EWE 8 1. Transmission Lines 1.1 Ideal Transmission Line Theory :series resistance per unit length in . :series inductance per unit length in . :shunt conductance per unit length in . :shunt capacitance per unit length in . By Kirchhoff’s voltage law: By Kirchhoff’s current law: As , For time-harmonic( ) circuits
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EWE 8
1. Transmission Lines1.1 Ideal Transmission Line Theory
:series resistance per unit length in .
:series inductance per unit length in .:shunt conductance per unit length in .:shunt capacitance per unit length in .
Special case:1. (short): .2. (open): .3. Half wavelength line:
4. Quarter wavelength line:
Two-transmission Line Junction
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At ,
: transmission coefficient.
Define Insertion loss: Conservation of energy
Incident power:
Reflected power:
Transmitted power:
Voltage Standing Wave Ratio (VSWR)
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Define Standing Wave Ratio
1.2 Coaxial LinesTEM modeLet be the inner radius of the coaxial line and be the outerradius of the coaxial line.Let be the potential function of the TEM mode, then satisfies Laplace’s equation . In polar coordinate
Note this result means maximum power delivered to theload under fixed . In reality, our concern is efficiency or howmuch portion of total power is delivered to the load which is
related to .
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1.6 Calculation of Transmission Line Parameters
Note:
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1.7 Criteria of Ideal Transmission LinesTEM WaveAssume and dependence of the form .Substitute to Maxwell’s equations, we have
where and . These lead to
1. The propagation constant of any TEM wave is the intrinsicpropagation constant of the media.
Also,
2. . The z-directed wave impedance of any TEM wave isthe intrinsic wave impedance of the medium.
Let , then from wave equation we have
.Similarly,
The boundary conditions at perfect conductors are
3. The boundary-value problem for and is the same asthe 2-dimensional electrostatic and magnetostatic
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problem. Thus, static capacitances and inductances canbe used for transmission lines even though the field istime-harmonic.
4. The conductor must be perfect, otherwise will exist.5. Voltage is uniquely defined on the cross-section of the
waveguide.
To sum up, two conditions must be satisfied to support idealtransmission lines:1. Homogeneous, i.e., or are independent of location.
(Why?)2. Two conductors. (Why?)
Multiplying both , we have
.
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Equating both , we have
1.8 Not Ideal Transmission Lines
Introduce mode functions , , mode voltages and mode currents according toTM:
TE:
We can choose forTM:
TE:
Also all modes are normalized according to
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Then, the characteristic impedance is
These is also the wave impedance. Also and willsatisfy transmission-line equations
The power transmitted is
Since
Then for
TE: TM:
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1.9 Composite Right-Left Hand (CRLH) Transmission Lines