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Electrical impedance PDF Print E-mail

Electrical impedance

Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. The concept of electrical impedance generalises Ohm's law to AC circuit analysis. Unlike electrical resistance, the impedance of an electric circuit can be a complex number, but the same unit, the ohm, is used for both quantities. Oliver Heaviside coined the term "impedance" in July of 1886.

Generalized impedances in a circuit can be drawn with the same symbol as a resistor or with a labeled box.
Generalized impedances in a circuit can be drawn with the same symbol as a resistor or with a labeled box.
 

AC steady state

In general, the solutions for the voltages and currents in a circuit containing resistors, capacitors and inductors (in short, all linearly behaving components) are solutions to a linear ordinary differential equation. It can be shown that if the voltage and current sources in the circuit are sinusoidal and of constant frequency, the solutions take a form referred to as AC steady state. Thus, all of the voltages and currents in the circuit are sinusoidal and have constant amplitude, frequency and phase.

In AC steady state, v(t) is a sinusoidal function of time with constant amplitude Vp, constant frequency f, and constant phase \varphi:

v(t) = V_\mathrm{p} \cos \left( 2 \pi f t + \varphi \right) = \Re \left( V_\mathrm{p} e^{j 2 \pi f t} e^{j \varphi} \right)

where

j represents the imaginary unit (\sqrt{-1})
\Re (z) means the real part of the complex number z.

The phasor representation of v(t) is the constant complex number V:

V = V_\mathrm{p} e^{j \varphi} \,

For a circuit in AC steady state, all of the voltages and currents in the circuit have phasor representations as long as all the sources are of the same frequency. That is, each voltage and current can be represented as a constant complex number. For DC circuit analysis, each voltage and current is represented by a constant real number. Thus, it is reasonable to suppose that the rules developed for DC circuit analysis can be used for AC circuit analysis by using complex numbers instead of real numbers.

[edit] Definition of electrical impedance

The impedance of a circuit element is defined as the ratio of the phasor voltage across the element to the phasor current through the element:

Z_\mathrm{R} = \frac{V_\mathrm{r}}{I_\mathrm{r}}

It should be noted that although Z is the ratio of two phasors, Z is not itself a phasor. That is, Z is not associated with some sinusoidal function of time.

For DC circuits, the resistance is defined by Ohm's law to be the ratio of the DC voltage across the resistor to the DC current through the resistor:

R = \frac{V_\mathrm{R}}{I_\mathrm{R}}

where

VR and IR above are DC (constant real) values.

Just as Ohm's law is generalized to AC circuits through the use of phasors, other results from DC circuit analysis such as voltage division, current division, Thevenin's theorem, and Norton's theorem generalize to AC circuits.

Impedance of different devices

For a resistor:

Z_\mathrm{resistor} = \frac{V_\mathrm{R}}{I_\mathrm{R}} = R \,

For a capacitor:

Z_\mathrm{capacitor} = \frac{V_\mathrm{C}}{I_\mathrm{C}} = \frac{1}{j \omega  C} \ = \frac{-j}{\omega  C} \,

For an inductor:

Z_\mathrm{inductor} = \frac{V_\mathrm{L}}{I_\mathrm{L}} = j \omega  L \,

For derivations, see Impedance of different devices (derivations).

Reactance

Main article: Reactance

The term reactance refers to the imaginary part of the impedance. Some examples:

A resistor's impedance is R (its resistance) and its reactance is 0.

A capacitor's impedance is j (-1/ωC) and its reactance is -1/ωC.

An inductor's impedance is j ω L and its reactance is ω L.

It is important to note that the impedance of a capacitor or an inductor is a function of the frequency ω and is an imaginary quantity - however is certainly a real physical phenomenon relating the shift in phases between the voltage and current phasors due to the existence of the capacitor or inductor. Earlier it was shown that the impedance of a resistor is constant and real, in other words a resistor does not cause a phase shift between voltage and current as do capacitors and inductors.

When resistors, capacitors, and inductors are combined in an AC circuit, the impedances of the individual components can be combined in the same way that the resistances are combined in a DC circuit. The resulting equivalent impedance is in general, a complex quantity. That is, the equivalent impedance has a real part and an imaginary part. The real part is denoted with an R and the imaginary part is denoted with an X. Thus:

Z_\mathrm{eq} = R_\mathrm{eq} + jX_\mathrm{eq} \,

where

Req is termed the resistive part of the impedance
Xeq is termed the reactive part of the impedance.

It is therefore common to refer to a capacitor or an inductor as a reactance or equivalently, a reactive component (circuit element). Additionally, the impedance for a capacitance is negative imaginary while the impedance for an inductor is positive imaginary. Thus, a capacitive reactance refers to a negative reactance while an inductive reactance refers to a positive reactance.

A reactive component is distinguished by the fact that the sinusoidal voltage across the component is 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. That is, unlike a resistance, a reactance does not dissipate power.

It is instructive to determine the value of the capacitive reactance at the frequency extremes. As the frequency approaches zero, the capacitive reactance grows without bound so that a capacitor approaches an open circuit for very low frequency sinusoidal sources. As the frequency increases, the capacitive reactance approaches zero so that a capacitor approaches a short circuit for very high frequency sinusoidal sources.

Conversely, the inductive reactance approaches zero as the frequency approaches zero so that an inductor approaches a short circuit for very low frequency sinusoidal sources. As the frequency increases, the inductive reactance increases so that an inductor approaches an open circuit for very high frequency sinusoidal sources.

Combining impedances

Main article: Series and parallel circuits

Combining impedances in series, parallel, or in delta-wye configurations, is the same as for resistors. The difference is that combining impedances involves manipulation of complex numbers.

In series

A diagram of several generalized impedances, connected end to end, with the same amount of current going through each

Combining impedances in series is simple:

Z_\mathrm{eq} = Z_1 + Z_2 = (R_1 + R_2) + j(X_1 + X_2) \!\ .

In parallel

A diagram of several generalized impedances, side by side, both leads of each connected to the same wires

Combining impedances in parallel is much more difficult than combining simple properties like resistance or capacitance, due to a multiplication term.

Z_\mathrm{eq} = Z_1 \| Z_2 = \left( {Z_\mathrm{1}}^{-1} + {Z_\mathrm{2}}^{-1}\right) ^{-1} = \frac{Z_\mathrm{1}Z_\mathrm{2}}{Z_\mathrm{1}+Z_\mathrm{2}} \!\ .

In rationalized form the equivalent resistance is:

Z_\mathrm{eq} = R_\mathrm{eq} + j X_\mathrm{eq} \!\ .
R_\mathrm{eq} = { (X_1 R_2 + X_2 R_1) (X_1 + X_2) + (R_1 R_2 - X_1 X_2) (R_1 + R_2) \over (R_1 + R_2)^2 + (X_1 + X_2)^2}
X_\mathrm{eq} = {(X_1 R_2 + X_2 R_1) (R_1 + R_2) - (R_1 R_2 - X_1 X_2) (X_1 + X_2) \over (R_1 + R_2)^2 + (X_1 + X_2)^2}

Circuits with general sources

Impedance is defined by the ratio of two phasors where a phasor is the complex peak amplitude of a sinusoidal function of time. For more general periodic sources and even non-periodic sources, the concept of impedance can still be used. It can be shown that virtually all periodic functions of time can be represented by a Fourier series. Thus, a general periodic voltage source can be thought of as a (possibly infinite) series combination of sinusoidal voltage sources. Likewise, a general periodic current source can be thought of as a (possibly infinite) parallel combination of sinusoidal current sources.

Using the technique of Superposition, each source is activated one at a time and an AC circuit solution is found using the impedances calculated for the frequency of that particular source. The final solutions for the voltages and currents in the circuit are computed as sums of the terms calculated for each individual source. However, it is important to note that the actual voltages and currents in the circuit do not have a phasor representation. Phasors can be added together only when each represents a time function of the same frequency. Thus, the phasor voltages and currents that are calculated for each particular source must be converted back to their time domain representation before the final summation takes place.

This method can be generalized to non-periodic sources where the discrete sums are replaced by integrals. That is, a Fourier transform is used in place of the Fourier series.

Magnitude and phase of impedance

Complex numbers are commonly expressed in two distinct forms. The rectangular form is simply the sum of the real part with the product of j and the imaginary part:

Z = R + jX \,

The polar form of a complex number the real magnitude of the number multiplied by the complex phase. This can be written with exponentials, or in phasor notation:

Z = \left|Z\right| e^ {j \varphi} = \left|Z\right|\angle \varphi

where

\left|Z\right| = \sqrt{R^2+X^2} = \sqrt{Z  Z^*} is the magnitude of Z (Z* denotes the complex conjugate of Z), and
\varphi = \arctan \bigg(\frac{X}{R} \bigg) is the angle.

Peak phasor versus rms phasor

A sinusoidal voltage or current has a peak amplitude value as well as an rms (root mean square) value. It can be shown that the rms value of a sinusoidal voltage or current is given by:

V_\mathrm{rms} = \frac{V_\mathrm{peak}}{\sqrt{2}}
I_\mathrm{rms} = \frac{I_\mathrm{peak}}{\sqrt{2}}

In many cases of AC analysis, the rms value of a sinusoid is more useful than the peak value. For example, to determine the amount of power dissipated by a resistor due to a sinusoidal current, the rms value of the current must be known. For this reason, phasor voltage and current sources are often specified as an rms phasor. That is, the magnitude of the phasor is the rms value of the associated sinusoid rather than the peak amplitude. Generally, rms phasors are used in electrical power engineering whereas peak phasors are often used in low-power circuit analysis.

In any event, the impedance is clearly the same. Whether peak phasors or rms phasors are used, the scaling factor cancels out when the ratio of the phasors is taken.

Matched impedances

Main article: Impedance matching

When fitting components together to carry electromagnetic signals, it is important to match impedance, which can be achieved with various matching devices. Failing to do so is known as impedance mismatch and results in signal loss and reflections. The existence of reflections allows the use of a time-domain reflectometer to locate mismatches in a transmission system.

For example, a conventional radio frequency antenna for carrying broadcast television in North America was standardized to 300 ohms, using balanced, unshielded, flat wiring. However cable television systems introduced the use of 75 ohm unbalanced, shielded, circular wiring, which could not be plugged into most TV sets of the era. To use the newer wiring on an older TV, small devices known as baluns were widely available. Today most TVs simply standardize on 75 ohm feeds instead.

Inverse quantities

The reciprocal of a non-reactive resistance is called conductance. Similarly, the reciprocal of an impedance is called admittance. The conductance is the real part of the admittance, and the imaginary part is called the susceptance. Conductance and susceptance are not the reciprocals of resistance and reactance in general, but only for impedances that are purely resistive or purely reactive; in the latter case a change of sign is required.

Origin of impedances

The origin of j was found by calculating an electrical circuit by the direct method, without using impedances or phasors. The circuit is formed by a resistance an inductance and a capacitor in series The circuit is connected to a sinusoidal voltage source and we have waited long enough so that all the transitory phenomena have faded away. It is now in steady sinusoidal state. As the system is linear, the steady state current will be also sinusoidal and of the same frequency of the voltage source. The only two quantities that we ignore are the amplitude of the current and its phase relative to the voltage source. If the voltage source is \scriptstyle{V=V_\circ\cos(\omega t)} the current will be of the form \scriptstyle{I=I_\circ\cos(\omega t+\varphi)}, where \scriptstyle{\varphi} is the relative phase of the current, which is unknown. The equation of the circuit is:

V_\circ\cos(\omega t)= V_R+V_L+V_C

where

\scriptstyle{V_R}, \scriptstyle{V_L} and \scriptstyle{V_C} are the voltages across the resistance, the inductance and the capacitor.
V_R\, is equal to RI_\circ\cos(\omega t+\varphi)

The definition of inductance says:

V_L=L\textstyle{{dI\over dt}}= L\textstyle{{d\left(I_\circ\cos(\omega t+\varphi)\right)\over dt}}= -\omega L I_\circ\sin(\omega t+\varphi).

The definition of capacitance says that \scriptstyle{I=C{dV_C\over dt}}. It is easy to verify (taking the expression derivative) that:

V_C=\textstyle{{1\over \omega C}} I_\circ\sin(\omega t+\varphi).

Then the equation to solve is:

V_\circ\cos(\omega t)= RI_\circ\cos(\omega t+\varphi) -\omega L I_\circ\sin(\omega t+\varphi)+ \textstyle{{1\over \omega C}} I_\circ\sin(\omega t+\varphi)

That is, we have to find the two values \scriptstyle{I_\circ} and \scriptstyle{\varphi} that makes this equation true for all values of time \scriptstyle{t}.

To do this, another circuit must be considered, identical to the former and fed by a voltage source whose only difference with the former is that it started with a lag of a quarter of a period. The voltage of this source is \scriptstyle{V=V_\circ\cos(\omega t - {\pi \over 2}  ) = V_\circ\sin(\omega t) }. The current in this circuit will be the same as in the former one but for a lag of a quarter of period:

I=I_\circ\cos(\omega t + \varphi - {\pi \over 2})= I_\circ\sin(\omega t + \varphi)  \,.

The voltage is given by:

V_\circ\sin(\omega t)= RI_\circ\sin(\omega t+\varphi) +\omega L I_\circ\cos(\omega t+\varphi)- \textstyle{{1\over \omega C}} I_\circ\cos(\omega t+\varphi)

Some of the signs have changed because a cosine becomes a sine, and a sine becomes a negative cosine.

The first equation is added with the second one multiplied by j, to try to replace expressions with the form \scriptstyle{\cos x+j\sin x} by \scriptstyle{e^{jx} }, using the les Euler's formula. This gives:

V_\circ e^{j\omega t} =RI_\circ e^{j\left(\omega t+\varphi\right)}+j\omega LI_\circ e^{j\left(\omega t+\varphi\right)} +\textstyle{{1\over j\omega C}}I_\circ e^{j\left(\omega t+\varphi\right)}

As \scriptstyle{e^{j\omega t} } is not zero we can divide all the equation by this factor:

V_\circ =RI_\circ e^{j\varphi}+j\omega LI_\circ e^{j\varphi} +\textstyle{{1\over j\omega C}}I_\circ e^{j\varphi}

This gives:

I_\circ e^{j\varphi}= \textstyle{V_\circ \over R + j\omega L + \scriptstyle{{1 \over j\omega C}}}

The left side of the equation contains the two values we are trying to deduce: the modulus and the phase of the current. The amplitude is the modulus of the complex number at the right and its phase is the argument of the complex number at the right.

The formula at right is the habitual formula which is written when doing circuit equations using phasors and impedances. The denominator of the equation is the impedances of the resistance, inductor and capacitor.

Even though the formula

I= \textstyle{V_\circ \over R + j\omega L + \scriptstyle{{1 \over j\omega C}}}

contains imaginary parts, at least some of the imaginary numbers will become real in the circuit (j*j = -1), which means that the previously stated formula can not be simplified to just

I= \textstyle{V_\circ \over R}

Analogous impedances

Electromagnetic impedance

In problems of electromagnetic wave propagation in a homogeneous medium, the intrinsic impedance of the medium is defined as:

\eta = \sqrt{\frac{\mu}{\varepsilon}}

where

μ and ε are the permeability and permittivity of the medium, respectively.

 Acoustic impedance

Main article: Acoustic impedance

In complete analogy to the electrical impedance discussed here, one also defines acoustic impedance, a complex number which describes how a medium absorbs sound by relating the amplitude and phase of an applied sound pressure to the amplitude and phase of the resulting sound flux.

Data-transfer impedance

Another analogous coinage is the use of impedance by computer programmers to describe how easy or difficult it is to pass data and flow of control between parts of a system, commonly ones written in different languages. The common usage is to describe two programs or languages/environments as having a low or high impedance mismatch.

Application to physical devices

Note that the equations above only apply to theoretical devices. Real resistors, capacitors, and inductors are more complex and each one may be modeled as a network of theoretical resistors, capacitors, and inductors. Rated impedances of real devices are actually nominal impedances, and are only accurate for a narrow frequency range, and are typically less accurate for higher frequencies. Even within its rated range, an inductor's resistance may be non-zero. Above the rated frequencies, resistors become inductive (power resistors more so), capacitors and inductors may become more resistive. The relationship between frequency and impedance may not even be linear outside of the device's rated range.

 External links

 References

  • Pohl R. W., Electrizitätslehre, Berlin-Göttingen-Heidelberg: Springer-Verlag, 1960.
  • Popov V. P., The Principles of Theory of Circuits, – M.: Higher School, 1985, 496 p. (Russian).
  • Küpfmüller K., Einführung in die theoretische Elektrotechnik, Springer-Verlag, 1959.




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