Editing Electrical impedance (section)

== Device examples ==

===Resistor===
[[File:VI phase.svg|thumb|right|250px|The phase angles in the equations for the impedance of capacitors and inductors indicate that the voltage across a capacitor ''lags'' the current through it by a phase of <math>\pi/2</math>, while the voltage across an inductor ''leads'' the current through it by <math>\pi/2</math>. The identical voltage and current amplitudes indicate that the magnitude of the impedance is equal to one.]]

The impedance of an ideal [[resistor]] is purely [[Real number|real]] and is called ''resistive impedance'':

:<math>\ Z_R = R</math>

In this case, the voltage and current waveforms are proportional and in phase.

===Inductor and capacitor (in the steady state)===
Ideal [[inductor]]s and [[capacitor]]s have a purely [[imaginary number|imaginary]] ''reactive impedance'':

the impedance of inductors increases as frequency increases;

:<math>Z_L = j\omega L</math>

the impedance of capacitors decreases as frequency increases;

:<math>Z_C = \frac{1}{j\omega C}</math>

In both cases, for an applied sinusoidal voltage, the resulting current is also sinusoidal, but in [[quadrature phase|quadrature]], 90 degrees out of phase with the voltage. However, the phases have opposite signs: in an inductor, the current is ''lagging''; in a capacitor the current is ''leading''.

Note the following identities for the imaginary unit and its reciprocal:

:<math>\begin{align}
                      j &\equiv \cos{\left( \frac{\pi}{2}\right)} + j\sin{\left( \frac{\pi}{2}\right)} \equiv e^{j \frac{\pi}{2}} \\
  \frac{1}{j} \equiv -j &\equiv \cos{\left(-\frac{\pi}{2}\right)} + j\sin{\left(-\frac{\pi}{2}\right)} \equiv e^{j\left(-\frac{\pi}{2}\right)}
\end{align}</math>

Thus the inductor and capacitor impedance equations can be rewritten in polar form:

:<math>\begin{align}
  Z_L &= \omega Le^{j\frac{\pi}{2}} \\
  Z_C &= \frac{1}{\omega C}e^{j\left(-\frac{\pi}{2}\right)}
\end{align}</math>

The magnitude gives the change in voltage amplitude for a given current amplitude through the impedance, while the exponential factors give the phase relationship.

=== Deriving the device-specific impedances ===
What follows below is a derivation of impedance for each of the three basic [[Electrical network|circuit]] elements: the resistor, the capacitor, and the inductor. Although the idea can be extended to define the relationship between the voltage and current of any arbitrary [[Signal (electrical engineering)|signal]], these derivations assume [[sinusoidal]] signals. In fact, this applies to any arbitrary periodic signals, because these can be approximated as a sum of sinusoids through [[Fourier analysis]].

====Resistor====
For a resistor, there is the relation
:<math>v_\text{R} \mathord\left( t \right) = i_\text{R} \mathord\left( t \right) R</math>

which is [[Ohm's law]].

Considering the voltage signal to be
:<math>v_\text{R}(t) = V_p \sin(\omega t)</math>

it follows that
:<math>\frac{v_\text{R} \mathord\left( t \right)}{i_\text{R} \mathord\left( t \right)} = \frac{V_p \sin(\omega t)}{I_p \sin \mathord\left( \omega t \right)} = R</math>

This says that the ratio of AC voltage amplitude to [[alternating current]] (AC) amplitude across a resistor is <math>R</math>, and that the AC voltage leads the current across a resistor by 0 degrees.

This result is commonly expressed as
:<math>Z_\text{resistor} = R</math>

==== Capacitor (in the steady state) ====
For a capacitor, there is the relation:
:<math>i_\text{C}(t) = C \frac{\mathrm{d}v_\text{C}(t)}{\mathrm{d}t}</math>

Considering the voltage signal to be
:<math>v_\text{C}(t) = V_p e^{j\omega t} </math>

it follows that
:<math>\frac{\mathrm{d}v_{\text{C}}(t)}{\mathrm{d}t} = j\omega V_p e^{j\omega t}</math>

and thus, as previously,
:<math>Z_\text{capacitor} = \frac{v_\text{C} \mathord\left( t \right)}{i_\text{C} \mathord\left( t \right)} = \frac{1}{j\omega C}.</math>

Conversely, if the current through the circuit is assumed to be sinusoidal, its complex representation being 
:<math>i_\text{C}(t) = I_p e^{j\omega t} </math>
then integrating the differential equation 
:<math>i_\text{C}(t) = C \frac{\mathrm{d}v_\text{C}(t)}{\mathrm{d}t}</math>
leads to 
:<math>v_C(t) = \frac{1}{j\omega C}I_p e^{j\omega t} + \text{Const.} = \frac{1}{j\omega C} i_C(t) + \text{Const.}</math>
The ''Const'' term represents a fixed potential bias superimposed to the AC sinusoidal potential, that plays no role in AC analysis. For this purpose, this term can be assumed to be 0, hence again the impedance  
:<math>Z_\text{capacitor} = \frac{1}{j\omega C}.</math>

==== Inductor (in the steady state) ====
For the inductor, we have the relation (from [[Faraday's law of induction|Faraday's law]]):
:<math>v_\text{L}(t) = L \frac{\mathrm{d}i_\text{L}(t)}{\mathrm{d}t}</math>

This time, considering the current signal to be:
:<math>i_\text{L}(t) = I_p \sin(\omega t)</math>

it follows that:
:<math>\frac{\mathrm{d}i_\text{L}(t)}{\mathrm{d}t} = \omega I_p \cos \mathord\left( \omega t \right)</math>

This result is commonly expressed in polar form as
:<math>Z_\text{inductor} = \omega L e^{j\frac{\pi}{2}}</math>

or, using Euler's formula, as
:<math>Z_\text{inductor} = j \omega L</math>

As in the case of capacitors, it is also possible to derive this formula directly from the complex representations of the voltages and currents, or by assuming a sinusoidal voltage between the two poles of the inductor. In the latter case, integrating the differential equation above leads to a constant term for the current, that represents a fixed DC bias flowing through the inductor. This is set to zero because AC analysis using frequency domain impedance considers one frequency at a time and DC represents a separate frequency of zero hertz in this context.