1
ECE 391 supplemental notes - #2
32 Oregon State University ECE391– Transmission Lines Spring Term 2014
Microstrip – Effective Dielectric Constant
whrr
1011
21
21
eff +−++≈ εεε
10-1 100 101 1020
2
4
6
8
10
w/h
ε ef
f
εr = 10
εr = 4
εr = 2
0→t
2
33 Oregon State University ECE391– Transmission Lines Spring Term 2014
Transmission Line Comparison
coaxial line
two-wire line (also twisted-pair)
microstrip
r s ( )rsZ 2cosh1 10
0−=
εµ
π
w t
h εr
ε0
⎟⎠⎞⎜
⎝⎛
++Ω=
twhZ
r 8.098.5ln
41.187
0 ε
a b
c
rε Z0 =µ
ε0εr
ln b a( )2π
34 Oregon State University ECE391– Transmission Lines Spring Term 2014
0 2 4 6 8 100
100
200
300
400
w/h or D/d or b/a
Z 0 ( Ω)
εr = 2.25 ___
εr = 1 - - -
Two-wire line
Coax
Microstrip
Characteristic Impedance of TLs
3
35 Oregon State University ECE391– Transmission Lines Spring Term 2014
Launching a Wave on an Infinitely Long Transmission Line
VS(t)
RS )()( 0 tuvtvs = i+
i+
v+
+
_
v+ (t) = v0 u(t)Z0
Z0 + Rs
Z0, vp
i+ (t) = v+ (t)Z0
= v0 u(t)1
Z0 + Rs
RS
Z0
i+
i+
v+
+
_ VS(t)
36 Oregon State University ECE391– Transmission Lines Spring Term 2014
Wave Propagation on Transmission Line RS
VS
Z0, vp!
v1+ (z, t) = v0
Z0Rs + Z0
u(t − z / vp ) = v0Z0
Rs + Z0u(z − vpt)
i1+ (z, t) = v0
1Rs + Z0
u(t − z / vp ) = v01
Rs + Z0u(z − vpt)
First traveling wave
)()( 0 tuvtvs =),(),( 101 tziZtzv ++ =
4
37 Oregon State University ECE391– Transmission Lines Spring Term 2014
Wave Propagation on Transmission Line RS
VS
Z0, vp!
)()( 0 tuvtvs =
zz1
v1+ (z, t = z1 vp ) = v0
Z0Rs + Z0
u(t1 − z / vp ) = v0Z0
Rs + Z0u((z1 − z) / vp )
i1+ (z, t = z1 vp ) = v0
1Rs + Z0
u(t1 − z / vp ) = v01
Rs + Z0u((z1 − z) / vp )
z20
v(z, t = z1 vp )vp
v0Z0
Rs + Z0t = z2 vp
38 Oregon State University ECE391– Transmission Lines Spring Term 2014
Transmission Line Circuit
Vr = ρL Vi
RS
RL VS
Z0, td!td = length/velocity
reflected wave
)/(1),(
)/(),(
001
0
001
ps
ps
vztuZR
vtzi
vztuZR
Zvtzv
−+
=
−+
=
+
+
First traveling wave
)()( 0 tuvtvs =
),(),( 101 tziZtzv ++ =
5
39 Oregon State University ECE391– Transmission Lines Spring Term 2014
Reflection Coefficient
Vr = ρL Vi
RS
RL VS reflected wave
{ })()(1)()(
)()(
110
11
11
dd
ddL
ddL
ttvttvZ
ttittiittvttvv
−−−=
−+−=
−+−=
−+
−+
−+
At load side at time t ≥ td"
)()( 0 tuvtvs =
0
0
ZRZR
VV
L
L
i
rL +
−==ρ
ρL
Solving for " LLL iRv =
Load-side reflection coefficient!
ρs
Z0, td!td = length/velocity
40 Oregon State University ECE391– Transmission Lines Spring Term 2014
Traveling Waves on TL
Outgoing waves:
Returning waves:
v1+ (z, t) = v1
+ (z − vpt) = v1+ (t − z vp )
v2+ (z, t) = ρsρLv1
+ (t − z vp − 2td )
1st
2nd
1st v1−(z, t) = v1
−(z+ vpt) = v1−(t + (z− zr ) vp )
v1− (z, t) = ρLv1
+ (t + (z − zr ) vp − td )zr is the length of the line
6
41 Oregon State University ECE391– Transmission Lines Spring Term 2014
Step Response of T-Line Circuit
Ls
L
Ls
L
s RRRVV
ZRZV
+=
−+
+=∞ 00
0
0
11
ρρρ
LsLs
L
s RRVV
ZRI
+=
−−
+=∞
1111
000 ρρ
ρ
V1+
42 Oregon State University ECE391– Transmission Lines Spring Term 2014
Reflection Diagram (also called Lattice or Bounce diagram)
7
43 Oregon State University ECE391– Transmission Lines Spring Term 2014
Load Reflection Coefficient ρL
V1+ +V1
− = RL I1+ + I1
−( )= RL
V1+
Z0+ V1
−
−Z0
⎛⎝⎜
⎞⎠⎟
ρL =VreflVinc
= V1−
V1+ = RL − Z0
RL + Z0
44 Oregon State University ECE391– Transmission Lines Spring Term 2014
VS − V1+ +V1
− +V2+( ) = RS I1
+ + I1− + I2
+( )
ρS =VreflVinc
= V2+
V1− = RS − Z0
RS + Z0
VS −V1+ = RS I1
+
− V1− +V2
+( ) = RS V1−
−Z0+ V2
+
Z0
⎛⎝⎜
⎞⎠⎟
For 1st outgoing wave:
For 2nd outgoing wave:
Source Reflection Coefficient ρS
8
45 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example 25Ω
83.333Ω 50Ω 1V
ρL =RL − Z0RL + Z0
= 14
ρS =RS − Z0RS + Z0
= − 13
-1/3 1/4
V1+ = 50
751V= 2
3V
2/3 V
1/6 V
-1/18 V
-1/72 V
1/216 V
2/150 A
-1/300 A
-1/900 A
1/3600 A
1/10800 A
46 Oregon State University ECE391– Transmission Lines Spring Term 2014
Voltage and Current Step Response
lz 41=
0.667
1.111
0.963
0.8642 0.8971
667.0=Lρ334.0−=Sρ
2
0.667
0.222 0.0741 0.1728 0.2058
V0
0.9091 V∞
0.1818 I∞
20ZRs = 05ZRL =
9
47 Oregon State University ECE391– Transmission Lines Spring Term 2014
Single-Line Transient Responses
-1 0 1 2 3 4 5 6 7 8 9 1000.20.40.60.8
11.21.41.6
Time/TD
Volta
ge (v
olts
)
sourcenear endfar end
-1 0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
Time/TD
Volta
ge (v
olts
)
sourcenear endfar end
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
Time/TD
Volta
ge (v
olts
)
sourcenear endfar end
-1 0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
Time/TD
Volta
ge (v
olts
)
sourcenear endfar end
48 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example RS
RL=0!VS
)()( 0 tuvtvs =
Z0, td!td = length/velocity
z
t/msec
z zr
ρL ρs
0
10
49 Oregon State University ECE391– Transmission Lines Spring Term 2014
Ideal Transmission Line – SPICE Model § Ideal Model
§ Spice Implementation
Load VS
Z0,TD
+ -
RS
0
,
0
,
)(
)()()(
)(
)()()(
ZTDtiTDtVTDtVtV
ZTDtiTDtVTDtVtV
in
inrefinout
out
outrefoutin
−+
+−−−=
−+
+−−−=
Z0 =LC
TD = zr LC
50 Oregon State University ECE391– Transmission Lines Spring Term 2014
Equivalent Circuit for t < 3td iL
iL
vL Z0 RL +
_
iL
iL
vL RL +
_
vL (t) = 1+ ρL( ) Z0Rs + Z0
Vs U t − td( )
vL (t) =RL
RL + Z02 Z0Rs + Z0
Vs U t − td( )
11
51 Oregon State University ECE391– Transmission Lines Spring Term 2014
Reactive and Nonlinear Terminations
• The equivalent circuit representation at the near or far end of a transmission line facilitates analysis of - reactive terminations (capacitive, inductive) - nonlinear terminations (see later).
52 Oregon State University ECE391– Transmission Lines Spring Term 2014
Reactive Terminations Reactive terminations occur e.g. as
- capacitive input of devices (buffers) - shunt capacitance of pads - inductance of bond wire - discontinuities in PCB traces (step,
bend, via …) - parasitics of packaging leads
Source: Google Images
12
53 Oregon State University ECE391– Transmission Lines Spring Term 2014
Capacitive Termination
RS = Z0
VS
Z0, td!td = length/velocity
C
vcap (t) = ?
)()( 0 tuvtvs =
54 Oregon State University ECE391– Transmission Lines Spring Term 2014
Equivalent Circuit at Termination
Rth = Z0
Vth=2Vinc(t-td)! !
=V0 u(t-td) C
{ } )(1)( /)(0cap d
tt ttueVtv d −−= −− τ
τ = ?
13
55 Oregon State University ECE391– Transmission Lines Spring Term 2014
Capacitive Termination
0 2 4 6 80
0.2
0.4
0.6
0.8
1
t/ td
vL(
t)/ V0
load-end source-end
RS = Z0
VS
Z0, td!td = length/velocity
C
vcap (t) =V0 1− e−(t−td )/τ{ } u(t − td )
CZ0=τ
s )()( 0 tuvtvs =
56 Oregon State University ECE391– Transmission Lines Spring Term 2014
RLC Series Termination (source matched)
pF0132.1nF10050 ==Ω= LLL CLR nsec22 ≈= LLCLT π
nsec1=TD
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
t/ td
Vol
tage
(vol
ts)
VsourceVloadVcap
14
57 Oregon State University ECE391– Transmission Lines Spring Term 2014
Effect of Source Resistance
driver too small!-> long delay
driver too large!-> ringing!-> long settling time
58 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example – Underdriven Line
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
t/ td
vL(
t)/ V0
50% threshold
∞→LR0ZRs >>
05ZRs =V1
+ =
)()( 0 tuVtvs =
15
59 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example – Overdriven Line
0 5 10 15 20 25 300
0.5
1
1.5
2
t/ td
vL(
t)/ V0
∞→LR0ZRs <<
5/0ZRs =V1
+ =
)()( 0 tuVtvs =
60 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example
+!- vs
Ω=10sourceR
Ω= K50LoadR
ns2,500 =Ω= DtZ
16
61 Oregon State University ECE391– Transmission Lines Spring Term 2014
Example
+!- vs
Ω= 250sourceR
Ω= K50LoadR
ns2,500 =Ω= DtZ
RsCtotal = 10 nsec!
62 Oregon State University ECE391– Transmission Lines Spring Term 2014
Basic TDR Principle Waveforms for resistive termination
ρL
measured load resistance totalinc
total
L
LL VV
VZZR−
=−+=
211
00 ρρ
TDR = Time-Domain Reflectometry
1+ ρL
1− ρL
=1+
VreflVinc
1−VreflVinc
=Vinc +VreflVinc −Vrefl
= VtotalVinc −Vrefl + Vinc −Vinc( ) =
Vtotal2Vinc −Vtotal
TDR Equipment
17
63 Oregon State University ECE391– Transmission Lines Spring Term 2014
TDR Waveforms for Resistive Terminations
V1
2t
d
V1
(1+ρL)V
1
ρLV
1
2td
V1
(1+ρ
L)V
1
-ρLV
1
2td
VTDR
VTDR
VTDR
64 Oregon State University ECE391– Transmission Lines Spring Term 2014
TDR Waveforms for Reactive Terminations
V1
(1+ρL)V
1
V1
2td
V1
(1+ρL)V
1
2td
[ ]( ){ } )'(1)'( /'initialfinalinitialload tueVVVtttv t
dτ−−−+=−=
LL
L CZRZR
0
0
+=τ
0ZRL
L
L
+=τ
VTDR
VTDR