Electrical impedance

This representation using complex exponentials may be justified by noting that (by Euler's formula):

${\displaystyle \ \cos(\omega t+\phi )={\frac {1}{2}}{\Big [}e^{j(\omega t+\phi )}+e^{-j(\omega t+\phi )}{\Big ]}}$

The real-valued sinusoidal function representing either voltage or current may be broken into two complex-valued functions. By the principle of superposition, we may analyse the behaviour of the sinusoid on the left-hand side by analysing the behaviour of the two complex terms on the right-hand side. Given the symmetry, we only need to perform the analysis for one right-hand term. The results are identical for the other. At the end of any calculation, we may return to real-valued sinusoids by further noting that

${\displaystyle \ \cos(\omega t+\phi )=\operatorname {\mathcal {R_{e}}} {\Big \{}e^{j(\omega t+\phi )}{\Big \}}}$

An AC supply applying a voltage ${\displaystyle V}$, across a load ${\displaystyle Z}$, driving a current ${\displaystyle I}$

The meaning of electrical impedance can be understood by substituting it into Ohm's law.[15][16] Assuming a two-terminal circuit element with impedance ${\displaystyle Z}$ is driven by a sinusoidal voltage or current as above, there holds

${\displaystyle \ V=IZ=I|Z|e^{j\arg(Z)}}$

The magnitude of the impedance ${\displaystyle |Z|}$ acts just like resistance, giving the drop in voltage amplitude across an impedance ${\displaystyle Z}$ for a given current ${\displaystyle I}$. The phase factor tells us that the current lags the voltage by a phase of ${\displaystyle \theta =\arg(Z)}$ (i.e., in the time domain, the current signal is shifted ${\textstyle {\frac {\theta }{2\pi }}T}$ later with respect to the voltage signal).

Just as impedance extends Ohm's law to cover AC circuits, other results from DC circuit analysis, such as voltage division, current division, Thévenin's theorem and Norton's theorem, can also be extended to AC circuits by replacing resistance with impedance.

A phasor is represented by a constant complex number, usually expressed in exponential form, representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. Phasors are used by electrical engineers to simplify computations involving sinusoids (such as in AC circuits[12]^: 53), where they can often reduce a differential equation problem to an algebraic one.

The impedance of a circuit element can be defined as the ratio of the phasor voltage across the element to the phasor current through the element, as determined by the relative amplitudes and phases of the voltage and current. This is identical to the definition from Ohm's law given above, recognising that the factors of ${\displaystyle e^{j\omega t}}$ cancel.

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 ${\displaystyle \pi /2}$, while the voltage across an inductor leads the current through it by ${\displaystyle \pi /2}$. 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 and is called resistive impedance:

${\displaystyle \ Z_{R}=R}$

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

Ideal inductors and capacitors have a purely imaginary reactive impedance:

the impedance of inductors increases as frequency increases;

${\displaystyle Z_{L}=j\omega L}$[lower-alpha 1]

the impedance of capacitors decreases as frequency increases;

${\displaystyle Z_{C}={\frac {1}{j\omega C}}}$

In both cases, for an applied sinusoidal voltage, the resulting current is also sinusoidal, but in 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:

${\displaystyle {\begin{aligned}j&\equiv \cos {\left({\frac {\pi }{2}}\right)}+j\sin {\left({\frac {\pi }{2}}\right)}\equiv e^{j{\frac {\pi }{2}}}\\{\frac {1}{j}}\equiv -j&\equiv \cos {\left(-{\frac {\pi }{2}}\right)}+j\sin {\left(-{\frac {\pi }{2}}\right)}\equiv e^{j\left(-{\frac {\pi }{2}}\right)}\end{aligned}}}$

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

${\displaystyle {\begin{aligned}Z_{L}&=\omega Le^{j{\frac {\pi }{2}}}\\Z_{C}&={\frac {1}{\omega C}}e^{j\left(-{\frac {\pi }{2}}\right)}\end{aligned}}}$

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

What follows below is a derivation of impedance for each of the three basic 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, 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.

For a resistor, there is the relation

${\displaystyle v_{\text{R}}{\mathord {\left(t\right)}}=i_{\text{R}}{\mathord {\left(t\right)}}R}$

which is Ohm's law.

Considering the voltage signal to be

${\displaystyle v_{\text{R}}(t)=V_{p}\sin(\omega t)}$

it follows that

${\displaystyle {\frac {v_{\text{R}}{\mathord {\left(t\right)}}}{i_{\text{R}}{\mathord {\left(t\right)}}}}={\frac {V_{p}\sin(\omega t)}{I_{p}\sin {\mathord {\left(\omega t\right)}}}}=R}$

This says that the ratio of AC voltage amplitude to alternating current (AC) amplitude across a resistor is ${\displaystyle R}$, and that the AC voltage leads the current across a resistor by 0 degrees.

This result is commonly expressed as

${\displaystyle Z_{\text{resistor}}=R}$

For a capacitor, there is the relation:

${\displaystyle i_{\text{C}}(t)=C{\frac {\mathrm {d} v_{\text{C}}(t)}{\mathrm {d} t}}}$

Considering the voltage signal to be

${\displaystyle v_{\text{C}}(t)=V_{p}e^{j\omega t}}$

it follows that

${\displaystyle {\frac {\mathrm {d} v_{\text{C}}(t)}{\mathrm {d} t}}=j\omega V_{p}e^{j\omega t}}$

and thus, as previously,

${\displaystyle Z_{\text{capacitor}}={\frac {v_{\text{C}}{\mathord {\left(t\right)}}}{i_{\text{C}}{\mathord {\left(t\right)}}}}={\frac {1}{j\omega C}}.}$

Conversely, if the current through the circuit is assumed to be sinusoidal, its complex representation being

${\displaystyle i_{\text{C}}(t)=I_{p}e^{j\omega t}}$

then integrating the differential equation

${\displaystyle i_{\text{C}}(t)=C{\frac {\mathrm {d} v_{\text{C}}(t)}{\mathrm {d} t}}}$

leads to

${\displaystyle v_{C}(t)={\frac {1}{j\omega C}}I_{p}e^{j\omega t}+{\text{Const.}}={\frac {1}{j\omega C}}i_{C}(t)+{\text{Const.}}}$

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

${\displaystyle Z_{\text{capacitor}}={\frac {1}{j\omega C}}.}$

For the inductor, we have the relation (from Faraday's law):

${\displaystyle v_{\text{L}}(t)=L{\frac {\mathrm {d} i_{\text{L}}(t)}{\mathrm {d} t}}}$

This time, considering the current signal to be:

${\displaystyle i_{\text{L}}(t)=I_{p}\sin(\omega t)}$

it follows that:

${\displaystyle {\frac {\mathrm {d} i_{\text{L}}(t)}{\mathrm {d} t}}=\omega I_{p}\cos {\mathord {\left(\omega t\right)}}}$

This result is commonly expressed in polar form as

${\displaystyle Z_{\text{inductor}}=\omega Le^{j{\frac {\pi }{2}}}}$

or, using Euler's formula, as

${\displaystyle Z_{\text{inductor}}=j\omega L}$

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.

The impedance ${\displaystyle Z}$ of an electrical component is defined as the ratio between the Laplace transforms of the voltage over it and the current through it, i.e.

${\displaystyle Z(s)={\frac {{\mathcal {L}}\{v(t)\}}{{\mathcal {L}}\{i(t)\}}}={\frac {V(s)}{I(s)}}\qquad {\text{(general impedance)}}}$

where ${\displaystyle s=\sigma +j\omega }$ is the complex Laplace parameter. As an example, according to the I-V-law of a capacitor, ${\displaystyle {\mathcal {L}}\{i(t)\}={\mathcal {L}}\{C\,\mathrm {d} v(t)/\mathrm {d} t\}=sC{\mathcal {L}}\{v(t)\}}$, from which it follows that ${\displaystyle Z_{C}(s)=1/sC}$.

In the phasor regime (steady-state AC, meaning all signals are represented mathematically as simple complex exponentials ${\displaystyle v(t)={\hat {V}}\,e^{j\omega t}}$ and ${\displaystyle i(t)={\hat {I}}\,e^{j\omega t}}$ oscillating at a common frequency ${\displaystyle \omega }$), impedance can simply be calculated as the voltage-to-current ratio, in which the common time-dependent factor cancels out:

${\displaystyle Z(\omega )={\frac {v(t)}{i(t)}}={\frac {{\hat {V}}\,e^{j\omega t}}{{\hat {I}}\,e^{j\omega t}}}={\frac {\hat {V}}{\hat {I}}}\qquad {\text{(phasor-regime impedance)}}}$

Again, for a capacitor, one gets that ${\displaystyle i(t)=C\,\mathrm {d} v(t)/\mathrm {d} t=j\omega C\,v(t)}$, and hence ${\displaystyle Z_{C}(\omega )=1/j\omega C}$. The phasor domain is sometimes dubbed the frequency domain, although it lacks one of the dimensions of the Laplace parameter.[17] For steady-state AC, the polar form of the complex impedance relates the amplitude and phase of the voltage and current. In particular:

• The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude;
• The phase of the complex impedance is the phase shift by which the current lags the voltage.

These two relationships hold even after taking the real part of the complex exponentials (see phasors), which is the part of the signal one actually measures in real-life circuits.

Resistance ${\displaystyle R}$ is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.

${\displaystyle \ R=|Z|\cos {\theta }\quad }$

Reactance ${\displaystyle X}$ is the imaginary part of the impedance; a component with a finite reactance induces a phase shift ${\displaystyle \theta }$ between the voltage across it and the current through it.

${\displaystyle \ X=|Z|\sin {\theta }\quad }$

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.

A capacitor has a purely reactive impedance that is inversely proportional to the signal frequency. A capacitor consists of two conductors separated by an insulator, also known as a dielectric.

${\displaystyle X_{\mathsf {C}}={\frac {-1\ ~}{\ \omega \ C\ }}={\frac {-1\ ~}{\ 2\pi \ f\ C\ }}~.}$

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 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 ${\displaystyle X_{L}}$ is proportional to the signal frequency ${\displaystyle f}$ and the inductance ${\displaystyle L}$.

${\displaystyle X_{L}=\omega L=2\pi fL\quad }$

An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf ${\displaystyle {\mathcal {E}}}$ (voltage opposing current) due to a rate-of-change of magnetic flux density ${\displaystyle B}$ through a current loop.

${\displaystyle {\mathcal {E}}=-{{d\Phi _{B}} \over dt}\quad }$

For an inductor consisting of a coil with ${\displaystyle N}$ loops this gives:

${\displaystyle {\mathcal {E}}=-N{d\Phi _{B} \over dt}\quad }$

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.

The total reactance is given by

${\displaystyle {X=X_{L}+X_{C}}}$ (note that ${\displaystyle X_{C}}$ is negative)

so that the total impedance is

${\displaystyle \ Z=R+jX}$

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

${\displaystyle \ Z_{\text{eq}}=Z_{1}+Z_{2}+\cdots +Z_{n}\quad }$

Or explicitly in real and imaginary terms:

${\displaystyle \ Z_{\text{eq}}=R+jX=(R_{1}+R_{2}+\cdots +R_{n})+j(X_{1}+X_{2}+\cdots +X_{n})\quad }$

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.

Hence the inverse total impedance is the sum of the inverses of the component impedances:

${\displaystyle {\frac {1}{Z_{\text{eq}}}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}+\cdots +{\frac {1}{Z_{n}}}}$

or, when n = 2:

${\displaystyle {\frac {1}{Z_{\text{eq}}}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}={\frac {Z_{1}+Z_{2}}{Z_{1}Z_{2}}}}$

${\displaystyle \ Z_{\text{eq}}={\frac {Z_{1}Z_{2}}{Z_{1}+Z_{2}}}}$

The equivalent impedance ${\displaystyle Z_{\text{eq}}}$ can be calculated in terms of the equivalent series resistance ${\displaystyle R_{\text{eq}}}$ and reactance ${\displaystyle X_{\text{eq}}}$.[18]

${\displaystyle {\begin{aligned}Z_{\text{eq}}&=R_{\text{eq}}+jX_{\text{eq}}\\R_{\text{eq}}&={\frac {(X_{1}R_{2}+X_{2}R_{1})(X_{1}+X_{2})+(R_{1}R_{2}-X_{1}X_{2})(R_{1}+R_{2})}{(R_{1}+R_{2})^{2}+(X_{1}+X_{2})^{2}}}\\X_{\text{eq}}&={\frac {(X_{1}R_{2}+X_{2}R_{1})(R_{1}+R_{2})-(R_{1}R_{2}-X_{1}X_{2})(X_{1}+X_{2})}{(R_{1}+R_{2})^{2}+(X_{1}+X_{2})^{2}}}\end{aligned}}}$

Consider an LC tank circuit. The complex impedance of the circuit is

${\displaystyle Z(\omega )={\frac {j\omega L}{1-\omega ^{2}LC}}.}$

It is immediately seen that the value of ${\textstyle {1 \over |Z|}}$ is minimal (actually equal to 0 in this case) whenever

${\displaystyle \omega ^{2}LC=1.}$

Therefore, the fundamental resonance angular frequency is

${\displaystyle \omega ={1 \over {\sqrt {LC}}}.}$