Inductance
and Partial
Inductance
What's it all
mean?
Bruce Archambeault, PhD
IEEE Fellow, IBM Distinguished
Engineer
Bruce.arch@ieee.org
Inductance
• Probably the most misunderstood concept in
electrical engineering
– Do not confuse ‘inductance’ with ‘inductors’
• Common Usage
–
–
–
–
–
–
–
–
Self inductance
Loop inductance
Mutual inductance
Equivalent inductance
Partial inductance
Partial self inductance
Partial mutual inductance
Apparent inductance
Bruce Archambeault, PhD
2
Inductance
• Current flow through metal =
inductance!
• Fundamental element in EVERYTHING
• Loop area first order concern
• Inductive impedance increases with
frequency and is MAJOR concern at
high frequencies
X L = 2πfL
Bruce Archambeault, PhD
3
Current Loop = Inductance
Courtesy of Elya Joffe
Bruce Archambeault, PhD
4
Inductance Definition
• Faraday’s Law
∂B
∫ E ⋅ dl = − ∫∫ ∂t ⋅ dS
• For a simple rectangular loop
Area = A
V
B
∂B
V = −A
∂t
Bruce Archambeault, PhD
5
Given the Definition of Inductance
• Do these have inductance?
“Ground Strap”
SMT Capacitor
PCB Via
Not until return path for current is identified!
Bruce Archambeault, PhD
6
Self Inductance
• Isolated circular loop
⎞
⎛ 8a
L ≈ μ0 a ln⎜⎜ − 2 ⎟⎟
⎠
⎝ r0
• Isolated rectangular loop
2
⎛
⎞
p
1
p
+
+
2μ0 a ⎜
1
1
2 ⎟
L=
ln
1+ p
+ −1 + 2 −
⎜ 1+ 2
⎟
p
p
π
⎝
⎠
Note that inductance is directly influenced
by loop AREA and less influenced by
conductor size!
Bruce Archambeault, PhD
p=
length of side
wire radius
7
Mutual Inductance
Φ 2 = M 21 I1
How much magnetic flux is
induced in loop #2 from a
current in loop #1?
Φ2
M 21 =
I1
Loop #1
Loop #2
Φ2 = ∫
S2
Bruce Archambeault, PhD
r
B1 (r ) ⋅ nˆ dS2
8
Flux from Current in Loop #1
Bruce Archambeault, PhD
9
Flux from Current in Loop #1
Bruce Archambeault, PhD
10
Change in mutual inductance with spacing
2
Mutual Inductance (nH)
X: 24
Y: 1.835
1.5
The magnetic field drops
off rapidly, so then does
the mutual inductance
1
X: 100
Y: 0.7312
0.5
X: 500
Y: 0.02507
0
0
200
400
600
800
Spacing between the coils(mils)
Bruce Archambeault, PhD
X: 1000
Y: 0.01955
1000
11
Mutual Inductance
Loop #1
Loop #1
Loop #2
Loop #2
Less loop area in loop #2
means less magnetic flux in
loop #2 and less mutual
inductance
Less loop area perpendicular to
the magnetic field in loop #2
means less magnetic flux in loop
#2 and less mutual inductance
Bruce Archambeault, PhD
12
Partial Inductance
• We now know that a loop of current has
inductance
• We now know that there must be a
complete loop to have inductance
• But where do we place this inductance in a
circuit?
Bruce Archambeault, PhD
13
Zero-to-One Transition
Where’s the Inductance Go??
Power
Supply
And how could you possibly calculate it?
Courtesy of Dr. Clayton Paul
Bruce Archambeault, PhD
14
Total Loop Inductance from Partial Inductance
L total=Lp1+ Lp2 + Lp3 + Lp4 – 2Mp1-3 – 2Mp2-4
Lp2
Mp2-4
Mp1-3
Lp3
Lp1
Lp4
Bruce Archambeault, PhD
Courtesy of Dr. Clayton Paul
15
Partial Inductance
• Simply a way to break the overall loop
into pieces in order to find total
inductance
L2
L1
L3
L4
L total=Lp11+ Lp22 + Lp33 + Lp44 - 2Lp13 - 2Lp24
Bruce Archambeault, PhD
16
Important Points About
Inductance
• Inductance is everywhere
• Loop area most important
• Inductance is everywhere
Bruce Archambeault, PhD
17
Example
Decoupling Capacitor Mounting
• Keep vias as close to capacitor pads as
possible!
Via Separation
Inductance Depends
on Loop AREA
Height above Planes
Bruce Archambeault, PhD
18
Via Configuration Can Change
Inductance
SMT Capacitor
Via
Best
The “Good”
Capacitor Pads
The “Bad”
Better
The “Ugly”
Really “Ugly”
Bruce Archambeault, PhD
19
High Frequency Capacitors
• Myth or Fact?
Bruce Archambeault, PhD
20
What is Capacitance?
Q = CV
Q
C=
V
• Capacitance is the
ability of a structure to
hold charge
(electrons) for a given
voltage
• Amount of charge
stored is dependant
on the size of the
capacitance (and
voltage)
Consider a capacitor as a bucket holding lot’s of electrons!
Bruce Archambeault, PhD
21
Comparison of Decoupling Capacitor Impedance
100 mil Between Vias & 10 mil to Planes
1000
1000pF
100
0.01uF
Impedance (ohms)
0.1uF
1.0uF
10
1
0.1
0.01
1.0E+06
1.0E+07
1.0E+08
1.0E+09
1.0E+10
Frequency (Hz)
Bruce Archambeault, PhD
22
0603 Size Cap Typical Mounting
9 mils
9 mils
10 mils*
20 mils
10 mils*
Via Barrel 10 mils
60 mils
108 mils minimum
128 mils typical
Bruce Archambeault, PhD
*Note: Minimum
distance is 10 mils but
more typical distance is
20 mils
23
0402 Size Cap Typical Mounting
8 mils
8 mils
10 mils*
20 mils
10 mils*
Via Barrel 10 mils
40 mils
86 mils minimum
106 mils typical
Bruce Archambeault, PhD
*Note: Minimum
distance is 10 mils but
more typical distance is
20 mils
24
Connection Inductance for Typical Capacitor Configurations
Distance into
board
to planes (mils)
0805
typical/minimum
(148 mils
between via
barrels)
0603
typical/minimu
m
(128 mils
between via
barrels)
0402
typical/minimum
(106 mils
between via
barrels)
10
1.2 nH
1.1 nH
0.9 nH
20
1.8 nH
1.6 nH
1.3 nH
30
2.2 nH
1.9 nH
1.6 nH
40
2.5 nH
2.2 nH
1.9 nH
50
2.8 nH
2.5 nH
2.1 nH
60
3.1 nH
2.7 nH
2.3 nH
70
3.4 nH
3.0 nH
2.6 nH
80
3.6 nH
3.2 nH
2.8 nH
90
3.9 nH
3.5 nH
3.0 nH
100
4.2 nH
3.7 nH
3.2 nH
Bruce Archambeault, PhD
25
Connection Inductance for Typical Capacitor
Configurations with 50 mils from Capacitor Pad to Via Pad
Distance into
board
to planes (mils)
0805
(208 mils
between via
barrels)
0603
(188 mils
between via
barrels)
0402
(166 mils
between via
barrels)
10
1.7 nH
1.6 nH
1.4 nH
20
2.5 nH
2.3 nH
2.0 nH
30
3.0 nH
2.8 nH
2.5 nH
40
3.5 nH
3.2 nH
2.8 nH
50
3.9 nH
3.5 nH
3.1 nH
60
4.2 nH
3.9 nH
3.5 nH
70
4.5 nH
4.2 nH
3.7 nH
80
4.9 nH
4.5 nH
4.0 nH
90
5.2 nH
4.7 nH
4.3 nH
100
5.5 nH
5.0 nH
4.6 nH
Bruce Archambeault, PhD
26
PCB Example for Return Current
Impedance
Trace
GND Plane
22” trace
10 mils wide, 1 mil thick, 10 mils above GND plane
Bruce Archambeault, PhD
27
PCB Example for Return Current
Impedance
Trace
GND Plane
Shortest DC path
For longest DC path, current returns under trace
Bruce Archambeault, PhD
28
MoM Results for Current Density
Frequency = 1 KHz
Bruce Archambeault, PhD
29
MoM Results for Current Density
Frequency = 1 MHz
Bruce Archambeault, PhD
30
U-shaped Trace Inductance
PowerPEEC Results
0.6
0.55
0.5
inductance (uH)
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
Frequency (Hz)
Bruce Archambeault, PhD
31
Two Wires in Parallel
• Reduce inductance by factor of two? NO!
LParallel =
L p1 L p 2 − M
2
p
L p1 + L p 2 − 2 M p
L p1 = L p 2 = L p
LParallel =
Lp + M p
2
Only if parallel wires are FAR APART!
Courtesy of Dr. Clayton Paul
Bruce Archambeault, PhD
32
Let’s Apply this to Decoupling
Capacitors
• Equivalent inductance
– Two capacitors vs one capacitor
– Relative location of two capacitors
– Use via between planes as ideal capacitor
Bruce Archambeault, PhD
33
What Happens if a 2nd Decoupling
Capacitor is placed near the First
Capacitor?
Via #1
distance
Observation
Point
Via #2 Moved in arc
around Observation
point while
maintaining 500 mil
distance to
observation point
500
mils
Bruce Archambeault, PhD
34
Second Via Around a circle
Port 3 ( x, y )
d1
Port 1
θ
R: distance between Port 1 and Port 2 in
mil
r: radius for all ports in mil
d2
d: thickness of dielectric layer in mil
R
d1: distance between Port 3 and Port 1
in mil
d1 = R
d 2 = 2 R sin
θ
Port 2
d2: distance between Port 2 and Port 3
in mil
2
μd ⎛⎜ (R + r )2 (d1 + r )2
ln
4π ⎜⎝
r 3 (d 2 + r )
⎛d +r⎞
ln 2 ⎜ 1
⎟
⎞ μd
+
R
r
⎝
⎠
⎟−
⎟ 4π
⎛d +r⎞
⎠
ln ⎜ 2
⎟
⎝ r ⎠
theta: angle as shown in the figure in
degree
Courtesy of Jingook Kim, Jun
Fan, Jim Drewniak
⎞
(R + r )
μd ⎛⎜
⎟
ln ⎜
3 ⎟
4π ⎝ (2 R sin(θ / 2) + r )r ⎠
4
Lequiv
=
Bruce Archambeault, PhD
Missouri University of Science
and Technology
35
Effective Inductance for Various Distances to Decoupling Capacitor
With Second Capacitor (Via) Equal Distance Around Circle
Plane Seperation = 35 mil -- Via Diameter = 20 mil
2100
2000
1900
1800
1700
250 mil
Inductnace (pH)
1600
500mil
1500
750 mil
1400
1000 mil
1300
1200
1100
1000
900
800
700
600
500
0
50
100
150
200
Angle (degrees)
Bruce Archambeault, PhD
36
Effective Inductance for Various Distances to Decoupling Capacitor
With Second Capacitor (Via) Equal Distance Around Circle
Plane Seperation = 10 mil -- Via Diameter = 20 mil
500
450
400
Inductnace (pH)
350
300
250
200
500mil
250 mil
150
750 mil
100
1000 mil
50
0
0
50
100
150
200
Angle (degrees)
Bruce Archambeault, PhD
37
Effective Inductance for Various Distances to Decoupling Capacitor
With Second Capacitor (Via) Equal Distance Around Circle
Plane Seperation = 5 mil -- Via Diameter = 20 mil
400
350
300
500mil
Inductnace (pH)
250 mil
250
750 mil
1000 mil
200
150
100
50
0
0
50
100
150
200
Angle (degrees)
Bruce Archambeault, PhD
38
Understanding Inductance Effects and
Proximity
1 via
10cm
2 via with degree 30°
10mm
20cm
10cm
2 via with degree 90°
2 via with degree 180°
20cm
Bruce Archambeault, PhD
39
Current Density
[m]
[m]
A/m2
A/m2
[m]
[m]
[m]
[m]
A/m2
A/m2
[m]
[m]
Bruce Archambeault, PhD
40
Current Density in Planes
0.12
8
2
40 324
8
6
4
64
16
8
16
24 16
32
40
6546
16 16
32
8
0.085
0.08
0.08 0.0850.09 0.095 0.1 0.1050.11 0.1150.12
0.08
0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12
0.12
0.115
0.085
24
56
54
6
6
48
8
48
16
16 1
6
16
8
8
0.09
24
3240
0.08
0.08 0.0850.09 0.095 0.1 0.1050.11 0.1150.12
0.095
6744208 2
55664 42304
48
0.085
0.1
48
56
6840
6472
808
4
40
324
2
8
0.105
8
8
4032
24
0.1
241 8
68450468
32
40
4782 60
0
782 6 7 564
7248 5 432
0 2 16
24
8
16
56408
5766244
16
0.11
4042
6
7
48
0.105
8
8
44302
0.11
21
56344206
5664
8
8
16
0.115
8
0.12
0.09
6840
7820 0
72 56448
24 32
16
0.09
8
0.085
0.095
0.095
204
564736422
56
0.09
8
8
0.095
16
0.1
80
8
80
72 50648
4
32
16 24
0.105
80
64
820
6754468
8
24
72
0.105
16
0.11
2
430
80
48
0.11
16
5468
64
80
8
24
24
0.115
0.115
0.1
8
8
48 16
0.12
0.08
0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12
Bruce Archambeault, PhD
41
Effect of Plane width on Inductance
Case1 : 10 inches
Case2 : 5 inches
Case3 : 2 inches
Port1
Port2
1 inch
Bruce Archambeault, PhD
42
Loop Inductance is Affected by Plane Width
Case2 : 5 inches
Case1 : 10 inches
~ 330pH
Case2 : 2 inches
~ 250pH
~ 560pH
Bruce Archambeault, PhD
43
Current Spreads in a Plane
Narrower planes means the multiple current paths are
limited therefore effect of mutual inductance between
parallel paths increases!
Bruce Archambeault, PhD
44
Observations
• Added via (capacitor) does not lower
effective inductance to 70-75% of original
single via case
• Thicker dielectric results in higher
inductance
• Normalizing inductance to single via case
gives same curve for all dielectric
thicknesses
Bruce Archambeault, PhD
45
Summary
• Inductance has meaning only for current
loops
• Size of the loop has the most impact on
amount of inductance
• Current density also impact inductance
• Partial inductance is a very useful concept
to understand which portions of the loop
have the largest impact on loop
inductance
Bruce Archambeault, PhD
46