1. Transmission Lines Neurons in the hippocampus Carbon nanotube
(Image courtesy Slice of Life Project)
Carbon nanotube

2. When should we worry about transmission lines?
LOAD l
Asin[wt-bx]
Reflection, Distortion, Loss
Phase difference between two ends: bl
Non-negligible if bl > 0.01 x 2p, i.e., l > 0.01l
Lumped Model  Distributed Model if long/high-speed
Optical Signal 1015 Hz, l = 0.3mm
Radio Wave 1 KHz, l = 300km!

3. Lumped model for TL L R G C
COPPER WIRE
COPPER MESH
INSULATION

4. Distributed model for TL
L’Dz
R’Dz
G’Dz
C’Dz
L’Dz
R’Dz
G’Dz
C’Dz
COPPER WIRE
INSULATION
COPPER MESH

5. Typical formulae for C’, G’, L’, R’
Two parallel plates
C’ = ew/d
L’ = md/w
R’ = 2Rs/w
G’ = sw/d d
Note !
L’C’ = me
G’/C’ = s/e

6. AC Circuit model Z’Dz ejwt Y’Dz Z’ = R’ + jwL’ (Impedances in series)
Y’ = G’ + jwC’ (Admittances in series)
ejwt

7. Kirchhoff’s Voltage Law
Y’Dz
Z’Dz
Z’ = R’ + jwL’ (Impedances in series)
Y’ = G’ + jwC’ (Admittances in series)
ejwt
V(z)
V(z+Dz)
KVL: V(z) - V(z+Dz) = IZ’Dz

8. Kirchhoff’s Voltage Law
Y’Dz
Z’Dz
Z’ = R’ + jwL’ (Impedances in series)
Y’ = G’ + jwC’ (Admittances in series)
ejwt
V(z)
V(z+Dz)
dV/dz = -Z’I

9. Kirchhoff’s Current Law
Y’Dz
Z’Dz
Z’ = R’ + jwL’ (Impedances in series)
Y’ = G’ + jwC’ (Admittances in series)
ejwt
I(z)
I(z+Dz)
KCL: I(z) - I(z+Dz) = -VY’Dz

10. Kirchhoff’s Current Law
Y’Dz
Z’Dz
Z’ = R’ + jwL’ (Impedances in series)
Y’ = G’ + jwC’ (Admittances in series)
ejwt
I(z)
I(z+Dz)
dI/dz = -Y’V

11. Telegrapher’s Equations
dV/dz = -Z’I
dI/dz = -Y’V
d2V/dz2 = g2V