﻿ Wave impedance

# Wave impedance

The characteristic impedance, the characteristic impedance or the impedance is a characteristic of a medium in which a wave is propagated. The ratio of reflected and transmitted amplitude of the wave at an interface is determined by the characteristic impedances of the two media.

This resistance One can vividly imagine about than the stiffness or hardness or softness, which opposes the medium of the propagating wave. This stand, for example, force and motion (for acoustic or mechanical waves ), voltage and current (with shaft on electric wires ) or electric and magnetic field component ( at electromagnetic field waves) in a certain ratio to each other.

• 2.1 characteristic impedance
• 2.2 The characteristic impedance, the line termination and the input impedance of a line
• 2.3 The line termination voltage pulses at
• 2.4 equivalent circuit diagram of an electric line
• 2.5 Definition of the line characteristic impedance in the general solution of the transmission line equations
• 2.6 Frequency dependence of the line characteristic impedance 2.6.1 Behaviour on DC
• 2.6.2 behavior at low frequencies
• 2.6.3 behavior at high and very high frequencies
• 3.1 Electrical Conduction
• 3.2 Acoustic waves in free space ( sound waves)
• 4.1 Acoustic waves in cylindrical tube
• 4.2 Acoustic waves with variable cross-section
• 4.3 reflections at interfaces
• 4.4 Acoustics: air -filled tube
• 5.1 Examples of attenuated reflection
• 5.2 Examples of complete reflection
• 5.3 Examples of non-reflective statements

## Electromagnetic waves in free space

### Wave impedance

In electrodynamics is wave impedance - engl. wave impedance - the ratio between the electric and magnetic field component of a transversely propagating electromagnetic wave in free space, far away from metal surfaces. The wave impedance is formed from the square root of the quotient, which is composed σ from the generally complex permeability μ, the complex permittivity ε in general and the electrical conductivity. He is generally a complex quantity. For the material through which the wave moves, we obtain the characteristic impedance of:

And mentions the ω is the angular frequency. If the wave is propagated in non-electrically conductive material, that is if σ = 0, the following applies:

The characteristic impedance of an electromagnetic wave in a vacuum is a universal constant. It is purely real. Its value is:

For the major in radio technology frequencies, the characteristic impedance of the air differs only slightly from this value.

The wave impedance is not equal to the known from the characteristic impedance transmission line theory, with which it is often confused.

## Current and voltage waves on lines

### Characteristic impedance

The characteristic impedance (also called nominal impedance or cable impedance, characteristic impedance English ) is a parameter along homogeneous lines; These include, for example, cable or single wire assemblies, which consist of at least two electrical conductors. The wave impedance of a waveguide is not considered here. The characteristic impedance is the ratio in a common direction propagating current and voltage waves. In an electric line of the characteristic impedance Z and the characteristic impedance Zw concerning the geometry of the boundary line are linked.

While on a homogeneous line piece of the signal propagation behavior itself is not determined by the characteristic impedance, but by the propagation constant, its abrupt changes by reflection and refraction at the joints of pipes and possibly present at the ends of mismatches affect the signal propagation behavior especially when the transmitted signals at high frequency in comparison are the inverse of the signal propagation delay on the line, or when the signals contain high frequency components. This is for example the case for

• High frequencies (e.g., RF signals or any high selective signals on lines )
• Long lines ( eg 50 Hz high- voltage lines across continents )
• Switching operations on lines ( see Pulse timetable)

The characteristic impedance of homogeneous high-frequency lines is often a real quantity (eg 50 Ω coaxial cables in common ) and independent of the line length, but usually slightly dependent on frequency ( dispersion). The frequency dependence is caused by the dielectric of the cable and must be taken into account for wideband signal transmission. The characteristic impedance is not to be confused with the ohmic line resistance, which describes the (heat) losses when the line is traversed by a stream. Rather, one can imagine the characteristic impedance as the input resistance of an infinitely long, homogeneous line, ie a line, at the end there is no signal reflection.

### The characteristic impedance, the line termination and the input impedance of a line

The characteristic impedance, there is not in the sense of a component. While shows an infinitely long line at its beginning as its input impedance impedance, in the real world, the characteristic impedance is however conveniently specified by the manufacturer or calculated from the geometry, since it can not be measured readily. One with a (possibly complex ) resistance, equal to their characteristic impedance, terminated line, however, shows regardless of their length at the beginning precisely this resistance. One calls this case " complete with the characteristic impedance ": At the end of the line, a resistor or other load with the resistance of the line characteristic impedance is connected. Which may be a resistor or an antenna. This adjustment is only possible with a real load impedance, if the characteristic impedance can be assumed to be real in the transmitted frequency band. That is, for example, at RF lines is virtually always the case.

If the line is not terminated with its characteristic impedance, the input impedance of the line varies widely depending on cable length, operating frequency, termination impedance and characteristic impedance and complex. The agreement between the impedances of source, load and line impedance is only necessary if annoying reflections or echoes of signals in both directions must be avoided ( for example, two-way data cable such as USB). Then there is power matching, the efficiency can not be greater than 50%. If a higher degree of efficiency required, it is sufficient, the line only at the end of a reflection-free (ie adapted) complete - the signal source may have any source impedance. When powerful transmitters therefore another, usually much smaller source impedance is always selected to provide a higher efficiency. The transmitter Wachenbrunn achieved in this way an efficiency of 85%.

The following three cases are distinguished in the high- frequency technology, of which the first two are often used to implement complex, frequency-dependent components, such as oscillator circuits, blocking circuits or high-pass filters. Such components are called line circuits.

### The line termination voltage pulses at

Applied to a homogeneous line, which is not completed at the output with the characteristic impedance, a voltage pulse is produced at the site of termination impedance reflection - comparable to an acoustic echo. By the mismatch differs from the characteristic impedance voltage-current relationship is forced which causes the proportional reflectance of the incoming wave. The reflected pulse portion depends on the degree of mismatch. It runs counter to the incoming voltage pulse. Corresponds to the source impedance of the signal source is not the impedance of the line, the signal at the source impedance is also reflected as an echo. The impulse then travels several times back and forth until its energy is converted into heat (see also Time Domain Reflectometry ).

A with its surge impedance line ( right) prevents the reflection of voltage pulses, as far as the impedance matches the line termination over the entire frequency spectrum of the pulse with the line impedance.

### Equivalent circuit diagram of an electric line

The left figure shows the equivalent circuit of a line section of the infinitesimal length dx. The sizes contained therein are related to the length of coverings: The inductance L ', the capacitance C', the resistance per unit length R 'and the conductance per unit length G'. Of sinusoidal signals can be determined, the two differential equations of the homogeneous line with the complex amplitudes of voltage U and current I on the line to this equivalent circuit:

(j is the imaginary unit here.)

Below the characteristic impedance defined by the abbreviation Z from the solution of the differential equation system.

### Definition of the line characteristic impedance in the general solution of the transmission line equations

Differentiating above first equation with respect to x and then sets the expression for dI / dx from the second equation, we obtain the following second order linear differential equation

By a solution of the form

Can be solved. By employing the approach and comparing coefficients can be determined γ:

Because of the occurring quadratic equation γ can be used with both positive and negative sign. These two solutions for the approach can be linearly superimposed ( provided with two constants). They produce the so-called " general solution " for the voltage U at a distance x from the beginning of the line

With the dependent on the boundary conditions coefficients a1 and a2. The complex propagation constant parameters or propagation constant γ is called. It is generally the frequency -dependent, and only if the line condition is satisfied the Heaviside, its real part is constant and the imaginary part of a linear function of the frequency.

The current intensity at the point x of the line can be determined from the line equations:

Substituting the above general solution for the voltage curve U ( x, ω ) is obtained for the current profile I (x, ω ) along the line as a function of the frequency ω:

The parameters occurring therein Zl is called characteristic impedance:

A calculation of unit length corresponding to the conductor geometry and insertion into the general shape of the line characteristic impedance shows that the closer the conductors lie down together, and the larger the wire size is, the lower the surge impedance.

### Frequency dependence of the line characteristic impedance

In the following, the behavior of the characteristic impedance of a line with direct current, low and high signal frequencies is explained. The diagrams in this section are intended to illustrate the frequency response of the line characteristic impedance, which corresponds to the input impedance of a lossy, infinitely long line. They show the frequency dependencies described in the following sections using the example of the line characteristic impedance of a real three-phase transmission line for 110 kV. In particular, the locus can be seen then discussed the frequency response of the line characteristic impedance well.

#### Behavior in DC

In DC (0 Hz) to disappear in the general formula of the line characteristic impedance, the two frequency- dependent terms, and thus the imaginary parts. Therefore, the characteristic impedance at the frequency of 0 Hz is large and real.

Ideally, G '= 0, it would be infinite. Typical values ​​for DC are between about 100 k and some 10 M.

#### Behavior at low frequencies

At low frequencies, mainly makes the capacitive covering noticeable, because it acts very quickly dominates the effect of the covering and discharge (especially for cables) the effect of the Induktivitätsbelags has not come yet in the order of magnitude of the line resistance coating. Then, L ' and G' in a first approximation, can be neglected, so that the impedance in a narrow frequency range

Amounts. The locus can be approximated by a straight line with a slope of -45 degrees.

Significance of this case for low- frequency transmission lines and telephone lines. The equation is for example also used to determine the values ​​of resistor and capacitor of the line termination ( the hybrid ) in analog telephones. With proper choice of the values ​​produced by the imaginary line be compensated. In this way the effect of the mismatch can be prevented, although it has a different resistance to the load ( small ) characteristic impedance.

#### Behavior at high and very high frequencies

At high frequencies, the ohmic resistance per unit length R 'and the conductance per unit length G ' with respect to the frequency-dependent terms of the capacitive and inductive pad jωC 'or jωL ' of the line are secondary. Then one can replace in the general equation for the characteristic impedance R 'and G' by zero, and the fracture within the root can be subsequently reduced by j.omega. Therefore, the characteristic impedance is obtained for high and very high frequencies approximated from capacitive and inductive line resistance and thus corresponds to the impedance of an ideal lossless line:

At very high frequencies of the order of GHz, although rising at a real line R ' due to the skin effect and G' due to the dielectric loss factor, but also have resistance per unit length R 'and conductance per unit length G' a still insignificant effect on the characteristic impedance.

However, the characteristic impedance of a lossless line can be somewhat frequency dependent also due to the dispersion ( qv) of the insulating material used ( dielectric ).

For the above reasons can often be expected with a constant, real characteristic impedance from frequencies greater than 20 kHz. This depends only on the line geometry and the dielectric characteristic impedance value is usually several 10 Ω ( coaxial eg 50 ... 75 Ohm ) to some 100 Ω ( two-wire line 150 ... 300 ohms). It is important for all high-frequency signals, and also for the transfer steep impulses.

### Management and field impedances of selected line forms

In a line made ​​in the same place a characteristic impedance and a characteristic impedance. The one featuring a natural current-voltage relationship of a wave, the other featuring the natural relation between electric and magnetic field component of an electromagnetic wave. The wave impedance in a line depends only on the material, the characteristic impedance of material and line geometry. Both values ​​exist in the same place in a line next to each other and generally take very different values ​​, which are, however, the geometry of the boundary line to each other.

The characteristic impedance can be calculated from the geometry of the conductor and the permittivity of its isolation. The characteristic impedance of a coaxial conductor ( coaxial cable ) is at high frequencies assuming ĩr = 1:

With the permittivity εr of the insulating material and the wave impedance of vacuum. Between the inner and outer conductors of the same coaxial line is the wave impedance:

This characteristic impedance applies to the transversal electromagnetic field inside the insulation of the coaxial line. He is only material dependent and independent of geometry. The characteristic impedance is dependent on the material and depends on the geometry of the conductor boundary. The geometry of the inner and outer conductors it is associated with the field characteristic impedance.

For the two-wire line or Lecher applies:

Or equivalent, but with the involvement of ĩr:

The characteristic impedance takes the same shape as in the coaxial cable because it does not depend on the line geometry, but also only from the insulation material. The relationship between Zw and Zl shows the following form of the above equation for the two-wire cable:

### Defaults

Conventional coaxial cables have a characteristic impedance between 50 and 75 Ω. Coaxial laboratory measurement cables usually have a characteristic impedance of 50 Ω. Coaxial TV antenna or cable TV lines have a characteristic impedance of 75 Ω. Also common in older radio reception systems two-wire lines as an antenna cables with a value of 240 Ω. Pair cables, shielded or unshielded, twisted or out in parallel, typically have line impedances on the order of 100 to 200 Ω. The signal attenuation of a line depends on the dielectric between the conductors and the metal cross-section and the selected conductor material of a line, but also from the outside diameter of a coaxial line. At a given frequency to adapt to different input impedance values ​​, for example with the aid of resonant transformers is carried out.

## Measuring the wave impedance

### Electrical conduction

One can determine the characteristic impedance ( characteristic impedance ) by the AC resistance of the open line Z0 ( capacitor ) and the AC resistance of the short-circuited line Zk (inductance ) measures and forms the geometric mean of the two measurements. The characteristic impedance Zl is then:

Instead of the name Zl the term is used Zw ( w wave) often. It should again be noted that this often leads to confusion or erroneous equating the two related but dissimilar sizes characteristic impedance and wave impedance.

Alternatively, time domain measurement methods are available. Tool for experimental verification are pulse generator and oscilloscope, the (TDR ) are also included in a time domain reflectometer.

In a third method ( for short-circuited end ) and L ( with open end ) C a short piece of cable is measured with an alternating voltage bridge and calculates the characteristic impedance of the formula. This method only provides a reliable result when the length of the cable piece is very much smaller than a quarter of the wavelength of the measured frequency in the cable.

### Acoustic waves in free space ( sound waves)

In acoustics the characteristic acoustic impedance equal to the characteristic impedance in the electrical dynamics - with the proviso that no limitations are present. In the far field pressure and velocity are in phase, and therefore the acoustic impedance is real-valued and can ρ from the density and sound velocity C of the transmitting medium can be calculated:

It is referred to as characteristic impedance - by analogy to the electric resistance R = V / I, since the voltage similarly to the sound pressure is related to the power and speed with a particle stream. Your SI derived unit is Ns / m or Pa · s / m or kg / (s · m²). In the near field we measure a residual phase angle between sound pressure and particle velocity, therefore, the IF is then a complex number.

## Acoustic impedance in the vicinity of waveguides

Once the shaft is in the vicinity of a boundary moves from other material, the wave impedance changes already at some distance to the border. The transition region is fluid and is of the order of a wavelength. Examples of high frequency technology and optics show that the waveguides do not have to be hollow. At evanescent and Goubau line, the direction of wave propagation is no longer a straight line, but appears to be curved.

### Acoustic waves in cylindrical tube

Spreads the sound in pipes from the wall inhibits the propagation of sound, because they have different wave impedances at the interface usually strong. We then speak no more of the acoustic impedance field, ignoring the effects of limitations, but by the acoustic flow impedance ZA. This is calculated as the ratio of sound pressure p and volume velocity q. If all of the particles of the transfer medium on a surface A have the same sound velocity (speed) have V, that is, when the rhythmic flow through the pipe cross-section A is in phase and all standing waves occur, can simplify the equation

The derived SI unit is Pa · s / m³.

### Acoustic waves with variable cross-section

For the case that the sound is not guided by a cylinder, but by a funnel, the above formula does not apply. With the cross-sectional area of the sound path changes the characteristic impedance, it is called an impedance transformer. Horn speaker, mouthpiece, trumpet and Makrofon transform the sound pressure is very effective in sound velocity to significantly increase the volume. A phonograph can without horns produce no appreciable volume: The stylus moves the diaphragm of a pressure chamber loudspeaker that would be far too quiet on its own. Even with electromagnetic waves transforms a horn, the wave impedance of a waveguide to field Zw0 impedance of the free space. Without this transformer would hardly energy radiated, would instead be in the waveguide form a standing wave. (see also Vivaldi antenna )

In brass instruments the bell shape affects some properties:

• Flat engmensurierte funnel give relatively little sound energy to the ambient air at the same time more energy is reflected by the instrument. This supports the formation of the standing wave, which very easily in these instruments.
• Instruments with weitmensurierten funnels sound louder because the impedance transformation is gleichmäßer. This, however, is reduced at the same time the reflected energy to the formation of the standing wave and the instrument responds relatively heavy.

The impedance transformation also works in reverse: a stethoscope, formerly called acoustic beam stop can collect sound waves and focus on the eardrum.

### Reflections at interfaces

At the interface of two materials with a large difference in impedance of the sound is strongly reflected. This difference between the air and, for example, Water particularly strong. Therefore, the probe is coupled with an ultrasonic examination by a more highly water -containing gel so that the sound is not reflected by air pockets between the probe head and the skin surface. Inside the body, however, impedance mismatch is desired in order to obtain high contrast images.

Illuminated objects can only be seen when light waves are reflected strongly enough to an impedance difference. This can lead to unwanted collisions with glass doors, with one-way mirrors on the other hand, the reflectivity is increased by vapor-deposited layers to simulate a lack of transparency.

The cross section of a sound channel does not change slowly enough, the pipe end acts as a point of discontinuity, which reflects a portion of the sound energy and runs in the opposite direction. In certain tube lengths, there may be standing waves and as a result, the acoustic flow impedance changes as a function of frequency is about a thousand times, as shown in the picture. This is the basic function of all instruments. Strictly speaking, one must reckon with complex numbers as in the transmission line theory because of the occurring phase shifts. Then here is omitted because of clarity.

### Acoustics: air -filled tube

If you measure at the end of both sides open, cylindrical tube with suitable microphones sound pressure and particle velocity, is the flow impedance, with knowledge of the pipe cross -section with the formula

Calculate. Since both ends are open, there is the special case of λ / 2, which is well known in the calculation of electromagnetic waves along wires. The measurement result in the picture shows several sharp minima of the flow impedance at multiples of the frequency 500 Hz A review with the tube length of 325 mm and the speed of sound in air provides the setpoint 528 Hz

Because the measured value of the deepest local minimum with about 40,000 Pa · s / m of the characteristic acoustic impedance of the surrounding air ( 413.5 Pa · s / m³) deviates, there is a significant mismatch and the vibrating air column in the pipe is only quiet place. It is the responsibility of the tube expansions in wind instruments like the trumpet, to reduce this mismatch, thereby increasing the volume. A corresponding mismatch is the cause of the very low efficiency of loudspeakers, which can be increased by a sufficiently large horn.

## Reflections by changes in the wave impedance

At the positions at which changes of the characteristic impedance, there will be reflections. The extreme cases of such changes in the characteristic impedance are open and closed ends. For this purpose, the following analogies can be found:

In these cases, an almost complete reflection occurs. However, the open waveguide radiates a portion of the electromagnetic wave. When a short-circuit line voltage component of the reflected wave changes sign on a line (also phase jump or 180 ° phase shift called ). By an electromagnetic wave incident perpendicularly on a conductive layer, this is for the electric field component of the case. The reflected wave then running, contrary to the incident wave. Reflections (eg, at the ends of a line ) are the cause of standing waves.

### Examples of attenuated reflection

• A sound wave encounters a soft box.
• Inhomogeneities in coaxial lines, v. a seamless change of the characteristic impedance at the connection points.
• Light shines on a dirty glass.
• Meet radar waves on a cloud.
• The end of a rope to swing excited is loaded with weights or attached to a spring at a fixed point.
• A water wave hits tetrapods.

### Examples of complete reflection

• A sound wave strikes from the air on a hard wall ( echo).
• A coaxial cable is short-circuited at the end or left open.
• An electromagnetic wave incident on an extended electrically conductive surface is ideal (see, radar cross-section).
• Light is incident on a mirror.
• A one-side fixed rope are excited into oscillation.
• A water wave strikes a cliff coast.

### Examples of non-reflective statements

Total reflection freedom is achieved with an exact correspondence of the wave impedance on both sides of the interface. The lack of suitable materials can be compensated by suitable shaping, as seen in the pictures.

• Walls of an anechoic chamber. ( Freedom from reflection by absorption )
• Horn of a gramophone. (Reflection freedom with impedance matching to the characteristic acoustic impedance)
• Exponentialtrichter or horn and spherical wave horn on the horn (speaker).
• The source impedance of a transmitter coincides with the characteristic impedance of the cable ( e.g., 50 Ω ) and the input impedance of an antenna or equivalent load match (see line matching ).
• Antireflection coating of optical components.
• A wave sink weakens the high-frequency wave in a waveguide with an absorbent material and reflects only a small proportion.
• Light shines on a matt black surface.
• A water wave travels on a flat coast with the correct slope angle.
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