LIMITS TO COMPUTATION SPEED
Devices:
Emitter
Gate
Carrier transit and diffusion times
(f = ma, v < c)
Field-effect
RC ≅ ε/σ; RL, LC time constants
transistors
Beyond scope of 6.013 (read Section 8.2)
Interconnect, short lines <<λ:
2r
Wire resistance R ∝ D/r2
Capacitance C = εA/d ∝ D2/d
τ = RC ∝ D3/r2d ≅ const. if D:r:d = const.
D
R is high for polysilicon, C is high for thin gaps
L/R and τ = LC scale well with size and do not limit speed
Drain
Charge
Carriers
D
d
D
Interconnect, long lines >~λ/8:
Propagation delay: c = 1/ με < 3 ×108 [m s-1] (ε might be ~2εo)
Reflections at wire and device junctions, unless carefully designed
Resistive loss
Radiation and cross-talk (3-GHz clocks imply 30-GHz harmonics)
L11-1
WIRED INTERCONNECTIONS
Transverse EM Transmission Lines:
TEM: Ez = Hz = 0
Parallel wires
Coaxial cable
z
Parallel
plates
Stripline
Arbitrary cross-section
≠ f(z)
L11-2
PARALLEL-PLATE TRANSMISSION LINE
Boundary Conditions:
H
×
E
=
S
E // = H⊥ = 0 at perfect conductors
Uniform Plane Wave Solution:
x-polarized wave propagating in
+z direction in free space
y
E = xˆ E+ t - z
c
H = yˆ ( 1 ) E+ t - z
ηo
c
( )
( )
z
H
z
I(z) = H(z)W, independent of path C
Js (z) = n̂ × H(z) [A m-1]
E
x
Currents in Plates:
v∫C H • ds =
∫∫A J •
da =
I(z)
Surface Currents Js(A m-1):
σ=∞
W
C
H
n̂
z
I(z)
I(z)
W
L11-3
TRANSMISSION LINE VOLTAGES
Voltages between plates:
Since Hz = 0 ⇒ v∫c E • ds = 0 at fixed z, Φ1
2
∫1 E • ds = Φ1 − Φ 2 = V(z)
y
V(z) is uniquely defined
σ=∞
+
E c
z V(z)
Φ2
x
ds
d
Surface charge density ρs(z) [C m-2]:
n̂ • εE(z) = ρs (z) (Boundary condition; from ∇ • D = ρ
Integrate E,H to find v(t,z),i(t,z)
)
2
(
)
ˆ + (t - z c)
v(t,z) = ∫1 E • ds = d × E+ t - z/c here, where E = xE
ˆ + ( t- z c )
i(t,z) = v∫c H • ds = (W/ηo )E+ ( t - z/c ) , where H = yE
v(t,z) = Zo i(t,z) [if there is no backward propagating wave]
Zo = ηod/W [ohms] "Characteristic impedance"
Note: v(z) violates KVL, and i(z) violates KCL
L11-4
TELEGRAPHER’S EQUATIONS
W
Equivalent Circuit:
LΔz i(z)
LΔz i(z+Δz) LΔz
z
y
CΔz
v(t,z)
CΔz
v(t,z+Δz)
CΔz
Δz
L [Henries m-1], C [Farads m-1]
+
v(t,z)
x
i(t,z)
E • zˆ = H • zˆ = 0 (TEM)
Difference Equations:
di(z)
dt
dv(z)
i(z+Δz) - i(z) = −CΔz
dt
v(z+Δz) - v(z) = −LΔz
d
Limit as Δz → 0:
Wave Equation:
dv(z)
di(z)
= −L
dz
dt
di(z)
dv(z)
= −C
dz
dt
d2 v = LC d2 v
dz2
dt 2
L11-5
SOLUTION:
TELEGRAPHER’S EQUATIONS
Wave Equation: d2v = LC d2 v
2
2
dz
Solution:
dt
v(z,t) = f+(t – z/c) + f-(t + z/c)
f+ and f- are arbitrary functions
Substituting into Wave Equation:
(1/c2) [f+″(t – z/c) + f-″(t + z/c)] = LC [f+″(t – z/c) + f-″(t + z/c)]
Therefore:
Current I(z,t):
Recall:
Therefore:
Therefore:
di(z)
dv(z)
= −C
= -C[f+′(t – z/c) + f-′(t + z/c)]
dz
dt
i(z,t) = cC [f+(t – z/c) – f-(t + z/c)]
cC = C LC = C L= Yo "Characteristic admittance"
Zo = 1/Yo =
ohms
“Characteristic impedance”
i(z,t) = Yo[f+(t – z/c) – f-(t + z/c)]
C
/
L
Where:
c = 1 LC = 1 με
L11-6
ARBITRARY TEM LINES
Can we estimate L and C?
s
≠ f(z)
s
Φi
E
d
1
L
LC
A
Zo =
=
where LC = με
C
C
d
εA
C = capacitance/m = d
ε
=
C = nC for n parallel square incremental capacitors (C = ε [Fm-1])
C = C /m for m capacitors C in series
m
Therefore:
C = nC /m = nε/m [Farads/m]
N cells
Zo = LC = με = ηo m ohms ηo = μ/ε]; Ζosingle cell = 377Ω
[
C
nε/m
n
L11-7
TRANSMISSION LINE VOLTAGES
v(z,t)
Velocity c
+
vs(t)
Zo ohms, c [m/s]
0
z
Matching boundary conditions:
v(t) and I(t) are continuous at z = 0
v(z,t) = vs(t – z/c)
1
i(z,t) =
v (t – z/c)
Zo s
L11-8
MIT OpenCourseWare
http://ocw.mit.edu
6.013 Electromagnetics and Applications
Spring 2009
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