2. transmission linesweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfexample 2-1: air line...
TRANSCRIPT
![Page 1: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/1.jpg)
2. TRANSMISSION LINES
![Page 2: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/2.jpg)
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.
![Page 3: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/3.jpg)
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:
![Page 4: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/4.jpg)
Dispersion and Attenuation
![Page 5: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/5.jpg)
Types of Transmission Modes
TEM (Transverse Electromagnetic): Electric and magnetic fields are orthogonal to one another, and both are orthogonal to direction of propagation
![Page 6: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/6.jpg)
Example of TEM Mode
Electric Field E is radial Magnetic Field H is azimuthal Propagation is into the page
![Page 7: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/7.jpg)
Transmission Line Model
![Page 8: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/8.jpg)
![Page 9: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/9.jpg)
![Page 10: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/10.jpg)
![Page 11: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/11.jpg)
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(
BACABjBAC
eezEzEAeAAeAAjAAe
j
j
j
j
j
z
+==→+=
=
=
=
=
+=
θ
θ
θ
θθ
θ
θ
θ
θ
θ
![Page 12: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/12.jpg)
Transmission-Line Equations
ac signals: use phasors
Transmission Line Equation
in Phasor Form
![Page 13: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/13.jpg)
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
![Page 14: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/14.jpg)
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!
![Page 15: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/15.jpg)
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
![Page 16: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/16.jpg)
Solution of Wave Equations (cont.)
Applet for standing wave: http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html
![Page 17: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/17.jpg)
Example
¨ Verify the solution to the wave equation for voltage in phasor form:
Note:
![Page 18: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/18.jpg)
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
+−⋅⋅=
+−⋅⋅=
ztzIztzV
π
π
![Page 19: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/19.jpg)
Section 2
![Page 20: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/20.jpg)
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
![Page 21: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/21.jpg)
Lossless Transmission Line
If
Then:
Non-dispersive line: All frequency components have the same speed!
What is Zo?
![Page 22: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/22.jpg)
Example
¨ Assume Lossless TL; ¨ Relative permittivity is 4 ¨ C’=10 pF/m
¤ Find phase velocity ¤ Find L’ ¤ Find Zo
Notes-1
![Page 23: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/23.jpg)
![Page 24: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/24.jpg)
The Big Idea….
Zin
ZL
Zo V+o
What is the voltage/current magnitude at different points of the line in the presence of load??
![Page 25: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/25.jpg)
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!
![Page 26: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/26.jpg)
Expressing wave in phasor form:
¨ Remember: ¨ If lossless
¤ no attenuation constant
All of these wave representations are along the
Transmission Line
![Page 27: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/27.jpg)
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!
![Page 28: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/28.jpg)
Voltage Reflection Coefficient
Normalized load impedance
Pay attention!
![Page 29: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/29.jpg)
Example
Example
Example
Notes
![Page 30: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/30.jpg)
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!
![Page 31: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/31.jpg)
Standing Wave http://www.falstad.com/circuit/e-tlstand.html
Due to standing wave the received wave at the load is now different
![Page 32: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/32.jpg)
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
![Page 33: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/33.jpg)
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!
![Page 34: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/34.jpg)
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
![Page 35: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/35.jpg)
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?
![Page 36: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/36.jpg)
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….
![Page 37: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/37.jpg)
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;
![Page 38: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/38.jpg)
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
![Page 39: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/39.jpg)
Example http://www.bessernet.com/Ereflecto/tutorialFrameset.htm
Notes
![Page 40: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/40.jpg)
Example
Notes
![Page 41: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/41.jpg)
Input Impedance
At input, d = l:
Zd
Wave Impedance
What is input voltage?
![Page 42: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/42.jpg)
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
![Page 43: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/43.jpg)
Input Impedance Special Cases - Lossless
What is Zin when matched?
![Page 44: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/44.jpg)
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
![Page 45: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/45.jpg)
![Page 46: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/46.jpg)
Example
¨ Check your notes!
![Page 47: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/47.jpg)
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)}
![Page 48: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/48.jpg)
Instantaneous Power Flow
![Page 49: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/49.jpg)
Average Power (Phasor Approach)
Fraction of power reflected!
Avg Power: ½ Re{I(z) * V_(z)}
![Page 50: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/50.jpg)
Example
¨ Assume Zo=50 ohm, ZL=100+i50 ohm; What fraction of power is reflected?
20 percent! This is |Γ|^2
Notes
![Page 51: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/51.jpg)
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
![Page 52: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/52.jpg)
Complex Plane
![Page 53: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/53.jpg)
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!
![Page 54: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/54.jpg)
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!
![Page 55: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/55.jpg)
Complete Smith Chart
rL Circles
Positive xL Circles
Negative xL Circles
![Page 56: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/56.jpg)
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
![Page 57: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/57.jpg)
Basic Rules
¨ Input impedance Yd=Yin (admittance) ¤ Once zin (normalized
Notes
ZL/Zo COR
dmin/dmax SWR
zin & Zin yin & Yin
![Page 58: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/58.jpg)
Reflection coefficient at the load
Example 1
1 2
3
![Page 59: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/59.jpg)
Input Impedance
![Page 60: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/60.jpg)
Maxima and Minima
Where Vmax is Where Vmin is
WTG Scale
![Page 61: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/61.jpg)
Impedance to Admittance Transformation
![Page 62: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/62.jpg)
(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
![Page 63: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/63.jpg)
Example 3
Normalized input admittance yin is 0.25l away from Zin (normalized) à Point E; Yin=yin*Yo=yin/Zo
![Page 64: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/64.jpg)
Given: S = 3 Z0 = 50 Ω first voltage min @ 5 cm from load Distance between adjacent minima = 20 cm Determine: ZL
![Page 65: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/65.jpg)
Matching Networks
![Page 66: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/66.jpg)
Examples of Matching Networks
![Page 67: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/67.jpg)
Lumped-Element Matching Choose d and Ys to achieve a match at MM’
![Page 68: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/68.jpg)
![Page 69: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/69.jpg)
Example 4
![Page 70: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/70.jpg)
Cont.
![Page 71: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/71.jpg)
Single-Stub Matching
![Page 72: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/72.jpg)
![Page 73: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/73.jpg)
Transients
Rectangular pulse is equivalent to the sum of two step functions
![Page 74: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/74.jpg)
Transient Response
Initial current and voltage
Reflection at the load
Second transient
Load reflection coefficient
Generator reflection coefficient
![Page 75: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/75.jpg)
T = l/up is the time it takes the wave to travel the full length of the line
Voltage Wave
![Page 76: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/76.jpg)
Steady State Response
![Page 77: 2. TRANSMISSION LINESweb.sonoma.edu/.../courses/es430/lectures/chap2_s11b.pdfExample 2-1: Air Line Draw the transmission line model and Find Cʼ and Lʼ ( , ) 0.2cos(2 700 10 20 5)](https://reader034.vdocuments.site/reader034/viewer/2022043023/5f3e9e610b27b517f17b4bd9/html5/thumbnails/77.jpg)
Bounce Diagrams