Lecture Notes: EEEC6430310 Electromagnetic Fields And Waves - Transmission Line
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This document discusses electromagnetic waves and transmission lines. It begins by introducing electromagnetic waves and how they propagate at the speed of light. Transmission lines guide electromagnetic waves from one place to another. The document then discusses transverse electromagnetic waves on transmission lines and how their characteristic impedance is determined. It provides examples of calculating properties of transmission lines like coaxial cable. Overall, the document provides an introduction to transmission lines and how they propagate electromagnetic waves through circuit analysis and wave equations.
Lecture Notes: EEEC6430310 Electromagnetic Fields And Waves - Transmission Line
- 1. EEEC6430310 ELECTROMAGNETIC FIELDS AND WAVES Transmission Line FACULTY OF ENGINEERING AND COMPUTER TECHNOLOGY BENG (HONS) IN ELECTRICALAND ELECTRONIC ENGINEERING Ravandran Muttiah BEng (Hons) MSc MIET
- 2. Introduction Radio signals propagating (travelling) through space are, like sunlight, periodic electromagnetic waves and propagate at the speed of light, c = 3 × 108 m s . If the period T of the wave is known, then the distance between equivalent points on the wave can be determined by the familiar 𝑑 = 𝑣𝑡. Thus, the length of one cycle of the waves called a wavelength, is determined from, 𝜆 = 𝑣𝑡 or, 𝜆 = c 𝑓 where 𝑓 = 1 𝑇 is the frequency of the periodic wave travelling through space at the velocity of light. Transmission lines and waveguides provide a structure for guiding electromagnetic waves from one place to another. The distinction between “circuit connection wires”, a transmission line, or a waveguide is very broad but relates to the structural dimensions as compared to the wavelength of the signal being propagated. 1
- 3. 2 Transverse Electro-Magnetic (TEM) Waves The electric and magnetic fields sketched in figure 1 are all points perpendicular to each other and the signal is propagating into the space. Seen broadside to the two-wire, lossless transmission line, both the electric and magnetic fields are always into or out of the space that is, transverse to the direction of propagation. The energy in these TEM waves propagates along the transmission line at a velocity, 𝑣p = 𝑐 𝜀r where 𝜀r is the dielectric constant of the transmission line relative to air (i.e. space). In free space, Radio Frequency (RF) signals also propagate as TEM waves and the ratio of electric field strength to magnetic field strength of any single wave is a constant with a value approximately equal to 377 volt/meter per amperes/meter.
- 4. 3 × • Figure 1: Electric (solid) and magnetic (dashed) fields in a twin-lead transmission line.
- 5. 4 This value has units of ohms, is called wave impedance, and is calculated from, 𝑍w = 𝜇o 𝜀o = 377 Ω where the permeability of space is 𝜇o = 4π × 10−7 H m and the permittivity (dielectric constant) is 𝜀o = 8.854 pF m . Similarly, a uniform lossless transmission line has a definite ratio of electric-to-magnetic field strength and also voltage-to-current ratio. This ratio is found from the electric circuit parameters of the transmission line and is called the characteristic, or surge impedance, 𝑍o of the line
- 6. 5 Transients And Reflections Figure 2 illustrates the distributed electrical parameters of a model transmission line. The incremental sections show the series inductance/meter L, resistance/meter R, the shunt capacitance/meter C, and conductance (leakage resistance)/meter G. The middle section shows the leakage components distributed along a balanced two-wire line. The transmission line thus reveals its electrical network characteristic, which may be analysed to yield an expression describing the behavior of voltage and current waves travelling along the line. Such waves are called travelling waves. From figure 2 a voltage wave travelling down the line (from left to right on the x-axis) will experience incremental voltage drops of d𝑣 d𝑥 ∆𝑥 , which by Kirchhoff’s law yields, 𝑅𝑖 ∆𝑥 + 𝐿 d𝑖 d𝑡 ∆𝑥 = − d𝑣 d𝑥 ∆𝑥
- 7. 6 Figure 2: Distributed circuit model of a transmission line. 𝑖 𝑣 𝑖 − ∆𝑖 ∆𝑖 𝑥 d𝑥 𝑣 − ∆𝑣 𝐿∆𝑥 𝑅∆𝑥 𝐶∆𝑥 𝐺∆𝑥
- 8. 7 Table 1: Transmission line (circuit) characteristics. 𝑍o Ω 𝐿 H m 𝐶 F m Twin Lead 120 𝜀r ln 2𝑠 𝑑 𝜇 π ln 2𝑠 𝑑 π𝜀 ln 2𝑠 𝑑 Coaxial 60 𝜀r ln 𝐷 𝑑 𝜇 2π ln 𝐷 𝑑 2π𝜀 ln 𝐷 𝑑 Microstrip 𝑍o = 377ℎ 𝜀r𝑊 1 + 1.74 𝜀r −0.07 𝑊 ℎ −0.836
- 9. 8 Also, the decrease in current due to ac shunting is, 𝐺𝑣 ∆𝑥 + 𝐶 d𝑣 d𝑡 ∆𝑥 = − d𝑖 d𝑥 ∆𝑥 Solving these equations under the generally justified assumption of very low power losses 𝑅 = 𝐺 = 0 yields, d2 𝑣 d𝑥2 = 𝐿𝐶 d2 𝑣 d𝑡2 and, d2 𝑖 d𝑥2 = 𝐿𝐶 d2 𝑖 d𝑡2 These equations describe travelling waves. The solutions of the wave equations above give the voltage across the line and current along the line as function of time and distance traveled.
- 10. 9 As an example, 𝑣 = F 𝐿𝐶𝑥 ± 𝑡 where the + or – gives the direction of travel (+ right, − left). And the velocity at which the voltage wave propagates (travels) is, 𝑣p = 1 𝐿𝐶 It also can be shown that the ratio of voltage to current for a single wave on the lossless transmission line network is given approximately by, 𝑍o = 𝐿 𝐶 and more generally by, 𝑍o = 𝐿 𝐶 = 𝑅 + 𝑠𝐿 𝐺 + 𝑠𝐶 where s is a frequency operator. For the particular case of sinusoidal waves, 𝑠 = j𝜔
- 11. 10 Example 1: A very low loss coaxial transmission line has 30 pF ft of distributed capacitance and 75 nH ft of inductance. Determine the following: (a) The capacitance of a 3 ft length of this line used as an oscilloscope probe. (b) The characteristic impedance, 𝑍o. (c) The velocity of propagation for a voltage and current transient (velocity relative to a TEM wave in free space). (d) The time required for an input transient to reach the oscilloscope. (e) The ratio of shield diameter to center conductor diameter of the coax.
- 12. 11 Solution: (a) The capacitance of a 3 ft length = 30 pF ft × 3 ft = 90 pF. (b) The characteristic impedance, 𝑍o = 𝐿 𝐶 = 75×10−9 30×10−12 = 50 Ω. (c) The velocity of propagation, 𝑣p = 1 𝐿𝐶 = 1 75 × 30 × 10−21 = 666.7 × 106 ft s 1 mi 5280 ft = 189.39 × 10−6 mi ft 𝑣p = 666.7 × 106 ft s × 189.39 × 10−6 mi ft = 126,263 mi s so that, 𝑣p 𝑐 = 126,263 186,000 = 0.679 a little more than two-thirds the speed of light.
- 13. 12 (d) Distance, 𝑑 = 𝑣p𝑡 Time required to reach the oscilloscope, 𝑡 = 𝑑 𝑣p = 3 ft 666.7×106ft s = 4.5 ns. (e) Table 1 gives, 𝑍o = 60 𝜀r ln 𝐷 𝑑 we know that, 𝜀r = 𝑐 𝑣p = 1 0.679 and, 𝑍o = 50 Ω. Therefore, 50 × 1.473 60 = 1.228 = ln 𝐷 𝑑 By the definition of logarithms, 𝐷 𝑑 = e1.228 = 3.41.
- 14. 13 To better understand the circuit behavior of Transmission Line (TL) with propagating waves and reflections from discontinuities, consider the voltage and current surges along an infinite length TL after closing the switch of figure 3. The same result can be achieved for a line of finite length 𝑙 if a load 𝑍L of value equal to 𝑍o is connected to the TL. Part (a) shows that the TL to the right of the separation is infinite in length and has a characteristic impedance of 𝑍o. There are no reflections when the TL is terminated with 𝑍L = 𝑍o (impedance matched). Suppose now that no load is attached to the end of the finite line. In this case the problem will start out the same as before, but when the voltage and current waves reach the open-circuited end, there will be reflection resulting in voltage and current waves travelling to the left toward the generator. The we have to analyse whether or not a reflection will occur at the generator end. If a reflection does occur, we will have to follow it back to the “load” and repeat the process until steady state occurs. We would expect that, for an open-circuited lossless TL, a steady state will be reached with 𝑉 along the line.
- 15. 14 ∞ 𝑍o 𝑍o 𝑍o 𝑉i 𝑍L = 𝑍o 𝐼i Figure 3: Equivalent transmission-line terminations. (a) (b) 𝑙
- 16. 15 To analyse reflections, a reflection coefficient, Γ=ρ∠𝜃 is calculated at each end of the line. The reflection coefficient is analysed as follows: At any point on the line, the resultant voltage and current must be determined by the sum of the incident and reflected waves passing the point. That is, 𝑉 = 𝑉+ + 𝑉− and 𝐼 = 𝐼+ + 𝐼− where 𝑉+ is the incident voltage wave and 𝑉− is the reflected voltage wave. The reflected wave 𝑉− is found from 𝑉− = Γ𝑉+ ; that is, Γ= 𝑉− 𝑉+ where the voltages and consequently Γ are, in general complex quantities (phasors) with magnitude and phase. Thus, at a load the percentage of the voltage arriving at the load that is reflected from the load is called the reflection coefficient at the load.
- 17. 16 Because of current reversal, the voltage reflection coefficient has the opposite sign to that of the current reflection coefficient ΓI. Consequently, ΓI=−Γ The incident current is 𝐼+ = 𝑉+ 𝑍o and at the same point the reflected current is 𝐼− = ΓI𝐼+ = −𝐼+ . Also, 𝑉 = 𝑉+ + 𝑉− = 𝑉+ + Γ𝑉+ 𝐼+ = 𝑉+ 𝑍o and 𝐼− = −𝑉− 𝑍o where the minus sign indicates current direction opposite to 𝐼+ .
- 18. The load impedance is, 𝑍L = 𝑉 𝐼 = 𝑉+ + 𝑉− 𝐼+ + 𝐼− 𝑍L = 𝑉+ + 𝑉− 𝑉+ 𝑍o − 𝑉− 𝑍o Solving this for 𝑉L − 𝑉L + at the load produces ΓL in terms of impedances as, 𝑉− 𝑉+ = ΓL = 𝑍L − 𝑍o 𝑍L + 𝑍o = 𝑍L 𝑍o − 1 𝑍L 𝑍o + 1 or ΓL = 𝑌o − 𝑌L 𝑌o + 𝑌L 17
- 19. 18 𝑍o = 50 Ω 𝑅g = 50 Ω 𝐸 𝑙 𝑉i 𝐼i Figure 4: Open-circuited transmission line.
- 20. 19 Turning now to the open circuited TL of figure 4, we see that 𝑍L = ∞, so that, ΓL = ∞ − 𝑍o ∞ + 𝑍o = 1 and ΓI = −Γ= − 1 When the reflected wave reaches the generator, it is a wave travelling on a 𝑍o = 50 Ω TL that is “terminated” by a “load” 𝑅g = 50 Ω. The reflection coefficient for this matched condition is, ΓL = 50 − 50 50 + 50 = 0 and no reflection occur.
- 21. 20 Sinusoidal Signals As Travelling Waves A sinusoidal signal travelling in the positive 𝑥 direction on a lossless transmission line can be expressed as, 𝑣 = 𝐴 cos 𝜔𝑡 − 𝛽𝑥 where 𝛽 is the transmission-line phase constant (radians/unit length). An observer travelling on a fixed point on wave moving at a constant velocity 𝑣p has a phase velocity found by differentiating 𝜔𝑡 − 𝛽𝑥 = constant. The result is, d𝑥 d𝑡 = 𝑣p = 𝜔 𝛽 𝑣p = 𝜆𝑓 So that, 𝛽 = 2π 𝜆
- 22. 21 cos 𝜔𝑡 − 𝛽𝑥 𝜔𝑡 = π 2 𝜔𝑡 = 0 𝛽𝑥 (radians) 𝑣p Figure 5: Sinusoidal travelling wave travelling in positive 𝑥 direction.
- 23. 22 If a constant loss in amplitude of 𝛼 (nepers/unit length) occurs all along the length of the line, then, 𝑣 (with losses) = 𝐴e−𝛼𝑥 cos 𝜔𝑡 − 𝛽𝑥 is the real part of the exponential expression, 𝑣 = 𝐴e−𝛼𝑥 e−j 𝜔𝑡−𝛽𝑥 = 𝐴e−𝛼𝑥 e−j𝛽𝑥 e−j𝜔𝑡 for which the phasor portion is, 𝑉 = 𝐴e− 𝛼+j𝛽 𝑥 A general solution of the wave equation for a voltage wave travelling in either direction is, 𝑉 = 𝐴1e𝛾𝑥 + 𝐴2e−𝛾𝑥 where 𝛾 is the propagation constant determined from transmission line parameters as 𝛾 = 𝑍𝑌, just as the characteristic impedance is determined from 𝑍o = 𝑍 𝑌 .
- 24. 23 For sinusoidal signals the calculations are made from, 𝑍o = 𝑅 + j𝜔𝐿 𝐺 + j𝜔𝑐 = 𝑅o + j𝑋o and 𝛾 = 𝛼 + j𝛽 = 𝑅 + j𝜔𝐿 𝐺 + j𝜔𝑐 To calculate the total signal attenuation over a distance travelled of 𝑥, solved 𝑉 = 𝐴e−𝛼𝑥 by taking the natural logarithm of both sides to give − 𝛼𝑥 = ln 𝑉 𝐴 nepers. To put the attenuation in decibels, note that 20 log10 e = 8.686, so that the voltage (or current) attenuation for a total distance traveled 𝑥 is, Attenuation (dB) = 8.686α𝑥
- 25. 24 Sinusoidal Signals Reflections And Standing Waves A transmission line with discontinuities and mismatched impedance conditions 𝑍L ≠ 𝑍o will have reflections that produce steady-state standing waves. This is the same wave phenomenon as for vibrating strings and pressure waves in musical instruments. When 𝑍L = 𝑍o , then no reflections occur and so standing waves appear. However larger values of load impedance will produce reflections from the load, causing the voltage measured along the line to vary, as shown figure 6(b) and 6(c); these patterns are called standing waves. In figure 6(c) for no load 𝑍L = ∞ , the voltage measured at the load is exactly twice the voltage along a lossless “flat” or matched line as shown in figure 6(a). This voltage value 𝐴 is also the open-circuited generator voltage. For 𝑍L < 𝑍o, 𝑉𝑙 will be less than 𝐴 2 and figure 6(d) illustrates the lower limit. The voltage at any point 𝑑 from the load is, in general, 𝑉 𝑑 = 𝑣i e𝛾𝑑 + Γe−𝛾𝑑 e𝛾𝑙 + Γe−𝛾𝑙 = 𝑣i 1 + Γe−2𝛾𝑑 1 + Γe−2𝛾𝑙 e𝛾𝑑 e𝛾𝑙
- 26. 25 Figure 6: Voltage measurements on line length 𝑙 for various loads. Standing waves create these voltage (and current) patterns. (a) (b) (d) (c) 𝑉i = 𝐴 2 𝑍L = 𝑍o 𝑍L → ∞ 𝐴 2 Open-circuit Short-circuit 𝑍L = 0 𝑍L = 2𝑍o Purely resistive 0 0 0 0 𝑥 = 0 𝑥 = 𝑙 𝑥 = 0 𝑥 = 𝑙 3𝜆 4 𝜆 4 𝜆 2 𝐴 𝐴 2 𝐴 2 𝜆 2 𝜆 4 𝐴 𝑉 𝑥 𝑉 max 𝑉min 𝐼min 𝐼max 𝑑min 𝐴
- 27. 26 For a lossless line 𝛼 = 0, 𝛾 = j𝛽 , 𝑉 𝑑 = 𝑣𝑖 𝑍Lcos 𝛽𝑑 + j 𝑍o sin𝛽𝑑 𝑍ocos 𝛽𝑙 + j 𝑍L sin𝛽𝑙 where Euler’s equalities are used to equate sinusoids and complex exponentials, Notice the discontinuity in the standing wave pattern of figure 6(c) at the generator output terminals (left side). The impedance seen by the generator is clearly not 𝑍o as it was for the matched load as shown in figure 6(c); it is less than 𝑍o and will be complex. If 𝑙 were equal to 𝜆 2 + 𝜆 4 = 3𝜆 4 , then the generator will be driving an apparent short circuit, as seen by the voltage null as in figure 6(c). The impedance at any point 𝑑 from the load can be calculated from the general expression, 𝑍 𝑑 = 𝑉 𝑑 𝐼 𝑑 = 𝑉+ 𝑑 + 𝑉− 𝑑 𝐼+ 𝑑 + 𝐼− 𝑑 = 𝑍o e𝛾𝑑 + Γe−𝛾𝑑 e𝛾𝑑 + Γe−𝛾𝑑
- 28. 27 For the lossless line 𝛼 = 0, , 𝑍 𝑑 = 𝑍o ej𝛽𝑑 + Γe−j𝛽𝑑 ej𝛽𝑑 − Γe−j𝛽𝑑 = 𝑍o 𝑍Lcos 𝛽𝑑 + j 𝑍osin 𝛽𝑑 𝑍ocos 𝛽𝑑 + j 𝑍Lsin 𝛽𝑑 = 𝑍o 𝑍o 𝑍o + j tan 𝛽𝑑 1 + j 𝑍L 𝑍o tan 𝛽𝑑 where the identity, e±j𝛽𝑑 = cos 𝛽𝑑 ± j sin 𝛽𝑑
- 29. 28 Slotted Line And Voltage Standing Wave Ratio (VSWR) A slotted line is a device with probe extending into a transmission line through a slot. The probe can be slid along the slot, and by means of a connection to a voltmeter, the voltage may be read at any point. Plots such as those illustrated in figure 6 can be produced from slotted-line measurements. This device also provides a very convenient means for computing the magnitude of the reflection coefficient and for determining the load impedance. The ratio of maximum voltage to minimum voltage of the standing wave is defined as the voltage standing-wave ratio, 𝑉𝑆𝑊𝑅 = 𝑉 max 𝑉min 𝑉 max occurs where the incident and reflected phasors are exactly in phase and add to a maximum. 𝑉min occurs where the phasors are exactly out of phase and subtract.
- 30. 29 The ratio is, 𝑉𝑆𝑊𝑅 = 𝑉+ 1 + 𝜌 𝑉+ 1 − 𝜌 𝑉𝑆𝑊𝑅 = 1 + 𝜌 1 − 𝜌 where 𝜌 is the magnitude of the reflection coefficient. Therefore, the magnitude of the reflection coefficient is, 𝜌 = 𝑉𝑆𝑊𝑅 − 1 𝑉𝑆𝑊𝑅 + 1 To determine the load impedance, 𝑍L, the distance, 𝑑min from the load to the first voltage minimum is used as 𝑑. Also, the voltage minimum occurs at an impedance minimum; consequently, the minimum impedance on the line can be calculated from, 𝑍min = 𝑍o 𝑉𝑆𝑊𝑅
- 31. 30 𝑍min can range in value from 𝑍o for a matched load impedance (Γ=0, 𝑉𝑆𝑊𝑅 = 1), to zero for a short or open load (𝜌=1, 𝑉𝑆𝑊𝑅 = ∞). Also the maximum impedance on the line can be determined by, 𝑍max = 𝑍o 𝑉𝑆𝑊𝑅
- 32. 31 Example 2: A lossless 50 Ω transmission line with 𝜀r = 2 is measured with a slotted line at 1 GHz to have 𝑉 max = 10 mV and 𝑉min = 2 mV . The first voltage minimum is 2 cm from the load. Determine: (a) The operating wavelength, 𝜆. (b) The Voltage Standing Wave Ratio (VSWR). (c) The magnitude of the reflection coefficient. (d) The distance in wavelengths between the load and first voltage minimum. (e) The value of minimum impedance. (f) The distance in millimeters between the voltage maximum and minimum.
- 33. 32 Solution: (a) Speed of light, 𝑐 = 3 × 108 ms Dielectric constant, 𝜀r = 2 Frequency, 𝑓 = 1 GHz The operating wavelength, 𝜆 = 𝑐 𝜀r 𝑓 = 3×108 2×109 = 212 mm. (b) VSWR = 𝑉max 𝑉min = 10 mV 2 mV = 5. (c) The magnitude of reflection coefficient, 𝜌 = 𝑉𝑆𝑊𝑅−1 𝑉𝑆𝑊𝑅+1 = 5−1 5+1 = 0.67. (d) The distance in wavelengths between the load and first voltage minimum, 𝑑min = 2 in 25.4 mm in = 50.8 mm. 𝑑min 𝜆 = 50.8 mm 212 mm = 0.24 𝑑min = 0.24𝜆.
- 34. 33 (e) Characteristic impedance, 𝑍o = 50 Ω The value of minimum impedance, 𝑍min = 𝑍o 𝑉𝑆𝑊𝑅 = 50 Ω 5 = 10 Ω. (f) Voltage maxima and minima are separated by a quarter-wavelength, 𝜆 4 . Thus, the distance in millimeters between the voltage maximum and minimum, 𝑑 = 𝜆 4 = 212 mm 4 = 53 mm.
- 35. (1) Paul H. Young, Electronic Communication Techniques, Prentice Hall, 1998. References 34