Electrical impedance
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In
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.[2] In general, it depends upon the frequency of the sinusoidal voltage.
Impedance extends the concept of
Impedance can be represented as a
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.[3]
The
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 (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 term impedance was coined by Oliver Heaviside in July 1886.[5][6] Heaviside recognised that the "resistance operator" (impedance) in his operational calculus was a complex number. In 1887 he showed that there was an AC equivalent to Ohm's law.[7]
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 (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.
Complex impedance
The impedance of a two-terminal circuit element is represented as a complex quantity . The
where the magnitude represents the ratio of the voltage difference amplitude to the current amplitude, while the argument (commonly given the symbol ) gives the phase difference between voltage and current. is the imaginary unit, and is used instead of in this context to avoid confusion with the symbol for electric current.[12]: 21
In
where the
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
Complex voltage and current
To simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as and .[13][14]
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 (by Euler's formula):
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.[15][16] 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 of (i.e., in the time domain, the current signal is shifted 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.
Phasors
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 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
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
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
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 (AC) 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
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
For the inductor, we have the relation (from Faraday's law):
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.
Generalised s-plane impedance
Impedance defined in terms of jω can strictly be applied only to circuits that are driven with a steady-state AC signal. The concept of impedance can be extended to a circuit energised with any arbitrary signal by using
Element | Impedance expression |
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Resistor | |
Inductor | |
Capacitor |
For a DC circuit, this simplifies to s = 0. For a steady-state sinusoidal AC signal s = jω.
Formal derivation
The impedance 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.
where is the complex Laplace parameter. As an example, according to the I-V-law of a capacitor, , from which it follows that .
In the
Again, for a capacitor, one gets that , and hence . 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 shiftby 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 vs reactance
Resistance and reactance together determine the magnitude and phase of the impedance through the following relations:
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 is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.
Reactance
Reactance is the imaginary part of the impedance; a component with a finite reactance induces a phase shift between the voltage across it and the current through it.
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
A capacitor has a purely reactive impedance that is
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
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
Inductive reactance is proportional to the signal frequency and the inductance .
An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf (voltage opposing current) due to a rate-of-change of
For an inductor consisting of a coil with loops this gives:
The back-emf is the source of the opposition to current flow. A constant
Total reactance
The total reactance is given by
- ( is negative)
so that the total impedance is
Combining impedances
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 are complex numbers. The general case, however, requires equivalent impedance transforms in addition to series and parallel.
Series combination
For components connected in series, the current through each circuit element is the same; the total impedance is the sum of the component impedances.
Or explicitly in real and imaginary terms:
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:
or, when n = 2:
The equivalent impedance can be calculated in terms of the equivalent series resistance and reactance .[18]
Measurement
The measurement of the impedance of devices and transmission lines is a practical problem in
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 the fast Fourier transform (FFT) to rapidly measure the electrical impedance of various electrical devices.[19]
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.
Example
Consider an LC tank circuit. The complex impedance of the circuit is
It is immediately seen that the value of is minimal (actually equal to 0 in this case) whenever
Therefore, the fundamental resonance angular frequency is
Variable impedance
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.,
See also
- Bioelectrical impedance analysis – Method for estimating body composition
- Characteristic impedance – Property of an electrical circuit
- Electrical characteristics of dynamic loudspeakers
- High impedance – Node in a circuit restricting current flow
- Immittance – Combined measure of impedance and admittance.
- Impedance analyzer – Type of electronic test equipment
- Impedance bridging
- Impedance cardiography – hemorheology technique for detecting the properties of the blood flow in the thorax
- Impedance control
- Impedance matching – Adjusting input/output impedances of an electrical circuit for some purpose
- Impedance microbiology
- Negative impedance converter – Active circuit which injects energy into circuits
- Resistance distance – Graph metric of electrical resistance between nodes
- Transmission line impedance
- Universal dielectric response – An emergent scaling behaviour in heterogeneous materials under alternating current
Notes
- ^ is the imaginary unit; i.e., used in electrical engineering. The character is not used as that is often used for current.
References
- ^ Slurzberg; Osterheld (1950). Essentials of Electricity for Radio and Television. 2nd ed. McGraw-Hill. pp. 360 - 362
- ^ Callegaro, L. (2012). Electrical Impedance: Principles, Measurement, and Applications. CRC Press, p. 5
- ^ Callegaro, Sec. 1.6
- ISBN 9780801842986, p. 78.
- ^ Science, p. 18, 1888[full citation needed][failed verification]
- ISBN 0-8218-3465-7
- ^ Kline, p. 79.
- ^ Kline, p. 81-2.
- ^ Kennelly, Arthur,"Impedance", Transactions of the American Institute of Electrical Engineers, vol. 10, pp. 175–232, 18 April 1893.
- ^ Kline, p. 85.
- ^ Kline, p. 90-1.
- ^ OCLC 863646311.
- ^ Complex impedance, Hyperphysics
- ISBN 978-0-521-37095-0.
- ^ AC Ohm's law, Hyperphysics
- ISBN 978-0-521-37095-0.
- ISBN 978-0-07-330115-0.
- ^ Parallel Impedance Expressions, Hyperphysics
- ^ PMID 19081773.
- Kline, Ronald R., Steinmetz: Engineer and Socialist, Plunkett Lake Press, 2019 (ebook reprint of Johns Hopkins University Press, 1992 ISBN 9780801842986).
External links
- ECE 209: Review of Circuits as LTI Systems – Brief explanation of Laplace-domain circuit analysis; includes a definition of impedance.