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Electrical impedance

From Wikipedia, the free encyclopedia
Opposition of a circuit to a current when a voltage is applied
Electromagnetism
Solenoid

Inelectrical engineering,impedance is the opposition toalternating current presented by the combined effect ofresistance andreactance in acircuit.[1]

Quantitatively, the impedance of a two-terminalcircuit element is the ratio of thecomplex representation of thesinusoidal voltage between its terminals, to the complex representation of the current flowing through it.[2] In general, it depends upon thefrequency of the sinusoidal voltage.

Impedance extends the concept ofresistance to alternating current (AC) circuits, and possesses bothmagnitude andphase, unlike resistance, which has only magnitude.

Impedance can be represented as acomplex number, with the same units as resistance, for which theSI unit is theohm (Ω).Its symbol is usuallyZ, and it may be represented by writing its magnitude and phase in thepolar form|Z|∠θ. However,Cartesian complex number representation is often more powerful for circuit analysis purposes.

The notion of impedance is useful for performing AC analysis ofelectrical networks, because it allows relating sinusoidal voltages and currents by a simple linear law. In multipleport networks, the two-terminal definition of impedance is inadequate, but the complex voltages at the ports and the currents flowing through them are stilllinearly related by theimpedance matrix.[3]

Thereciprocal of impedance isadmittance, whoseSI unit is thesiemens.

Instruments used to measure the electrical impedance are calledimpedance analyzers.

History

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Perhaps the earliest use of complex numbers in circuit analysis was by Johann Victor Wietlisbach in 1879 in analysing theMaxwell bridge. Wietlisbach avoided using differential equations by expressing AC currents and voltages asexponential functions withimaginary exponents (see§ Validity of complex representation). Wietlisbach found the required voltage was given by multiplying the current by a complex number (impedance), although he did not identify this as a general parameter in its own right.[4]

The termimpedance was coined byOliver Heaviside in July 1886.[5][6] Heaviside recognised that the "resistance operator" (impedance) in hisoperational calculus was a complex number. In 1887 he showed that there was an AC equivalent toOhm's law.[7]

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 (showing resistance, reactance, and impedance as the lengths of the sides of a right angle triangle) developed byJohn Ambrose Fleming in 1889. Impedances could thus be addedvectorially. Kennelly realised that this graphical representation of impedance was directly analogous to graphical representation of complex numbers (Argand diagram). Problems in impedance calculation could thus be approached algebraically with a complex number representation.[8][9] Later that same year, Kennelly's work was generalised to all AC circuits byCharles 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.[10] Steinmetz's work was highly influential in spreading the technique amongst engineers.[11]

Introduction

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In addition to resistance as seen in DC circuits, impedance in AC circuits includes the effects of the induction of voltages in conductors by themagnetic fields (inductance), and the electrostatic storage of charge induced by voltages between conductors (capacitance). The impedance caused by these two effects is collectively referred to asreactance and forms theimaginary part of complex impedance whereas resistance forms thereal part.

Complex impedance

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A graphical representation of thecomplex impedance plane

The impedance of a two-terminal circuit element is represented as acomplex quantityZ{\displaystyle Z}. Thepolar form conveniently captures both magnitude and phase characteristics as

 Z=|Z|ejarg⁡(Z){\displaystyle \ Z=|Z|e^{j\arg(Z)}}

where the magnitude|Z|{\displaystyle |Z|} represents the ratio of the voltage difference amplitude to the current amplitude, while the argumentarg⁡(Z){\displaystyle \arg(Z)} (commonly given the symbolθ{\displaystyle \theta }) gives the phase difference between voltage and current. In electrical engineering, the letteri{\displaystyle i} is used forelectric current, so theimaginary unit is instead represented by the letterj{\displaystyle j}.[12]: 21 

InCartesian form, impedance is defined as

 Z=R+jX{\displaystyle \ Z=R+jX}

where thereal part of impedance is the resistanceR and theimaginary part is thereactanceX.

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 normalconversion rules of complex numbers.

Complex voltage and current

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Generalized impedances in a circuit can be drawn with the same symbol as a resistor (US ANSI or DIN Euro) or with a labeled box.

To simplify calculations,sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted asV{\displaystyle V} andI{\displaystyle I}.[13][14]

V=|V|ej(ωt+ϕV),I=|I|ej(ωt+ϕI).{\displaystyle {\begin{aligned}V&=|V|e^{j(\omega t+\phi _{V})},\\I&=|I|e^{j(\omega t+\phi _{I})}.\end{aligned}}}

The impedance of a bipolar circuit is defined as the ratio of these quantities:

Z=VI=|V||I|ej(ϕV−ϕI).{\displaystyle Z={\frac {V}{I}}={\frac {|V|}{|I|}}e^{j(\phi _{V}-\phi _{I})}.}

Hence, denotingθ=ϕV−ϕI{\displaystyle \theta =\phi _{V}-\phi _{I}}, we have

|V|=|I||Z|,ϕV=ϕI+θ.{\displaystyle {\begin{aligned}|V|&=|I||Z|,\\\phi _{V}&=\phi _{I}+\theta .\end{aligned}}}

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

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

 cos⁡(ωt+ϕ)=12[ej(ωt+ϕ)+e−j(ωt+ϕ)]{\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 ofsuperposition, 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

 cos⁡(ωt+ϕ)=Re⁡{ej(ωt+ϕ)}{\displaystyle \ \cos(\omega t+\phi )=\operatorname {Re} {\Big \{}e^{j(\omega t+\phi )}{\Big \}}}

Ohm's law

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An AC supply applying a voltageV{\displaystyle V}, across aloadZ{\displaystyle Z}, driving a currentI{\displaystyle I}
Main article:Ohm's law

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

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

The magnitude of the impedance|Z|{\displaystyle |Z|} acts just like resistance, giving the drop in voltage amplitude across an impedanceZ{\displaystyle Z} for a given currentI{\displaystyle I}. Thephase factor tells us that the current lags the voltage by a phaseθ=arg⁡(Z){\displaystyle \theta =\arg(Z)} (i.e., in thetime domain, the current signal is shiftedθ2πT{\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 asvoltage division,current division,Thévenin's theorem andNorton's theorem, can also be extended to AC circuits by replacing resistance with impedance.

Phasors

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Main article:Phasor
Further information:Analytic representation

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 fromOhm's law given above, recognising that the factors ofejωt{\displaystyle e^{j\omega t}} cancel.

Device examples

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Resistor

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The phase angles in the equations for the impedance of capacitors and inductors indicate that the voltage across a capacitorlags the current through it by a phase ofπ/2{\displaystyle \pi /2}, while the voltage across an inductorleads the current through it byπ/2{\displaystyle \pi /2}. The identical voltage and current amplitudes indicate that the magnitude of the impedance is equal to one.

The impedance of an idealresistor is purelyreal and is calledresistive impedance:

 ZR=R{\displaystyle \ Z_{R}=R}

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

Inductor and capacitor (in the steady state)

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Idealinductors andcapacitors have a purelyimaginaryreactive impedance:

the impedance of inductors increases as frequency increases;

ZL=jωL{\displaystyle Z_{L}=j\omega L}

the impedance of capacitors decreases as frequency increases;

ZC=1jωC{\displaystyle Z_{C}={\frac {1}{j\omega C}}}

In both cases, for an applied sinusoidal voltage, the resulting current is also sinusoidal, but inquadrature, 90 degrees out of phase with the voltage. However, the phases have opposite signs: in an inductor, the current islagging; in a capacitor the current isleading.

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

j≡cos⁡(π2)+jsin⁡(π2)≡ejπ21j≡−j≡cos⁡(−π2)+jsin⁡(−π2)≡ej(−π2){\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:

ZL=ωLejπ2ZC=1ωCej(−π2){\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.

Deriving the device-specific impedances

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What follows below is a derivation of impedance for each of the three basiccircuit 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 arbitrarysignal, these derivations assumesinusoidal signals. In fact, this applies to any arbitrary periodic signals, because these can be approximated as a sum of sinusoids throughFourier analysis.

Resistor

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For a resistor, there is the relation

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

which isOhm's law.

Considering the voltage signal to be

vR(t)=Vpsin⁡(ωt){\displaystyle v_{\text{R}}(t)=V_{p}\sin(\omega t)}

it follows that

vR(t)iR(t)=Vpsin⁡(ωt)Ipsin⁡(ωt)=R{\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 toalternating current (AC) amplitude across a resistor isR{\displaystyle R}, and that the AC voltage leads the current across a resistor by 0 degrees.

This result is commonly expressed as

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

Capacitor (in the steady state)

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For a capacitor, there is the relation:

iC(t)=CdvC(t)dt{\displaystyle i_{\text{C}}(t)=C{\frac {\mathrm {d} v_{\text{C}}(t)}{\mathrm {d} t}}}

Considering the voltage signal to be

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

it follows that

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

and thus, as previously,

Zcapacitor=vC(t)iC(t)=1jωC.{\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

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

then integrating the differential equation

iC(t)=CdvC(t)dt{\displaystyle i_{\text{C}}(t)=C{\frac {\mathrm {d} v_{\text{C}}(t)}{\mathrm {d} t}}}

leads to

vC(t)=1jωCIpejωt+Const.=1jωCiC(t)+Const.{\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.}}}

TheConst 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

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

Inductor (in the steady state)

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

vL(t)=LdiL(t)dt{\displaystyle v_{\text{L}}(t)=L{\frac {\mathrm {d} i_{\text{L}}(t)}{\mathrm {d} t}}}

This time, considering the current signal to be:

iL(t)=Ipsin⁡(ωt){\displaystyle i_{\text{L}}(t)=I_{p}\sin(\omega t)}

it follows that:

diL(t)dt=ωIpcos⁡(ωt){\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

Zinductor=ωLejπ2{\displaystyle Z_{\text{inductor}}=\omega Le^{j{\frac {\pi }{2}}}}

or, using Euler's formula, as

Zinductor=jωL{\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.

Generalised s-plane impedance

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The impedance of capacitors and inductors were defined above in terms ofjω{\displaystyle j\omega }, but this simplification can strictly be applied only to circuits that are driven withsteady state AC and DC signals, a condition called thesteady state. The mere act of turning a signal source ON or OFF violates this steady state condition. To allow evaluation of thetransient response in addition to the steady state of a circuit, the concept of impedance can be extended to a circuit energised with any arbitrary signal by usingcomplex frequencys=σ+jω{\displaystyle s{=}\sigma {+}j\omega } instead of justjω{\displaystyle j\omega }. Signals are expressed in terms of complex frequency by taking theLaplace transform of thetime domain expression of the signal. The impedance of the basic circuit elements in this more general notation is as follows:

ElementImpedance expression
ResistorR{\displaystyle R\,}
InductorsL{\displaystyle sL\,}
Capacitor1sC{\displaystyle {\frac {1}{sC}}\,}

For a steady state signal,s{\displaystyle s} simplifies tojω{\displaystyle j\omega }. This further simplifies tos=0{\displaystyle s{=}0} for a DC-only circuit, in which case every inductor will behave as a short circuit and every capacitor will behave as an open circuit.

Formal derivation

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

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

wheres=σ+jω{\displaystyle s=\sigma +j\omega } is the complex Laplace parameter.

Inductor

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Using thecurrent–voltage relationship of an inductor[v(t)=Ldi(t)dt]{\displaystyle [v(t){=}L{\tfrac {\mathrm {d} i(t)}{\mathrm {d} t}}]}, the Laplace transform of its voltage is:

L{v(t)}=L{Ldi(t)dt}=sLL{i(t)}.{\displaystyle {\mathcal {L}}\{v(t)\}={\mathcal {L}}\{L\,{\tfrac {\mathrm {d} i(t)}{\mathrm {d} t}}\}=sL\,{\mathcal {L}}\{i(t)\}\,.}

Rearranging so thatL{v(t)}{\displaystyle {\mathcal {L}}\{v(t)\}} is divided byL{i(t)}{\displaystyle {\mathcal {L}}\{i(t)\}} provides the inductor's impedance:ZL(s)=sL{\displaystyle Z_{L}(s){=}sL}.

Capacitor

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Using thecurrent–voltage relationship of a capacitor[i(t)=Cdv(t)dt]{\displaystyle [i(t){=}C{\tfrac {\mathrm {d} v(t)}{\mathrm {d} t}}]}, the Laplace transform of its current is:

L{i(t)}=L{Cdv(t)dt}=sCL{v(t)}.{\displaystyle {\mathcal {L}}\{i(t)\}={\mathcal {L}}\{C\,{\tfrac {\mathrm {d} v(t)}{\mathrm {d} t}}\}=sC\,{\mathcal {L}}\{v(t)\}\,.}

Rearranging so thatL{v(t)}{\displaystyle {\mathcal {L}}\{v(t)\}} is divided byL{i(t)}{\displaystyle {\mathcal {L}}\{i(t)\}} provides the capacitor's impedance:ZC(s)=1sC{\displaystyle Z_{C}(s){=}{\tfrac {1}{sC}}}.

Relation to phasors

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In thephasor regime (steady state, meaning all signals are represented mathematically as simplecomplex exponentialsv(t)=V^ejωt{\displaystyle v(t)={\hat {V}}\,e^{j\omega t}} andi(t)=I^ejωt{\displaystyle i(t)={\hat {I}}\,e^{j\omega t}} oscillating at a common frequencyω{\displaystyle \omega }), impedance can simply be calculated as the voltage-to-current ratio, in which the common time-dependent factor cancels out:

Z(ω)=v(t)i(t)=V^ejωtI^ejωt=V^I^(phasor-regime impedance){\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)}}}

The phasor domain is sometimes dubbed the frequency domain, although it lacks one of the dimensions of the Laplace parameter.[17] For steady-state, thepolar 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 thephase shift by which the current lags the voltage.

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

Resistance vs reactance

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

|Z|=ZZ∗=R2+X2θ=arctan⁡(XR){\displaystyle {\begin{aligned}|Z|&={\sqrt {ZZ^{*}}}={\sqrt {R^{2}+X^{2}}}\\\theta &=\arctan {\left({\frac {X}{R}}\right)}\end{aligned}}}

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

Resistance

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Main article:Electrical resistance

ResistanceR{\displaystyle 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⁡θ{\displaystyle \ R=|Z|\cos {\theta }\quad }

Reactance

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Main article:Electrical reactance

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

 X=|Z|sin⁡θ{\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.

Capacitive reactance

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Main article:Capacitance

A capacitor has a purely reactive impedance that isinversely proportional to the signalfrequency. A capacitor consists of twoconductors separated by aninsulator, also known as adielectric.

XC=−1   ω C =−1   2π f C  .{\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 causescharge to accumulate on one side; theelectric field due to the accumulated charge is the source of the opposition to the current. When thepotential 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

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Main article:Inductance

Inductive reactanceXL{\displaystyle X_{L}} isproportional to the signalfrequencyf{\displaystyle f} and theinductanceL{\displaystyle L}.

XL=ωL=2πfL{\displaystyle X_{L}=\omega L=2\pi fL\quad }

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

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

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

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

The back-emf is the source of the opposition to current flow. A constantdirect current has a zero rate-of-change, and sees an inductor as ashort-circuit (it is typically made from a material with a lowresistivity). Analternating current has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

Total reactance

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The total reactance is given by

X=XL+XC{\displaystyle {X=X_{L}+X_{C}}} (XC{\displaystyle X_{C}} is negative)

so that the total impedance is

 Z=R+jX{\displaystyle \ Z=R+jX}

Combining impedances

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Main article: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 arecomplex numbers. The general case, however, requiresequivalent impedance transforms in addition to series and parallel.

Series combination

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

 Zeq=Z1+Z2+⋯+Zn{\displaystyle \ Z_{\text{eq}}=Z_{1}+Z_{2}+\cdots +Z_{n}\quad }

Or explicitly in real and imaginary terms:

 Zeq=R+jX=(R1+R2+⋯+Rn)+j(X1+X2+⋯+Xn){\displaystyle \ Z_{\text{eq}}=R+jX=(R_{1}+R_{2}+\cdots +R_{n})+j(X_{1}+X_{2}+\cdots +X_{n})\quad }

Parallel combination

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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:

1Zeq=1Z1+1Z2+⋯+1Zn{\displaystyle {\frac {1}{Z_{\text{eq}}}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}+\cdots +{\frac {1}{Z_{n}}}}

or, when n = 2:

1Zeq=1Z1+1Z2=Z1+Z2Z1Z2{\displaystyle {\frac {1}{Z_{\text{eq}}}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}={\frac {Z_{1}+Z_{2}}{Z_{1}Z_{2}}}}
 Zeq=Z1Z2Z1+Z2{\displaystyle \ Z_{\text{eq}}={\frac {Z_{1}Z_{2}}{Z_{1}+Z_{2}}}}

The equivalent impedanceZeq{\displaystyle Z_{\text{eq}}} can be calculated in terms of the equivalent series resistanceReq{\displaystyle R_{\text{eq}}} and reactanceXeq{\displaystyle X_{\text{eq}}}.[18]

Zeq=Req+jXeqReq=(X1R2+X2R1)(X1+X2)+(R1R2−X1X2)(R1+R2)(R1+R2)2+(X1+X2)2Xeq=(X1R2+X2R1)(R1+R2)−(R1R2−X1X2)(X1+X2)(R1+R2)2+(X1+X2)2{\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}}}

Measurement

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The measurement of the impedance of devices and transmission lines is a practical problem inradio technology and other fields. 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 aradio antenna, thestanding wave ratio orreflection coefficient may be more useful than the impedance alone. The measurement of impedance requires the measurement of the magnitude of voltage and current, and the phase difference between them. Impedance is often measured by"bridge" methods, similar to the direct-currentWheatstone 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.[19]

The use of an impulse response may be used in combination with thefast Fourier transform (FFT) to rapidly measure the electrical impedance of various electrical devices.[19]

TheLCR meter measures a component's inductance (L), capacitance (C), and resistance (R); from these values, the impedance at any frequency can be calculated.

Example

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Consider a parallel LCtank circuit. The complex impedance of the circuit is

Z(ω)=jωL1−ω2LC.{\displaystyle Z(\omega )={\frac {j\omega L}{1-\omega ^{2}LC}}.}

It is immediately seen that the value of1|Z|{\textstyle {1 \over |Z|}} is minimal (actually equal to 0 in this case) whenever

ω2LC=1.{\displaystyle \omega ^{2}LC=1.}

Therefore, the fundamental resonance angular frequency is

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

Variable impedance

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See also:Network analysis (electrical circuits) § Time-varying components

In general, neither impedance nor admittance can vary with time, since they are defined for complex exponentials in which−∞ <t < +∞. If the complex exponential voltage to current ratio changes over time or amplitude, the circuit element cannot be described using the frequency domain. However, many components and systems (e.g.,varicaps that are used inradio tuners) may exhibit non-linear or time-varying voltage to current ratios that seem to belinear time-invariant (LTI) for small signals and over small observation windows, so they can be roughly described as if they had a time-varying impedance. This description is an approximation: Over large signal swings or wide observation windows, the voltage to current relationship will not be LTI and cannot be described by impedance.

See also

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Notes

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References

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  1. ^Slurzberg; Osterheld (1950).Essentials of Electricity for Radio and Television. 2nd ed. McGraw-Hill. pp. 360 - 362
  2. ^Callegaro, L. (2012). Electrical Impedance: Principles, Measurement, and Applications. CRC Press, p. 5
  3. ^Callegaro, Sec. 1.6
  4. ^Kline, Ronald R.,Steinmetz: Engineer and Socialist, Johns Hopkins University Press, 1992ISBN 9780801842986, p. 78.
  5. ^Science, p. 18, 1888[full citation needed][failed verification]
  6. ^Oliver Heaviside,The Electrician, p. 212, 23 July 1886, reprinted asElectrical Papers, Volume II, p 64, AMS Bookstore,ISBN 0-8218-3465-7
  7. ^Kline, p. 79.
  8. ^Kline, p. 81-2.
  9. ^Kennelly, Arthur,"Impedance",Transactions of the American Institute of Electrical Engineers, vol. 10, pp. 175–232, 18 April 1893.
  10. ^Kline, p. 85.
  11. ^Kline, p. 90-1.
  12. ^abGross, Charles A. (2012).Fundamentals of electrical engineering. Thaddeus Adam Roppel. Boca Raton, FL: CRC Press.ISBN 978-1-4398-9807-9.OCLC 863646311.
  13. ^Complex impedance, Hyperphysics
  14. ^Horowitz, Paul; Hill, Winfield (1989)."1".The Art of Electronics. Cambridge University Press. pp. 31–32.ISBN 978-0-521-37095-0.
  15. ^AC Ohm's law, Hyperphysics
  16. ^Horowitz, Paul; Hill, Winfield (1989)."1".The Art of Electronics. Cambridge University Press. pp. 32–33.ISBN 978-0-521-37095-0.
  17. ^Alexander, Charles; Sadiku, Matthew (2006).Fundamentals of Electric Circuits (3, revised ed.). McGraw-Hill. pp. 387–389.ISBN 978-0-07-330115-0.
  18. ^Parallel Impedance Expressions, Hyperphysics
  19. ^abGeorge Lewis Jr.; George K. Lewis Sr. & William Olbricht (August 2008)."Cost-effective broad-band electrical impedance spectroscopy measurement circuit and signal analysis for piezo-materials and ultrasound transducers".Measurement Science and Technology.19 (10) 105102.Bibcode:2008MeScT..19j5102L.doi:10.1088/0957-0233/19/10/105102.PMC 2600501.PMID 19081773.
  • Kline, Ronald R.,Steinmetz: Engineer and Socialist, Plunkett Lake Press, 2019 (ebook reprint of Johns Hopkins University Press, 1992ISBN 9780801842986).

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