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Chapter 2: Transmission Lines 

General case:

Telegrapher’s Equations (time-domain representation)

      

      

Telegrapher’s Equations (Phasor domain representation)

 

 

{}  ; {} 

Wave Equations (time-domain representation)

|| ||  

Wave Equations (Phasor domain representation)

 

 

      *+

 

* + 

Solution to the Wave Equation (Phasor domain)

   

 

 

 

 

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Complex Amplitudes

 

 

 

 

The time delay associated with the length of the line l  manifests itself as a constant phase shift φ0 

 

Propagation velocity: velocity of propagation of a travelling wave

 

   

 μ: magnetic permeability  

ε: electric permeability  

σ: electrical conductivity  

 

 

Relationship in TEM lines

TEM lines are characterized by two factors:

Characteristic Impedance

       

Voltage Reflection Coefficient

 

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Lossless Case: 

√    

√    √   

  

   √  √   

√  

  

√ 

√ 

 

√  

Solution to Wave Equations

 

 

 

     

 

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| |  gamma is a complex quantity

Note: ||  

 

(4) Standing wave voltage amplitude –maxima/minima

(5) Standing wave ratio

Load is “matched” 

     

|| |

| 

 

Load is OC

         

 

( )  

( )  

 

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Load is SC

           

(   )  

(   )  

 

 

Load is purely reactive (Rin = 0)

Acts like an inductive source:

    

 

 

 

l = minimum length that would result in an input impedance Zin_sc equivalent to that of an inductor

Acts like a capacitive source

    

 

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l = minimum length that would result in an input impedance Zin_sc equivalent to that of an capacitor 

Solution to wave equation when line is SC (above)

Solution to Wave equations (lossless line)

  ( ) 

( ) 

Standing wave pattern

Total magnitude of V at any point along the transmission line = Incident Wave + Reflected Wave

Maximum value: incident wave and reflected wave are in-phase

Minimum value: incident wave and reflected wave are in phase-opposition

|| ||   ||  

|| || ||  

Note: Repetition Period for individual incident & reflected wave: λd

Repetition Period of standing wave: λ/2 

When no reflected wave is present, there will be no interference and no standing wave

λ/4: pattern of SC & OC circuit of standing wave are apart by this amount of wavelength

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Maximum Point

|| ||   

 

n = 0 or a positive integer

   

 

Minimum Point

|

| || 

  ⁄ ⁄ ⁄ ⁄  

Standing Wave Ratio

Provides measure of the mismatch between the load and the transmission line

|||| | | | | 

Wave Impedance

 

 

Gamma-d: phase shifted voltage reflection coefficient

Z(d) = total voltage (incident + reflected)/ Total current at any point d on the line

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Input Impedance

(located at the source end of the transmission line d = l)

 

|| 

 

 

 

( ) 

 

Difference between wave impedance(V(d)) and input impedance (Vi)?

What exactly is the wave impedance?

Useful Relation (between SC and OC lossless line)

   

   

Lines of length

 

n is an integer

⁄ ⁄  

This gives us   for  

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A half-wavelength line (or any integer multiple of lambda/2) does NOT modify the load

impedance

Quarter Wavelength transformer (   n = 0 or a positive integer)

⁄ ⁄  

  for l =

 

Matched Transmission Line

     

Input Impedance ZIN = Z0 for all locations d on the line

All incident power is delivered to the load, regardless of line length l

Power Flow on a lossless Transmission Line

z = -d

( ) 

( ) 

{}  ||, || - 

|

| , || - 

 

|| , || -

 

||

 

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|| ||

 

||

 

|| |

|

 

Instantaneous power consists of a DC (non-time-varying) term and an AC term that oscillates at an

angular frequency of 2ω 

Time Average Power

∫ ∫ ⁄  

  ||

 

|| || ||

 

 

|

|

|| 

Average reflected power is equal to the average incident power diminished by a multiplicative factor of

|gamma|^2

Pav_i and Pav_r are independent of d.

Time average power carried by the incident & reflected waves do not change as they travel along the

transmission line

Pav is the net average power flowing towards (i.e. absorbed by) the load 

--DONE--

--Next: Smith Chart— 

--Organize the formulas… put them where they belong--

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Telegrapher’s Equations (time-domain representation)

      

      

Telegrapher’s Equations (Phasor representation) 

{}  {} 

   

Wave Equations (time-domain representation)

Wave Equations (Phasor representation)

 

 

      *+

 *+ 

Solution to the Wave Equation (time-domain representation)

Solution to the Wave Equation (Phasor domain)

   

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Lossless case