Transmission line

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Transmission_line 

Schematic of a wave moving rightward down a lossless two-wire transmission line. Black dots represent electrons, and the arrows show the electric field.

 

One of the most common types of transmission line, coaxial cable.

In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmission must be taken into account. This applies especially to radio-frequency engineering because the short wavelengths mean that wave phenomena arise over very short distances (this can be as short as millimetres depending on frequency). However, the theory of transmission lines was historically developed to explain phenomena on very long telegraph lines, especially submarine telegraph cables.

Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas (they are then called feed lines or feeders), distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses. RF engineers commonly use short pieces of transmission line, usually in the form of printed planar transmission lines, arranged in certain patterns to build circuits such as filters. These circuits, known as distributed-element circuits, are an alternative to traditional circuits using discrete capacitors and inductors.

Overview

Ordinary electrical cables suffice to carry low frequency alternating current (AC), such as mains power, which reverses direction 100 to 120 times per second, and audio signals. However, they cannot be used to carry currents in the radio frequency range, above about 30 kHz, because the energy tends to radiate off the cable as radio waves, causing power losses. Radio frequency currents also tend to reflect from discontinuities in the cable such as connectors and joints, and travel back down the cable toward the source. These reflections act as bottlenecks, preventing the signal power from reaching the destination. Transmission lines use specialized construction, and impedance matching, to carry electromagnetic signals with minimal reflections and power losses. The distinguishing feature of most transmission lines is that they have uniform cross sectional dimensions along their length, giving them a uniform impedance, called the characteristic impedance, to prevent reflections. Types of transmission line include parallel line (ladder line, twisted pair), coaxial cable, and planar transmission lines such as stripline and microstrip. The higher the frequency of electromagnetic waves moving through a given cable or medium, the shorter the wavelength of the waves. Transmission lines become necessary when the transmitted frequency's wavelength is sufficiently short that the length of the cable becomes a significant part of a wavelength.

At microwave frequencies and above, power losses in transmission lines become excessive, and waveguides are used instead, which function as "pipes" to confine and guide the electromagnetic waves. Some sources define waveguides as a type of transmission line; however, this article will not include them. At even higher frequencies, in the terahertz, infrared and visible ranges, waveguides in turn become lossy, and optical methods, (such as lenses and mirrors), are used to guide electromagnetic waves.

History

Mathematical analysis of the behaviour of electrical transmission lines grew out of the work of James Clerk Maxwell, Lord Kelvin, and Oliver Heaviside. In 1855, Lord Kelvin formulated a diffusion model of the current in a submarine cable. The model correctly predicted the poor performance of the 1858 trans-Atlantic submarine telegraph cable. In 1885, Heaviside published the first papers that described his analysis of propagation in cables and the modern form of the telegrapher's equations.

The four terminal model

Variations on the schematic electronic symbol for a transmission line.

For the purposes of analysis, an electrical transmission line can be modelled as a two-port network (also called a quadripole), as follows:

In the simplest case, the network is assumed to be linear (i.e. the complex voltage across either port is proportional to the complex current flowing into it when there are no reflections), and the two ports are assumed to be interchangeable. If the transmission line is uniform along its length, then its behaviour is largely described by a single parameter called the characteristic impedance, symbol Z₀. This is the ratio of the complex voltage of a given wave to the complex current of the same wave at any point on the line. Typical values of Z₀ are 50 or 75 ohms for a coaxial cable, about 100 ohms for a twisted pair of wires, and about 300 ohms for a common type of untwisted pair used in radio transmission.

When sending power down a transmission line, it is usually desirable that as much power as possible will be absorbed by the load and as little as possible will be reflected back to the source. This can be ensured by making the load impedance equal to Z₀, in which case the transmission line is said to be matched.

A transmission line is drawn as two black wires. At a distance x into the line, there is current I(x) travelling through each wire, and there is a voltage difference V(x) between the wires. If the current and voltage come from a single wave (with no reflection), then V(x) / I(x) = Z₀, where Z₀ is the characteristic impedance of the line.

Some of the power that is fed into a transmission line is lost because of its resistance. This effect is called ohmic or resistive loss (see ohmic heating). At high frequencies, another effect called dielectric loss becomes significant, adding to the losses caused by resistance. Dielectric loss is caused when the insulating material inside the transmission line absorbs energy from the alternating electric field and converts it to heat (see dielectric heating). The transmission line is modelled with a resistance (R) and inductance (L) in series with a capacitance (C) and conductance (G) in parallel. The resistance and conductance contribute to the loss in a transmission line.

The total loss of power in a transmission line is often specified in decibels per metre (dB/m), and usually depends on the frequency of the signal. The manufacturer often supplies a chart showing the loss in dB/m at a range of frequencies. A loss of 3 dB corresponds approximately to a halving of the power.

High-frequency transmission lines can be defined as those designed to carry electromagnetic waves whose wavelengths are shorter than or comparable to the length of the line. Under these conditions, the approximations useful for calculations at lower frequencies are no longer accurate. This often occurs with radio, microwave and optical signals, metal mesh optical filters, and with the signals found in high-speed digital circuits.

Telegrapher's equations

Main article: Telegrapher's equations

See also: Reflections on copper lines

The telegrapher's equations (or just telegraph equations) are a pair of linear differential equations which describe the voltage (${\displaystyle V}$) and current (${\displaystyle I}$) on an electrical transmission line with distance and time. They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's equations.

Schematic representation of the elementary component of a transmission line.

The transmission line model is an example of the distributed-element model. It represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

• The distributed resistance ${\displaystyle R}$ of the conductors is represented by a series resistor (expressed in ohms per unit length).
• The distributed inductance ${\displaystyle L}$ (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (in henries per unit length).
• The capacitance ${\displaystyle C}$ between the two conductors is represented by a shunt capacitor (in farads per unit length).
• The conductance ${\displaystyle G}$ of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (in siemens per unit length).

The model consists of an infinite series of the elements shown in the figure, and the values of the components are specified per unit length so the picture of the component can be misleading. ${\displaystyle R}$, ${\displaystyle L}$, ${\displaystyle C}$, and ${\displaystyle G}$ may also be functions of frequency. An alternative notation is to use ${\displaystyle R'}$, ${\displaystyle L'}$, ${\displaystyle C'}$ and ${\displaystyle G'}$ to emphasize that the values are derivatives with respect to length. These quantities can also be known as the primary line constants to distinguish from the secondary line constants derived from them, these being the propagation constant, attenuation constant and phase constant.

The line voltage ${\displaystyle V(x)}$ and the current ${\displaystyle I(x)}$ can be expressed in the frequency domain as

${\displaystyle {\frac {\partial V(x)}{\partial x}}=-(R+j\,\omega \,L)\,I(x)}$

${\displaystyle {\frac {\partial I(x)}{\partial x}}=-(G+j\,\omega \,C)\,V(x)~\,.}$

(see differential equation, angular frequency ω and imaginary unit j)

Special case of a lossless line

When the elements ${\displaystyle R}$ and ${\displaystyle G}$ are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the ${\displaystyle L}$ and ${\displaystyle C}$ elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:

${\displaystyle {\frac {\partial ^{2}V(x)}{\partial x^{2}}}+\omega ^{2}L\,C\,V(x)=0}$

${\displaystyle {\frac {\partial ^{2}I(x)}{\partial x^{2}}}+\omega ^{2}L\,C\,I(x)=0~\,.}$

These are wave equations which have plane waves with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.

General case of a line with losses

In the general case the loss terms, ${\displaystyle R}$ and ${\displaystyle G}$, are both included, and the full form of the Telegrapher's equations become:

${\displaystyle {\frac {\partial ^{2}V(x)}{\partial x^{2}}}=\gamma ^{2}V(x)\,}$

${\displaystyle {\frac {\partial ^{2}I(x)}{\partial x^{2}}}=\gamma ^{2}I(x)\,}$

where ${\displaystyle \gamma }$ is the (complex) propagation constant. These equations are fundamental to transmission line theory. They are also wave equations, and have solutions similar to the special case, but which are a mixture of sines and cosines with exponential decay factors. Solving for the propagation constant ${\displaystyle \gamma }$ in terms of the primary parameters ${\displaystyle R}$, ${\displaystyle L}$, ${\displaystyle G}$, and ${\displaystyle C}$ gives:

${\displaystyle \gamma ={\sqrt {(R+j\,\omega \,L)(G+j\,\omega \,C)\,}}}$

and the characteristic impedance can be expressed as

${\displaystyle Z_{0}={\sqrt {{\frac {R+j\,\omega \,L}{G+j\,\omega \,C}}\,}}~\,.}$

The solutions for ${\displaystyle V(x)}$ and ${\displaystyle I(x)}$ are:

${\displaystyle V(x)=V_{(+)}e^{-\gamma \,x}+V_{(-)}e^{+\gamma \,x}\,}$

${\displaystyle I(x)={\frac {1}{Z_{0}}}\,\left(V_{(+)}e^{-\gamma \,x}-V_{(-)}e^{+\gamma \,x}\right)~\,.}$

The constants ${\displaystyle V_{(\pm )}}$ must be determined from boundary conditions. For a voltage pulse ${\displaystyle V_{\mathrm {in} }(t)\,}$, starting at ${\displaystyle x=0}$ and moving in the positive ${\displaystyle x}$ direction, then the transmitted pulse ${\displaystyle V_{\mathrm {out} }(x,t)\,}$ at position ${\displaystyle x}$ can be obtained by computing the Fourier Transform, ${\displaystyle {\tilde {V}}(\omega )}$, of ${\displaystyle V_{\mathrm {in} }(t)\,}$, attenuating each frequency component by ${\displaystyle e^{-\operatorname {Re} (\gamma )\,x}\,}$, advancing its phase by ${\displaystyle -\operatorname {Im} (\gamma )\,x\,}$, and taking the inverse Fourier Transform. The real and imaginary parts of ${\displaystyle \gamma }$ can be computed as

${\displaystyle \operatorname {Re} (\gamma )=\alpha =(a^{2}+b^{2})^{1/4}\cos(\psi )\,}$

${\displaystyle \operatorname {Im} (\gamma )=\beta =(a^{2}+b^{2})^{1/4}\sin(\psi )\,}$

with

${\displaystyle a~\equiv ~R\,G\,-\omega ^{2}L\,C\ ~=~\omega ^{2}L\,C\,\left[\left({\frac {R}{\omega L}}\right)\left({\frac {G}{\omega C}}\right)-1\right]}$

${\displaystyle b~\equiv ~\omega \,C\,R+\omega \,L\,G~=~\omega ^{2}L\,C\,\left({\frac {R}{\omega \,L}}+{\frac {G}{\omega \,C}}\right)}$

the right-hand expressions holding when neither ${\displaystyle L}$, nor ${\displaystyle C}$, nor ${\displaystyle \omega }$ is zero, and with

${\displaystyle \psi ~\equiv ~{\tfrac {1}{2}}\operatorname {atan2} (b,a)\,}$

where atan2 is the everywhere-defined form of two-parameter arctangent function, with arbitrary value zero when both arguments are zero.

Alternatively, the complex square root can be evaluated algebraically, to yield:

${\displaystyle \alpha ={\frac {\pm b}{\sqrt {2\left(-a+{\sqrt {a^{2}+b^{2}}}\right)~}}},}$

and

${\displaystyle \beta =\pm {\sqrt {{\tfrac {1}{2}}\left(-a+{\sqrt {a^{2}+b^{2}}}\right)~}},}$

with the plus or minus signs chosen opposite to the direction of the wave's motion through the conducting medium. (Note that a is usually negative, since ${\displaystyle G}$ and ${\displaystyle R}$ are typically much smaller than ${\displaystyle \omega C}$ and ${\displaystyle \omega L}$, respectively, so −a is usually positive. b is always positive.)

Special, low loss case

For small losses and high frequencies, the general equations can be simplified: If ${\displaystyle {\tfrac {R}{\omega \,L}}\ll 1}$ and ${\displaystyle {\tfrac {G}{\omega \,C}}\ll 1}$ then

${\displaystyle \operatorname {Re} (\gamma )=\alpha \approx {\tfrac {1}{2}}{\sqrt {L\,C\,}}\,\left({\frac {R}{L}}+{\frac {G}{C}}\right)\,}$

${\displaystyle \operatorname {Im} (\gamma )=\beta \approx \omega \,{\sqrt {L\,C\,}}~.\,}$

Since an advance in phase by ${\displaystyle -\omega \,\delta }$ is equivalent to a time delay by ${\displaystyle \delta }$, ${\displaystyle V_{out}(t)}$ can be simply computed as

${\displaystyle V_{\mathrm {out} }(x,t)\approx V_{\mathrm {in} }(t-{\sqrt {L\,C\,}}\,x)\,e^{-{\tfrac {1}{2}}{\sqrt {L\,C\,}}\,\left({\frac {R}{L}}+{\frac {G}{C}}\right)\,x}.\,}$

Heaviside condition

Main article: Heaviside condition

The Heaviside condition is a special case where the wave travels down the line without any dispersion distortion. The condition for this to take place is

${\displaystyle {\frac {G}{C}}={\frac {R}{L}}}$

Input impedance of transmission line

Looking towards a load through a length ${\displaystyle \ell }$ of lossless transmission line, the impedance changes as ${\displaystyle \ell }$ increases, following the blue circle on this impedance Smith chart. (This impedance is characterized by its reflection coefficient, which is the reflected voltage divided by the incident voltage.) The blue circle, centred within the chart, is sometimes called an SWR circle (short for constant standing wave ratio).

The characteristic impedance ${\displaystyle Z_{0}}$ of a transmission line is the ratio of the amplitude of a single voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line.

The impedance measured at a given distance ${\displaystyle \ell }$ from the load impedance ${\displaystyle Z_{\mathrm {L} }}$ may be expressed as

${\displaystyle Z_{\mathrm {in} }\left(\ell \right)={\frac {V(\ell )}{I(\ell )}}=Z_{0}{\frac {1+{\mathit {\Gamma }}_{\mathrm {L} }e^{-2\gamma \ell }}{1-{\mathit {\Gamma }}_{\mathrm {L} }e^{-2\gamma \ell }}}}$,

where ${\displaystyle \gamma }$ is the propagation constant and ${\displaystyle {\mathit {\Gamma }}_{\mathrm {L} }={\frac {\,Z_{\mathrm {L} }-Z_{0}\,}{Z_{\mathrm {L} }+Z_{0}}}}$ is the voltage reflection coefficient measured at the load end of the transmission line. Alternatively, the above formula can be rearranged to express the input impedance in terms of the load impedance rather than the load voltage reflection coefficient:

${\displaystyle Z_{\mathrm {in} }(\ell )=Z_{0}\,{\frac {Z_{\mathrm {L} }+Z_{0}\tanh \left(\gamma \ell \right)}{Z_{0}+Z_{\mathrm {L} }\,\tanh \left(\gamma \ell \right)}}}$.

Input impedance of lossless transmission line

For a lossless transmission line, the propagation constant is purely imaginary, ${\displaystyle \gamma =j\,\beta }$, so the above formulas can be rewritten as

${\displaystyle Z_{\mathrm {in} }(\ell )=Z_{0}{\frac {Z_{\mathrm {L} }+j\,Z_{0}\,\tan(\beta \ell )}{Z_{0}+j\,Z_{\mathrm {L} }\tan(\beta \ell )}}}$

where ${\displaystyle \beta ={\frac {\,2\pi \,}{\lambda }}}$ is the wavenumber.

In calculating ${\displaystyle \beta ,}$ the wavelength is generally different inside the transmission line to what it would be in free-space. Consequently, the velocity factor of the material the transmission line is made of needs to be taken into account when doing such a calculation.

Special cases of lossless transmission lines

Half wave length

For the special case where ${\displaystyle \beta \,\ell =n\,\pi }$ where n is an integer (meaning that the length of the line is a multiple of half a wavelength), the expression reduces to the load impedance so that

${\displaystyle Z_{\mathrm {in} }=Z_{\mathrm {L} }\,}$

for all ${\displaystyle n\,.}$ This includes the case when ${\displaystyle n=0}$, meaning that the length of the transmission line is negligibly small compared to the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.

Quarter wave length

Main article: quarter-wave impedance transformer

For the case where the length of the line is one quarter wavelength long, or an odd multiple of a quarter wavelength long, the input impedance becomes

${\displaystyle Z_{\mathrm {in} }={\frac {Z_{0}^{2}}{Z_{\mathrm {L} }}}~\,.}$

Matched load

Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is matched), in which case the impedance reduces to the characteristic impedance of the line so that

${\displaystyle Z_{\mathrm {in} }=Z_{\mathrm {L} }=Z_{0}\,}$

for all ${\displaystyle \ell }$ and all ${\displaystyle \lambda }$.

Short

Standing waves on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). Black dots represent electrons, and the arrows show the electric field.

For the case of a shorted load (i.e. ${\displaystyle Z_{\mathrm {L} }=0}$), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)

${\displaystyle Z_{\mathrm {in} }(\ell )=j\,Z_{0}\,\tan(\beta \ell ).\,}$

Open

For the case of an open load (i.e. ${\displaystyle Z_{\mathrm {L} }=\infty }$), the input impedance is once again imaginary and periodic

${\displaystyle Z_{\mathrm {in} }(\ell )=-j\,Z_{0}\cot(\beta \ell ).\,}$

Practical types

Coaxial cable

Main article: coaxial cable

Coaxial lines confine virtually all of the electromagnetic wave to the area inside the cable. Coaxial lines can therefore be bent and twisted (subject to limits) without negative effects, and they can be strapped to conductive supports without inducing unwanted currents in them. In radio-frequency applications up to a few gigahertz, the wave propagates in the transverse electric and magnetic mode (TEM) only, which means that the electric and magnetic fields are both perpendicular to the direction of propagation (the electric field is radial, and the magnetic field is circumferential). However, at frequencies for which the wavelength (in the dielectric) is significantly shorter than the circumference of the cable other transverse modes can propagate. These modes are classified into two groups, transverse electric (TE) and transverse magnetic (TM) waveguide modes. When more than one mode can exist, bends and other irregularities in the cable geometry can cause power to be transferred from one mode to another.

The most common use for coaxial cables is for television and other signals with bandwidth of multiple megahertz. In the middle 20th century they carried long distance telephone connections.

Planar lines

Main article: Planar transmission line

Planar transmission lines are transmission lines with conductors, or in some cases dielectric strips, that are flat, ribbon-shaped lines. They are used to interconnect components on printed circuits and integrated circuits working at microwave frequencies because the planar type fits in well with the manufacturing methods for these components. Several forms of planar transmission lines exist.

Microstrip

A type of transmission line called a cage line, used for high power, low frequency applications. It functions similarly to a large coaxial cable. This example is the antenna feed line for a longwave radio transmitter in Poland, which operates at a frequency of 225 kHz and a power of 1200 kW.

 

Main article: microstrip

A microstrip circuit uses a thin flat conductor which is parallel to a ground plane. Microstrip can be made by having a strip of copper on one side of a printed circuit board (PCB) or ceramic substrate while the other side is a continuous ground plane. The width of the strip, the thickness of the insulating layer (PCB or ceramic) and the dielectric constant of the insulating layer determine the characteristic impedance. Microstrip is an open structure whereas coaxial cable is a closed structure.

Stripline

Main article: Stripline

A stripline circuit uses a flat strip of metal which is sandwiched between two parallel ground planes. The insulating material of the substrate forms a dielectric. The width of the strip, the thickness of the substrate and the relative permittivity of the substrate determine the characteristic impedance of the strip which is a transmission line.

Coplanar waveguide

Main article: Coplanar waveguide

A coplanar waveguide consists of a center strip and two adjacent outer conductors, all three of them flat structures that are deposited onto the same insulating substrate and thus are located in the same plane ("coplanar"). The width of the center conductor, the distance between inner and outer conductors, and the relative permittivity of the substrate determine the characteristic impedance of the coplanar transmission line.

Balanced lines

Main article: Balanced line

A balanced line is a transmission line consisting of two conductors of the same type, and equal impedance to ground and other circuits. There are many formats of balanced lines, amongst the most common are twisted pair, star quad and twin-lead.

Twisted pair

Main article: Twisted pair

Twisted pairs are commonly used for terrestrial telephone communications. In such cables, many pairs are grouped together in a single cable, from two to several thousand. The format is also used for data network distribution inside buildings, but the cable is more expensive because the transmission line parameters are tightly controlled.

Star quad

Main article: Star quad cable

Star quad is a four-conductor cable in which all four conductors are twisted together around the cable axis. It is sometimes used for two circuits, such as 4-wire telephony and other telecommunications applications. In this configuration each pair uses two non-adjacent conductors. Other times it is used for a single, balanced line, such as audio applications and 2-wire telephony. In this configuration two non-adjacent conductors are terminated together at both ends of the cable, and the other two conductors are also terminated together.

When used for two circuits, crosstalk is reduced relative to cables with two separate twisted pairs.

When used for a single, balanced line, magnetic interference picked up by the cable arrives as a virtually perfect common mode signal, which is easily removed by coupling transformers.

The combined benefits of twisting, balanced signalling, and quadrupole pattern give outstanding noise immunity, especially advantageous for low signal level applications such as microphone cables, even when installed very close to a power cable. The disadvantage is that star quad, in combining two conductors, typically has double the capacitance of similar two-conductor twisted and shielded audio cable. High capacitance causes increasing distortion and greater loss of high frequencies as distance increases.

Twin-lead

Main article: Twin-lead

Twin-lead consists of a pair of conductors held apart by a continuous insulator. By holding the conductors a known distance apart, the geometry is fixed and the line characteristics are reliably consistent. It is lower loss than coaxial cable because the characteristic impedance of twin-lead is generally higher than coaxial cable, leading to lower resistive losses due to the reduced current. However, it is more susceptible to interference.

Lecher lines

Main article: Lecher lines

Lecher lines are a form of parallel conductor that can be used at UHF for creating resonant circuits. They are a convenient practical format that fills the gap between lumped components (used at HF/VHF) and resonant cavities (used at UHF/SHF).

Single-wire line

Unbalanced lines were formerly much used for telegraph transmission, but this form of communication has now fallen into disuse. Cables are similar to twisted pair in that many cores are bundled into the same cable but only one conductor is provided per circuit and there is no twisting. All the circuits on the same route use a common path for the return current (earth return). There is a power transmission version of single-wire earth return in use in many locations.

General applications

Signal transfer

Electrical transmission lines are very widely used to transmit high frequency signals over long or short distances with minimum power loss. One familiar example is the down lead from a TV or radio aerial to the receiver.

Transmission line circuits

Main article: Distributed-element circuit

A large variety of circuits can also be constructed with transmission lines including impedance matching circuits, filters, power dividers and directional couplers.

Stepped transmission line

See also: Waveguide filter § Impedance matching

 

A simple example of stepped transmission line consisting of three segments.

A stepped transmission line is used for broad range impedance matching. It can be considered as multiple transmission line segments connected in series, with the characteristic impedance of each individual element to be ${\displaystyle Z_{\mathrm {0,i} }}$. The input impedance can be obtained from the successive application of the chain relation

${\displaystyle Z_{\mathrm {i+1} }=Z_{\mathrm {0,i} }\,{\frac {\,Z_{\mathrm {i} }+j\,Z_{\mathrm {0,i} }\,\tan(\beta _{\mathrm {i} }\ell _{\mathrm {i} })\,}{Z_{\mathrm {0,i} }+j\,Z_{\mathrm {i} }\,\tan(\beta _{\mathrm {i} }\ell _{\mathrm {i} })}}\,}$

where ${\displaystyle \beta _{\mathrm {i} }}$ is the wave number of the ${\displaystyle \mathrm {i} }$-th transmission line segment and ${\displaystyle \ell _{\mathrm {i} }}$ is the length of this segment, and ${\displaystyle Z_{\mathrm {i} }}$ is the front-end impedance that loads the ${\displaystyle \mathrm {i} }$-th segment.

The impedance transformation circle along a transmission line whose characteristic impedance ${\displaystyle Z_{\mathrm {0,i} }}$ is smaller than that of the input cable ${\displaystyle Z_{0}}$. And as a result, the impedance curve is off-centred towards the ${\displaystyle -x}$ axis. Conversely, if ${\displaystyle Z_{\mathrm {0,i} }>Z_{0}}$, the impedance curve should be off-centred towards the ${\displaystyle +x}$ axis.

Because the characteristic impedance of each transmission line segment ${\displaystyle Z_{\mathrm {0,i} }}$ is often different from the impedance ${\displaystyle Z_{0}}$ of the fourth, input cable (only shown as an arrow marked ${\displaystyle Z_{0}}$ on the left side of the diagram above), the impedance transformation circle is off-centred along the ${\displaystyle x}$ axis of the Smith Chart whose impedance representation is usually normalized against ${\displaystyle Z_{0}}$.

Stub filters

See also: Distributed-element filter § Stub band-pass filters

If a short-circuited or open-circuited transmission line is wired in parallel with a line used to transfer signals from point A to point B, then it will function as a filter. The method for making stubs is similar to the method for using Lecher lines for crude frequency measurement, but it is 'working backwards'. One method recommended in the RSGB's radiocommunication handbook is to take an open-circuited length of transmission line wired in parallel with the feeder delivering signals from an aerial. By cutting the free end of the transmission line, a minimum in the strength of the signal observed at a receiver can be found. At this stage the stub filter will reject this frequency and the odd harmonics, but if the free end of the stub is shorted then the stub will become a filter rejecting the even harmonics.

Wideband filters can be achieved using multiple stubs. However, this is a somewhat dated technique. Much more compact filters can be made with other methods such as parallel-line resonators.

Pulse generation

Transmission lines are used as pulse generators. By charging the transmission line and then discharging it into a resistive load, a rectangular pulse equal in length to twice the electrical length of the line can be obtained, although with half the voltage. A Blumlein transmission line is a related pulse forming device that overcomes this limitation. These are sometimes used as the pulsed power sources for radar transmitters and other devices.

Sound

The theory of sound wave propagation is very similar mathematically to that of electromagnetic waves, so techniques from transmission line theory are also used to build structures to conduct acoustic waves; and these are called acoustic transmission lines.

Subliminal stimuli

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Subliminal_stimuli 

Subliminal stimuli (/sʌbˈlɪmɪnəl/; the prefix sub- literally means "below" or "less than") are any sensory stimuli below an individual's threshold for conscious perception, in contrast to supraliminal stimuli (above threshold). A 2012 review of functional magnetic resonance imaging (fMRI) studies showed that subliminal stimuli activate specific regions of the brain despite participants' unawareness. Visual stimuli may be quickly flashed before an individual can process them, or flashed and then masked to interrupt processing. Audio stimuli may be played below audible volumes or masked by other stimuli.

Effectiveness

Applications of subliminal stimuli are often based on the persuasiveness of a message. Research on action priming has shown that subliminal stimuli can only trigger actions a receiver of the message plans to perform anyway. However, consensus of subliminal messaging remains unsubstantiated by other research. Most actions can be triggered subliminally only if the person is already prepared to perform a specific action.

The context that the stimulus is presented in affects their effectiveness. For example, if the target is thirsty then a subliminal stimulus for a drink is likely to influence the target to purchase that drink if it is readily available. The stimuli can also influence the target to choose the primed option over other habitually chosen options. If the subliminal stimuli are for a product that is not quickly accessible or if there is no need for it within a specific context then the stimuli will have little to no effect. Subliminal priming can direct people's actions even when they believe they are making free choices. When primed to push a button with their off-hand, people will use that hand even if they are given a free choice between using their off-hand and their dominant hand. However, a meta analysis of many strong articles displaying effectiveness of subliminal messaging revealed its effects on actual consumer purchasing choices between two alternatives are not statistically significant; subliminal messaging is only effective in very specific contexts.

Method

In subliminal stimuli research, the threshold is the level at which the participant is not aware of the stimulus being presented. Researchers determine a threshold for the stimulus that is used as the subliminal stimulus. That stimulus is then presented during the study at some point and measures are taken to determine the effects of the stimulus. The way in which studies operationally define thresholds depends on the methods of the particular article. The methodology of the research also varies by the type of subliminal stimulus (auditory or visual) and the dependent variables they measure.

Objective threshold

The objective threshold is found using a forced-choice procedure, in which participants must choose which stimulus they saw from options given to them. For example, participants are flashed a stimulus (like the word orange) and then given a few choices and asked which one they saw. Participants must choose an answer in this design. The objective threshold is obtained when participants are at the chance level of performance in this task. The length of presentation that causes chance performance on the forced-choice task is used later in the study for the subliminal stimuli.

Subjective threshold

The subjective threshold is determined when the participant reports that their performance on the forced-choice procedure approximates chance. The subjective threshold is 30 to 50 ms slower than the objective threshold, demonstrating that participants' ability to detect the stimuli is earlier than their perceived accuracy ratings would indicate; that is, stimuli presented at a subjective threshold have a longer presentation time than those presented at an objective threshold. When using the objective threshold, priming stimuli neither facilitated nor inhibited the recognition of a color. However, the longer the duration of the priming stimulus, the greater effect it had on subsequent responding. These findings indicate that the results of some studies may be due to their definition of below threshold.

Direct and indirect measures

Perception without awareness can be demonstrated through the comparison of direct and indirect measures of perception. Direct measures use responses to task definitions in accordance to the explicit instructions given to the subjects, while indirect measures use responses that are not a part of the task definition given to subjects. Both direct and indirect measures are displayed under comparable conditions except for the direct or indirect instruction. For example, in a typical Stroop test, subjects are asked to name the color of a patch of ink. A direct measure is accuracy—true to the instructions given to the participants. The popular indirect measure used in the same task is response time—subjects are not told that they are being measured for response times.

Similarly, a direct effect is the effect of a task stimulus on the instructed response to it, and is usually measured as accuracy. An indirect effect is an uninstructed effect of the task stimulus on behavior, sometimes measured by including an irrelevant or distracting component in the task stimulus and measuring its effect on accuracy. These effects are then compared on their relative sensitivity: an indirect effect that is greater than a direct effect indicates that unconscious cognition exists.

Visual stimuli

In order to study the effects of subliminal stimuli, researchers often prime participants with specific visual stimuli, and determine if those stimuli elicit different responses. Subliminal stimuli have mostly been studied in the context of emotion; in particular, researchers have focused a lot of attention to the face perception and how subliminal presentation to different facial expression affects emotion. Visual subliminal stimuli have also been used to study emotion eliciting stimuli and simple geometric stimuli. A significant amount of research has been produced throughout the years to demonstrate the effects of subliminal visual stimuli.

Images

Attitudes can develop without being aware of their antecedents. Individuals viewed slides of people performing familiar daily activities after being exposed to either an emotionally positive scene, such as a romantic couple or kittens, or an emotionally negative scene, such as a werewolf or a dead body between each slide and the next. After exposure from something which the individuals consciously perceived as a flash of light, the participants exhibited more positive personality traits to those people whose slides were associated with an emotionally positive scene and vice versa. Despite the statistical difference, the subliminal messages had less of an impact on judgment than the slide's inherent level of physical attractiveness.

Individuals show right amygdala activity in response to subliminal fear, and a greater left amygdala response to supraliminal fear. In a 2005 study, participants were exposed to a subliminal image flashed for 16.7 milliseconds that could signal a potential threat and again with a supraliminal image flashed for half a second. Furthermore, supraliminal fear showed more sustained cortical activity, suggesting that subliminal fear may not entail conscious surveillance while supraliminal fear entails higher-order processing.

Emotion eliciting stimuli

A subliminal sexual stimulus has a different effect on men compared to women. In a study by Omri Gilliath et al., men and women were subliminally exposed to either a sexual or a neutral picture, and their sexual arousal was recorded. Researchers examined the accessibility of sex-related thoughts after following the same procedure with either a pictorial judgment task or lexical decision task. The results revealed that the subliminal sexual stimuli did not have an effect on men, but for women, lower levels of sexual arousal were reported. However, in conditions related to accessibility of sex-related thoughts, the subliminal sexual stimuli led to higher accessibility for both men and women.

Subliminal stimuli can elicit significant emotional changes, but these changes are not valuable for a therapeutic effect. Spider-fearful and non-fearful undergraduates experienced either a positive, negative, or neutral subliminal priming stimulus followed immediately by a picture of a spider or a snake. Using visual analogue scales, the participants rated the affective quality of the picture. No evidence was found to support that the unpleasantness of the pictures can be modulated by subliminal priming. Non-fearful participants rated the spiders as being more frightening after being primed with a negative stimulus, but the event was not found in fearful participants.

Simple geometric stimuli

Laboratory research on unconscious perception often employs simple stimuli (e.g., geometric shapes or colors) in which visibility is controlled by visual masking. Masked stimuli are then used to prime the processing of subsequently presented target stimuli. For instance, in the response priming paradigm, participants have to respond to a target stimulus (e.g. by identifying whether it is a diamond or a square) which is immediately preceded by a masked priming stimulus (also a diamond or a square). The prime has large effects on responses to the target: it speeds responses when it is consistent with the target, and slows responses when it is inconsistent. Response priming effects can be dissociated from visual awareness of the prime, such as when prime identification performance is at chance, or when priming effects increase despite decreases in prime visibility.

The presentation of geometric figures as subliminal stimuli can result in below threshold discriminations. The geometric figures were presented on slides of a tachistoscope followed by a supraliminal shock for a given slide every time it appeared. The shock was administered after a five-second interval. Electrical skin changes of the participants that occurred before the reinforcement (shock) or non-reinforcement were recorded. The findings indicate that the proportion of electrical skin changes that occurred following subliminal visual stimuli was significantly greater than expected, while the proportion of electrical skin changes that occurred in response to the stimuli which were not reinforced was significantly less. As a whole, participants were able to make below threshold discriminations.

Word and non-word stimuli

Another form of visual stimuli is words and non-words. In a set of experiments, words and non-words were used as subliminal primes. Priming stimuli that work best as subliminal stimuli are words that have been classified several times before they are used to prime. Word primes can also be made from parts of practiced words to create new words. In this case, the actual word used to prime can have the opposite meaning of the words it came from (its "parents"), but it will still prime for the meaning of the parent words. Non-words created from previously practiced stimuli have a similar effect, even when they are unpronounceable (e.g. made of all consonants). These primes generally only increase response times for later stimuli for a very short period of time (milliseconds).

Masking visual stimuli

Visual stimuli are often masked by forward and backward masks so that they can be displayed for longer periods of time without the subject being able to recognize the priming stimuli. A forward mask is briefly displayed before the priming stimulus and a backward mask usually follows it to prevent the subject from recognizing the stimulus.

Auditory stimuli

Auditory masking

One method for creating subliminal auditory stimuli is masking, which involves hiding the target auditory stimulus in some way. Auditory subliminal stimuli are shown to have some effect on the participant, but not a large one. For example, one study used other speechlike sounds to cover up the target words, and it found evidence of priming in the absence of awareness of the stimuli. The effects of these subliminal stimuli were only seen in one of the outcome measures of priming, while the effects of conscious stimuli were seen in multiple outcome measures. However, the empirical evidence for the assumption of an impact of auditory subliminal stimuli on human behavior remains weak; in an experimental study on the influence of subliminal target words (embedded into a music track) on choice behavior for a drink, authors found no evidence for a manipulative effect.

Self-help audio recordings

A study investigated the effects on self-concept of rational emotive behavior therapy and auditory subliminal stimulation (separately and in combination) on 141 undergraduate students with self-concept problems. They were randomly assigned to one of four groups receiving either rational-emotive therapy, subliminal stimulation, both, or a placebo treatment. Rational-emotive therapy significantly improved scores on all dependent measures (cognition, self-concept, self-esteem, anxiety) except behavior. Results for the subliminal stimulation group were similar to those of the placebo treatment except for a significant self-concept improvement and a decline in self-concept-related irrational cognitions. The combined treatment yielded results similar to those of rational-emotive therapy, with tentative indications of continued improvement in irrational cognitions and self-concept from posttest to follow-up.

Consumption and television

Some studies looked at the efficacy of subliminal messaging in television. Subliminal messages produce only one-tenth of the effects of detected messages and the findings related to the effects of subliminal messaging were relatively ambiguous. Participants’ ratings of positive responses to commercials were not affected by subliminal messages in the commercials.

Johan Karremans suggests that subliminal messages have an effect when the messages are goal-relevant. In a study, researchers made half of the 105 volunteers feel thirsty by giving them food with lots of salt before performing the experiment. At the end, as predicted, they found that the subliminal message had succeeded among the thirsty. 80% of them chose a certain ice tea brand versus the 20% of the control group that were not exposed to the message. Those who were not thirsty did not choose the drink in question, despite the subliminal message. The experiment showed that in certain circumstances subliminal advertising worked.

Karremans did a study assessing whether subliminal priming of a brand name of a drink would affect a person's choice of drink, and if this effect was caused by the individual's feelings of being thirsty. In another study, participant's ratings of thirst were higher after viewing an episode of The Simpsons that contained single frames of the word thirsty or of a picture of a Coca-Cola can. Some studies showed greater effects of subliminal messaging with as high as 80% of participants showing a preference for a particular rum when subliminally primed by the name placed in an ad backward. Martin Gardner, however, criticizes claims, such as those by Wilson Bryan Key, by pointing out that "recent studies" serving as the basis for his claims were not identified by place or experimenter. He also suggests that claims about subliminal images are due to the "tendency of chaotic shapes to form patterns vaguely resembling familiar things". In 2009, the American Psychological Association defended that subliminal stimuli are subordinated to previously structured associative stimuli and that their only role is to reinforce a certain behavior or a certain previous attitude, without there being conclusive evidence that the stimulus that provokes these behaviors is properly subliminal.

Currently, there is still speculation about this effect. Many authors have continued to argue for the effectiveness of subliminal cues in changing consumption behavior, citing environmental cues as a main culprit of behavior change. Authors who support this line of reasoning cite findings such as Ronald Millman's research that showed slow-paced music in a supermarket was associated with more sales and customers moving at a slower pace. Findings such as these support the notion that external cues can affect behavior, although the stimulus may not fit into a strict definition of subliminal stimuli because although the music may not be attended to or consciously affecting the customers, they are certainly able to perceive it.

Subliminal messaging is prohibited in advertising in the United Kingdom.

Studies on advertising with subliminal stimuli in still images

Among the researchers in favor of subliminal stimuli is Wilson Bryan Key. One of Key's most cited studies is a whisky ad in which he found several hidden figures in ice cubes. However, Cecil Adams wrote that Key is someone with a sexual fixation.

Luís Bassat suggests an interesting observation by indicating that the current objective of advertising is "to get the consumer to take into account the brand when making the decision", a trend opposed to the objective of subliminal advertising. In turn, Fernando Ocaña showed that the essential thing in the field of media planning is to obtain the greatest possible memory, which implies a conscious perception and not an unconscious one as it should be the case.