* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Transmission Line Eq..
Pulse-width modulation wikipedia , lookup
Mains electricity wikipedia , lookup
Stray voltage wikipedia , lookup
Scattering parameters wikipedia , lookup
Chirp spectrum wikipedia , lookup
Resistive opto-isolator wikipedia , lookup
Loading coil wikipedia , lookup
Opto-isolator wikipedia , lookup
Electrical grid wikipedia , lookup
Power engineering wikipedia , lookup
Rectiverter wikipedia , lookup
Overhead power line wikipedia , lookup
Zobel network wikipedia , lookup
Nominal impedance wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Three-phase electric power wikipedia , lookup
Amtrak's 25 Hz traction power system wikipedia , lookup
Telecommunications engineering wikipedia , lookup
Alternating current wikipedia , lookup
Electrical substation wikipedia , lookup
Impedance matching wikipedia , lookup
Electric power transmission wikipedia , lookup
_____ Notes _____ Lossless Transmission Line Signal loss occurs by two basic mechanisms: signal power can be dissipated in a resistor [or conductance] or signal currents may be shunted to an AC ground via a reactance. In transmission line theory, a lossless transmission line does not dissipate power. Signals, will still gradually diminish however, as shunt reactances return the current to the source via the ground path. For the power loss to equal zero, R = G = 0. This condition occurs when the transmission line is very short. An oscilloscope probe is an example of a very short transmission line. The transmission line equation reduces to: 2 v 2v LC x 2 t 2 15 – 6 2 i 2 i 2 LC x t 2 To determine how sinusoidal signals are affected by this type of line, we simply substitute a sinusoidal voltage or current into the above expressions and solve as before, or we could take a much simpler approach. We could start with the solution for the general case: R jLG jC j Let R = G = 0, and simplify: j jL jC LC j 2 Equating the real and imaginary parts: 0 LC 15 – 7 This expression tells us that a signal travelling down a lossless transmission line, experiences a phase shift directly proportional to its frequency. Wireless Communications Systems 15 - 1 Transmission Line Theory _____ Notes _____ PHASE VELOCITY A new parameter, known as phase velocity, can be extracted from these variables: Vp 1 meters/second LC 15 – 8 Phase velocity is the speed at which a fixed point on a wavefront, appears to move. In the case of wire transmission lines, it is also the velocity of propagation., typically: 0.24c Vp 0.9c . The distance between two identical points on a wavefront is its wavelength () and since one cycle is defined as 2 radians: 2 and 2f therefore: Vp f In free space, the phase velocity is 3 x 108 meters/sec, the speed of light. In a cable, the phase velocity is somewhat lower because the signal is carried by electrons. In a waveguide transmission line, the phase velocity exceeds the speed of light. Substituting 13 – 8 into 13 – 5, we obtain: v vo cos t LC x Or 15 – 9 Distortionless Transmission Line A distortionless transmission line is not the same as a lossless one. A lossless line modifies the phase characteristics of the signal but does not consume power. A distortionless line does not distort the signal phase, but does introduce a signal loss. Since common transmission lines are not super conductors, a distortionless line does produce attenuation distortion. Phase distortion does not occur if the phase velocity Vp is constant at all frequencies. By definition, a phase shift of 2 radians occurs over one wavelength . 2 Transmission Line Theory _____ Notes _____ Since: Vp f Then: Vp 2 and f 2 2 2 This tells us that in order for phase velocity Vp to be constant, the phase shift coefficient must vary directly with frequency . Recall: R jLG jC j The problem now is to find . This can be done as follows: R jL G jC jL jC jL jC j LC 1 R G 1 jL jC The 2nd and 3rd roots can be expanded by means of the Binomial Expansion. Recall: 1 x n 1 nx nn 1 2 nn 1n 2 3 x x ... 2! 3! In this instance n = 1/2. Since the contribution of successive terms diminishes rapidly, is expanded to only 3 terms: 2 2 1 R 1 R 1 G 1 G j LC 1 1 2 jL 8 jL 2 jC 8 jC 2 1 G 1 G 1 R 1 RG 1 2 2 jC 8 jC 2 jL 4 LC 1 R G 2 1 R 2 j LC 16 jL jC 8 jL 2 2 2 R G 1 R G 1 16 jL jC 64 jL jC Since j equate the imaginary terms to find . 3 Transmission Line Theory _____ Notes _____ 1 G 2 1 RG 1 R 2 1 R 2 G 2 LC 1 2 L L C 8 64 8 C 4 LC Difference of Squares Very Small 1 R G 2 LC 1 8 L C Note that if R G L C LC then: From this we observe that is directly proportional to . This means that the requirement for distortionless transmission is: RC LG If we equate the real terms, we obtain: RG The Frequency Domain Signal analysis is often performed the frequency domain. This tells us how the transmission line affects the spectral content of the signals they are carrying. To determine this, it is necessary to find the Fourier Transform of the transmission line equation. 2 v v 2 v RGv RC LG LC 2 t t 2 Recall: x and recall the Fourier Transform: Ff t F e jt f t dt To prevent this analysis from ‘blowing up’, we must put a stipulation on the voltage function namely, that it vanishes to zero at an infinite distance down the line. This comprises a basic boundary condition. let 4 v 0 as x Transmission Line Theory _____ Notes _____ This stipulation is in agreement with actual laboratory experiments. It is well known that the signal magnitude diminishes as the path lengthens. Likewise, a time boundary condition, that the signal was zero at some time in the distant past and will be zero at some time in the distant future, must be imposed. let v 0 as t Although engineers have no difficulty imposing these restrictions, mathematical purists, are somewhat offended. For this and other reasons, other less restrictive transforms have been developed. The most notable in this context, is the Laplace transform, which does not have the same boundary conditions. Having made the necessary concessions in order to continue our analysis, we must find the Fourier Transform corresponding to the following terms: v and t Fv , F Let: 2v 2 t F Fv V Then applying the transform on the derivative, we obtain: v t F e jt v dt t This equation can be solved by using integration by parts: u dv uv v du u e j t du je jt dt v and dv v v t let v jt F e jt v | dt v je t Applying the boundary conditions when t goes to infinity makes the 1st term disappear. 5 Transmission Line Theory _____ Notes _____ v F j e jwt v dt t Note that the resulting integral is simply the Fourier Transform. In other words: v j Fv jV t F 2 v 2 2 j t F Similarly: Fv j2 V We can now write the transmission line equation in the frequency domain: 2V 2 2 RGV RC LG j V LC j V x Where: V V Fvt Rearranging the terms, we obtain: 2V RG RC LGj jL jCV x 2 Or Since 2V R jLG jCV x 2 R jLG jC j Then Or 2V 2V x 2 2V 2V 0 x 2 This represents the most general form of the transmission line equation in the frequency domain. This equation must now be solved for V to observe how voltage (or current) varies with distance and frequency. This can be done by assuming a solution of the form: V Ae x Be x forward wave 6 reverse wave Transmission Line Theory _____ Notes _____ These terms represent an exponential decay as the signal travels down the transmission line. If we ignore any reflections, assuming that the cable is infinitely long or properly terminated, this simplifies to: V Vo e x To verify whether this assumption is correct, substitute it into the equation, and see if a contradiction occurs. If there is no contradiction, then our assumption constitutes a valid solution. 2 V e x 2 Vo e x 0 x 2 o Vo e x 2 Vo e x 0 x 2 Vo e x 2 Vo e x 0 00 Thus we validate the assumed solution. This tells us that in the frequency domain, the voltage or current on a transmission line decays exponentially: V Vo e x where R jL G jC j propagation constant = attenuation coefficient = phase coefficient Where: R2 2 L2 G 2 2 C2 1 1 L C tan tan 1 R G 2 R jLG jC cos Im R jL G jC sin Re and: In exponential notation, a sinusoid may be represented by a rotating unity vector, of some frequency: e jt cost j sint Note that the magnitude of this function is 1, but the phase angle is changing as a function of t. 7 Transmission Line Theory _____ Notes _____ Vo e jt If we let: Then t x j V e jt e x e jt e jx e x e attenuation phase vs. x vs. t and x This result is quite interesting because it is the same solution for the transmission line equation in the time domain. The term x e represents an exponential decay. The signal is attenuated as length x increases. The amount of attenuation is defined as: Attenuation in Nepers = lne x Attenuation in dB x = 20loge x 8.68589x This allows us to determine the attenuation at any frequency at any point in a transmission line, if we are given the basic line parameters of R, L, G, & C. jt x The term e represents a rotating unity vector since: e jt x cost x j sint x The phase angle of this vector is x radians. Characteristic Impedance The characteristic impedance of a transmission line is also known as its surge impedance, and should not be confused with its resistance. If a line is infinitely long, electrical signals will still propagate down it, even though the resistance approaches infinity. The characteristic impedance is determined from its AC attributes, not its DC ones. Recall from our earlier analysis: v i Ri L x t and i v Gv C x t Taking the Fourier Transform of these expressions, we obtain: V RI jLI and x 8 I GV jCV x Transmission Line Theory _____ Notes _____ If voltage and current are forcing functions of frequency, they are not functions of distance. i.e. the frequency of a signal remains constant regardless of how long the transmission line is, therefor: V V and x I I x Consequently, we may write the above equations as: V RI jLI R jLI I GV jCV G jCV Taking the ratio of these two attributes in order to obtain impedance: V R jL I I G jC V V 2 R jL I G jC Zo R jL G jC This can be simplified for the low and high frequency case: Z olow freq R G Zo high freq L C Skin Effect Under DC conditions, electrons are uniformly spread throughout the cross-section of the conductor. However, as frequency increases, electrons have a tendency to redistribute themselves, and migrate towards the outer surface. The skin depth is the distance from the surface, where the current density has dropped to 1/e of its surface value. [The conductor is assumed to have a thickness of at least three times the skin depth]. Under such conditions, the solid conductor can be replaced by a hollow one. 9 Transmission Line Theory _____ Notes _____ skin depth = 1 meters f where permeability conductivity Conductivity of copper: copper = 5.81 x 107 mhos per meter. Permeability of copper: copper = 4 x 10-7 henries per meter. -2 10 Skin Depth vs. Frequency -3 Depth in Meters 10 -4 10 Copper Aluminium -5 10 -6 10 3 10 4 10 5 10 6 10 Frequency 7 10 8 9 10 10 RELATIVE PERMEABILITY Material Copper Silver Gold Bismuth Plastics Aluminum Titanium Palladium Nickel Cobalt r o Comments r 0.99999 Diamagnetic 0.99998 “ 0.99996 “ 0.99983 “ ~1.0 “ 1.000021 Paramagnetic 1.00018 “ 1.00082 “ 250 Ferromagnetic 600 “ where : o 4 107 henries/meter For most transmission line dielectrics: o and since 10 o o o then = o r o 377 r Transmission Line Theory _____ Notes _____ RELATIVE DIELECTRIC CONSTANTS r 1.0 1.0006 2.0 2.25 2.5 3.0 5.0 7.5 Material Vacuum Air Teflon Polyethylene Paraffin paper Rubber Mica Glass r o where : o 1 9 10 farads/meter 36 RELATIVE CONDUCTIVITY AND RESISTIVITY Material Aluminum Brass Copper (annealed) Gallium Gold Iron Lead Mercury Nichrome I Nickel Silver Steel Tantalum Tin Titanium Tungsten Zinc r 0.610 0.256 1.0 0.0176 0.7062 0.178 0.0782 0.01789 0.01538 0.198 1.05 0.0189 0.111 0.149 0.0209 0.307 0.294 r 1.64 3.9 1.0 56.8 1.416 5.6 12.78 55.6 65.0 5.05 0.95 52.8 9.0 6.7 47.8 3.25 3.4 rc c 1.7241 108 ohms/meter r c c 5.81 10 7 mhos/meter 1 L ohms A L length R A cross -sectional area 11 Transmission Line Theory _____ Notes _____ resistance per square = Rsq 1 ohms Twin Lead Cable Twin lead cables are used to connect television receivers to set or roof mounted antennas. The typical characteristic impedance of these cables is 300Ω. d D 2 D 1 d for: L 2D ln h/m d 1 .2 1 0 In d u ctan ce 1 .1 1 0 1 10 9 10 8 10 7 10 2D ln d C -6 -6 Twin Lead In d u ctan ce/m v s. D/d -6 -7 -7 -7 3 4 5 6 7 8 9 10 9 10 D/d 3 .5 1 0 -11 Cap acitan ce p er Meter Twin Lead Cap acitan ce/m v s. D/d 3 10 2 .5 1 0 2 10 -11 -11 -11 3 12 4 5 6 D/d 7 8 f/m Transmission Line Theory _____ Notes _____ TWIN LEAD FIELDS Coaxial Cable Coax is used primarily in cable TV applications. The typical characteristic impedance of these cables is 75Ω. D d 2 for: L D ln h/m 2 d C D 1 d 2 f/m D ln d G 2 mhos/m D ln d 13 Transmission Line Theory _____ Notes _____ 5 10 In d u ctan ce p er Meter 4 .5 1 0 4 10 3 .5 1 0 3 10 2 .5 1 0 2 10 -7 -7 Co ax In d u ctan ce/m v s. D/d -7 -7 -7 -7 -7 3 1 .2 1 0 Cap acitan ce p er Meter 1 .1 1 0 1 10 9 10 8 10 7 10 6 10 5 10 4 5 6 D/d 7 8 9 10 8 9 10 -10 -10 Co ax Cap acitan ce/m v s. D/d -10 -11 -11 -11 -11 -11 3 4 5 6 D/d 7 Transient Analysis When a DC source is attached to a transmission line, a voltage pulse travels down the line the far end towards the load. If the voltage surge meets any discontinuity or change in impedance, a portion of the signal will be reflected back to the source. Equilibrium will eventually be achieved as the resulting bouncing signals diminish to zero. Assuming a lossless transmission line, the fraction of the voltage reflected back to its origin, is known as the reflection coefficient, and is given by: o D D D o where D reflection at the discontinuity D impedance at the discontinuity o characteristic impedance of the transmission line The instantaneous voltage on a transmission line can be thought of as being composed of three components: • The initial condition 14 Transmission Line Theory _____ Notes _____ • The incoming signal • The outgoing reflection EXAMPLE In the circuit illustrated below, let: Vs = 10 volts Rs = source resistance of 150 Ω Zo = transmission line characteristic impedance of 50 Ω ZL = load impedance of 25 Ω Vs = source voltage of 10 volts T = propagation time down the transmission line Rs Vs Zo Vin VL RL If the switch is closed at t = 0, the instantaneous voltage Vin is given by: Vin t 0 VS o 50 10 2.5v RS o 150 50 The reflection coefficient at the load is given by: L Z L Zo 25 50 1 Z L Zo 25 50 3 The reflection coefficient at the source is given by: S ZS Zo 150 50 1 ZS Zo 150 50 2 Since it takes T seconds for the input signal to travel down the line, the initial voltage at VL = 0. When t = T, the input signal arrives and a voltage appears at VL. and the total result is: 15 Transmission Line Theory _____ Notes _____ 1 VL V V V 2.5 1.6667v i in in L 0 2.5 t T 3 initial condition incoming signal reflected signal -.83333 Notice that in this example, the reflected signal is negative and subtracts from the incoming signal. Had RL been greater than Zo, the reflected signal would have been positive. The -0.8333 v reflected signal component is sent back to the source where it is reflected again. At t = 2T, the voltage at Vin is given by: 1 VL V Vin V 1.2501v i in S 2.5 .8333 .8333 t 2T 2 initial condition incoming signal reflected signal -.4167 The -0.4167 volt reflection is sent back down to the load: VL t 3T 1.6667 .4167 .4167 1 1.3889v 3 This process continues until equilibrium is reached. Both ends of this lossless loop will converge at a final value of: V final Vin VL VS Time 0 T 2T 3T 4T 5T 6T 7T RL 25 10 1.429v RS RL 150 25 Vin 2.5 2.5 1.25 1.25 1.4583 1.4583 1.4236 1.4236 VL 0 1.6667 1.6667 1.3889 1.3889 1.4352 1.4352 1.4275 The derivation of these numbers is often more readily apparent from a ‘bounce diagram’. This type of sketch shows the size of the signal that is reflected, back and forth between the two ends of the transmission line. 16 Transmission Line Theory _____ Notes _____ Time Vin 2.5 0 VL 0 2.5 1.6666 T -.8333 1.25 2T Reflected Signal -.4166 1.3889 3T .1389 1.4583 4T .0694 1.4352 5T -.0231 1.4236 6T -.0116 1.4274 7T .0039 2.5 2 Vin 1.458 1.4236 1.25 1 T 2T 3T 4T 5T 6T 7T 8T t 2 1.6667 VL 1.4352 1.3889 1.4275 1 0 T 2T 3T 4T 5T 6T 7T 8T t From the forgoing analysis, we can conclude that: Termination Open circuit ZL > Zo ZL = Zo ZL < Zo Short circuit Voltage Reflection Total, in phase, positive Current Reflection Total, out of phase, negative Partial, in phase positive Partial, out of phase, negative None None Partial, out of phase, Partial, in phase positive negative Total, out of phase, Total, in phase, positive negative 17 Transmission Line Theory _____ Notes _____ Reflections are undesirable. They send power back to the source, which may ultimately cause damage, and they can lead to the development of standing waves when the transmission line is continuously excited. Standing Waves If waves are allowed to reflect back and forth on a transmission line, the incident and reflected waves will interact to create a standing wave. This is somewhat similar to the vibration of a stringed instrument. The ratio of maximum to minimum voltage of a standing wave is known as the VSWR† . VSWR Emax I max 1 Emin Imin 1 The VSWR can take on any value between 1 and ∞. If the load is purely resistive, this expression may be simplified to: Zo RL VSWR or RL Z L whichever is larger If VSWR = ∞, total reflection occurs. Ideally for a matched load, VSWR = 1 and there are no reflections. The energy contained in the reflected wave may be dissipated in the cable itself as I2R losses or absorbed by the generator. A VSWR ≠ 1may be acceptable if the transmission line is used as a tuned circuit or a reactive load. Quarter Wavelength Impedance Transformer To terminate a transmission line with a resistive load RL not equal to its characteristic impedance, a /4 section with a characteristic impedance of Z Zo RL can be placed between the two as an impedance transformer. † 18 Voltage Standing Wave Ratio Transmission Line Theory _____ Notes _____ Two reflections occur; one at the input of the matching section, and one at the load, but they are equal and anti-phase and therefore cancel out. The impedance of any point along a transmission line is given by: Z jZo tanh x Z Z o L Z o jZ L tanh x Z jZ o tanx Z Z o L Z o jZ L tanx This expression simplifies to lossless transmission line [R = G = 0]. for the Smith Chart All values on the Smith chart are normalized to the characteristic impedance of the line. Z Z Zo The location of all impedances along the transmission line are described by circles on a Smith chart. 19 Transmission Line Theory _____ Notes _____ Inductance Loci [j] Resistance Loci Capacitance Loci [-j] Smith Chart 20 Transmission Line Theory _____ Notes _____ Assignment Questions Quick Quiz 1. Signals are attenuated [linearly, exponentially] as they propagate down a transmission line. Analytical Questions 5. Show that the attenuation coefficient for a distortionless transmission line is given by: RG 5. Given the following circuit: RS 50 10 volts Zo 91 RL 50 Transmission Line a) Calculate the reflection coefficient b) Sketch the voltage waveform going into the transmission line and at the load for 5 propagation time constants, when the switch is closed 6. A coaxial cable with a .75 mm center conductor diameter, using Teflon as the dielectric (r = 2), has a characteristic impedance of 75 . Stating all assumptions, find: a) The diameter of the outer conductor b) The inductance per meter c) The capacitance per meter d) The phase velocity 7. A 150 MHz, 1 Vrms signal is injected into a transmission line with the following characteristic values: R = .025 /m 21 Transmission Line Theory _____ Notes _____ L = .02 nH/m C = .15 pfd/m G = 1 S/m a) Find the signal amplitude and phase angle after 1000 meters b) Calculate the low and high characteristic impedance c) Find the magnitude and phase angle of the cable impedance at 250 MHz d) Find the magnitude of the impedance half way along the 1 Km cable, if it is terminated in 50 Ω. 8. Explain why a lumped model of a transmission line does not give the same results as a distributed model. 9. What is another name for characteristic impedance? 10. Fill in the reflection polarity in the following table: Voltage Reflection Current Reflection Open Circuit Short Circuit 11. Given the following circuit: RS 50 10 volts Zo 75 RL 100 Transmission Line a) Sketch the approximate waveforms going into the transmission line and at the load for 3 propagation time constants, when the switch is closed. b) Find the final quiescent voltage on the transmission line. c) Calculate the VSWR if the circuit is modified as follows: 22 Transmission Line Theory _____ Notes _____ RS 75 1 Vp-p 100 M Hz Z o 75 RL 100 20 meter Transmission Line 12. A coaxial cable with a 2 mm center conductor, uses Teflon as the dielectric, and has a characteristic impedance of 91 . Stating all assumptions, find: a) the diameter of the outer conductor b) The inductance per meter c) The capacitance per meter Composition Questions 1. What is a lossless transmission line? 2. Sketch and label the field patterns for a twin lead cable. 3. What conditions must be met for distortionless transmission? 23 Transmission Line Theory _____ Notes _____ For Further Research Anderson, Edwin M., Electric Transmission Line Fundamentals, Reston Publishing, Reston, 1985 Chipman, Robert A., Transmission Lines, Kreyszig, Erwin, Advanced Engineering Mathematics Magnusson, Philip C., Transmission Lines and Wave Propagation, Allyn and Bacon, Boston, 1970 Poularikas & Seely, Signals & Systems Reed & Reed, Mathematical Methods in Electrical Engineering Sinnema, William, Electronic Transmission Technology, Prentice-Hall, Englewood Cliffs, 1979 Reference Data for Radio Engineers, IT&T 24