Inductance and Partial Inductance What's it all mean? Bruce - - PowerPoint PPT Presentation

Bruce Archambeault, PhD

IEEE Fellow, IBM Distinguished Engineer

Bruce.arch@ieee.org

Inductance and Partial Inductance What's it all mean?

Bruce Archambeault, PhD 2

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 3

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

fL X L π 2 =

Bruce Archambeault, PhD 4

Current Loop = Inductance

Courtesy of Elya Joffe

Bruce Archambeault, PhD 5

Inductance Definition

• Faraday’s Law

∫ ∫∫

⋅ ∂ ∂ − = ⋅ S d t B dl E

t B A V ∂ ∂ − =

V B Area = A

• For a simple rectangular loop

Bruce Archambeault, PhD 6

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 7

Self Inductance

• Isolated circular loop

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ≈ 2 8 ln r a a L μ

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + − + + + + =

2 2

1 1 2 1 1 2 1 1 ln 2 p p p p p a L π μ

• Isolated rectangular loop

Note that inductance is directly influenced by loop AREA and less influenced by conductor size!

radius wire side

• f

length p =

Bruce Archambeault, PhD 8

Mutual Inductance

1 2 21 1 21 2

I M I M Φ = = Φ

Loop #1 Loop #2

( )

∫

⋅ = Φ

2

2 1 2

dS ˆ r B

S

n r How much magnetic flux is induced in loop #2 from a current in loop #1?

Bruce Archambeault, PhD 9

Flux from Current in Loop #1

Bruce Archambeault, PhD 10

Flux from Current in Loop #1

Bruce Archambeault, PhD 11

200 400 600 800 1000 0.5 1 1.5 2

X: 24 Y: 1.835

Spacing between the coils(mils) Mutual Inductance (nH) Change in mutual inductance with spacing

X: 100 Y: 0.7312 X: 500 Y: 0.02507 X: 1000 Y: 0.01955

The magnetic field drops

• ff rapidly, so then does

the mutual inductance

Bruce Archambeault, PhD 12

Mutual Inductance

Loop #1 Loop #2

Less loop area in loop #2 means less magnetic flux in loop #2 and less mutual inductance

Loop #1 Loop #2

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 13

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 14

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 15

Lp4 Lp1 Lp3 Lp2 Mp1-3 Mp2-4

Total Loop Inductance from Partial Inductance

L total=Lp1+ Lp2 + Lp3 + Lp4 – 2Mp1-3 – 2Mp2-4

Courtesy of Dr. Clayton Paul

Bruce Archambeault, PhD 16

Partial Inductance

• Simply a way to break the overall loop

into pieces in order to find total inductance

L3 L4 L2 L1

L total=Lp11+ Lp22 + Lp33 + Lp44 - 2Lp13 - 2Lp24

Bruce Archambeault, PhD 17

Important Points About Inductance

• Inductance is everywhere
• Loop area most important
• Inductance is everywhere

Bruce Archambeault, PhD 18

Example

Decoupling Capacitor Mounting

• Keep vias as close to capacitor pads as

possible!

Height above Planes Via Separation

Inductance Depends

• n Loop AREA

Bruce Archambeault, PhD 19

Via Configuration Can Change Inductance

Via Capacitor Pads SMT Capacitor

The “Good” The “Bad” The “Ugly” Really “Ugly” Better Best

Bruce Archambeault, PhD 20

High Frequency Capacitors

• Myth or Fact?

Bruce Archambeault, PhD 21

What is Capacitance?

• Capacitance is the

ability of a structure to hold charge (electrons) for a given voltage

V Q C =

CV Q =

• Amount of charge

stored is dependant

• n the size of the

capacitance (and voltage)

Consider a capacitor as a bucket holding lot’s of electrons!

Bruce Archambeault, PhD 22 Comparison of Decoupling Capacitor Impedance 100 mil Between Vias & 10 mil to Planes

0.01 0.1 1 10 100 1000 1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 Frequency (Hz) Impedance (ohms) 1000pF 0.01uF 0.1uF 1.0uF

Bruce Archambeault, PhD 23

0603 Size Cap Typical Mounting

Via Barrel 10 mils 60 mils 20 mils 10 mils* 9 mils 9 mils 10 mils* 108 mils minimum 128 mils typical *Note: Minimum distance is 10 mils but more typical distance is 20 mils

Bruce Archambeault, PhD 24

0402 Size Cap Typical Mounting

Via Barrel 10 mils 40 mils 20 mils 10 mils* 8 mils 8 mils 10 mils* 86 mils minimum 106 mils typical *Note: Minimum distance is 10 mils but more typical distance is 20 mils

Bruce Archambeault, PhD 25

3.2 nH 3.7 nH 4.2 nH 100 3.0 nH 3.5 nH 3.9 nH 90 2.8 nH 3.2 nH 3.6 nH 80 2.6 nH 3.0 nH 3.4 nH 70 2.3 nH 2.7 nH 3.1 nH 60 2.1 nH 2.5 nH 2.8 nH 50 1.9 nH 2.2 nH 2.5 nH 40 1.6 nH 1.9 nH 2.2 nH 30 1.3 nH 1.6 nH 1.8 nH 20 0.9 nH 1.1 nH 1.2 nH 10 0402 typical/minimum (106 mils between via barrels) 0603 typical/minimu m (128 mils between via barrels) 0805 typical/minimum (148 mils between via barrels) Distance into board to planes (mils)

Connection Inductance for Typical Capacitor Configurations

Bruce Archambeault, PhD 26

Connection Inductance for Typical Capacitor Configurations with 50 mils from Capacitor Pad to Via Pad

4.6 nH 5.0 nH 5.5 nH 100 4.3 nH 4.7 nH 5.2 nH 90 4.0 nH 4.5 nH 4.9 nH 80 3.7 nH 4.2 nH 4.5 nH 70 3.5 nH 3.9 nH 4.2 nH 60 3.1 nH 3.5 nH 3.9 nH 50 2.8 nH 3.2 nH 3.5 nH 40 2.5 nH 2.8 nH 3.0 nH 30 2.0 nH 2.3 nH 2.5 nH 20 1.4 nH 1.6 nH 1.7 nH 10 0402 (166 mils between via barrels) 0603 (188 mils between via barrels) 0805 (208 mils between via barrels) Distance into board to planes (mils)

Bruce Archambeault, PhD 27

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 28

PCB Example for Return Current Impedance

Trace GND Plane Shortest DC path For longest DC path, current returns under trace

Bruce Archambeault, PhD 29

MoM Results for Current Density Frequency = 1 KHz

Bruce Archambeault, PhD 30

MoM Results for Current Density Frequency = 1 MHz

Bruce Archambeault, PhD 31 U-shaped Trace Inductance PowerPEEC Results

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 Frequency (Hz) inductance (uH)

Bruce Archambeault, PhD 32

Two Wires in Parallel

• Reduce inductance by factor of two? NO!

p p p p p p Parallel

M L L M L L L 2

2 1 2 2 1

− + − =

2

2 1 p p Parallel p p p

M L L L L L + = = =

Only if parallel wires are FAR APART!

Courtesy of Dr. Clayton Paul

Bruce Archambeault, PhD 33

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 34

What Happens if a 2nd Decoupling Capacitor is placed near the First Capacitor?

Observation Point Via #1 Via #2 Moved in arc around Observation point while maintaining 500 mil distance to

• bservation point

500 mils distance

Bruce Archambeault, PhD 35

Second Via Around a circle

theta: angle as shown in the figure in degree d2: distance between Port 2 and Port 3 in mil d1: distance between Port 3 and Port 1 in mil d: thickness of dielectric layer in mil r: radius for all ports in mil R: distance between Port 1 and Port 2 in mil Courtesy of Jingook Kim, Jun Fan, Jim Drewniak Missouri University of Science and Technology

Port 2 Port 1

θ

( )

y x,

Port 3

( ) ( ) ( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + r r d r R r d d r d r r d r R d

2 1 2 2 3 2 1 2

ln ln 4 ln 4 π μ π μ 2 sin 2

2 1

θ R d R d = =

( ) ( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + =

3 4

) 2 / sin( 2 ln 4 r r R r R d θ π μ

1

d

2

d

R

Port 2 Port 1

θ

( )

y x,

Port 3

( ) ( ) ( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + r r d r R r d d r d r r d r R d

2 1 2 2 3 2 1 2

ln ln 4 ln 4 π μ π μ 2 sin 2

2 1

θ R d R d = =

( ) ( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + =

3 4

) 2 / sin( 2 ln 4 r r R r R d θ π μ

1

d

2

d

R

Lequiv

Bruce Archambeault, PhD 36 Effective Inductance for Various Distances to Decoupling Capacitor With Second Capacitor (Via) Equal Distance Around Circle Plane Seperation = 35 mil -- Via Diameter = 20 mil

500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 50 100 150 200 Angle (degrees) Inductnace (pH)

250 mil 500mil 750 mil 1000 mil

Bruce Archambeault, PhD 37 Effective Inductance for Various Distances to Decoupling Capacitor With Second Capacitor (Via) Equal Distance Around Circle Plane Seperation = 10 mil -- Via Diameter = 20 mil

50 100 150 200 250 300 350 400 450 500 50 100 150 200 Angle (degrees) Inductnace (pH) 500mil 250 mil 750 mil 1000 mil

Bruce Archambeault, PhD 38 Effective Inductance for Various Distances to Decoupling Capacitor With Second Capacitor (Via) Equal Distance Around Circle Plane Seperation = 5 mil -- Via Diameter = 20 mil

50 100 150 200 250 300 350 400 50 100 150 200 Angle (degrees) Inductnace (pH) 500mil 250 mil 750 mil 1000 mil

Bruce Archambeault, PhD 39

Understanding Inductance Effects and Proximity

1 via

10mm

2 via with degree 30° 2 via with degree 90° 2 via with degree 180°

20cm 20cm 10cm 10cm

Bruce Archambeault, PhD 40

Current Density

A/m2

[m] [m]

A/m2

[m] [m]

A/m2

[m] [m]

A/m2

[m] [m]

Bruce Archambeault, PhD 41

Current Density in Planes

8 8 8 8 8 8 1 6 1 6 16 1 6 16 2 4 24 2 4 24 32 3 2 32 4 4 40 48 48 48 56 56 56 64 64 64 64 72 72 80 80 8 8 0.08 0.0850.09 0.095 0.1 0.1050.11 0.1150.12 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 8 8 8 8 8 8 1 6 16 1 6 16 1 6 24 24 2 4 24 32 32 32 32 4 40 40 40 40 40 48 48 48 4 8 48 48 56 56 5 6 5 6 5 6 56 6 4 64 64 6 4 7 2 72 72 72 7 2 80 8 0.08 0.0850.09 0.095 0.1 0.1050.11 0.1150.12 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 8 8 8 8 8 8 16 1 6 1 6 16 16 16 24 24 24 24 32 32 32 32 4 40 40 48 4 8 48 56 56 56 56 64 6 4 6 4 64 7 2 7 2 7 2 7 2 80 80 80 80 0.08 0.0850.09 0.095 0.1 0.1050.11 0.1150.12 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 8 8 8 8 8 8 16 1 6 1 6 16 1 6 2 4 24 24 24 32 32 3 2 32 40 40 40 40 40 40 4 8 4 8 48 4 8 4 8 4 8 5 6 5 6 5 6 5 6 56 5 6 6 4 64 64 64 7 2 72 7 2 80 80 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12

Bruce Archambeault, PhD 42

Effect of Plane width on Inductance

Case1 : 10 inches Case2 : 5 inches Case3 : 2 inches

1 inch Port1 Port2

Bruce Archambeault, PhD 43

Loop Inductance is Affected by Plane Width

~ 250pH ~ 330pH ~ 560pH

Case1 : 10 inches Case2 : 5 inches Case2 : 2 inches

Bruce Archambeault, PhD 44

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 45

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 46

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