Electrical impedance

In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.
Quantitatively, the impedance of a two-terminal circuit element is the ratio of the complex representation of the sinusoidal voltage between its terminals, to the complex representation of the current flowing through it. In general, it depends upon the frequency of the sinusoidal voltage.
Impedance extends the concept of resistance to alternating current circuits, and possesses both magnitude and phase, unlike resistance, which has only magnitude.
Impedance can be represented as a complex number, with the same units as resistance, for which the SI unit is the ohm.
Its symbol is usually, and it may be represented by writing its magnitude and phase in the polar form. However, Cartesian complex number representation is often more powerful for circuit analysis purposes.
The notion of impedance is useful for performing AC analysis of electrical networks, because it allows relating sinusoidal voltages and currents by a simple linear law.
In multiple port networks, the two-terminal definition of impedance is inadequate, but the complex voltages at the ports and the currents flowing through them are still linearly related by the impedance matrix.
The reciprocal of impedance is admittance, whose SI unit is the siemens.
Instruments used to measure the electrical impedance are called impedance analyzers.

History

Perhaps the earliest use of complex numbers in circuit analysis was by Johann Victor Wietlisbach in 1879 in analysing the Maxwell bridge. Wietlisbach avoided using differential equations by expressing AC currents and voltages as exponential functions with imaginary exponents. Wietlisbach found the required voltage was given by multiplying the current by a complex number, although he did not identify this as a general parameter in its own right.
The term impedance was coined by Oliver Heaviside in July 1886. Heaviside recognised that the "resistance operator" in his operational calculus was a complex number. In 1887 he showed that there was an AC equivalent to Ohm's law.
Arthur Kennelly published an influential paper on impedance in 1893. Kennelly arrived at a complex number representation in a rather more direct way than using imaginary exponential functions. Kennelly followed the graphical representation of impedance developed by John Ambrose Fleming in 1889. Impedances could thus be added vectorially. Kennelly realised that this graphical representation of impedance was directly analogous to graphical representation of complex numbers. Problems in impedance calculation could thus be approached algebraically with a complex number representation. Later that same year, Kennelly's work was generalised to all AC circuits by Charles Proteus Steinmetz. Steinmetz not only represented impedances by complex numbers but also voltages and currents. Unlike Kennelly, Steinmetz was thus able to express AC equivalents of DC laws such as Ohm's and Kirchhoff's laws. Steinmetz's work was highly influential in spreading the technique amongst engineers.

Introduction

In addition to resistance as seen in DC circuits, impedance in AC circuits includes the effects of the induction of voltages in conductors by the magnetic fields, and the electrostatic storage of charge induced by voltages between conductors. The impedance caused by these two effects is collectively referred to as reactance and forms the imaginary part of complex impedance whereas resistance forms the real part.

Complex impedance

The impedance of a two-terminal circuit element is represented as a complex quantity. The polar form conveniently captures both magnitude and phase characteristics as
where the magnitude represents the ratio of the voltage difference amplitude to the amplitude of the current, while the argument gives the phase difference between voltage and current. In electrical engineering, the letter is used for electric current, so the imaginary unit is instead represented by the letter.
In Cartesian form, impedance is defined as
where the real part of impedance is the resistance and the imaginary part is the reactance.
Where it is needed to add or subtract impedances, the cartesian form is more convenient; but when quantities are multiplied or divided, the calculation becomes simpler if the polar form is used. A circuit calculation, such as finding the total impedance of two impedances in parallel, may require conversion between forms several times during the calculation. Conversion between the forms follows the normal conversion rules of complex numbers.

Complex voltage and current

To simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as and.
The impedance of a bipolar circuit is defined as the ratio of these quantities:
Hence, denoting, we have
The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.

Validity of complex representation

This representation using complex exponentials may be justified by noting that :
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

Ohm's law

The meaning of electrical impedance can be understood by substituting it into Ohm's law. Assuming a two-terminal circuit element with impedance is driven by a sinusoidal voltage or current as above, there holds
The magnitude of the impedance acts just like resistance, giving the drop in voltage amplitude across an impedance for a given current. The phase factor tells us that the current lags the voltage by a phase .
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.

Phasors

A phasor is represented by a constant complex number, usually expressed in exponential form, representing the complex amplitude of a sinusoidal function of time. Phasors are used by electrical engineers to simplify computations involving sinusoids, 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 cancel.

Device examples

Resistor

The impedance of an ideal resistor is purely real and is called resistive impedance:
In this case, the voltage and current waveforms are proportional and in phase.

Inductor and capacitor (in the steady state)

Ideal inductors and capacitors have a purely imaginary reactive impedance:
the impedance of inductors increases as frequency increases;
the impedance of capacitors decreases as frequency increases;
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:
Thus the inductor and capacitor impedance equations can be rewritten in polar form:
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 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.

Resistor

For a resistor, there is the relation
which is Ohm's law.
Considering the voltage signal to be
it follows that
This says that the ratio of AC voltage amplitude to alternating current amplitude across a resistor is, and that the AC voltage leads the current across a resistor by 0 degrees.
This result is commonly expressed as

Capacitor (in the steady state)

For a capacitor, there is the relation:
Considering the voltage signal to be
it follows that
and thus, as previously,
Conversely, if the current through the circuit is assumed to be sinusoidal, its complex representation being
then integrating the differential equation
leads to
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

Inductor (in the steady state)

For the inductor, we have the relation :
This time, considering the current signal to be:
it follows that:
This result is commonly expressed in polar form as
or, using Euler's formula, as
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.