedance is the sum of the inverses of the component impedances:
:<math>\frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots + \frac{1}{Z_n}</math>

or, when n = 2:
:<math>\frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} = \frac{Z_1 + Z_2}{Z_1 Z_2}</math>
:<math>\ Z_{\text{eq}} = \frac{Z_1 Z_2}{Z_1 + Z_2}</math>

The equivalent impedance <math>Z_{\text{eq}}</math> can be calculated in terms of the equivalent series resistance <math>R_{\text{eq}}</math> and reactance <math>X_{\text{eq}}</math>.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html#c3 Parallel Impedance Expressions], Hyperphysics</ref>

:<math>\begin{align}
  Z_{\text{eq}} &= R_{\text{eq}} + j X_{\text{eq}} \\
  R_{\text{eq}} &= \frac{(X_1 R_2 + X_2 R_1) (X_1 + X_2) + (R_1 R_2 - X_1 X_2) (R_1 + R_2)}{(R_1 + R_2)^2 + (X_1 + X_2)^2} \\
  X_{\text{eq}} &= \frac{(X_1 R_2 + X_2 R_1) (R_1 + R_2) - (R_1 R_2 - X_1 X_2) (X_1 + X_2)}{(R_1 + R_2)^2 + (X_1 + X_2)^2}
\end{align}</math>