# What Really is Characteristic Impedance?

```Slide -1
What Really is Characteristic Impedance?
or
Developing Engineering Intuition about Interconnect Design
Eric Bogatin
Signal Integrity Evangelist
www.BeTheSignal.com
PCB West March 27, 2007
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -2
www.BeTheSignal.com
Feature articles and columns
Signal integrity public classes
Online lectures (30 hours of web
based training)
Resources
Contact : [email protected] 913-393-1305
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -3
What is the Designer’s Most
Important Tool?
To Here
From Here
Or Here
?
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -4
Creativity is the key ingredient to the
design process
Intuition is the designer’s most
important tool
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -5
What Does a Signal See as it
Propagates Down an Interconnect?
“…be the signal”
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -6
A Propagating Signal
• A real, physical transmission line
• A signal
• A propagating signal
Vsignal
V
Signal path
Vin
V
Return path
GROUND
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -7
What Determines the Speed of a Signal?
Ans: the dielectric constant of the laminate
v
signal
Dk
return
in air: v = 186,000 miles per sec
v=
12 inches
n sec
4
=
v = 12 inches/nsec
12 inches
n sec
= 6 inches
n sec
2
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -8
What is the Most Important Electrical
Ans: the impedance
Vsignal
V
Signal path
Vin
V
Return path
GROUND
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -9
Electrical Model of a Lossless
Transmission Line
Telegraphers’ equation
∂
∂
V (x, t ) = −L I(x, t )
∂x
∂t
∂
∂
I(x, t ) = −C V (x, t )
∂x
∂t
Wave equation
∂2
1 ∂2
V (x, t ) =
V (x, t )
2
∂t
LC ∂x 2
∂2
1 ∂2
I(x, t ) =
I(x, t )
2
∂t
LC ∂x 2
derive
Z0 =
L
C
 Eric Bogatin 2007
v=
1
LC
www.BeTheSignal.com
Slide -10
“…be the signal”
Charging up a transmission line
+++++
- - - - -
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -11
What is Impedance?
Z=
Voltage applied
Current through
I
C
∆x
V
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -12
A Propagating Signal
I
+++++
V
------Charged line
after 1 nsec
I
+++++
+++++
V
-------
-------
Charged line after 2 nsec
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -13
Geometry, Current and Impedance
Line width increases, capacitance ______________,
impedance______________
increases
decreases
Line width decreases, capacitance ______________,
impedance ______________
decreases
increases
Dielectric thickness increases, capacitance ___________,
___________
decreases impedance, increases
www.BeTheSignal.com
 Eric Bogatin 2007
Slide -14
Instantaneous Impedance
the Signal Sees
Z=
Voltage applied
Current through
C
∆x
instantaneous impedance of the transmission line
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -15
0th Order Model of Transmission Line
CL= Capacitance per length [pF/in]
∆x
Vin
C
C
C
C
C
C
C
C
C
C = C L ∆x
∆x
every ∆t = v
∆Q = CV,
What is the current into the line?
www.BeTheSignal.com
 Eric Bogatin 2007
Slide -16
Current Into Transmission Line
CL= Capacitance per length [pF/in]
∆x
Vin
C
C
C
C
C
C
C
C
C
C = C L ∆x
∆Q = CV,
∆x
every ∆t = v
I, V definition of
Transmission Line:
I = ∆∆Qt =
vCL ∆ x V
= vCLV
∆x
What’s the impedance?
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -17
Instantaneous Impedance of a
Transmission Line
I = v C LV
V = 1
Z = V = vC
vCL
I
LV
Features of the instantaneous impedance:
•
•
•
•
varies inversely with capacitance
looks like a resistor
dependant on intrinsic properties
independent of length
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -18
Characteristic Impedance
Z0 =
1
v CL
only applies to uniform transmission lines
the one instantaneous impedance that characterizes a
uniform transmission line
independent of length
is the instantaneous impedance a signal will see when
propagating down a uniform section
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -19
The Ideal, Lossless
Transmission Line Model
A “transmission line” is the name we use to refer
to the ideal electrical circuit element
a new, fundamental, ideal circuit element: T
• Characteristic impedance, Z0,
• Time delay: TD
A “transmission line” is also the name we
use to refer to a real, physical interconnect
www.BeTheSignal.com
 Eric Bogatin 2007
Slide -20
Controlled Impedance
Interconnects
• A uniform transmission line
• Same instantaneous impedance everywhere on the line
• Any value Z0 can be controlled impedance
Controlled impedance structures
twisted pair
coax
microstrip
embedded
microstrip
stripline
asymmetric
stripline
coplanar
Famous Characteristic Impedances:
RG174
50 Ω
RG58
52 Ω
RG59
75 Ω
RG62
93 Ω
TV Antenna
300 Ω
Cable TV
75 Ω
Twisted pairs
100-130 Ω
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -21
What Does it Mean to Refer to a
Cable as a “50 Ohm Cable”?
hm
ng 50 O
3 foot lo
coax
Ω
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -22
What Does it Mean to Have a 50 Ohm Line?
Ve
0
ggggg 5
rrrry lon
ax
Ohm co
Ω
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -23
open
Characteristic
impedance
Impedance (Ohms)
The Input Impedance of a Transmission
Line is Time Dependent
Round trip
time of
flight
Many round trip
time of flights
 Eric Bogatin 2007
Time
www.BeTheSignal.com
Slide -24
“…the impedance” of a
Transmission Line is Ambiguous
• The input impedance of the transmission line - may be
time dependent
• The instantaneous impedance of the transmission line
• The Characteristic impedance of the transmission line
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -25
What Influences Z0 of Microstrip?
• First order terms:
w
h
Dk
• Second order terms:
Mixed dielectrics
t
Etch back
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -26
Calculating Characteristic
Impedance
• 2D field solvers
• Approximations
• Rules of thumb
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -27
Calculating Characteristic
Impedance: Approximations
w
Z0 =
h
b
h2
87
h

ln 7.5 
(ε r + 1.41)  w 
w
Z0 =
60 
b
ln 2.35 Ω
εr 
w
w
Z0 =
80 
h1
h1 
ln 4.75  1 −

εr 
w  4h 2 
h1
IPC-2141
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -28
Using a Field Solver Does Not Require
Knowing Maxwell’s Equations!
Time Domain
∇ • εE =
ρ
ε0
∇•B = 0
∂B
=0
∂t
&micro;ε ∂E
∇x B − 2
= &micro;0 J
c ∂t
∇x E +
Frequency Domain
∇ • εE =
ρ
ε0
∇ • &micro;H = 0
∇x E + jω&micro; H = 0
∇x H − jωε E = J
“But this is the simplified version for the general public”
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -29
Characteristic Im pedance
(Ohm s)
IPC Microstrip Accuracy
120
110
100
90
80
70
60
50
40
30
20
10
0
Ansoft
Polar
IPC
0
2
4
6
8
10
12
14
16
18
20
Line W idth (m ils)
0%
Polar
-10%
IPC
-15%
-20%
-25%
-30%
-35%
0
2
4
6
8
10
12
14
16
18
20
Line W idth, m ils
www.BeTheSignal.com
 Eric Bogatin 2007
Slide -30
Stripline Example
60.0
50.0
40.0
Zo
30.0
20.0
10.0
7.
00
10
.0
0
13
.0
0
16
.0
0
19
.0
0
22
.0
0
25
.0
0
4.
00
0.0
1.
00
H1 = 5 mils
Er1 = 4
T1 = 1.4 mils
W1=W2 = changing
Absolute Error
-5%
Height (H1)
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -31
Rules of Thumb
w
• Microstrip in FR4:
w=2xh
• Dual stripline in FR4:
b=3xh
h=w
h
wh
h
h
w
b
www.BeTheSignal.com
 Eric Bogatin 2007
Slide -32
The Input Impedance of a
Transmission Line
TD
If rise time &lt; 2 x TD, what impedance does driver see during transition?
Rsource
Rsource
Z0
=
 Eric Bogatin 2007
Z0
www.BeTheSignal.com
Slide -33
Important Principles
• A signal is a changing voltage AND a current loop
• The signal sees an instantaneous impedance each step
along a uniform transmission line
• Characteristic impedance depends inversely on the
capacitance per length of the interconnect
• The instantaneous impedance the signal sees depends
as much on the signal path as the return path
• The input impedance of a uniform transmission line is
the characteristic impedance for times short compared
to the round trip delay
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -34
Thanks for
listening!
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -35
www.BeTheSignal.com
Feature articles and columns
Signal integrity public classes
Online lectures (30 hours of web
based training)
Resources
Contact : [email protected] 913-393-1305
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -36
Current Return Path in T Lines
Current into signal line
TD = 1 nsec
When does the return current return?
For DC currents:
For high speed currents? When does current come out return path?
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -37
How Does Current Get Through A
Capacitor?
I
insulating dielectric
At DC, the current = 0
www.BeTheSignal.com
 Eric Bogatin 2007
Slide -38
+++
Capacitance is ALWAYS constantjust depends on geometry
V=
∆V =
---
Q
C
∆Q
C
∆Q = C x ∆V
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -39
Equivalent Views
+++
+++
=
I
---
+++
Do actual + charges move through
the insulating dielectric?
www.BeTheSignal.com
 Eric Bogatin 2007
Slide -40
Return Current Paths in
Transmission Lines
signal
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
To control impedance, manage the return
path as carefully as the signal path
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -41
Current Distributions
50 Ohm microstrip, FR4
t = 3 mils
1 MHz
10 MHz
100 MHz
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -42
Microstrip vs. Frequency
• Above 1 MHz, current in return path is mostly under
signal path
• As frequency increases, current in return path
concentrates more at the surface
Skin depth ~ 2 microns x sqrt(1/f), f in GHz
• What drives current re-distribution?
The path of lowest loop inductance
Current within each conductor spreads out
Current between conductors get as close together as possible
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -43
Symmetrical Stripline
50 Ohms,
FR4,
w = 5 mils,
t = 3 mils
b = 19 mils
b
@ 100 MHz
www.BeTheSignal.com
 Eric Bogatin 2007
Slide -44
Asymmetrical Stripline, @ 100 MHz
Observations:
symmetrical
1.
Doesn't matter what the DC
voltage is on the planes- any
plane can be a return path
2.
Return current is symmetrical
when trace is symmetrical
3.
The closer trace is to a plane,
the more return current is in it
asymmetrical
 Eric Bogatin 2007
•
Driven by lowest loop
inductance
www.BeTheSignal.com
Slide -45
Important Principles
• A signal is a changing voltage AND a current loop
• The signal sees an instantaneous impedance each step
along a uniform transmission line
• Characteristic impedance depends inversely on the
capacitance per length of the interconnect
• The instantaneous impedance the signal sees depends
as much on the signal path as the return path
• The input impedance of a uniform transmission line is
the characteristic impedance
• Above 10 MHz the current through a transmission line is
limited to near the conductor surface
 Eric Bogatin 2007
www.BeTheSignal.com
Slide -46
Thanks for
listening!
 Eric Bogatin 2007
www.BeTheSignal.com
```