Electrical impedance
Electromagnetism 

Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied.
In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current (AC) circuit. Impedance extends the concept of resistance to AC circuits, and possesses both magnitude and phase, unlike resistance, which has only magnitude. When a circuit is driven with direct current (DC), there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero phase angle.
It is necessary to introduce the concept of impedance in AC circuits because there are two additional impeding mechanisms to be taken into account besides the normal resistance of DC circuits: the induction of voltages in conductors selfinduced by the magnetic fields of currents (inductance), and the electrostatic storage of charge induced by voltages between conductors (capacitance). 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.
The symbol for impedance is usually Z and it may be represented by writing its magnitude and phase in the form Z∠θ. However, complex number representation is often more powerful for circuit analysis purposes. The term impedance was coined by Oliver Heaviside in July 1886.^{[1]}^{[2]} Arthur Kennelly was the first to represent impedance with complex numbers in 1893.^{[3]}
Impedance is defined as the frequency domain ratio of the voltage to the current.^{[4]} In other words, it is the voltage–current ratio for a single complex exponential at a particular frequency ω. In general, impedance will be a complex number, with the same units as resistance, for which the SI unit is the ohm (Ω). For a sinusoidal current or voltage input, 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.
The reciprocal of impedance is admittance (i.e., admittance is the currenttovoltage ratio, and it conventionally carries units of siemens, formerly called mhos).
Contents
 Complex impedance 1
 Ohm's law 2

Complex voltage and current 3
 Validity of complex representation 3.1
 Phasors 3.2

Device examples 4

Deriving the devicespecific impedances 4.1
 Resistor 4.1.1
 Capacitor 4.1.2
 Inductor 4.1.3

Deriving the devicespecific impedances 4.1
 Generalised splane impedance 5

Resistance vs reactance 6
 Resistance 6.1

Reactance 6.2
 Capacitive reactance 6.2.1
 Inductive reactance 6.2.2
 Total reactance 6.2.3

Combining impedances 7
 Series combination 7.1
 Parallel combination 7.2
 Measurement 8
 Variable impedance 9
 See also 10
 References 11
 External links 12
Complex impedance
Impedance is represented as a complex quantity \scriptstyle Z and the term complex impedance may be used interchangeably; the polar form conveniently captures both magnitude and phase characteristics,
 \ Z = Z e^{j\arg (Z)}
where the magnitude \scriptstyle Z represents the ratio of the voltage difference amplitude to the current amplitude, while the argument \scriptstyle \arg (Z) (commonly given the symbol \scriptstyle \theta ) gives the phase difference between voltage and current. \scriptstyle j is the imaginary unit, and is used instead of \scriptstyle i in this context to avoid confusion with the symbol for electric current. In Cartesian form,
 \ Z = R + jX
where the real part of impedance is the resistance \scriptstyle R and the imaginary part is the reactance \scriptstyle X.
Where it is required 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.
Ohm's law
The meaning of electrical impedance can be understood by substituting it into Ohm's law.^{[5]}^{[6]}
 \ V = I Z = I Z e^{j \arg (Z)}
The magnitude of the impedance \scriptstyle Z acts just like resistance, giving the drop in voltage amplitude across an impedance \scriptstyle Z for a given current \scriptstyle I. The phase factor tells us that the current lags the voltage by a phase of \scriptstyle \theta \;=\; \arg (Z) (i.e., in the time domain, the current signal is shifted \scriptstyle \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.
Complex voltage and current
In order to simplify calculations, sinusoidal voltage and current waves are commonly represented as complexvalued functions of time denoted as \scriptstyle V and \scriptstyle I.^{[7]}^{[8]}
 \begin{align} V &= Ve^{j(\omega t + \phi_V)} \\ I &= Ie^{j(\omega t + \phi_I)} \end{align}
Impedance is defined as the ratio of these quantities.
 \ Z = \frac{V}{I}
Substituting these into Ohm's law we have
 \begin{align} V e^{j(\omega t + \phi_V)} &= I e^{j(\omega t + \phi_I)} Z e^{j\theta} \\ &= I Z e^{j(\omega t + \phi_I + \theta)} \end{align}
Noting that this must hold for all t, we may equate the magnitudes and phases to obtain
 \begin{align} V &= I Z \\ \phi_V &= \phi_I + \theta \end{align}
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 (by Euler's formula):
 \ \cos(\omega t + \phi) = \frac{1}{2} \Big[ e^{j(\omega t + \phi)} + e^{j(\omega t + \phi)}\Big]
The realvalued sinusoidal function representing either voltage or current may be broken into two complexvalued functions. By the principle of superposition, we may analyse the behaviour of the sinusoid on the lefthand side by analysing the behaviour of the two complex terms on the righthand side. Given the symmetry, we only need to perform the analysis for one righthand term; the results will be identical for the other. At the end of any calculation, we may return to realvalued sinusoids by further noting that
 \ \cos(\omega t + \phi) = \Re \Big\{ e^{j(\omega t + \phi)} \Big\}
Phasors
A phasor is 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, 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 \scriptstyle e^{j\omega t} cancel.
Device examples
The impedance of an ideal resistor is purely real and is referred to as a resistive impedance:
 \ 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;
 \ Z_L = j\omega L
the impedance of capacitors decreases as frequency increases;
 \ 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:
 \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(\frac{\pi}{2})} \end{align}
Thus the inductor and capacitor impedance equations can be rewritten in polar form:
 \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}
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 devicespecific 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 will assume sinusoidal signals, since any arbitrary signal can be approximated as a sum of sinusoids through Fourier analysis.
Resistor
For a resistor, there is the relation:
 v_{\text{R}} \left( t \right) = {i_{\text{R}} \left( t \right)}R
This is Ohm's law.
Considering the voltage signal to be
 v_{\text{R}}(t) = V_p \sin(\omega t)
it follows that
 \frac{v_{\text{R}} \left( t \right)}{i_{\text{R}} \left( t \right)} = \frac{V_p \sin(\omega t)}{I_p \sin \left( \omega t \right)} = R
This says that the ratio of AC voltage amplitude to alternating current (AC) amplitude across a resistor is \scriptstyle R, and that the AC voltage leads the current across a resistor by 0 degrees.
This result is commonly expressed as
 Z_{\text{resistor}} = R
Capacitor
For a capacitor, there is the relation:
 i_{\text{C}}(t) = C \frac{\operatorname{d}v_{\text{C}}(t)}{\operatorname{d}t}
Considering the voltage signal to be
 v_{\text{C}}(t) = V_p \sin(\omega t) \,
it follows that
 \frac{\operatorname{d}v_{\text{C}}(t)}{\operatorname{d}t} = \omega V_p \cos \left( \omega t \right)
And thus
 \frac{v_{\text{C}} \left( t \right)}{i_{\text{C}} \left( t \right)} = \frac{V_p \sin(\omega t)}{\omega V_p C \cos \left( \omega t \right)}= \frac{\sin(\omega t)}{\omega C \sin \left( \omega t + \frac{\pi}{2}\right)}
This says that the ratio of AC voltage amplitude to AC current amplitude across a capacitor is \scriptstyle \frac{1}{\omega C}, and that the AC voltage lags the AC current across a capacitor by 90 degrees (or the AC current leads the AC voltage across a capacitor by 90 degrees).
This result is commonly expressed in polar form, as
 \ Z_{\text{capacitor}} = \frac{1}{\omega C} e^{j \frac{\pi}{2}}
or, by applying Euler's formula, as
 \ Z_{\text{capacitor}} = j\frac{1}{\omega C} = \frac{1}{j \omega C}
Inductor
For the inductor, we have the relation:
 v_{\text{L}}(t) = L \frac{\operatorname{d}i_{\text{L}}(t)}{\operatorname{d}t}
This time, considering the current signal to be:
 i_{\text{L}}(t) = I_p \sin(\omega t)
it follows that:
 \frac{\operatorname{d}i_{\text{L}}(t)}{\operatorname{d}t} = \omega I_p \cos \left( \omega t \right)
And thus:
 \frac{v_{\text{L}} \left( t \right)}{i_{\text{L}} \left( t \right)} = \frac{\omega I_p L \cos(\omega t)}{I_p \sin \left( \omega t \right)} = \frac{\omega L \sin \left( \omega t + \frac{\pi}{2}\right)}{\sin(\omega t)}
This says that the ratio of AC voltage amplitude to AC current amplitude across an inductor is \scriptstyle \omega L, and that the AC voltage leads the AC current across an inductor by 90 degrees.
This result is commonly expressed in polar form, as
 \ Z_{\text{inductor}} = \omega L e^{j \frac{\pi}{2}}
or, using Euler's formula, as
 \ Z_{\text{inductor}} = j \omega L
Generalised splane impedance
Impedance defined in terms of jω can strictly only be applied to circuits which are driven with a steadystate AC signal. The concept of impedance can be extended to a circuit energised with any arbitrary signal by using complex frequency instead of jω. Complex frequency is given the symbol s and is, in general, a complex number. Signals are expressed in terms of complex frequency by taking the Laplace transform of the time domain expression of the signal. The impedance of the basic circuit elements in this more general notation is as follows:
Element  Impedance expression 

Resistor  R \, 
Inductor  sL \, 
Capacitor  \frac{1}{sC} \, 
For a DC circuit this simplifies to s = 0. For a steadystate sinusoidal AC signal s = jω.
Resistance vs reactance
Resistance and reactance together determine the magnitude and phase of the impedance through the following relations:
 Z = \sqrt{Z Z^*} = \sqrt{R^2 + X^2}
 \theta = \arctan{\left(\frac{X}{R}\right)}
In many applications the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant.
Resistance
Resistance \scriptstyle R is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.
 \ R = Z \cos{\theta} \quad
Reactance
Reactance \scriptstyle X is the imaginary part of the impedance; a component with a finite reactance induces a phase shift \scriptstyle \theta between the voltage across it and the current through it.
 \ 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 will not dissipate any power.
Capacitive reactance
A capacitor has a purely reactive impedance which is inversely proportional to the signal frequency. A capacitor consists of two conductors separated by an insulator, also known as a dielectric.
 X_C = (\omega C)^{1} = (2\pi f C)^{1}\quad
At low frequencies a capacitor is open circuit, as no charge flows in the dielectric. 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 will only accumulate a limited amount of charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.
Inductive reactance
Inductive reactance \scriptstyle{X_L} is proportional to the signal frequency \scriptstyle{f} and the inductance \scriptstyle{L}.
 X_L = \omega L = 2\pi f L\quad
An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf \scriptstyle{\mathcal{E}} (voltage opposing current) due to a rateofchange of magnetic flux density \scriptstyle{B} through a current loop.
 \mathcal{E} =  = Z_1 + Z_2 + \cdots + Z_n \quad
Or explicitly in real and imaginary terms:
 \ Z_{\text{eq}} = R + jX = (R_1 + R_2 + \cdots + R_n) + j(X_1 + X_2 + \cdots + X_n) \quad
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.
Hence the inverse total impedance is the sum of the inverses of the component impedances:
 \frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots + \frac{1}{Z_n}
or, when n = 2:
 \frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} = \frac{Z_1 + Z_2}{Z_1 Z_2}
 \ Z_{\text{eq}} = \frac{Z_1 Z_2}{Z_1 + Z_2}
The equivalent impedance \scriptstyle Z_{\text{eq}} can be calculated in terms of the equivalent series resistance \scriptstyle R_{\text{eq}} and reactance \scriptstyle X_{\text{eq}}.^{[9]}
 \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}
Measurement
The measurement of the impedance of devices and transmission lines is a practical problem in radio technology and others. Measurements of impedance may be carried out at one frequency, or the variation of device impedance over a range of frequencies may be of interest. The impedance may be measured or displayed directly in ohms, or other values related to impedance may be displayed; for example in a radio antenna the standing wave ratio or reflection coefficient may be more useful than the impedance alone. Measurement of impedance requires measurement of the magnitude of voltage and current, and the phase difference between them. Impedance is often measured by "bridge" methods, similar to the directcurrent Wheatstone bridge; a calibrated reference impedance is adjusted to balance off the effect of the impedance of the device under test. Impedance measurement in power electronic devices may require simultaneous measurement and provision of power to the operating device.
The impedance of a device can be calculated by complex division of the voltage and current. The impedance of the device can be calculated by applying a sinusoidal voltage to the device in series with a resistor, and measuring the voltage across the resistor and across the device. Performing this measurement by sweeping the frequencies of the applied signal provides the impedance phase and magnitude.^{[10]}
The use of an impulse response may be used in combination with the fast Fourier transform (FFT) to rapidly measure the electrical impedance of various electrical devices.^{[10]}
The LCR meter (Inductance (L), Capacitance (C), and Resistance (R)) is a device commonly used to measure the inductance, resistance and capacitance of a component; from these values the impedance at any frequency can be calculated.
Variable impedance
In general, neither impedance nor admittance can be time varying as they are defined for complex exponentials for –∞ < t < +∞. If the complex exponential voltage–current ratio changes over time or amplitude, the circuit element cannot be described using the frequency domain. However, many systems (e.g., varicaps that are used in radio tuners) may exhibit nonlinear or timevarying voltage–current ratios that appear to be linear timeinvariant (LTI) for small signals over small observation windows; hence, they can be roughly described as having a timevarying impedance. That is, this description is an approximation; over large signal swings or observation windows, the voltage–current relationship is nonLTI and cannot be described by impedance.
See also
 Characteristic impedance
 Electrical characteristics of dynamic loudspeakers
 High impedance
 Immittance
 Impedance bridging
 Impedance cardiography
 Impedance matching
 Negative impedance converter
 Resistance distance
References
 ^ Science, p. 18, 1888
 ^ Oliver Heaviside, The Electrician, p. 212, 23 July 1886, reprinted as Electrical Papers, p 64, AMS Bookstore, ISBN 0821834657
 ^ (AIEE, 1893)ImpedanceKennelly, Arthur.
 ^ Alexander, Charles; Sadiku, Matthew (2006). Fundamentals of Electric Circuits (3, revised ed.). McGrawHill. pp. 387–389.
 ^ AC Ohm's law, Hyperphysics
 ^ Horowitz, Paul; Hill, Winfield (1989). "1". The Art of Electronics. Cambridge University Press. pp. 32–33.
 ^ Complex impedance, Hyperphysics
 ^ Horowitz, Paul; Hill, Winfield (1989). "1". The Art of Electronics. Cambridge University Press. pp. 31–32.
 ^ Parallel Impedance Expressions, Hyperphysics
 ^ ^{a} ^{b} George Lewis Jr., George K. Lewis Sr. and William Olbricht (August 2008). "Costeffective broadband electrical impedance spectroscopy measurement circuit and signal analysis for piezomaterials and ultrasound transducers". Measurement Science and Technology 19 (10): 105102.
External links
 Explaining Impedance
 Antenna Impedance
 ECE 209: Review of Circuits as LTI Systems – Brief explanation of Laplacedomain circuit analysis; includes a definition of impedance.
