I'jaz

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A page of the Qur'an,16th century: "They would never produce its like not though they backed one another" written at the center.

In Islam, ’i‘jāz (Arabic: اَلْإِعْجَازُ, romanized: al-ʾiʿjāz) or inimitability^{[citation needed]} of the Qur’ān is the doctrine which holds that the Qur’ān has a miraculous quality, both in content and in form, that no human speech can match. According to this doctrine the Qur'an is a miracle and its inimitability is the proof granted to Muhammad in authentication of his prophetic status. It serves the dual purpose of proving the authenticity of its divineness as being a source from the creator as well as proving the genuineness of Muhammad's prophethood to whom it was revealed as he was the one bringing the message.

Today, works continue to be written, especially about the scientific and hurufic/numerolologic miraculousness of the Quran, and arouse interest in certain segments of Islamic society. (Quran code)

History and sociology

The concept of “I'jaz” (lit; challenging) existed in preislamic Arabic poetry as a tradition in the sense of challenging one's rivals and rendering them incapable of creating a similar one, and a large part of the Quran was in the "nature of poetry".

The first works about the I'jaz of the Quran began to appear in the 9th century in the Mu'tazila circles, which emphasized only its literary aspect, and were adopted by other religious groups. The scientific miraculousness of the Quran began to be claimed in recent times. The claim that it was a miracle was reinforced by the emphasis that, despite some rumors to the contrary, Muhammad could not have achieved these feats without being able to read and write, and that this success could only come with Divine help.

Angelika Neuwirth lists the factors that led to the emergence of the doctrine of I'jaz: The necessity of explaining some challenging verses in the Quran; In the context of the emergence of the theory of "proofs of prophecy" (dâ'il an-nubûvva) in Islamic theology, proving that the Quran is a work worthy of the emphasized superior place of Muhammad in the history of the prophets, thus gaining polemical superiority over Jews and Christians; Preservation of Arab national pride in the face of confrontation with the Iranian Shu'ubiyya movement, etc.

The poetic structure of the Quran also means that it can contain many allegories or literal mysteries that cause problems in Quran translations, and that some literary arts and exaggerations are used in the Quran to increase impressiveness.

Heinz Grotzfeld talks about the advantages of metaphorical interpretations. Thus, some Muslims may adopt a more flexible lifestyle in the face of the rules imposed by religious leaders on society based on the apparent meaning of the expressions of the Quran, and some religious leadersowner of great claims such as being mahdi, mujaddid, or "being chosen" such as Said Nursi, may claim that some verses of the Quran are actually talking about themselves or their works and giving good news to them.

Qur'anic basis

The concept of inimitability originates in the Qur'an. In five different verses, opponents are challenged to produce something like the Qur'an. The suggestion is that those who doubt the divine authorship of the Qur'an should try to disprove it by demonstrating that a human being could have created it:

• "If men and Jinn banded together to produce the like of this Qur'an they would never produce its like not though they backed one another." (17:88) 
• "Say, Bring you then ten chapters like unto it, and call whomsoever you can, other than God, if you speak the truth!" (11:13) 
• "Or do they say he has fabricated it? Say bring then a chapter like unto it, and call upon whom you can besides God, if you speak truly!" (10:38) 
• "Or do they say he has fabricated it? Nay! They believe not! Let them then produce a recital like unto it if they speak the truth." (52:34) 
• "And if you are in doubt concerning that which We have sent down to our servant, then produce a chapter of the like." (2:23) 

In the verses cited, Muhammad's opponents are invited to try to produce a text like the Qur'an, or even ten chapters, or even a single chapter. It is thought among Muslims that the challenge has not been met.

Study

Folio from a section of the Qur'an, 14th century

The literary quality of the Qur'an has been praised by Muslim scholars and by many non-Muslim scholars. Some Muslim scholars claim that early Muslims accepted Islam on the basis of evaluating the Qur'an as a text that surpasses all human production. Whilst western views typically ascribe social, ideological, propagandistic, or military reasons for the success of early Islam, Muslim sources view the literary quality of the Qur'an as a decisive factor for the adoption of the Islamic creed and its ideology, resulting in its spread and development in the 7th century. A thriving poetic tradition existed at the time of Muhammad, but Muslim scholars such as Afnan Fatani contend that Muhammad had brought, despite being unlettered, something that was superior to anything that the poets and orators had ever written or heard. The Qur'an states that poets did not question this, what they rejected was the Qur'an's ideas, especially monotheism and resurrection. Numerous Muslim scholars devoted time to finding out why the Qur'an was inimitable. The majority of opinions was around eloquence of the Qur'an are in both wording and meaning as its speech does not form to poetry nor prose commonly expressed in all languages. However, some Muslims differed, claiming that after handing down the Qur'an, God performed an additional miracle which rendered people unable to imitate the Qur'an, and that this is the source of I'jaz. This idea was less popular, however.

Nonlinguistic approaches focus on the inner meanings of the Qur'an. Oliver Leaman, favoring a nonlinguistic approach, criticizes the links between aesthetic judgment and faith and argues that it is possible to be impressed by something without thinking that it came about supernaturally and vice versa it is possible to believe in the divine origin of the Qur'an without agreeing to the aesthetic supremacy of the text. He thinks that it is the combination of language, ideas, and hidden meanings of the Qur'an that makes it an immediately convincing product.

Classic works

There are numerous classical works of Islamic literary criticism which have studied the Qur'an and examined its style:

The most famous works on the doctrine of inimitability are two medieval books by the grammarian Al Jurjani (d. 1078 CE), Dala’il al-i'jaz ('the Arguments of Inimitability') and Asraral-balagha ('the Secrets of Eloquence'). Al Jurjani argued that the inimitability of the Qur'an is a linguistic phenomenon and proposed that the Qur'an has a degree of excellence unachievable by human beings. Al Jurjani believed that Qur'an's eloquence must be a certain special quality in the manner of its stylistic arrangement and composition or a certain special way of joining words. He studied the Qur'an with literary proofs and examined the various literary features and how they were utilized. He rejected the idea that the words (alfaz) and meaning (ma'ani) of a literary work can be separated. In his view the meaning was what determined the quality of the style and that it would be absurd to attribute qualities of eloquence to a text only by observing its words. He explains that eloquence does not reside in the correct application of grammar as these are only necessary not sufficient conditions for the quality of a text. The originality of Al Jurjani is that he linked his view on meaning as the determining factor in the quality of a text by considering it not in isolation but as it is realized within a text. He wished to impress his audience with the need to study not only theology but also grammatical details and literary theory in order to improve their understanding of the inimitability of the Qur'an. For Al Jurjani the dichotomy much elaborated by earlier critics between 'word' and 'meaning' was a false one. He suggested considering not merely the meaning but 'the meaning of the meaning'. He defined two types of meaning one that resorts to the 'intellect' the other to the 'imagination'.

A page of the Qur'an with illumination, 16th century

Al-Baqillani (d. 1013 CE) wrote a book named I'jaz al-Qur'an ('inimitability of the Qur'an') and emphasized that the style of the Qur'an cannot be classified, and eloquence sustains throughout the Qur'an in spite of dealing with various themes. Al Baqillani's point was not that the Qur'an broke the custom by extraordinary degree of eloquence but that it broke the custom of the existing literary forms by creating a new genre of expression.

Ibrahim al-Nazzam of Basra (d. 846 CE) was among the first to study the doctrine. According to Al Nazzam, the Qur'an's inimitability is due to the information in its content which as divine revelation contains divine knowledge. Thus, Qur'an's supremacy lies in its content rather than its style. A- Murtaza (d. 1044 CE) had similar views, turning to divine intervention as the only viable explanation as to why the challenge was not met.

Al-Qadi Abd al-Jabbar (d. 1025 CE), in his book Al-Mughni ("the sufficient book"), insists on the hidden meanings of the Qur'an along with its eloquence and provides some counter-arguments against the criticism leveled at Muhammad and the Qur'an. Abd al-Jabbar studies the doctrine in parts 15 and 16 of his book series. According to Abd al-Jabbr, Arabs chose not to compete with Muhammad in the literary field but on the battlefield and this was another reason that they recognized the superiority of the Qur'an. Abd al-Jabbar rejected the doctrine of sarfah (the prohibition from production) because according to him sarfah makes a miracle of something other than the Qur'an and not the Qur'an itself. The doctrine of sarfah means that people can produce a rival to the Qur'an but due to some supernatural or divine cause decide against doing so. Therefore, according to Abd al-Jabbar, the correct interpretation of sarfah is that the motives to rival the Qur'an disappears because of the recognition of the impossibility of doing so.

Yahya ibn Ziyad al-Farra (d. 822 CE), Abu Ubaydah (d. 824 CE), Ibn Qutaybah (d. 889 CE), Rummani (d. 994 CE), Khattabi (d. 998 CE), and Zarkashi (d. 1392 CE) are also among notable scholars in this subject. Ibn Qutaybah considered 'brevity' which he defined as "jam' al-kathir mi ma'anih fi l-qalil min lafzih" (collection of many ideas in a few words) as one aspect of Qur'anic miraculousness. Zarkashi in his book Al-Burhan stated that miraculousness of the Qur'an can be perceived but not described.

Scientific I'jaz Literature

Main article: Islamic attitudes towards science

Some hold that certain verses of the Qur'an contain scientific theories that have been discovered only in modern times, confirming Qur'an's miraculousness. This has been criticized by the scientific community. Critics argue that verses which allegedly explain modern scientific facts, about subjects such as plate tectonics, the expansion of the universe, subterranean oceans, biology, human evolution, the beginnings and origin of human life, or the history of Earth, for example, contain fallacies and are unscientific.

Maurice Bucaille argued that some Quranic verses are agreement with modern science and contain information that had not been known in the past. He stated that he examined the degree of compatibility between the Qur'an and modern scientific data and concluded that the Qur'an did not contradict modern science. He argued that it is inconceivable that the scientific statements of the Qur'an could have been the work of man. Bucaille's arguments have been criticized by both Muslim and non-Muslim scientists.

The methodology of scientific I'jaz has not gained full approval by Islamic scholars and is the subject of ongoing debate. According to Ziauddin Sardar, the Qur'an does not contain many verses that point towards nature, however, it constantly asks its readers to reflect on the wonders of the cosmos. He refers to verse 29:20 which says "Travel throughout the earth and see how He brings life into being" and 3:190 which says "In the creation of the heavens and the earth and the alternation of night and day there are indeed signs for men of understanding" and concludes that these verses do not have any specific scientific content, rather they encourage believers to observe natural phenomena and reflect on the complexity of the universe. According to Nidhal Guessoum some works on miracles in the Qur'an follow a set pattern; they generally begin with a verse from the Qur'an, for example, the verse "So verily I swear by the stars that run and hide . . ." (81:15-16) and quickly declare that it refers to black holes, or take the verse "I swear by the Moon in her fullness, that ye shall journey on from stage to stage" (84:18-19) and decide it refers to space travel, and so on. "What is meant to be allegorical and poetic is transformed into products of science".

I'jaz has also been examined from the vantage point of its contribution to literary theory by Rebecca Ruth Gould, Lara Harb, and others.

Muhammad's illiteracy

In Islamic theology, Muhammad's illiteracy is a way of emphasizing that he was a transparent medium for divine revelation and a sign of the genuineness of his prophethood since the illiterate prophet could not have composed the eloquent poetry and prose of the Qur'an. According to Tabatabaei (d. 1981), a Muslim scholar, the force of this challenge becomes clear when we realize that it is issued for someone whose life should resemble that of Muhammad namely the life of an orphan, uneducated in any formal sense, not being able to read or write and grew up in the unenlightened age of the jahiliyah period (the age of ignorance) before Islam.

The references to illiteracy are found in verses 7:158, 29:48, and 62:2. The verse 25:5 also implies that Muhammad was unable to read and write. The Arabic term "ummi" in 7:158 and 62:2 is translated to 'illiterate' and 'unlettered'. The medieval exegete Al Tabari (d. 923 CE) maintained that the term induced two meanings: firstly, the inability to read or write in general and secondly, the inexperience or ignorance of the previous books or scriptures.

The early sources on the history of Islam provide that Muhammad especially in Medina used scribes to correspond with the tribes. Likewise, though infrequently rather than constantly, he had scribes write down, on separate pages not yet in one single book, parts of the Qur'an. Collections of prophetic tradition occasionally mention Muhammad having basic knowledge of reading and writing, while others deny it. For example, in the book Sahih al-Bukhari, a collection of early sayings, it is mentioned that when Muhammad and the Meccans agreed to conclude a peace treaty, Muhammad made a minor change to his signature or in one occasion he asked for a paper to write a statement. On another occasion, the Sira of Ibn Ishaq records that Muhammad wrote a letter with secret instructions to be opened after two days on the expedition to Nakhla in 2 A.H. Alan Jones has discussed these incidents and the use of Arabic writing in the earliest Islamic period in some detail.

Fakhr Al-Razi, the 12th century Islamic theologian, has expressed his idea is his book Tafsir Al Razi:

...Most arabs were not able to read or write and the prophet was one of them. The prophet recited a perfect book to them again and again without editing or changing the words, in contrast when arab orators prepared their speech they added or deleted large or small parts of their speech before delivering it. But the Prophet did not write down the revelation and recited the book of God without addition, deletion, or revision...If he had mastered writing and reading, people would have suspected that he had studied previous books but he brought this noble Qur'an without learning and education...the Prophet had not learned from a teacher, he had not studied any book, and did not attend any classroom of a scholar because Mecca was not a place of scholars. And he was not absent from Mecca for a long period of time which would make it possible to claim that he learned during that absence.

Contrary views

Imitators

Towards the end of Muhammad's life and after his death several men and a woman appeared in various parts of Arabia and claimed to be prophets. Musaylimah, a contemporary of Muhammad, claimed that he received revelations; some of his revelations are recorded. Ibn al-Muqaffa' was a critic of the Qur'an and reportedly made attempts to imitate it. Bashshar ibn Burd (d. 784), Abul Atahiya (d. 828), Al-Mutanabbi (d. 965), and Al-Maʿarri (d. 1058) claimed that their writings surpassed Qur'an in eloquence.

Critics

German orientalist Theodor Nöldeke criticized the Qur'anic text as careless and imperfect, pointing out claimed linguistic defects. His argument was countered by Muslim scholar Muhammad Mohar Ali in his book "The Qur'an and the Orientalists". Orientalist scholars Friedrich Schwally and John Wansbrough held a similar opinion to Nöldeke. Some writers have questioned Muhammad's illiteracy. Ruthven states that "The fact of Muhammad's illiteracy would in no way constitute proof of the Qur'an miraculous origin as the great pre-Islamic poets were illiterate." Peters writes: "We do not know where this minor merchant of Mecca learned to make poetry...most oral poets and certainly the best have been illiterate." Others believe that Muhammad hired poets or that the Qur'an was translated into Arabic from another language.

Religious skepticism

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

Religious skepticism is a type of skepticism relating to religion. Religious skeptics question religious authority and are not necessarily anti-religious but rather are skeptical of either specific or all religious beliefs and/or practices. Socrates was one of the most prominent and first religious skeptics of whom there are records; he questioned the legitimacy of the beliefs of his time in the existence of the Greek gods. Religious skepticism is not the same as atheism or agnosticism, and some religious skeptics are deists (or theists who reject the prevailing organized religion they encounter, or even all organized religion).

Overview

The word skeptic (sometimes sceptic) is derived from the middle French sceptique or the Latin scepticus, literally "sect of the sceptics". Its origin is in the Greek word skeptikos, meaning inquiring, which was used to refer to members of the Hellenistic philosophical school of Pyrrhonism, which doubted the possibility of knowledge. As such, religious skepticism generally refers to doubting or questioning something about religion. Although, as noted by Schellenberg the term is sometimes more generally applied to anyone that has a negative view of religion.

The majority of skeptics are agnostics and atheists, but there are also a number of religious people that are skeptical of religion. The religious are generally skeptical about claims of other religions, at least when the two denominations conflict concerning some stated belief. Some philosophers put forth the sheer diversity of religion as a justification for skepticism by theists and non-theists alike. Theists are also generally skeptical of the claims put forth by atheists.

Michael Shermer wrote that religious skepticism is a process for discovering the truth rather than general non-acceptance. For this reason a religious skeptic might believe that Jesus existed while questioning claims that he was the messiah or performed miracles (see historicity of Jesus). Thomas Jefferson's The Life and Morals of Jesus of Nazareth, a literal cut and paste of the New Testament that removes anything supernatural, is a prominent example.

History

Ancient history

Ancient Greece was a polytheistic society in which the gods were not omnipotent and required sacrifice and ritual. The earliest beginnings of religious skepticism can be traced back to Xenophanes. He critiqued popular religion of his time, particularly false conceptions of the divine that are a byproduct of the human propensity to anthropomorphize deities. He took the scripture of his time to task for painting the gods in a negative light and promoted a more rational view of religion. He was very critical of religious people privileging their belief system over others without sound reason.

Socrates' conception of the divine was that the gods were always benevolent, truthful, authoritative, and wise. Divinity was to operate within the standards of rationality. This critique of established religion ultimately resulted in his trial for impiety and corruption as documented in The Apology. The historian Will Durant writes that Plato was "as skeptical of atheism as of any other dogma."

Democritus was the father of materialism in the West, and there is no trace of a belief in any afterlife in his work. Specifically, in Those in Hades he refers to constituents of the soul as atoms that dissolve upon death. This later inspired the philosopher Epicurus and the philosophy he founded, who held a materialist view and rejected any afterlife, while further claiming the gods were also uninterested in human affairs. In the poem De rerum natura Lucretius proclaimed Epicurean philosophy, that the universe operates according to physical principles and guided by fortuna, or chance, instead of the Roman gods.

In De Natura Deorum, the Academic Skeptic philosopher Cicero presented arguments against the Stoics calling into question the character of the gods, whether or not they participate in earthly affairs, and questions their existence. 

In ancient India, there was a materialist philosophical school called the Cārvāka, who were known as being skeptical of the religious claims of Vedic religion, its rituals and texts. A forerunner to the Charvaka school, philosopher Ajita Kesakambali, did not believe in reincarnation.

Early modern history

Thomas Hobbes took positions that strongly disagreed with orthodox Christian teachings. He argued repeatedly that there are no incorporeal substances, and that all things, even God, heaven, and hell are corporeal, matter in motion. He argued that "though Scripture acknowledge spirits, yet doth it nowhere say, that they are incorporeal, meaning thereby without dimensions and quantity".

Voltaire, although himself a deist, was a forceful critic of religion and advocated for acceptance of all religions as well as separation of church and state. In Japan, Yamagata Bantō (d. 1821) declared that "in this world there are no gods, Buddhas, or ghosts, nor are there strange or miraculous things".

Modern religious skepticism

The term has morphed into one that typically emphasizes scientific and historical methods of evidence. There are some skeptics that question whether religion is a viable topic for criticism given that it doesn't require proof for belief. Others, however, insist it is as much as any other knowledge, especially when it makes claims that contradict those made by science.

There has been much work since the late 20th century by philosophers such as Schellenburg and Moser, and both have written numerous books pertaining to the topic. Much of their work has focused on defining what religion is and specifically what people are skeptical of about it. The work of others have argued for the viability of religious skepticism by appeal to higher-order evidence (evidence about our evidence and our capacities for evaluation), what some call meta-evidence.

There are still echoes of early Greek skepticism in the way some current thinkers question the intellectual viability of belief in the divine. In modern times there is a certain amount of mistrust and lack of acceptance of religious skeptics, particularly towards those that are also atheists. This is coupled with concerns many skeptics have about the government in countries, such as the US, where separation of church and state are central tenets.

Telegrapher's equations

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https://en.wikipedia.org/wiki/Telegrapher%27s_equations

The telegrapher's equations (or just telegraph equations) are a set of two coupled, linear equations that predict the voltage and current distributions on a linear electrical transmission line. The equations are important because they allow transmission lines to be analyzed using circuit theory. The equations and their solutions are applicable from 0 Hz (i.e. direct current) to frequencies at which the transmission line structure can support higher order non-TEM modes. The equations can be expressed in both the time domain and the frequency domain. In the time domain the independent variables are distance and time. The resulting time domain equations are partial differential equations of both time and distance. In the frequency domain the independent variables are distance ${\displaystyle x}$ and either frequency, ${\displaystyle \omega }$, or complex frequency, ${\displaystyle s}$. The frequency domain variables can be taken as the Laplace transform or Fourier transform of the time domain variables or they can be taken to be phasors. The resulting frequency domain equations are ordinary differential equations of distance. An advantage of the frequency domain approach is that differential operators in the time domain become algebraic operations in frequency domain.

The equations come from Oliver Heaviside who developed the transmission line model starting with an August 1876 paper, On the Extra Current. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can form along the line. Originally developed to describe telegraph wires, the theory can also be applied to radio frequency conductors, audio frequency (such as telephone lines), low frequency (such as power lines), and pulses of direct current.

Distributed components

Schematic representation of the elementary components of a transmission line

The telegrapher's equations, like all other equations describing electrical phenomena, result from Maxwell's equations. In a more practical approach, one assumes that the conductors are composed of 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). In practical conductors, at higher frequencies, ${\displaystyle R}$ increases approximately proportional to the square root of frequency due to the skin effect.
• The distributed inductance ${\displaystyle L}$ (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).
• The capacitance ${\displaystyle C}$ between the two conductors is represented by a shunt capacitor ${\displaystyle C}$ (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 (siemens per unit length). This resistor in the model has a resistance of ${\displaystyle R_{\text{shunt}}=\frac{1}{G}\mathrm{o}\mathrm{h}\mathrm{m}\mathrm{s}}$. ${\displaystyle G}$ accounts for both bulk conductivity of the dielectric and dielectric loss. If the dielectric is an ideal vacuum, then ${\displaystyle G\equiv 0}$.

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

Role of different components

Schematic showing a wave flowing rightward down a lossless transmission line. Black dots represent electrons, and the arrows show the electric field.

The role of the different components can be visualized based on the animation at right.

Inductance L

The inductance couples current to energy stored in the magnetic field. It makes it look like the current has inertia – i.e. with a large inductance, it is difficult to increase or decrease the current flow at any given point. Large inductance L makes the wave move more slowly, just as waves travel more slowly down a heavy rope than a light string. Large inductance also increases the line's surge impedance (more voltage needed to push the same AC current through the line).

Capacitance C

The capacitance couples voltage to the energy stored in the electric field. It controls how much the bunched-up electrons within each conductor repel, attract, or divert the electrons in the other conductor. By deflecting some of these bunched up electrons, the speed of the wave and its strength (voltage) are both reduced. With a larger capacitance, C, there is less repulsion, because the other line (which always has the opposite charge) partly cancels out these repulsive forces within each conductor. Larger capacitance equals weaker restoring forces, making the wave move slightly slower, and also gives the transmission line a lower surge impedance (less voltage needed to push the same AC current through the line).

Resistance R

Resistance corresponds to resistance interior to the two lines, combined. That resistance R couples current to ohmic losses that drop a little of the voltage along the line as heat deposited into the conductor, leaving the current unchanged. Generally, the line resistance is very low, compared to inductive reactance ωL at radio frequencies, and for simplicity is treated as if it were zero, with any voltage dissipation or wire heating accounted for as corrections to the "lossless line" calculation, or just ignored.

Conductance G

Conductance between the lines represents how well current can "leak" from one line to the other. Conductance couples voltage to dielectric loss deposited as heat into whatever serves as insulation between the two conductors. G reduces propagating current by shunting it between the conductors. Generally, wire insulation (including air) is quite good, and the conductance is almost nothing compared to the capacitive susceptance ωC, and for simplicity is treated as if it were zero.

All four parameters L, C, R, and G depend on the material used to build the cable or feedline. All four change with frequency: R, and G tend to increase for higher frequencies, and L and C tend to drop as the frequency goes up. The figure at right shows a lossless transmission line, where both R and G are zero, which is the simplest and by far most common form of the telegrapher's equations used, but slightly unrealistic (especially regarding R).

Values of primary parameters for telephone cable

Representative parameter data for 24-gauge telephone polyethylene insulated cable (PIC) at 70 °F (294 K)

Frequency	R		L		G		C
Hz	Ω⁄km	Ω⁄1000 ft	μH⁄km	μH⁄1000 ft	μS⁄km	μS⁄1000 ft	nF⁄km	nF⁄1000 ft
1 Hz	172.24	52.50	612.9	186.8	0.000	0.000	51.57	15.72
1 kHz	172.28	52.51	612.5	186.7	0.072	0.022	51.57	15.72
10 kHz	172.70	52.64	609.9	185.9	0.531	0.162	51.57	15.72
100 kHz	191.63	58.41	580.7	177.0	3.327	1.197	51.57	15.72
1 MHz	463.59	141.30	506.2	154.3	29.111	8.873	51.57	15.72
2 MHz	643.14	196.03	486.2	148.2	53.205	16.217	51.57	15.72
5 MHz	999.41	304.62	467.5	142.5	118.074	35.989	51.57	15.72

This data is from Reeve (1995). The variation of ${\displaystyle R}$ and ${\displaystyle L}$ is mainly due to skin effect and proximity effect. The constancy of the capacitance is a consequence of intentional design.

The variation of G can be inferred from Terman: "The power factor ... tends to be independent of frequency, since the fraction of energy lost during each cycle ... is substantially independent of the number of cycles per second over wide frequency ranges." A function of the form $${\displaystyle G(f)=G_1\cdot {\left(\frac{f}{f_1}\right)}^{g_{\mathrm{e}}}}$$ with ${\displaystyle g_{\mathrm{e}}}$ close to 1.0 would fit Terman's statement. Chen gives an equation of similar form. Whereas G(·) is conductivity as a function of frequency, ${\displaystyle G_1,f_1}$, and ${\displaystyle g_e}$ are all real constants.

Usually the resistive losses grow proportionately to ${\displaystyle f^{1/2}}$ and dielectric losses grow proportionately to ${\displaystyle f^{g_{\mathrm{e}}}}$ with ${\displaystyle g_{\mathrm{e}}\approx 1}$ so at a high enough frequency, dielectric losses will exceed resistive losses. In practice, before that point is reached, a transmission line with a better dielectric is used. In long distance rigid coaxial cable, to get very low dielectric losses, the solid dielectric may be replaced by air with plastic spacers at intervals to keep the center conductor on axis.

The equations

Time domain

The telegrapher's equations in the time domain are: $${\displaystyle \begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }x}V(x,t)& =-L\frac{\mathrm{\partial }}{\mathrm{\partial }t}I(x,t)-RI(x,t)\\ \frac{\mathrm{\partial }}{\mathrm{\partial }x}I(x,t)& =-C\frac{\mathrm{\partial }}{\mathrm{\partial }t}V(x,t)-GV(x,t)\end{array}}$$

They can be combined to get two partial differential equations, each with only one dependent variable, either ${\displaystyle V}$ or ${\displaystyle I}$: $${\displaystyle \begin{array}{rl}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}V(x,t)-LC\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}V(x,t)& =\left(RC+GL\right)\frac{\mathrm{\partial }}{\mathrm{\partial }t}V(x,t)+GRV(x,t)\\ \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}I(x,t)-LC\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}I(x,t)& =\left(RC+GL\right)\frac{\mathrm{\partial }}{\mathrm{\partial }t}I(x,t)+GRI(x,t)\end{array}}$$

Except for the dependent variable (${\displaystyle V}$ or ${\displaystyle I}$) the formulas are identical.

Frequency domain

The telegrapher's equations in the frequency domain are developed in similar forms in the following references: Kraus, Hayt, Marshall, Sadiku, Harrington, Karakash, Metzger. $${\displaystyle \begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }x}{\mathbf{V}}_{\omega }(x)& =-\left(j\omega L_{\omega }+R_{\omega }\right){\mathbf{I}}_{\omega }(x),\\ \frac{\mathrm{\partial }}{\mathrm{\partial }x}{\mathbf{I}}_{\omega }(x)& =-\left(j\omega C_{\omega }+G_{\omega }\right){\mathbf{V}}_{\omega }(x).\end{array}}$$ The first equation means that ${\displaystyle {\mathbf{V}}_{\omega }(x)}$, the propagating voltage at point ${\displaystyle x}$, is decreased by the voltage loss produced by ${\displaystyle {\mathbf{I}}_{\omega }(x)}$, the current at that point passing through the series impedance ${\displaystyle R+j\omega L}$. The second equation means that ${\displaystyle {\mathbf{I}}_{\omega }(x)}$, the propagating current at point ${\displaystyle x}$, is decreased by the current loss produced by ${\displaystyle {\mathbf{V}}_{\omega }(x)}$, the voltage at that point appearing across the shunt admittance ${\displaystyle G+j\omega C}$.

The subscript ω indicates possible frequency dependence. ${\displaystyle {\mathbf{I}}_{\omega }(x)}$ and ${\displaystyle {\mathbf{V}}_{\omega }(x)}$ are phasors.

These equations may be combined to produce two, single-variable partial differential equations. $${\displaystyle \begin{array}{rl}\frac{d^2}{d{x}^2}{\mathbf{V}}_{\omega }(x)& ={\gamma }^2{\mathbf{V}}_{\omega }(x)\\ \frac{d^2}{d{x}^2}{\mathbf{I}}_{\omega }(x)& ={\gamma }^2{\mathbf{I}}_{\omega }(x)\end{array}}$$ where $\gamma \equiv \alpha +j\beta \equiv \sqrt{\left(R_{\omega }+j\omega L_{\omega }\right)\left(G_{\omega }+j\omega C_{\omega }\right)}$
${\displaystyle \alpha }$ is called the attenuation constant and ${\displaystyle \beta }$ is called the phase constant.

Homogeneous solutions

Each of the preceding partial differential equations have two homogeneous solutions in an infinite transmission line.

For the voltage equation $${\displaystyle \begin{array}{r}{\mathbf{V}}_{\omega ,F}(x)={\mathbf{V}}_{\omega ,F}(a)e^{+\gamma (a-x)};\\ {\mathbf{V}}_{\omega ,R}(x)={\mathbf{V}}_{\omega ,R}(b)e^{-\gamma (b-x)};\end{array}}$$ $${\displaystyle {\mathbf{V}}_{\omega }(x)={\mathbf{V}}_{\omega ,F}(x)+{\mathbf{V}}_{\omega ,R}(x).}$$

For the current equation $${\displaystyle \begin{array}{r}{\mathbf{I}}_{\omega ,F}(x)={\mathbf{I}}_{\omega ,F}(a)e^{+\gamma (a-x)};\\ {\mathbf{I}}_{\omega ,R}(x)={\mathbf{I}}_{\omega ,R}(b)e^{-\gamma (b-x)};\end{array}}$$ $${\displaystyle {\mathbf{I}}_{\omega }(x)={\mathbf{I}}_{\omega ,F}(x)-{\mathbf{I}}_{\omega ,R}(x).}$$

The negative sign in the previous equation indicates that the current in the reverse wave is traveling in the opposite direction.

Note: $${\displaystyle \begin{array}{r}{\mathbf{V}}_{\omega ,F}(x)={\mathbf{Z}}_c{\mathbf{I}}_{\omega ,F}(x),\\ {\mathbf{V}}_{\omega ,R}(x)={\mathbf{Z}}_c{\mathbf{I}}_{\omega ,R}(x),\end{array}}$$ $${\displaystyle {\mathbf{Z}}_c=\sqrt{\frac{R_{\omega }+j\omega L_{\omega }}{G_{\omega }+j\omega C_{\omega }}},}$$ where the following symbol definitions hold:

Symbol definitions

Symbol	Definition
${\displaystyle a}$	point at which the values of the forward waves are known
${\displaystyle b}$	point at which the values of the reverse waves are known
${\displaystyle {\mathbf{V}}_{\omega }(x)}$	value of the total voltage at point x
${\displaystyle {\mathbf{V}}_{\omega ,F}(x)}$	value of the forward voltage wave at point x
${\displaystyle {\mathbf{V}}_{\omega ,R}(x)}$	value of the reverse voltage wave at point x
${\displaystyle {\mathbf{V}}_{\omega ,F}(a)}$	value of the forward voltage wave at point a
${\displaystyle {\mathbf{V}}_{\omega ,R}(b)}$	value of the reverse voltage wave at point b
${\displaystyle {\mathbf{I}}_{\omega }(x)}$	value of the total current at point x
${\displaystyle {\mathbf{I}}_{\omega ,F}(x)}$	value of the forward current wave at point x
${\displaystyle {\mathbf{I}}_{\omega ,R}(x)}$	value of the reverse current wave at point x
${\displaystyle {\mathbf{I}}_{\omega ,F}(a)}$	value of the forward current wave at point a
${\displaystyle {\mathbf{I}}_{\omega ,R}(b)}$	value of the reverse current wave at point b
${\displaystyle {\mathbf{Z}}_c}$	Characteristic impedance

Finite length

Coaxial transmission line with one source and one load

Johnson gives the following solution, $${\displaystyle \begin{array}{rl}\frac{{\mathbf{V}}_{\mathsf{L}}}{{\mathbf{V}}_{\mathsf{S}}}& ={\left[\left(\frac{{\mathbf{H}}^{-1}+\mathbf{H}}{2}\right)\left(1+\frac{{\mathbf{Z}}_{\mathsf{S}}}{{\mathbf{Z}}_{\mathsf{L}}}\right)+\left(\frac{{\mathbf{H}}^{-1}-\mathbf{H}}{2}\right)\left(\frac{{\mathbf{Z}}_{\mathsf{S}}}{{\mathbf{Z}}_{\mathsf{C}}}+\frac{{\mathbf{Z}}_{\mathsf{C}}}{{\mathbf{Z}}_{\mathsf{L}}}\right)\right]}^{-1}\\ & =\frac{{\mathbf{Z}}_{\mathsf{L}}{\mathbf{Z}}_{\mathsf{C}}}{{\mathbf{Z}}_{\mathsf{C}}\left({\mathbf{Z}}_{\mathsf{L}}+{\mathbf{Z}}_{\mathsf{S}}\right)\cosh \left(\mathit{\gamma }x\right)+\left({\mathbf{Z}}_{\mathsf{L}}{\mathbf{Z}}_{\mathsf{S}}+{\mathbf{Z}}_{\mathsf{C}}^2\right)\sinh \left(\mathit{\gamma }x\right)}\end{array}}$$ where ${\displaystyle \mathbf{H}\equiv e^{-\mathit{\gamma }x},}$ and ${\displaystyle x}$ is the length of the transmission line.

In the special case where all the impedances are equal, ${\displaystyle {\mathbf{Z}}_{\mathsf{L}}={\mathbf{Z}}_{\mathsf{S}}={\mathbf{Z}}_{\mathsf{C}},}$ the solution reduces to ${\displaystyle \frac{{\mathbf{V}}_{\mathsf{L}}}{{\mathbf{V}}_{\mathsf{S}}}=\frac{1}{2}{e}^{-\mathit{\gamma }x}}$.

Lossless transmission

When ${\displaystyle \omega L\gg R}$ and ${\displaystyle \omega C\gg G}$, wire resistance and insulation conductance can be neglected, and the transmission line is considered as an ideal lossless structure. In this case, the model depends only on the L and C elements. The telegrapher's equations then describe the relationship between the voltage V and the current I along the transmission line, each of which is a function of position x and time t: $${\displaystyle \begin{array}{rl}V& =V(x,t)\\ I& =I(x,t)\end{array}}$$

The equations for lossless transmission lines

The equations themselves consist of a pair of coupled, first-order, partial differential equations. The first equation shows that the induced voltage is related to the time rate-of-change of the current through the cable inductance, while the second shows, similarly, that the current drawn by the cable capacitance is related to the time rate-of-change of the voltage. $${\displaystyle \frac{\mathrm{\partial }V}{\mathrm{\partial }x}=-L\frac{\mathrm{\partial }I}{\mathrm{\partial }t}}$$ $${\displaystyle \frac{\mathrm{\partial }I}{\mathrm{\partial }x}=-C\frac{\mathrm{\partial }V}{\mathrm{\partial }t}}$$

These equations may be combined to form two exact wave equations, one for voltage ${\displaystyle V}$, the other for current ${\displaystyle I}$: $${\displaystyle \begin{array}{rl}\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{t}^2}-{\tilde{v}}^2\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{x}^2}& =0\\ \frac{{\mathrm{\partial }}^{2}I}{\mathrm{\partial }{t}^2}-{\tilde{v}}^2\frac{{\mathrm{\partial }}^{2}I}{\mathrm{\partial }{x}^2}& =0\end{array}}$$ where $${\displaystyle \tilde{v}\equiv \frac{1}{\sqrt{LC}}}$$ is the propagation speed of waves traveling through the transmission line. For transmission lines made of parallel perfect conductors with vacuum between them, this speed is equal to the speed of light.

Sinusoidal steady-state

In the case of sinusoidal steady-state (i.e., when a pure sinusoidal voltage is applied and transients have ceased), the voltage and current take the form of single-tone sine waves: $${\displaystyle \begin{array}{rl}V(x,t)& ={\mathcal{R}}_{\mathcal{e}}\{V(x)e^{j\omega t}\},\\ I(x,t)& ={\mathcal{R}}_{\mathcal{e}}\{I(x)e^{j\omega t}\},\end{array}}$$ where ${\displaystyle \omega }$ is the angular frequency of the steady-state wave. In this case, the telegrapher's equations reduce to $${\displaystyle \begin{array}{r}\frac{dV}{dx}=-j\omega LI=-L\frac{dI}{dt}\\ \frac{dI}{dx}=-j\omega CV=-C\frac{dV}{dt}\end{array}}$$

Likewise, the wave equations reduce to $${\displaystyle \begin{array}{rl}\frac{d^{2}V}{d{x}^2}+k^{2}V& =0\\ \frac{d^{2}I}{d{x}^2}+k^{2}I& =0\end{array}}$$ where k is the wave number: $${\displaystyle k\equiv \omega \sqrt{LC}=\frac{\omega }{\tilde{v}}.}$$

Each of these two equations is in the form of the one-dimensional Helmholtz equation.

In the lossless case, it is possible to show that $${\displaystyle V(x)=V_1{e}^{-jkx}+V_2{e}^{+jkx}}$$ and $${\displaystyle I(x)=\frac{V_1}{Z_{\mathsf{o}}}{e}^{-jkx}-\frac{V_2}{Z_{\mathsf{o}}}{e}^{+jkx}}$$ where in this special case, ${\displaystyle k}$ is a real quantity that may depend on frequency and ${\displaystyle Z_{\mathsf{o}}}$ is the characteristic impedance of the transmission line, which, for a lossless line is given by $${\displaystyle Z_{\mathsf{o}}=\sqrt{\frac{L}{C}}}$$ and ${\displaystyle V_1}$ and ${\displaystyle V_2}$ are arbitrary constants of integration, which are determined by the two boundary conditions (one for each end of the transmission line).

This impedance does not change along the length of the line since L and C are constant at any point on the line, provided that the cross-sectional geometry of the line remains constant.

The lossless line and distortionless line are discussed in Sadiku (1989) and Marshall (1987).

Loss-free case, general solution

In the loss-free case (${\displaystyle R=G=0}$), the most general solution of the wave equation for the voltage is the sum of a forward traveling wave and a backward traveling wave: $${\displaystyle V(x,t)=f_1(x-\tilde{v}t)+f_2(x+\tilde{v}t)}$$ where

• ${\displaystyle f_1}$ and ${\displaystyle f_2}$ can be any two analytic functions, and
• ${\displaystyle \tilde{v}\equiv \frac{1}{\sqrt{LC}}}$ is the waveform's propagation speed (also known as phase velocity).

Here, ${\displaystyle f_1}$ represents the amplitude profile of a wave traveling from left to right – in a positive ${\displaystyle x}$ direction – whilst ${\displaystyle f_2}$ represents the amplitude profile of a wave traveling from right to left. It can be seen that the instantaneous voltage at any point ${\displaystyle x}$ on the line is the sum of the voltages due to both waves.

Using the current ${\displaystyle I}$ and voltage ${\displaystyle V}$ relations given by the telegrapher's equations, we can write $${\displaystyle I(x,t)=\frac{1}{Z_{\mathsf{o}}}[f_1(x-\tilde{v}t)-f_2(x+\tilde{v}t)].}$$

Lossy transmission line

In the presence of losses the solution of the telegrapher's equation has both damping and dispersion, as visible when compared with the solution of a lossless wave equation.

When the loss elements ${\displaystyle R}$ and ${\displaystyle G}$ are too substantial to ignore, the differential equations describing the elementary segment of line are $${\displaystyle \begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }x}V(x,t)& =-L\frac{\mathrm{\partial }}{\mathrm{\partial }t}I(x,t)-RI(x,t),\\ \frac{\mathrm{\partial }}{\mathrm{\partial }x}I(x,t)& =-C\frac{\mathrm{\partial }}{\mathrm{\partial }t}V(x,t)-GV(x,t).\end{array}}$$

By differentiating both equations with respect to x, and some algebra, we obtain a pair of hyperbolic partial differential equations each involving only one unknown: $${\displaystyle \begin{array}{rl}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}V& =LC\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}V+\left(RC+GL\right)\frac{\mathrm{\partial }}{\mathrm{\partial }t}V+GRV,\\ \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}I& =LC\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}I+\left(RC+GL\right)\frac{\mathrm{\partial }}{\mathrm{\partial }t}I+GRI.\end{array}}$$

These equations resemble the homogeneous wave equation with extra terms in V and I and their first derivatives. These extra terms cause the signal to decay and spread out with time and distance. If the transmission line is only slightly lossy (${\displaystyle R\ll \omega L}$ and ${\displaystyle G\ll \omega C}$), signal strength will decay over distance as ${\displaystyle e^{-\alpha x}}$ where ${\displaystyle \alpha \approx \frac{R}{2Z_0}+\frac{G{Z}_0}{2}}$.

Solutions of the telegrapher's equations as circuit components

Equivalent circuit of an unbalanced transmission line (such as coaxial cable) where: 2/Z_o is the trans-admittance of VCCS (Voltage Controlled Current Source), x is the length of transmission line, Z(s) ≡ Z_o(s) is the characteristic impedance, T(s) is the propagation function, γ(s) is the propagation "constant", s ≡ j ω, and j² ≡ −1.

The solutions of the telegrapher's equations can be inserted directly into a circuit as components. The circuit in the figure implements the solutions of the telegrapher's equations.

The solution of the telegrapher's equations can be expressed as an ABCD two-port network with the following defining equations $${\displaystyle \begin{array}{rl}{V}_1& =V_2\cosh (\gamma x)+I_2{Z}_{\mathsf{o}}\sinh (\gamma x),\\ I_1& =\frac{V_2}{Z_{\mathsf{o}}}\sinh (\gamma x)+I_2\cosh (\gamma x).\end{array}}$$ where $${\displaystyle Z_{\mathsf{o}}\equiv \sqrt{\frac{R_{\omega }+j\omega L_{\omega }}{G_{\omega }+j\omega C_{\omega }}},}$$ and $${\displaystyle \gamma \equiv \sqrt{\left(R_{\omega }+j\omega L_{\omega }\right)\left(G_{\omega }+j\omega C_{\omega }\right)},}$$ just as in the preceding sections. The line parameters R_ω, L_ω, G_ω, and C_ω are subscripted by ω to emphasize that they could be functions of frequency.

The ABCD type two-port gives ${\displaystyle V_1}$ and ${\displaystyle I_1}$ as functions of ${\displaystyle V_2}$ and ${\displaystyle I_2}$. The voltage and current relations are symmetrical: Both of the equations shown above, when solved for ${\displaystyle V_1}$ and ${\displaystyle I_1}$ as functions of ${\displaystyle V_2}$ and ${\displaystyle I_2}$ yield exactly the same relations, merely with subscripts "1" and "2" reversed, and the ${\displaystyle \sinh }$ terms' signs made negative ("1"→"2" direction is reversed "1"←"2", hence the sign change).

Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which is called the shield, sheath, common, earth, or ground. So every two-wire balanced transmission line has two modes which are nominally called the differential mode and common mode. The circuit shown in the bottom diagram only can model the differential mode.

In the top circuit, the voltage doublers, the difference amplifiers, and impedances Z_o(s) account for the interaction of the transmission line with the external circuit. This circuit is a useful equivalent for an unbalanced transmission line like a coaxial cable.

These are not unique: Other equivalent circuits are possible.