introduction to transmission lines...part ii dr. farid farahmand fall 2012 . transmission line model...
TRANSCRIPT
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INTRODUCTION TO TRANSMISSION LINES PART II DR. FARID FARAHMAND FALL 2012
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Transmission Line Model
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Perfect Conductor and Perfect Dielectric (notes)
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Simulation Example
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Transmission Line Model
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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
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Transmission-Line Equations
ac signals: use phasors
Transmission Line Equation
in Phasor Form
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Derivation of Wave Equations
Combining the two equations leads to:
Second-order differential equation
complex propagation constant
attenuation constant (Neper/m)
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
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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
Note that Zo is NOT V(z)/I(z)
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Solution of Wave Equations (cont.)
Applet for standing wave: http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html
So, V(z) and I(z) have two parts:
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Assume the following waves:
Assume having perfect dielectric insulator and the wire have
perfect conductivity with no loss
Example: Air-Line
Draw the transmission line model and Find C’ and L’; Assume perfect conductor and perfect dielectric materials are used!
)520107002cos(2.0),()520107002cos(10),(
6
6
+−⋅⋅=
+−⋅⋅=
ztzIztzV
π
π
Perfect Conductorà Rs=0àR’ = 0 Perfect Dielec à COND=0 à
G’=0
Note: If atten. Is zero àreal part MUST be zero!
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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
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Lossless Transmission Line
If
Then:
Non-dispersive line: All frequency components have the same speed!
What is Zo?
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The Big Idea…. Impedance is measured at difference points in the circuit!
Zin
ZL
Zo V+o
What is the voltage/current magnitude at different points on the line in the presence of load??
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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!
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Expressing wave in phasor form:
¨ Remember: ¨ If lossless
¤ no attenuation constant
All of these wave representations are along the
Transmission Line
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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
Remember: Everything is with respect
to the load so far!
Vref= Vreflected ; Vinc = Vincident
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Voltage Reflection Coefficient
Normalized load impedance
Pay attention!
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Example
Example
Example
Notes
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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!
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Standing Wave http://www.falstad.com/circuit/e-tlstand.html
Due to standing wave the received wave at the load is now different
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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 (indicating the two waves are
interfering destructively)!
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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!
No reflection, No standing wave
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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!
No reflection, No standing wave
BUT WHEN DO MAX & MIN
Voltages Occur?
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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;
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Example Measuring ZL with a Slotted Line
Slotted Coaxial Line
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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
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Example http://www.bessernet.com/Ereflecto/tutorialFrameset.htm
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Example
Notes
http://www.amanogawa.com/archive/Trans1/Trans1-2.html
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Input Impedance
At input, d = l:
Zd
Wave Impedance
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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
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Standing Wave Properties
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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)}
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Average Power (Phasor Approach)
Fraction of power reflected!
Avg Power: ½ Re{I(z) * V_(z)}
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Summary
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Practice 1- Assume the load is 100 + j50 connected to a 50 ohm line. Find coefficient of reflection (mag, & angle) and SWR. Is it matched well?
2- For a 50 ohm lossless transmission line terminated in a load impedance ZL=100 + j50 ohm, determine the fraction of the average incident power reflected by the load. Also, what is the magnitude of the average reflected power if |Vo|=1?
3- Make sure you understand the slotted line problem.
4- Complete the Simulation Lab answer the following questions:
- Remove the MLOC so the TEE will be open. How does the result change? Take a snapshot. Briefly explain.
- In the original circuit, what happen if we use paper as the dielectric (paper has er of 3.85). Take a snapshot. Briefly explain.
- For the obtained Zo in your Smith Chart calculate the admittance. You must show all your work.
- What exactly is mag(S11)? How is it different from coefficient of reflection? Is the reflection of coefficient measured at the source or load?
- What happens if the impedance of the source (TERM1) is changed to 25 ohm? How does the impedance on the smith chart change?
- How do you calculate the effective length?