高功率電子學(xué)

1、TransmissionLines2Cross-sectional view of typical transmission lines (a) coaxial line, (b) two-wire line, (c) planar line, (d) wire above conducting plane, (e) microstrip line.(a) Coaxial line connecting the generator to the load; (b) E and H fields on the coaxial lineTransmission Lines3Electric and

2、 magnetic fields around single-phase transmission lineStray fieldTriplate lineTransmission Lines4Transmission LinesTransmission Line Equations for a Lossless LineThe transmission line consists of two parallel and uniform conuductors, not necessarily identical. Where L and C are the inductance and ca

3、pacitance per unit length of the line, respectively. Transmission Lines5By applying Kirchhoffs voltage law to N - (N + 1) - (N + 1) - N loop, we obtainIf node N is at the position z, node (N +1) is at position z + h, and Definitions of currents and voltages for the lumped-circuit transmission-line m

4、odel.Transmission LinesN(N+1)iNS6Since h is an arbitrary small distance, we can let h approach zeroApplying Kirchhoffs current law to node N we get from whichTransmission Lines7TelegraphersEquationsAll cross-sectional information about the particular line is contained in L and CWave EquationTransmis

5、sion Lines8Waves on the Lossless Transmission LineRoughly speaking, a wave is a disturbance that moves away from its source as time passes. Suppose that the voltage on a transmission line as a function of position z and time t has the formV(z,t) = f(z-Ut) U = constThis is the same function as f(z),

6、but shifted to the right a distance of Ut along the z axis. The displacement increases as time increases. The velocity of motion is U. f(x) has its maximum where x = z Ut = 0, and the position of maximum Zmax at t = to is given by Zmax = Utox = Z-UtAny function of the argument (z-Ut) keeps its shape

7、 and moves as a unit in the +z direction. For example, let f(x) be the triangular function shown in (a). Then at time t=0 f(z-Ut)=f(z) is the function of z shown in (b). At a later time to , f(z-Ut)=f(z-Uto) is the function of z shown in (c). Note that the pulse is moving to the right with velocity

8、U. Transmission Lines9The function V(z,t) = f(z-Ut) describes undistorted propogation in the +z direction and represents a solution of the wave equation for a lossless transmission line:The wave equation is satisfied provided that The leftward-traveling wave v(z,t) = f(z+Ut) is also a solution. Wave

9、 EquationTransmission Lines10The wavelength of the wave is defined as the distance between the maxima at any fixed instant of time. V(z,t) has maxima when its argument (kz-t) is zero,2, 4, etc. At t = 0, there is a maximum at z = 0. The next one occurs when kz = 2 , or z = 2 / k. = 2 / k U = 2f / k

10、= ft = 0t = tov(z,t)=Acos(kz-t) (U= /k)An important special case is that in which the function f is a sinusoid. Fig (a) shows the function v(z,t)=Acos(kz-t) as it appears if photographed with a flash camera at time t=0. In (b) it is seen at the later time toTransmission Lines11The separation of time

11、 and space dependence for sinusoidal (time harmonic) waves is achieved by the use of phasors. Phasors are the complex quantities (in polar form) representing the magnitude and the phase of sinusoidal functions. Phasors are independent of time. Time-harmonic function expressed as a cosine wavePhasor“

12、The real part of”Time FactorFor a wavemoving in the +z direction, The phasor representing this positive going wave isFor a wavemoving to the left,Transmission Lines12A = const at all z since we are dealing with a lossless line. However, the phase does vary with z. For the leftward moving wave, the p

13、hasor would rotate in the counter clockwise direction. Right-ward moving waveTransmission Lines13Characteristic ImpedanceThe positive - going voltage wave:Instantaneous voltageVoltage phasor(A=constant, = constant)The second telegraphers equation in phasor formFor the positive going wave, Characteri

14、stic impedance (independent of position)- real number (50-400)SinceandTransmission Lines14For a negative going wave, Power transmitted by a single waveCharacteristic Impedance continued(average power; the instantaneous power oscillates at twice the fundamental frequency)Transmission Lines15Reflectio

15、n and TransmissionAt z = 0,Assuming that the incident waveis known and solving for , we obtainLoads Reflection CoefficientTransmission Lines16ExampleSuppose ZL = (open circuit). Find the distribution of the voltage on the line if the incident wave is Assume that A is real ( = 0)Reflection and Transm

16、ission continuedThe total voltage on the line is:atThe instantaneous voltage is:Atat all times.The total voltage is the sum of the two waves of equal amplitude moving in opposite directions. The positions of zero total voltage stand still. This phenomenon is referred to as a standing wave.In the cas

17、e of a single traveling wave, there are positionswhere the voltage vanishes, but these positions move at the velocity of the waveTransmission Lines17Reflection and Transmission continuedIf (short circuit),andAgain we have a standing wave but with the nulls at If(resistive),When n=1 (, i.e. the line

18、is terminated in its characteristic impedance),the reflected wave vanishesSuppose that one more transmission line is connected at the load terminals (z=0)Z01Z02Transmission Lines18Reflection and Transmission continuedThe voltage at z=0, if we approach from the left, is . If we approach from the righ

19、t, it is . Thus we can write:Applying Kirchhoffs current law we get Further,The currentsand,can be expressed in term of, and,respectively:Now, assuming that is known, we can findand(Transmission Coefficient)Transmission Lines(Reflection Coefficient)19Standing-Wave Ratio(losseless transmission line)T

20、he total phasor voltage as a function of position on a line connected to a load at z=0 isThe magnitude of the reflected voltage phasor isAt any position, the instantaneous voltage on the line is a sinusoidal function of time, with the amplitude given by the above expression. The amplitude regularly

21、increases and decreases as the cosine function varies. The positions of voltage amplitude maxima and minima are stationary (independent of time). This phenomenon is referred to as a standing wave. The amplitude of voltage as a function of zTransmission Lines20Standing-Wave Ratio continued(losseless

22、transmission line)In the special case of , the reflected wave vanishes and there is only a single traveling wave moving to the right. In this case the voltage amplitude is independent of position (“flat” voltage profile).If there are two (or more) traveling waves on the line, they will interact to p

23、roduce a standing wave.Transmission Lines21Standing-Wave Ratio continued(losseless transmission line)an integerThe standing-wave ratio (SWR) is defined as SWR = 1 whenFor two adjacent maxima at, say, N=1 and N=0 we can writeVoltage maxima and minima repeat every half wavelength.Transmission Lines22T

24、ransmission Line Equations for a Lossy Line(sinusoidal waves)From Kirchhoffs laws in their phasor form, we haveProceeding as before (for a lossless lines), we obtain the phasor form of the telegrapher equations,where L, R, C, and G are, respectively, the series inductance, series resistance, shunt c

25、apacitance, and shunt conductance per unit length. The corresponding (voltage) wave equation isThe two solutions of the wave equation arewhere and are constants describing the waves amplitude and phase and is the propagation constant. +z-zTransmission Lines23Transmission Line Equations for a Lossy L

26、ine continued(sinusoidal waves)The propagation constant of a lossy transmission line is (complex number)Inserting R=0, G=0 (lossless line) we obtainwhere and are real numbers.Causes negative phase shift (phasor rotates clockwise as z increases)Causes attenuation (amplitude es smaller exponentially a

27、s z increases)Transmission LinesFor the positive-going wave24Transmission Line Equations for a Lossy Line continued(sinusoidal waves)A phase shift of equal to corresponds to the wave travel distance z equal to the wavelength : is the phase constant (measured in radians per meter) is the attenuation

28、constant (measured in Nepers per meter) is the attenuation length (amplitude decreases 1/e over z= )The corresponding instantaneous voltage is The position of a maximum is given byAs t increases, the maximum moves to the right with velocity(A is assumed to be real)- Phase Velocity (Up)In general, no

29、nlinear functions of Transmission Lines25DispersionIn general, the phase velocity Up is a function of frequency; that is, a signal containing many frequencies tends to e dispersed (some parts of the signal arrived sooner and others later.)Up is independent of frequency for (1) lossless lines (R=0, G

30、=0) and (2) distortionless lines (R/L=G/C) because for those lines is a linear function of .Up at any frequency is equal to the slope of a line drawn from the origin to the corresponding point on the graph. For = radians/second Up = . In general, Up can be either greater or less then c. Information

31、in a wave travels at a different velocity known as the group velocity is equal to the slope of the tangent to the - curve at the frequency in question (for for this particular system). always remains less then c. Transmission LinesCut-off frequencyExample of dispersiondiagram for an arbitrary system

32、 that ischaracterized by Upc26Non-Sinusoidal Waves(lossless transmission line)Transmission LinesReflection of a rectangular pulse of a short circuit. (a) Shows the incident pulse moving to the right.In (b) it is striking the short-circuit termination, note that the sum of the incident and the reflec

33、ted voltages must always be zero at that position. In (c) the reflected pulse is moving to the left. 27Multiple ReflectionsExample Suppose t2 = (an infinitely long pulse or a step function) and , so that . Find the total voltage on the line after a very long time. The initial (incident) wave moving

34、to the right has amplitudeThe first reflected wave moving to the left has amplitude+z-z+zwhereTransmission LinesV0t1t228Multiple Reflections continuedThe second reflected wave moving to the right has amplitudeThe total voltage at is given by the infinite seriesInserting the values of and we find tha

35、t (simply results from the voltage divider of Rs and RL, as if the line were not there. and so onTransmission Lines29Lattice (bounce) diagramThis is a space/time diagram which is used to keep track of multiple reflections.Transmission LinesVoltage at the receiving endIdeal voltage sourcezz30Points t

36、o RememberIn this chapter we have surveyed several different types of waves on transmission lines. It is important that these different cases not be confused. When approaching a transmission-line problem, the student should begin by asking, “Are the waves in this problem sinusoidal, or rectangular p

37、ulses? Is the line ideal, or does it have losses?” Then the proper approach to the problem can be taken. The ideal lossless line supports waves of any shape (sinusoidal or non-sinusoidal), and transmits them without distortion. The velocity of these waves is . The ratio of the voltage to current is

38、, provided that only one wave is present. Sinusoidal waves are treated using phasor analysis. (A common error is that of attempting to analyze non-sinusoidal waves with phasors. Beware! This makes no sense at all.)When the line contains series resistance and or shunt conductance it is said to be los

39、sy. Lossy lines no longer exhibit undistorted propagation; hence a rectangular pulse launched on such a line will not remain rectangular, instead evolving into irregular, messy shapes. However, sinusoidal waves, because of their unique mathematical properties, do continue to be sinusoidal on lossy l

40、ines. The presence of losses changes the velocity of propagation and causes the wave to be attenuated ( e smaller) as it travels. Transmission Lines31Points to Remember continuedFor lines other than the simple ideal lossless lines, the velocity of propagation usually is a function of frequency. This

41、 velocity, the speed of voltage maxima on the line, is properly called the phase velocity Up. The change of Up with frequency is called dispersion. The velocity with which information travels on the line is not Up, but a different velocity, known as the group velocity . The phase velocity is given b

42、y . However5.Examples of non-sinusoidal waves are short rectangular pulses, and also infinitely long rectangular pulses, which are the same as step functions. Problems involving sudden voltage steps differ from sinusoidal problems, just as in ordinary circuits, problems involving transients differ f

43、rom the sinusoidal steady state. Pulse problems are usually approached by superposition; that is, one tracks the pulses that propagate back and forth, adding up the waves to obtain the total voltage at any place and time. Transmission Lines32Points to Remember continuedAll kinds of waves are reflect

44、ed at discontinuities in the line. If the line continues beyond the discontinuity, a portion of the wave is transmitted as well. The reflected and transmitted waves are described by the reflection coefficient and the transmission coefficient. For sinusoidal waves there is a simple formula giving the reflection coefficient for any load impedance ZL. For non-sinusoidal waves, the same formula can be used, but only if the load impedance is purely resistive. Otherwise the reflected wave has a different shape from the inci