1. Transmission lines I 1

2. 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:

3. 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

4. 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

5. Governing equations for transmission line
Or, equivalently:

6. General solution Perform Fourier transformation on both sides:
We obtain:
likewise:
+: natural materials
-: meta-materials
where:
propagation constant

7. 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.

8. 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!

9. 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

10. 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