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Transmission lines I 1
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E/M fields in transmission line: TEM wave
In a transmission line system that supports TEM wave, we have: x y z r φ In cylindrical system, Maxwell’s equation take the form of: Or:
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From E/M fields to V/I Selected area/path for integration We obtain:
z z+Δz z distance: l radius: a Selected area/path for integration We obtain: Or: distributive resistance distributive inductance
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From E/M fields to V/I Selected area/path for integration We obtain:
z+Δz z z Selected area/path for integration We obtain: Or: distributive conductance ~ 0 distributive capacitance
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Governing equations for transmission line
Or, equivalently:
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General solution Perform Fourier transformation on both sides:
We obtain: likewise: +: natural materials -: meta-materials where: propagation constant
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General solution Use either one of the 1st order equations to find the relation between V and I: For example, take the 1st equation: Perform Fourier transform to obtain: Or: We find: Home work – use the other 1st order equation: characteristic impedance of the transmission line to find the same result.
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General solution – a final form
forward propagation current term backward propagation current term in backward propagation term z is negative to make sure only attenuation exists Time domain expression? Taking inverse Fourier transform!
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Lossless propagation If there is no loss: speed of light
in vacuum material refractive index material factor geometric (size/shape) factor V/I solution in lossless transmission line: no distortion, no loss, ideal transmission
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Low loss propagation ideal (lossless) term – loss term –
provides transmission dispersion term – distorts the waveform loss term – brings in attenuation V/I solution in low loss transmission line: with distortion and loss, non-ideal transmission
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