ECE 391
supplemental notes - #1
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
1
Lumped vs. Distributed Circuits
Lumped-Element Circuits:
• Physical dimensions of circuit are such that voltage
across and current through conductors connecting
elements does not vary.
• Current in two-terminal lumped circuit element does not
vary (phase change or transit time are neglected)
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
2
1
Lumped vs. Distributed Circuits
Distributed Circuits:
• Current varies along conductors and elements;
• Voltage across points along conductor or within element
varies
è phase change or transit time cannot be neglected
25 cm
Example:
λ=
c
3 ×108 ms
=
= 1m
f 300 ×10 6 1s
wavelength λ
= 1 period in space
è∞
vp=c
f = 300MHz
current
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distance
ECE391– Transmission Lines
Spring Term 2014
3
Example: Hybrid Microwave
Integrated Circuit (MIC)
distributed element
transmission line
interconnection
lumped element
èlow loss
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
4
2
When Do We Need to Consider
Transmission Lines Efects
digital domain
analog domain
λ
100%
90%
10%
0%
distance
tf
tr
risetime
falltime
flight time (delay time) td=d/vp
maximum dimension d
d vs λ
tr, tf vs td
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
5
Lumped vs. Distributed
td = length/velocity
Z0, td
RL
VS
vs (t ) = v0 u (t )
§ Rule-of-thumb (heuristic)
Delay time td = length of line / velocity
Rise time of signal tr (fall time tf)
Signal path can be treated as
lumped element"
if! tr
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td > 6
distributed element"
if! t r
ECE391– Transmission Lines
td < 2.5
Spring Term 2014
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3
Example
• CMOS Buffer with tr = tf = 0.5 ns
• FR-4 PCB (velocity ≈ 0.45 c)
• Note:
vp ≈ c
VDD
signal
gnd
εr
εr
Determine maximum distance d so that tr/td > 6
tr/td = tr/(d/vp) > 6
è
d < tr vp/6
d < 0.5ns * (0.45*30cm/ns)/6 = 1.125 cm ≈ 11 mm
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
7
TL Problem Classification
“faster board”
0
risetime
2.5td
6td
simulate and check after
layout of ckt
simulate and
check after
layout of ckt
simulate and
check after
layout of ckt
no problems
go for it!
consider quick
hand analysis
no problems
go for it!
no problems expected
clock reset signals
with critical timing (or
edge sensitivity)
non-critical signals
(e.g. data lines with
tsu ≈ 6 td)
status signals; slow,
insensitive lines
In this region,
other consid.
begin to surface
(EMI, crosstalk)
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
8
4
Sinusoidal Signals
V0 cos(2π ft)
= V0 cos(ω t)
z
phase constant
ω 2π
β= =
vp
λ
V0 cos(ω (t − td )
= V0 cos(ω t − ω td )
= V0 cos(ω t − ω
z
) = V0 cos(ω t − β z)
vp
In practice: lumped if td < 0.1 T (d < 0.1 λ)
safely lumped if d < 0.01λ
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
9
Illustration
CH2
CH1
time
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ECE391– Transmission Lines
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5
Example: Lumped vs. Distributed
z of the current
i(z, t) = I 0 cos(ω t − β z)
For example, the spatial dependence
in a conductor can be neglected if
Normalized current I(z,t=0)
Example:
z λ << 1.
1.25
1
0.75
0.5
0.25
f = 100 MHz
f = 1GHz
f = 10 GHz
0
-0.25
-0.5
-0.75
-1
-1.25
-4
-3
-2
-1 0 1 2
Distance (cm)
3
4
resistor
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
11
Concept: Electrical Length
• Electrical length (θ, E) is a measure of the
physical length expressed in terms of
wavelength λ
θ = E = 2π
z
λ
z
θ = E = 360 0
λ
z
E=
λ
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(in radians)
(in degrees)
(as fraction of wavelength)
ECE391– Transmission Lines
Spring Term 2014
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6
Transmission Line Examples
coaxial line
two-wire line (also twisted-pair)
microstrip
stripline
coplanar strip (CPS)
coplanar waveguide (CPW)
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ECE391– Transmission Lines
Spring Term 2014
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Transmission Lines/Interconnects
§ What is a transmission line?
– lumped vs. distributed circuit?
– how many conductors?
§ Types of transmission lines
–
–
–
–
–
on-chip (à transmission line?)
off chip
PCB, packages
cables (coax, twisted pair, …)
etc.
§ Applications of transmission lines
– interconnections
§ signal transmission
§ power transmission
– circuit elements/functional components (at high frequencies)
§
§
§
§
filters (e.g. coupled lines, stubs, …)
couplers
power dividers
matching networks
– …
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
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7
Interconnect Technologies
Interconnection
Type
Line
Line
Width Thickness
(µm)
(µm)
Max.
Length
(cm)
On-Chip
0.5-2
0.5-2
0.3-1.5
Package
Thin-Film
10-25
5-8
20-45
PCB
Ceramic
75-10
0
16-25
20-50
PCB
60-10
0
8-70
40-70
On-chip

cables, etc.
Source: IBM (plus changes)
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
15
Transmission Line –
Characteristic Dimensions
Cross-sectional Dimensions
<< wavelength
D << λ
D
L
D
L
L < λ  L >> λ
Lengths vary from fractions of a
wavelength to many wavelengths
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ECE391– Transmission Lines
Spring Term 2014
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8
Model for Transmission Line
Note: here R,L,G,C
represent total values
per section
generic equivalent
circuit model
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ECE391– Transmission Lines
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Transmission Line Parameters
Cross-sectional view of typical uniform interconnects:
§
§
§
§
Capacitance between conductors, C (F/m)
Inductance of conductor loop, L (H/m)
Resistance of conductors (conductor loss), R (Ω/m)
Shunt conductance (dielectric loss), G (S/m)
R,L,G,C are specified as per-unit-length parameters
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
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9
Derivation of Transmission Line Equations
∂i(z, t )
∂t
∂v(z + Δz , t )
− [i (z + Δz, t ) − i(z, t )] = CΔz
∂t
− [v(z + Δz, t ) − v(z , t )] = LΔz
Note: here L,C are per-unit-length parameters
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
19
Transmission Line Equations
Lossless transmission line
i
∂i(z, t )
∂t
∂v(z + Δz , t )
− [i (z + Δz, t ) − i(z, t )] = CΔz
∂t
− [v(z + Δz, t ) − v(z , t )] = LΔz
after taking
Δz → 0
Telegrapher’s Equations
−
∂i ( z , t )
∂v( z , t )
∂v( z , t )
∂i ( z , t )
−
=C
=L
∂z
∂t
∂z
∂t
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ECE391– Transmission Lines
Spring Term 2014
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10
Wave Equations
−
∂v( z , t )
∂i ( z , t )
=L
∂z
∂t
−
∂i ( z , t )
∂v( z , t )
=C
∂z
∂t
Wave Equations
∂ 2 v( z , t )
∂ 2 v( z , t )
1 ∂ 2 v( z , t )
=
LC
=
∂z 2
∂t 2
v 2p
∂t 2
∂ 2i ( z , t )
∂ 2i ( z , t )
1 ∂ 2i ( z , t )
= LC
= 2
∂z 2
∂t 2
vp
∂t 2
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
21
General Wave Solutions
∂ 2v
∂ 2v
1 ∂ 2v
=
LC
=
∂z 2
∂t 2
v 2p ∂t 2
General Solution for Voltage
v( z , t ) = v + ( z − v p t ) + v − ( z + v p t )
= v + (t − z / v p ) + v − (t + z / v p )
+z direction
Velocity of Propagation
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vp =
-z direction
1
(m/s )
LC
ECE391– Transmission Lines
Spring Term 2014
22
11
Illustration of Space–Time Variation of
Single Traveling Wave
v( z , t ) = v + ( z − v p t ) + v − ( z + v p t )
= v + (t − z / v p ) + v − (t + z / v p )
vp
vp
time
distance
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
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Wave Solutions for Current
i( z, t ) = i + (t − z / v p ) + i − (t + z / v p )
−
∂v( z, t )
∂i( z, t )
=L
∂z
∂t
−
∂i( z, t )
∂v( z, t )
=C
∂z
∂t
{
} {
}
{
} {
}
1 +
v (t − z / v p ) − v − (t + z / v p ) = L i + (t − z / v p ) + i − (t + z / v p )
vp
1 −
i (t − z / v p ) − i − (t + z / v p ) = C v + (t − z / v p ) + v − (t + z / v p )
vp
i( z, t ) =
v + (t − z / v p )
Z0
Characteristic Impedance
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−
Z0 = v p L =
ECE391– Transmission Lines
v − (t + z / v p )
Z0
1
=
v pC
L
(Ω)
C
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12
Summary of
Transmission Line Parameters
• Capacitance per-unit-length C (F/m)
• Inductance per-unit-length L (H/m)
• Characteristic impedance Z 0 (Ω)
• Velocity of propagation v p (m/s)
• Per-unit-length delay time t p = 1 v p (s/m)
• Delay time (TD) t d = l v p = l t p (sec)
L
C
Z0 =
vp =
1
LC
L = Z0 v p = Z0 t p
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t p = LC
td = l LC
C = 1 (Z 0v p ) = t p Z 0
ECE391– Transmission Lines
Spring Term 2014
25
Properties of Ideal Transmission Lines
§ C from 2D electro-static solution
§ L from 2D magneto-static solution
§ Velocity of propagation
vp =
c
1
1
=
= 0
LC
µ0 µ r ε 0ε r
εr
c0 ≈ 30 cm/nsec
(neglecting magnetic materials)
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
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13
Propagation Speeds for
Typical Dielectrics
Dielectric
Rel. Dielectric Constant
Polyimide
Silicon dioxide
Epoxy glass (PCB)
Alumina (ceramic)
εr
Propagation
speed
(cm/nsec)
Delay time
per unit length
(ps/cm)
2.5 – 3.5
3.9
5.0
9.5
16-19
15
13
10
53 - 62
66
75
103
Oregon State University
ECE391– Transmission Lines
Spring Term 2014
27
Example T-Line Structures
Parallel-Plate Line
w
εr
t
εr
µ s
ε 0ε r w
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a
s
b
c
w
s
C = ε 0ε r
L = µ0
s
w
Z0 =
Coaxial Line
RDC =
C=
2πε 0ε r
ln(b a )
Z0 =
2ρ
wt
RDC =
ECE391– Transmission Lines
L=
µ0 ⎛ b ⎞
ln⎜ ⎟
2π ⎝ a ⎠
µ ln(b a )
ε 0ε r 2π
ρ
ρ
+
2
2
π a π (c − b 2 )
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Examples (cont’d)
r
r
s
RDC =
C=
s
2ρ
π r2
RDC =
πε
cosh
−1
(s
C=
2r )
µ0
cosh −1 (s 2r )
π
1 µ0
Z0 =
cosh −1 (s 2r )
π ε
2πε
cosh −1 (s r )
µ0
cosh −1 (s r )
2π
1 µ0
Z0 =
cosh −1 (s r )
2π ε
L=
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ρ
π r2
L=
ECE391– Transmission Lines
Spring Term 2014
29
Examples (cont’d)
stripline
microstrip
ε0
W
εr
Z0 =
t
1 µ ⎛ 1+ w b ⎞
⎟
ln⎜
4 ε ⎜⎝ w b + t b ⎟⎠
C = εr
1
c0 Z 0
L = εr
Z0
c0
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W
t
εr
b
Z0 =
1
2π
h
µ
⎛ 4h ⎞
ln⎜ ⎟
ε 0ε eff ⎝ d ⎠
d = 0.536w + 0.67t
ε eff ≈ 0.475ε r + 0.67
(very approximate!)
ECE391– Transmission Lines
Spring Term 2014
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Alternative Formulas
A
B
Z 0,sym =
ε0
w
εr
t
t
εr
b
⎞
60Ω ⎛
4b
⎟
ln⎜
ε r ⎜⎝ 0.67π (t + 0.8w) ⎟⎠
Z0 =
for w/b < 0.35 and
t/b < 0.25
Z 0,offset ≅ 2
w
h
87 Ω
⎛ 5.98h ⎞
ln⎜
⎟
ε r + 1.41 ⎝ 0.8w + t ⎠
for 0.1 < w/h < 2
and 1 < εr < 15
Z 0, sym (2 A, w, t , ε r ) * Z 0, sym (2 B, w, t , ε r )
Z 0, sym (2 A, w, t , ε r ) + Z 0, sym (2 B, w, t , ε r )
more accurate if 2A # 2A+t and 2B # 2B+t
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ECE391– Transmission Lines
Spring Term 2014
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