16.360 Lecture 4 • Transmission lines Transmission line parameters, equations Wave propagations Lossless line, standing wave and reflection coefficient Input impedence Special cases of lossless line Power flow Smith chart Impedence matching Transients on transmission lines
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16.360 Lecture 4
• Transmission lines
1. Transmission line parameters, equations2. Wave propagations3. Lossless line, standing wave and reflection coefficient 4. Input impedence5. Special cases of lossless line6. Power flow7. Smith chart8. Impedence matching9. Transients on transmission lines
1. Transmission line parameters, equations
Vg(t) VBB’(t)VAA’(t)
A
A’ B’
B
L
VAA’(t) = Vg(t) = V0cos(t),
VBB’(t) = VAA’(t-td) = VAA’(t-L/c) = V0cos((t-L/c)),
VBB’(t) = VAA’(t)
Low frequency circuits:
Approximate result
VBB’(t) = VAA’(t)
16.360 Lecture 4
1. Transmission line parameters, equations
Vg(t) VBB’(t)VAA’(t)
A
A’ B’
B
L
VBB’(t) = VAA’(t-td) = VAA’(t-L/c) = V0cos((t-L/c)) = V0cos(t- 2L/),
Recall: =c, and = 2
If >>L, VBB’(t) V0cos(t) = VAA’(t),
If <= L, VBB’(t) VAA’(t), the circuit theory has to be replaced.
16.360 Lecture 4
1. Transmission line parameters, equations
Vg(t) VBB’(t)VAA’(t)
A
A’ B’
B
L
= 2ft = 0.06
e. g: = 1GHz, L = 1cm
Time delay t = L/c = 1cm /3x1010 cm/s = 30 ps
Phase shift VBB’(t) = VAA’(t)
= 2ft = 0.6
= 10GHz, L = 1cm
Time delay t = L/c = 1cm /3x1010 cm/s = 30 ps
Phase shift VBB’(t) VAA’(t)
16.360 Lecture 4
• Transmission line parameters
Vg(t) VBB’(t)VAA’(t)
A
A’ B’
B
L
• time delay
VBB’(t) = VAA’(t-td) = VAA’(t-L/vp),
• Reflection: the voltage has to be treat as wave, some bounce back
• power loss: due to reflection and some other loss mechanism,
• Dispersion: in material, Vp could be different for different wavelength
16.360 Lecture 4
• Types of transmission lines
• Transverse electromagnetic (TEM) transmission lines
B
E
E
B
a) Coaxial line b) Two-wire line c) Parallel-plate line
d) Strip line e) Microstrip line
16.360 Lecture 4
• Types of transmission lines
• Higher-order transmission lines
a) Optical fiber
b) Rectangular waveguide c) Coplanar waveguide
16.360 Lecture 4
• Lumped-element Model
• Represent transmission lines as parallel-wire configuration
Vg(t) VBB’(t)VAA’(t)
A
A’ B’
B
z z z
Vg(t)
R’z L’z
G’z C’z
R’z L’z
G’z
R’z
C’z
L’z
G’z
C’z
16.360 Lecture 4
Expressions will be derived in later chapters
Definitions of TL dimensions
TEM (Transverse Electromagnetic): Electric and magnetic fields are orthogonal to one another, and both are orthogonal to direction of propagation
16.360 Lecture 4
• Lumped-element Model
• Represent transmission lines as parallel-wire configuration
Vg(t) VBB’(t)VAA’(t)
A
A’ B’
B
z z z
Vg(t)
R’z L’z
G’z C’z
R’z L’z
G’z
R’z
C’z
L’z
G’z
C’z
• Transmission line equations
• Represent transmission lines as parallel-wire configuration
V(z,t)R’z L’z
G’z C’z V(z+ z,t)
V(z,t) = R’z i(z,t) + L’z i(z,t)/ t + V(z+ z,t), (1)
i(z,t) i(z+z,t)
i(z,t) = G’z V(z+ z,t) + C’z V(z+ z,t)/t + i(z+z,t), (2)
16.360 Lecture 4
• Transmission line equations
V(z,t) = R’z i(z,t) + L’z i(z,t)/ t + V(z+ z,t), (1)
V(z,t)R’z L’z
G’z C’z V(z+ z,t)
i(z,t) i(z+z,t)
-V(z+ z,t) + V(z,t) = R’z i(z,t) + L’z i(z,t)/ t
- V(z,t)/z = R’ i(z,t) + L’ i(z,t)/ t, (3)
Rewrite V(z,t) and i(z,t) as phasors, for sinusoidal V(z,t) and i(z,t):
V(z,t) = Re( V(z) ejt
), i (z,t) = Re( i (z) ejt
),
16.360 Lecture 4
• Transmission line equations
V(z,t)R’z L’z
G’z C’z V(z+ z,t)
i(z,t) i(z+z,t)
Recall:
di(t)/dt = Re(d i e jt )/dt ), = Re(ijt
ej
- V(z,t)/z = R’ i(z,t) + L’ i(z,t)/ t, (3)
- dV(z)/dz = R’ i(z) + jL’ i(z), (4)
16.360 Lecture 4
• Transmission line equations
• Represent transmission lines as parallel-wire configuration
V(z,t)R’z L’z
G’z C’z V(z+ z,t)
V(z,t) = R’z i(z,t) + L’z i(z,t)/ t + V(z+ z,t), (1)
i(z,t) i(z+z,t)
i(z,t) = G’z V(z+ z,t) + C’z V(z+ z,t)/t + i(z+z,t), (2)
16.360 Lecture 4
• Transmission line equations
V(z,t)R’z L’z
G’z C’z V(z+ z,t)
i(z,t) i(z+z,t)
- i (z+ z,t) + i (z,t) = G’z V(z + z ,t) + C’z V(z + z,t)/ t
- i(z,t)/z = G’ V(z,t) + C’ V(z,t)/ t, (5)
Rewrite V(z,t) and i(z,t) as phasors, for sinusoidal V(z,t) and i(z,t):
V(z,t) = Re( V(z) ejt
), i (z,t) = Re( i (z) ejt
),
i(z,t) = G’z V(z+ z,t) + C’z V(z+ z,t)/t + i(z+z,t), (2)
16.360 Lecture 4
• Transmission line equations
V(z,t)R’z L’z
G’z C’z V(z+ z,t)
i(z,t) i(z+z,t)
Recall:
dV(t)/dt = Re(d V e jt )/dt ), = Re(Vjt
ej
- i(z,t)/z = G’ V(z,t) + C’ V(z,t)/ t, (6)
- d i(z)/dz = G’ V(z) + jC’ V(z), (7)
16.360 Lecture 4
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