Chap2 s11b
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Transmission lines connect generators to loads and include parallel wires, coaxial cable, and optical fiber. They can experience effects like delay, dispersion, and attenuation. Different transmission modes include TEM where electric and magnetic fields are orthogonal to the direction of propagation. The transmission line is modeled as a distributed circuit with inductance and capacitance. This leads to transmission line equations that can be solved as wave equations. The characteristic impedance of the line determines how waves propagate on the line, with standing wave patterns forming for mismatched loads. The input impedance of the line depends on the load impedance and distance along the line.
![Transmission-Line Equations
Kirchhoff Voltage Law:
Vin-Vout – VR – VL =0
Kirchhoff Current Law:
Iin – Iout – Ic – IG =0
Note:
VL=L . di/dt
Ic=C . dv/dt
Remember:
22
||;tan
1||
|)(|)(
]Im[)sin(
]Re[)cos(
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BAC
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B
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e
ezEzE
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j
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z
+==→+=
=
=
=
=
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θ
θ
θ
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![Basic Rules
¨ Given ZL find the coefficient of reflection (COR)
¤ Find ZL on the chart (Pt. P) [1] – Normalized Load
¤ Extend it and find the angle of COR [3]
¤ Use ruler to measure find OP/OR ; OR is simply unity circle - This will be the magnitude of COR
¨ Find dmin and dmax
¤ From the extended OP to
¨ Find VSWR (or S)
¤ Draw a circle with radius of ZL (OP)
¤ Find Pmin and Pmax=S along the circle (where |Vmin| and |Vmax| are)
¨ Input impedance Zd=Zin
¤ Find S on the chart (OP)
¤ Extend ZL all the way to hit a point on the outer circle
¤ Then move away in the direction of WL TOWARD GENERATOR by d=xλ
¤ Draw a line toward the center of the circle
n The intersection of the S circle and this line will be the input load (Zin)
Notes
ZL/Zo
COR
dmin/dmax
SWR
zin & Zin
yin & Yin](https://image.slidesharecdn.com/chap2s11b-160430073703/85/Chap2-s11b-56-320.jpg)
Chap2 s11b
- 2. Transmission Lines A transmission line connects a generator to a load Transmission lines include: • Two parallel wires • Coaxial cable • Microstrip line • Optical fiber • Waveguide • etc.
- 3. Transmission Line Effects Delayed by l/c At t = 0, and for f = 1 kHz , if: (1) l = 5 cm: (2) But if l = 20 km:
- 5. Types of Transmission Modes TEM (Transverse Electromagnetic): Electric and magnetic fields are orthogonal to one another, and both are orthogonal to direction of propagation
- 6. Example of TEM Mode Electric Field E is radial Magnetic Field H is azimuthal Propagation is into the page
- 11. Transmission-Line Equations Kirchhoff Voltage Law: Vin-Vout – VR – VL =0 Kirchhoff Current Law: Iin – Iout – Ic – IG =0 Note: VL=L . di/dt Ic=C . dv/dt Remember: 22 ||;tan 1|| |)(|)( ]Im[)sin( ]Re[)cos( )sin()cos( BAC A B jBAC e ezEzE AeA AeA AjAAe j j j j j z +==→+= = = = = += θ θ θ θθ θ θ θ θ θ
- 12. Transmission-Line Equations ac signals: use phasors Transmission Line Equation in Phasor Form
- 13. Derivation of Wave Equations Combining the two equations leads to: Second-order differential equation complex propagation constant attenuation constant Phase constant Transmission Line Equation First Order Coupled Equations! WE WANT UNCOUPLED FORM! Pay Attention to UNITS! Wave Equations for Transmission Line Impedance and Shunt Admittance of the line
- 14. Solution of Wave Equations (cont.) Proposed form of solution: Using: It follows that: Characteristic Impedance of the Line (ohm) So What does V+ and V- Represent? Pay att. To Direction Make sure you know how we got this!
- 15. Solution of Wave Equations (cont.) In general (each component has Magnitude and Phase): wave along +z because coefficients of t and z have opposite signs wave along –z because coefficients of t and z have the same sign So, V(z) and I(z) have two parts: But what are Vo+ and Vo- ? ß We are interested in Sinusoidal Steady-state Condition Refer to Notes
- 16. Solution of Wave Equations (cont.) Applet for standing wave: http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html
- 17. Example ¨ Verify the solution to the wave equation for voltage in phasor form: Note:
- 18. Assume the following waves: Assume having perfect dielectric insolator and the wire have perfect conductivity with no loss Example 2-1: Air Line Draw the transmission line model and Find C and L )520107002cos(2.0),( )520107002cos(10),( 6 6 +−⋅⋅= +−⋅⋅= ztzI ztzV π π
- 19. Section 2
- 20. Transmission Line Characteristics ¨ Line characterization ¤ Propagation Constant (function of frequency) ¤ Impedance (function of frequency) n Lossy or Losless ¨ If lossless (low ohmic losses) ¤ Very high conductivity for the insulator ¤ Negligible conductivity for the dielectric
- 21. Lossless Transmission Line If Then: Non-dispersive line: All frequency components have the same speed! What is Zo?
- 22. Example ¨ Assume Lossless TL; ¨ Relative permittivity is 4 ¨ C =10 pF/m ¤ Find phase velocity ¤ Find L ¤ Find Zo Notes-1
- 24. The Big Idea…. Zin ZL ZoV+o What is the voltage/current magnitude at different points of the line in the presence of load??
- 25. Voltage Reflection Coefficient Consider looking from the Load point of view At the load (z = 0): Reflection coefficient Normalized load impedance The smaller the better!
- 26. Expressing wave in phasor form: ¨ Remember: ¨ If lossless ¤ no attenuation constant All of these wave representations are along the Transmission Line
- 27. Special Line Conditions (Lossless) ¨ Matching line ¤ ZL=Zo àΓ=0; Vref=0 ¨ Open Circuit ¤ ZL=INF àΓ=1; Vref=Vinc ¨ Short Circuit ¤ ZL=0 àΓ=-1; Vref=-Vinc Notes Remember: Everything is with respect to the load so far!
- 28. Voltage Reflection Coefficient Normalized load impedance Pay attention!
- 30. Standing Waves Finding Voltage Magnitude Note: When there is no REFLECTION Coef. Of Ref. = 0 à No standing wave! Remember: Standing wave is created due to interference between the traveling waves (incident & reflected) When lossless! We are interested to know what happens to the magnitude of the |V| as such interference is created!
- 31. Standing Wave http://www.falstad.com/circuit/e-tlstand.html Due to standing wave the received wave at the load is now different
- 32. Standing Waves Finding Voltage Magnitude voltage magnitude Conjugate! is the magnitude at the load? What Z=-d This is standing wave! Each position has a different value! voltage magnitude due to interference
- 33. Standing Waves Finding Voltage Magnitude voltage magnitude at z= -d current magnitude at the source Let s see how the magnitude looks like at different z values! Remember max current occurs where minimum voltage occurs!
- 34. Standing Wave Patterns for 3 Types of Loads (Matched, Open, Short) ¨ Matching line ¤ ZL=Zo àΓ=0; Vref=0 ¨ Short Circuit ¤ ZL=0 àΓ=-1; Vref=-Vinc (angle –/+π) ¨ Open Circuit ¤ ZL=INF àΓ=1; Vref=Vinc (angle is 0) Remember max current occurs where minimum voltage occurs! Notes No reflection, No standing wave
- 35. Standing Wave Patterns for 3 Types of Loads (Matched, Open, Short) ¨ Matching line ¤ ZL=Zo àΓ=0; Vref=0 ¨ Short Circuit ¤ ZL=0 àΓ=-1; Vref=-Vinc (angle –/+π) ¨ Open Circuit ¤ ZL=INF àΓ=1; Vref=Vinc (angle is 0) Remember max current occurs where minimum voltage occurs! Notes No reflection, No standing wave BUT WHEN DO MAX & MIN Voltages Occur?
- 36. Standing Wave Pattern ¨ For Voltage: ¤ Max occurs when cos( ) = 1à ¤ In this case n=0,1,2,… ¤ NOTE that the FIRST & SECOND dmax are λ/2 apart ¤ Thus, First MIN happens λ/4 after first dmax ¤ And so on….
- 37. Finding Maxima & Minima Of Voltage Magnitude S = Voltage Standing Wave Ratio (VSWR) For a matched load: S = 1 For a short, open, or purely reactive load: S(open)=S(short) = INF where |Γ|=1;
- 38. What is the Reflection Coefficient (Γd) at any point away from the load? (assume lossless line) At a distance d from the load: Wave impedance
- 40. Example Notes
- 41. Input Impedance At input, d = l: Zd Wave Impedance What is input voltage?
- 42. Short-Circuited Line For the short-circuited line: At its input, the line appears like an inductor or a capacitor depending on the sign of ZL=0
- 43. Input Impedance Special Cases - Lossless What is Zin when matched?
- 44. Short-Circuit/Open-Circuit Method ¨ For a line of known length l, measurements of its input impedance, one when terminated in a short and another when terminated in an open, can be used to find its characteristic impedance Z0 and electrical length
- 46. Example ¨ Check your notes!
- 47. Power Flow ¨ How much power is flowing and reflected? ¤ Instantaneous P(d,t) = v(d,t).i(d,t) n Incident n Reflected ¤ Average power: Pav = Pavi + Pavr n Time-domain Approach n Phasor-domain Approach (z and t independent) n ½ Re{I*(z) . V(z)}
- 49. Average Power (Phasor Approach) Fraction of power reflected! Avg Power: ½ Re{I(z) * V_(z)}
- 50. Example ¨ Assume Zo=50 ohm, ZL=100+i50 ohm; What fraction of power is reflected? 20 percent! This is |Γ|^2 Notes
- 51. The Smith Chart ¨ Developed in 1939 by P. W. Smith as a graphical tool to analyze and design transmission-line circuits ¨ Today, it is used to characterize the performance of microwave circuits
- 52. Complex Plane
- 53. Smith Chart Parametric Equations Equation for a circle For a given Coef. Of Reflection various load combinations can be considered. These combinations can be represented by different circuits! Smith Chart help us see these variations! Parameteric Equation!
- 54. Smith Chart Parametric Equations rL circles xL circles Imag. Part of ZL rL circles are contained inside the unit circle Only parts of the xL circles are contained within the unit circle Each node on the chart will tell us about the load characteristics and coef. of ref. of the line!
- 55. Complete Smith Chart rL Circles Positive xL Circles Negative xL Circles
- 56. Basic Rules ¨ Given ZL find the coefficient of reflection (COR) ¤ Find ZL on the chart (Pt. P) [1] – Normalized Load ¤ Extend it and find the angle of COR [3] ¤ Use ruler to measure find OP/OR ; OR is simply unity circle - This will be the magnitude of COR ¨ Find dmin and dmax ¤ From the extended OP to ¨ Find VSWR (or S) ¤ Draw a circle with radius of ZL (OP) ¤ Find Pmin and Pmax=S along the circle (where |Vmin| and |Vmax| are) ¨ Input impedance Zd=Zin ¤ Find S on the chart (OP) ¤ Extend ZL all the way to hit a point on the outer circle ¤ Then move away in the direction of WL TOWARD GENERATOR by d=xλ ¤ Draw a line toward the center of the circle n The intersection of the S circle and this line will be the input load (Zin) Notes ZL/Zo COR dmin/dmax SWR zin & Zin yin & Yin
- 57. Basic Rules ¨ Input impedance Yd=Yin (admittance) ¤ Once zin (normalized Notes ZL/Zo COR dmin/dmax SWR zin & Zin yin & Yin
- 58. Reflection coefficient at the load Example 1 12 3
- 59. Input Impedance
- 60. Maxima and Minima Where Vmax is Where Vmin is WTG Scale
- 61. Impedance to Admittance Transformation
- 62. (c) (d) (a) (b) The generator is at (0.135+0.3)λ = .435λ –this is pt. D àZin normalized is the intersection of D and S (3.3)λà(0.3)λ
- 63. Example 3 Normalized input admittance yin is 0.25l away from Zin (normalized) à Point E; Yin=yin*Yo=yin/Zo
- 64. Given: S = 3 Z0 = 50 Ω first voltage min @ 5 cm from load Distance between adjacent minima = 20 cm Determine: ZL
- 66. Examples of Matching Networks
- 67. Lumped-Element Matching Choose d and Ys to achieve a match at MM
- 69. Example 4
- 70. Cont.
- 73. Transients Rectangular pulse is equivalent to the sum of two step functions
- 74. Transient Response Initial current and voltage Reflection at the load Second transient Load reflection coefficient Generator reflection coefficient
- 75. T = l/up is the time it takes the wave to travel the full length of the line Voltage Wave
- 77. Bounce Diagrams