ECE 107: Electromagnetism

Transcription

1 ECE 107: Electromagnetism Set 2: Transmission lines Instructor: Prof. Vitaliy Lomakin Department of Electrical and Computer Engineering University of California, San Diego, CA

2 Outline Transmission Lines for Communications General description Lumped element model Transmission line equations Wave propagation in transmission lines Lossless transmission lines Reflection from loads and standing waves Input impedance and concepts of matching 2

3 General description (1) Electromagnetic Transmission Lines (TLs) are structures or media that transfer energy/information between two points by means of electromagnetic fields. TL is a two-port network, where each port consists of two terminals 3

4 General description (2) Examples of transmission lines 4

5 General description (3) Why 50 Ohms?

6 General description (4) Close the switch attaching the battery to two wires v(t) ~c Close switch at t = 0 As the switch is closed a voltage (and current) pulse moves down the wire. Time to travel length of wire is l/c. If voltage is not varying at this time scale then the system equilibrates such that the wire is at a constant voltage and no current flowing. However if the voltage varies at a frequency comparable to this time then the system is not in equilibrium. l

7 General description (5) Propagating signal can be thought of in terms of fields E and H or voltage V and current I. V 2 = E d 1,2 l 1 C = Q/V electostatics magnetostatics

8 General description (8) How is a TL different from an electric circuit? Consider the following structure: When ω 0 VBB = VAA, i.e. the voltage does not depend on the length of the connecting wires When ω is not too small, VBB = VAA ( t lc) = V0 cos( ω( t lc)) i.e. the voltage depends on the length of the wires! What is low and high frequency regime? Look at! If low frequency (no delay). If high frequency (large delay). response delay 8

9 General description (7) UCT) ATE Data rate 1 GHz (30 cm) Rise times 100 psec 60 Hz λ = 5000 km λ ~ 1 30 cm

10 General description (9) Types of TLs Transverse electromagnetic (TEM) TLs: Electric AND magnetic field are normal to the propagation direction Higher-order TLs: There are electric or/and magnetic field components in the propagation direction 10

11 General description (10) Example of TEM Mode Electric Field E is radial Magnetic Field H is azimuthal Propagation is into the page

12 Lumped element model (1) Only TEM TLs will be considered in the course A TL will be presented by an equivalent. parallel-wire configuration regardless of the specific shape. This is allowed because the field propagates in TEM TLs independently of the cross-sectional field distribution. The parameters of this configuration will be different for different types of TLs. 12

13 Lumped element model (2) Procedure subdividing the TL into differential sections represent each section by an equivalent circuit all parameters are per unit length! The representation is the same for any TEM TL The parameters are different for different TLs 13

14 and

15 Lumped element model (3) Lumped element parameters for typical TLs Depend on the cross-sectional geometry Depend on the filling medium parameters 15

16 Transmission line equations (1) What is the relation between the voltage and current izt (,) in a TL? First, consider a single cell Apply Kirchhoff s voltage law vzt (,) Divide by z and re-arrange z 0 - diff. equation! Apply Kirchhoff s current law and repeat the procedure 16

17 Transmission line equations (2) Time domain TL equations or telegrapher s equations: Frequency domain TL equations Consider phasors: Substitute above 17

18 Wave propagation in a TL (1) Second order equations for Differentiate the first TL equation and Substitute from the second TL equation Alternatively: Similar steps 18

19 Wave propagation in a TL (2) Wave equations for and Propagation constant: Attenuation constant: Phase constant: The square roots above are chosen so that α and β are positive 19

20 Wave propagation in a TL (3) Solutions of the wave equations General solutions: z e γ z and e γ describe the propagation in +z and z direction + + ( V0, I0 ) and ( V0, I0 ) are amplitudes of the +z and z waves Alternative form for : Characteristic impedance The solution can be written as 20

21 Wave propagation in a TL (4) Waves propagating in the +z and z directions Complex amplitudes: Time domain voltage: attenuation +z propagation (opposite signs in front of t and z) attenuation -z propagation (positive signs in front of t and z) α = 0 γ = α + jβ = jβ lossless TL 21

22 Lossless TLs (1) Description γ = α + jβ u p does not depend on ω lossless TEM TLs are nondispersive! This is important for communications 22

23 Lossless TLs (2) Parameters of certain TEM TLs 23

24 Lossless TLs (3) TL equations for a lossless TL two unknowns V 0 & V 0 are contained, which are the amplitudes of the +z and z traveling waves V 0 & V 0 are not related for an infinite TL V 0 & V 0 are related if a TL is terminated by a load impedance Z L 24

25 Lossless TLs (4) Terminated TL load impedance V & V length l origin is at the load Load :, where are total voltage & current at the location of the load, i.e. at From TL equations: ( Z impedance of the TL) 0 the relation between V + 0 & V 0 :! 25

26 Lossless TLs (5) Voltage reflection coefficient Reflection coefficient is the ratio between For z traveling waves - complex valued quantity even if Z 0 is real. This is because can be complex Particular loads: ZL = Z0 Γ= 0 (matched load) Z = Γ= 1 (open circuit) Z L L = 0 Γ= 1 (short circuit) 26

27 Standing waves (1) Voltage and current in a TL: What are effects of Γ on the distributions of? Consider 27

28 Standing waves (2) Voltage and current magnitudes: note the signs These are standing waves They appear due to interference The pattern is a periodic function The pattern has maxima and minima Note that these expressions are given for phasors! The actual field oscillates in time with amplitude that depend on the spatial coordinate z 28

29 Standing waves (3) Standing wave characteristics: Maxima are obtained when the incident and reflected components are in phase, i.e. add constructively 2βz+ θr = 2πn Minima are obtained when the incident and reflected components are in opposite phase, i.e. add destructively 2 βz+ θ = (2n+ 1) π r Repetition period is λ 2 (not λ!) Maxima of V are minima of I 29

30 Standing waves (4) Voltage maxima: Constructive interference is at 2βz+ θ = 2πn r Maximum is First maximum (for n = 0) (for n = 1) Maxima of are minima of! 30

31 Standing waves (5) Voltage minima: Destructive interference is at 2 βz+ θ = (2n+ 1) π λθr λ(2n + 1) z = + 4π 4 Minimum is First minimum r Min and max distance is λ 4 Minima of are maxima of! 31

32 Standing waves (6) Voltage standing wave ratio: Often used acronym: VSWR Provides the measure of mismatch between the TL and the loading impedance Larger VSWR corresponds to stronger mismatch VSWR is an important parameters that is widely used in the engineering community to characterize TLs, antennas, medium interfaces, etc 32

33 Standing waves (7) Special cases: Matched TL: Short-circuited TL: Z L = 0 Open-circuited TL: Z L = shifted by π 2 33

34 Input impedance (1) Definition: Input impedance is the ratio of the total voltage to the total current at a point z (depends on z!) resistance reactance Input impedance allows replacing a TL by a lumped impedance and helps solve many problems 34

35 Input impedance (2) Solution for a TL excited by a generator: Replace the TL by Z in to find the voltage across Z in looking from the generator Find the voltage across Z in using the TL equations in the TL Note that all unknowns in the TL are known now! 35

36 Input impedance (3) Short circuited lossless TLs: ZL = 0 Γ= 1, S = reactance Z sc is purely imaginary (reactive) sc Zin goes from 0 to, then jumps to, etc 36

37 Input impedance (4) Short circuited lossless TLs (cont d): Positive reactance X > in 0 : X in > 0 - operates as an equivalent inductance Equivalent inductance: jωleq = jz0 tan βz Leq = ( Z0 ω) tan βz The smallest l to lead to L eq is Negative reactance X < 0 : in X < in 0 - operates as an equivalent capacitor Equivalent capacitance: 1 1 = jz0 tan β z Ceq = jωc Z ωtan βz eq 0 The smallest l to lead to C eq is 37

38 Input impedance (5) Open circuited lossless TLs: ZL = Γ= 1, S = reactance Z in is purely imaginary (reactive) Z in goes from to, then jumps to, etc 38

39 Input impedance (6) Applications of SC and OC to measure the characteristic impedance and wavenumber Consider a TL line of length l sc Shorten it from one end and measure Z at the other end in Open it from one end and measure Z at the other end in The characteristic impedance and wavenumber are found from 39

40 Input impedance (7) Matched (lossless) TL ZL = Z0 ZL Z0 Γ= = Z Half-wavelength TLs L + Z 0 0 no reflections are obtained! Z = Z Half-wavelength section of a TLs have no effect on the input impedance, reflection coefficient, and reflected wave in L 40

41 Input impedance (8) Quarter wave (lossless) TL λ nλ l = + βl = π λ λ = π 4 2 (2 )( 4) 2 Quarter wave transformer: Z = in Z Z 2 0 L Zin = Z Z 2 02 L Choose Z = Z Z L Zin = Z01 Γ=0 i.e. the quarter wave transformer can be used to eliminate reflections entirely! 41

42 Input impedance (9) General properties of input impedance Purely reactive (imaginary) impedance 1 ZL Z0 jx L Z 0 jπ j2tan XL Z0 ZL = jx L Γ= = = e Z + Z jx + Z jφ Γ= e Γ = 1 Reactive loads lead to total reflection with phase shift Input impedance depends on frequency E.g. L L 0 Z = jz tan βz = jz tan( ωz u ) in The dependence is periodic For lossless TLs ANY impedance ALWAYS increases with an increase of the frequency! This is called the Foster theorem. The impedance can, however, jump from + to to start 42 increasing again. p

43 Input impedance (10) Summary of standing waves in lossless TLs 43

44 Impedance matching (1) What is impedance matching TL are typically connected to a generator and a load. When ZL = Z0 or YL = Y0, then the TL is matched to the load. For a perfectly matched TL, no signal is reflected. The simplest matching is to take Z L = Z0 but it is often impossible practically. Alternative solution to eliminate the reflections is to use a matching network between the TL and the load. Example: a piece of TL, lumped elements, etc. 44

45 Impedance matching (2) Example: Single stub matching Consider a port at MM Introduce two pieces of TL: One connected to Z L and one shortened connected in parallel (called a stub) Two degrees of freedom are required to match ZL Impedance at MM is obtained as based on the Re{ }&Im{ Z } input impedances of these 1 1 TLs as Zin Y = in = ( Yd + Ys ) L 45

46 Impedance matching (3) Goal: Step 1: Y = Y Re{ Y } = Re{ Y } in 0 in 0 & Im{ Y } = Im{ Y } Match the real part, i.e. in 0 Step 2: Match the imaginary part, i.e. 46

47 Impedance matching (4) General comments Many more options for matching networks exist Lumped element matching is a good option as well In most situations, perfect matching ( Γ=0) occurs only for the matched frequency. When the frequency is modified, the matching is not perfect anymore, i.e. Γ 0. The rate of mismatch depends on the shift of frequency and type of the matching network 47

48 Power flow Instantaneous power flow Incident: Reflected 48

49 Power flow Time average power flow Incident and reflected: Net average power delivered to the load Phasor representation 49