Technology & Engineering
Transmission Line Equation
The transmission line equation describes the behavior of electrical signals as they travel along a transmission line, taking into account factors such as resistance, inductance, capacitance, and conductance. It is used to analyze and design transmission lines in various electronic systems, such as telecommunications networks and high-speed digital circuits. The equation helps engineers optimize signal integrity and minimize signal distortion.
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11 Key excerpts on "Transmission Line Equation"
- Wenquan Sui(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
The equations that describe the transmission line can be derived from a field equation rigorously under quasi-static assumption or for the field with transverse electromagnetic (TEM) mode, when both electric and magnetic waves are transverse to the direction of propagation. Figure 3-16 shows a typical microstrip line that can be modeled as a transmission line where an electromagnetic field is mostly confined inside the dielectric region between the strip and the ground plane. The electric and magnetic field isopotential lines indicate the nonuniform distribution of the field, but it can be considered a quasi-TEM case when certain approximation is made. As will be seen soon, the solution to the transmission line includes the effects of time delay, or phase shift in frequency domain, and it is influenced by both the impedance and load of the line. The transmission line model can be considered a special device between the field and circuit theory; it could be thought as a bridging component between the distributed and lumped domains. In fact, some numerical techniques use the transmission line theory as the foundation for building an equivalent circuit model to solve field distribution of general distributive systems. On one hand, the transmission line has the nature of a distributed element with time delay and many other effects; in fact, it indeed mathematically represents a group of distributed structures, like parallel-plate transmission, coaxial cable and microstrip line. On the other hand, under quasi-static approximation or TEM-mode propagation, transmission-line behavior can be described by voltage and current, and the transmission line model is included in most of the SPICE-like analog circuit simulator, as will be shown later.
eBook - PDF- Rajeev Bansal(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
The chapter concludes with a brief summary of more advanced transmission-line concepts and gives a brief discussion of current technological developments and future directions. 6.2. BASIC TRANSMISSION-LINE CHARACTERISTICS A transmission line is inherently a distributed system that supports propagating electromagnetic waves for signal transmission. One of the main characteristics of a transmission line is the delayed-time response due to the finite wave velocity. The transmission characteristics of a transmission line can be rigorously determined by solving Maxwell’s equations for the corresponding electromagnetic problem. For an ‘‘ideal’’ transmission line consisting of two parallel perfect conductors embedded in a homogeneous dielectric medium, the fundamental transmission mode is a transverse electromagnetic (TEM) wave, which is similar to a plane electromagnetic wave described in the previous chapter [2]. The electromagnetic field formulation for TEM waves on a transmission line can be converted to corresponding voltage and current circuit quantities by integrating the electric field between the conductors and the magnetic field around a conductor in a given plane transverse to the direction of wave propagation [3,4]. Alternatively, the transmission-line characteristics may be obtained by considering the transmission line directly as a distributed-parameter circuit in an extension of the traditional circuit theory [5]. The distributed circuit parameters, however, need to be determined from electromagnetic field theory. The distributed-circuit approach is followed in this chapter. 6.2.1. Transmission-line Parameters A transmission line may be described in terms of the following distributed-circuit parameters, also called line parameters : the inductance parameter L (in H/m), which represents the series (loop) inductance per unit length of line, and the capacitance parameter C (in F/m), which is the shunt capacitance per unit length between the two conductors.
eBook - PDF- David M. Pozar(Author)
- 2012(Publication Date)
- Wiley(Publisher)
C h a p t e r T w o Transmission Line Theory Transmission line theory bridges the gap between field analysis and basic circuit theory and therefore is of significant importance in the analysis of microwave circuits and devices. As we will see, the phenomenon of wave propagation on transmission lines can be approached from an extension of circuit theory or from a specialization of Maxwell’s equations; we shall present both viewpoints and show how this wave propagation is described by equations very similar to those used in Chapter 1 for plane wave propagation. 2.1 THE LUMPED-ELEMENT CIRCUIT MODEL FOR A TRANSMISSION LINE The key difference between circuit theory and transmission line theory is electrical size. Circuit analysis assumes that the physical dimensions of the network are much smaller than the electrical wavelength, while transmission lines may be a considerable fraction of a wavelength, or many wavelengths, in size. Thus a transmission line is a distributed- parameter network, where voltages and currents can vary in magnitude and phase over its length, while ordinary circuit analysis deals with lumped elements, where voltage and current do not vary appreciably over the physical dimension of the elements. As shown in Figure 2.1a, a transmission line is often schematically represented as a two-wire line since transmission lines (for transverse electromagnetic [TEM] wave propa- gation) always have at least two conductors. The piece of line of infinitesimal length z of Figure 2.1a can be modeled as a lumped-element circuit, as shown in Figure 2.1b, where R, L , G, and C are per-unit-length quantities defined as follows: R = series resistance per unit length, for both conductors, in /m. L = series inductance per unit length, for both conductors, in H/m. G = shunt conductance per unit length, in S/m. C = shunt capacitance per unit length, in F/m. 48
- Tze-Chuen Toh(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
It is then demonstrated that a pair of transmis-sion lines can be approximated by a distributed line model commonly found in the literature. This thus justifies the use of a distributed line model in deriving the Transmission Line Equation. Transmission Line Equations are also known as the telephone equations . The following assumptions are made in the derivation. • The conductors are imperfect; that is, 1 << σ < ∞ , where σ is the con-ductivity of the conductors, and without loss of generality, the con-ductors are assumed to have the same conductivity. • The dielectric medium is imperfect; that is, < σ << 0 1 0 , where σ 0 is the conductivity of the surrounding homogeneous dielectric medium. • The TEM solution holds between the pair of conductors, as < σ << << σ 0 1 0 . • The conductors are arbitrarily long with uniform cross-sections. 103 Transmission Line Theory Figure 4.1 shows a TEM wave (or equivalently, some oscillating voltage source not shown) incident on an infinitely long pair of conductors. The closed path Γ = γ ∪ ∪ ′ γ ∪ ′ C C is oriented as shown in the figure and S is the rectangular surface that spans the oriented loop Γ . That is, Γ (0) = γ (0) and Γ (1) = γ (1). The unit vector n normal to S is directed into the page, consistent with the orientation of Γ (right-hand rule). Assume for simplicity that γ is oriented in the e y direction along the y -axis, C is oriented in the e z direction along the z -axis and n is directed in the + x -axis direction, and suppose that the angular frequency of the incident TEM wave is ω . Suppose also that the conductors are very good conductors (viz., σ >> ωε for the ω in question). Finally, recall that the following approxi-mations are employed: (a) the general solution is a TEM solution and (b) a very small E z -field on the surface of the conductors is assumed in order to overcome the ohmic loss due to finite conductivity of the conductors.
- Franco Di Paolo(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
1.2 “TELEGRAPHIST” AND “TRANSMISSION LINE” EQUATIONS Let us examine Figure 1.2.1. In part (a) of this figure we have indicated a general representation of a transmission line. The two long rectangular bars represent two conductors, one of which is called “hot conductor” (or simply “hot”) and the other “cold conductor” (or simply “cold”). The reader who is familiar with microstrip or stripline* circuits should not confuse the representation in Figure 1.2.1 with two coupled lines.** Similarly, the reader who knows the waveguide mechanics can be dubious about this representation, but we know that modes in waveguides can also be represented with an equivalent transmission line.*** So, Figure 1.2.1 can be used to generically represent any transmission line. Let us define a positive direction “x” and take into consideration an infinitesimal piece ”dx” of this coordinate. Let us consider the t.l. to be lossless, so that the line will only have a series inductance “L” for u.l. and a shunt, or parallel, capacitance for u.l. With these assumptions, a variation “di” in the time “dt” of the series current “i” will produce a voltage drop “dv” given by: (1.2.1) where the minus sign is a consequence of the coordinate system of Figure 1.2.1. This signal also means that a positive variation “di” of current produces a variation “dv” that contrasts such “di.” Similarly, we can note that a variation “dv” in the time “dt” of the parallel voltage “v” will produce a current variation “di” given by: (1.2.2) where the minus sign means that a positive variation “dv” of voltage produces a variation “di”, which is in a direction opposite to the positive one. From the previous two equations we can recognize how “v” and “i” can be set as functions of coordinates and time, and so they can be written more appropriately as: (1.2.3) (1.2.4) These last two equations are called “telegraphist’s equations,” and relate time variation of voltage and current along a t.l.
eBook - PDFTransmission Lines and Lumped Circuits
Fundamentals and Applications
- Giovanni Miano, Antonio Maffucci(Authors)
- 2001(Publication Date)
- Academic Press(Publisher)
In conclusion, the importance of the transmission line models resides in their ability to describe a large variety of guiding structures in the low frequency regime, offering a powerful unified approach to their study. Transmission lines model guiding structures such as cables, wires, power lines, printed circuit board traces, buses for carrying digital data or control signals in modern electronic circuits, VLSI interconnections, coupled microstrips, and microwave circuits. 1.2 Two-Conductor Transmission Line Equations 21 Even if transmission line models describe only approximately the electromagnetic behavior of these systems they are particularly important for the engineering application in view of its inherent simplicity, physical intuition, and scalar description of the problem. 1.2. TWO-CONDUCTOR Transmission Line EquationS Let us consider a transmission line with two conductors, Fig. 1.2. Let the x-axis be oriented along the line axis, with x = 0 corresponding to the left end of the line and x = d corresponding to the right end. Let v- v(x,t) represent the voltage between the two conductors and i = i(x,t) the current flowing through the upper conductor, at the abscissa x (with 0 ~< x < d) and time t. The references for the current and voltage directions are those shown in Fig. 1.2. According to the assumptions, the current distribution along the other wire is equal to - i(x, t). 1.2.1. Ideal Transmission Lines Ideal transmission lines model ideal guiding structures, that is, interconnections without losses, uniform in space and with param- eters independent of frequency. The equations for the voltage and the current distributions are (e.g., Heaviside, 1893; Franceschetti, 1997), 8v 8i 0x = L 8~, (1.5) 8i 8v =C-- (1.6) 8x 8t ' i (x,t) > + v(x,t) i ill II I I ..... I ) x =0 x =d x Figure 1.2. Sketch of a two-conductor transmission line.
eBook - PDF- David M. Pozar(Author)
- 2021(Publication Date)
- Wiley(Publisher)
As shown in Figure 2.1a, a transmission line is often schematically represented as a two-wire line since transmission lines (for transverse electromagnetic [TEM] wave propagation) always have at least two conductors. The piece of line of infinitesimal length Δz of Figure 2.1a can be modeled as a lumped-element circuit, as shown in Figure 2.1b, where R, L, G, and C are per-unit-length quantities defined as follows: R = series resistance per unit length, for both conductors, in Ω/m. L = series inductance per unit length, for both conductors, in H/m. G = shunt conductance per unit length, in S/m. C = shunt capacitance per unit length, in F/m. The series inductance L represents the total self-inductance of the two conductors, and the shunt ca- pacitance C is due to the close proximity of the two conductors. The series resistance R represents the resistance due to the finite conductivity of the individual conductors, and the shunt conductance G is due to dielectric loss in the material between the conductors. R and G, therefore, represent loss. A finite length of transmission line can be viewed as a cascade of sections of the form shown in Figure 2.1b. From the circuit of Figure 2.1b, Kirchhoff’s voltage law can be applied to give v(z, t) − RΔzi(z, t) − LΔz i(z, t) t − v(z + Δz, t) = 0, (2.1a) 47 48 Chapter 2: Transmission Line Theory Δz Δz i (z, t) i (z, t) i (z +Δz, t) z (a) (b) RΔz LΔz GΔz CΔz v(z + Δz, t) v (z, t) + + – – v (z, t) + – FIGURE 2.1 | Voltage and current definitions and equivalent circuit for an incremental length of transmission line. (a) Voltage and current definitions. (b) Lumped-element equivalent circuit. and Kirchhoff’s current law leads to i(z, t) − GΔzv(z + Δz, t) − CΔz v(z + Δz, t) t − i(z + Δz, t) = 0. (2.1b) Dividing (2.1a) and (2.1b) by Δz and taking the limit as Δz → 0 gives the following differential equations: v(z, t) z = −Ri(z, t) − L i(z, t) t , (2.2a) i(z, t) z = −Gv(z, t) − C v(z, t) t .
eBook - PDFElements Of Microwave Networks, Basics Of Microwave Engineering
Basics of Microwave Engineering
- Carmine Vittoria(Author)
- 1998(Publication Date)
- World Scientific(Publisher)
We may re-write eq.(14) as follows V i2 = Ax(jcoL + R)/, (15) where now L = n' and R = coi . Thus, we can represent the transmission line from x, to x 2 as an inductor, L, in series with a resistor, R. However, L is in units of henrys/m and R in ohms/m, see figure 3.4. Fig.3.4 Symbolic representation of medium in terms of R and L between x, and x 2 . Telegraph's Equations 49 From equation (1) we may approximate the voltage at x 2 as (assuming time harmonic variation of voltage). e(x 2 -x l ) sbx eAx jmsAx (16) The reader is reminded that V 2 is measured relative to a ground plane. We may also write equation (16) as follows V= I 2 Ax(jati+axr) (17) Eq.(17) implies that the impedance from a pointy relative to the ground plane is Ax(jB+G) or the admittance, Y, as Y- Ax(jB+G) , where B =wC = we', C = £', and G = coe . Thus, at;c 2 we may represent the transmission line as shown in figure 3.5. The units for C is in farads/m and G in mhos/m. Thus, the transmission line may be described mathematically in terms of an equivalent circuit containing four parameters: R, L, G and C. There are other equivalent circuit representations, but they must contain no more than four circuit parameters. So far, we have obtained an equivalent circuit assuming TEM mode of wave propagation. For TE or TM modes of propagation the equivalent circuit will necessarily be different from the one derived here. Whatever equivalent circuit one hypothesizes it must represent the wave equation derived in eqs. (12) or (13). Otherwise, there is no basis to describe the electrical properties of a transmission line. The parameters R, L, G and C are mathematical parameters. For example, if we were to construct or fabricate a transmission line from.lumped circuit elements, the 50 Elements of Microwave. Networks composite circuit may not even allow propagation! The introduction of the equivalent circuit parameters is a convenient way to allow us to use our intuition developed from circuit analysis.
eBook - PDFAnalog Circuit Techniques
With Digital Interfacing
- T. H. Wilmshurst(Author)
- 2001(Publication Date)
- Butterworth-Heinemann(Publisher)
22 Transmission lines and transformers Summary A transmission line is essentially any pair of conductors conveying a signal from one location to another, with the following as examples. • Twisted-pair telephone line. • Coaxial cable. • Ribbon cable, as frequently used to carry digital signals from one printed circuit board (PCB) to another. • Tracks from one part of a PCB to another, with the ground-plane often the return. • Tracks from one part of an on-chip design to another. For a uniform structure such as coaxial cable the distrib-uted series inductance L and shunt capacitance C combine to form a ‘travelling wave structure’. Here a voltage step applied as in Fig. 22.1 travels down the line at a speed approaching that of light. Apart from the delay T x , there is ideally no change in the wave reaching the load. While such waves may travel in either direction, for the case shown there is only the ‘forward wave’, comprising the voltage and current components V f and I f . The line has a ‘characteristic impedance’ R o such that for the forward wave R o =V f /I f , with R o normally in the approximate range from 30 Ω to 600 Ω . Unless the load impedance R L is equal to R o , there is a reflection when the forward wave reaches the load, giving the ‘reflected wave’ V r with I r , as in Fig. 22.2 for R L = ∞ . Similarly, unless the source impedance R s =R o , there is a further reflection when the reflected wave reaches the source. For a high degree of ‘mismatch’ at each end of the line the resulting sequence of reflections can be as in Fig. 22.3(b). This is a point of particular importance for the digital circuit designer, with the time taken for the re-ceived signal to settle to the intended value a good deal increased from the inevitable T x . The above is the ‘time-domain’ view of the line func-tion. Equally important is the response to a sine wave input, for example in the coupling the output of a radio transmitter to the antenna.
- Robert C. Scully, Mark A. Steffka, Clayton R. Paul(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
The transmission-line equations were derived for a lossless line from the equivalent circuit in Fig. 4.4 and are given in Eq. (4.2). The per-unit-length equivalent circuit for a lossy line is shown in Fig. 4.37 where we have added the per-unit-length resistance r and conductance g. The per-unit-length resistance r represents losses in the line conductors, whereas the per-unit-length conductance g represents losses in the dielectric surrounding the conductors, From this, we can obtain the transmission-line equations in the limit as Δz → 0 in phasor form as d ̂ V (z) dz = −z ̂ I (z) (4.105a) d ̂ I (z) dz = −y ̂ V (z) (4.105b) where the per-unit-length impedance ̂ z and admittance ̂ y are given by ̂ z = r(f ) + j𝜔l (4.106a) ̂ y = g + j𝜔c (4.106b) We have shown the per-unit-length conductor resistance as a function of frequency to emphasize its dependence on skin effect. Equations (4.105) are again a set of coupled differential equations. They may be uncoupled by differentiating one and substituting the other to give d 2 ̂ V (z) dz 2 − ̂ z ̂ y ̂ V (z) = 0 (4.107a) d 2 ̂ I (z) dz 2 − ̂ ŷ z ̂ I (z) = 0 (4.107b) 4.5 SINUSOIDAL EXCITATION OF THE LINE AND THE PHASOR SOLUTION 201 FIGURE 4.37 The per-unit-length equivalent circuit of a two-conductor lossy line: (a) the equivalent circuit for a Δz section; (b) modeling the entire line as a cascade of Δz sections from which the transmission-line equations are derived in the limit as Δz → 0. The general solution of these equations is quite similar in form to that of the lossless case: ̂ V (z) = ̂ V + e −𝛼z e −j𝛽z + ̂ V − e 𝛼z e j𝛽z (4.108a) ̂ I (z) = ̂ V + ̂ Z C e −𝛼z e −j𝛽z − ̂ V − ̂ Z C e 𝛼z e j𝛽z (4.108b) where the characteristic impedance is ̂ Z C = √ ̂ z ̂ y = √ r(f ) + j𝜔l g + j𝜔c (4.109) and the propagation constant is ̂ 𝛾 = √ ̂ z ̂ y = 𝛼 + j𝛽 (4.110) 202 TRANSMISSION LINES AND SIGNAL INTEGRITY The solution process for lossy lines is virtually unchanged from that for lossless lines.
eBook - PDFElectrical Energy Conversion and Transport
An Interactive Computer-Based Approach
- George G. Karady, Keith E. Holbert(Authors)
- 2013(Publication Date)
- Wiley-IEEE Press(Publisher)
: . . = = ⋅ = ⋅ = 2 0 583 3 238 Finally, the capacitance per unit length is: C GMD r nF mi bun 345 0 345 2 18 73 : ln . . = ⋅ ⋅ = ⋅ π ε TRANSMISSIONLINENETWORKS 277 Figure5.58. Equivalentnetworkofa dxlongsectionofalongtransmissionline. , G, 9 G9 9 , 9 5 , 5 [ [ / G[ & G[ 5 G[ 9 6 , 6 G, G9 G[ 6HQGLQJ RU VXSSO\ HQG 5HFHLYLQJ RU ORDG HQG [ Ɛ The corresponding capacitive reactance is: X C k mi C345 345 1 141 6 : . . = - ⋅ = - ⋅ ⋅ ω Ω Notice how the inductive reactance is multiplied by the line length, whereas the capaci- tive reactance must be divided by the line length. 5.8.2. Long Transmission Lines The transmission line resistance, inductance, and capacitance are distributed along the line. A model of the transmission line is created by dividing the line into small, dx long sections. Each section has resistance and reactance connected in series and capacitance connected in parallel. Figure 5.58 shows a dx long section of a long transmission line with distributed parameters. The resistance (R) is measured in ohm per mile, the induc- tance (L) is given as henry per mile, and the capacitance (C) as farad per mile. In the steady-state condition, the impedance of the unit length line section is z RL = R + jωL. The capacitive reactance of the unit length section is z C = 1/(jωC). Referring to Figure 5.58, the voltage loop (Kirchhoff’s voltage law, KVL) equation for the dx length equivalent circuit is: ( ) . V V Iz V + - - = d dx RL 0 (5.107) The rearrangement of this formula yields a first-order differential equation for the voltage: d dx RL V z I = . (5.108) The node point (Kirchhoff’s current law, KCL) equation using the equivalent network is: ( ) , I I V z I + - - = d dx C 0 (5.109) 278 TRANSMISSIONLINESANDCABLES since 1/(jωCdx) = z C /dx. The rearrangement of this expression yields a first-order dif- ferential equation for the current d dx C I V z = .
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